Principal Components Analysis of
Trading Economics Data
Michael Hall
August 7, 2017
Introduction
This study engages in some unsupervised learning on a collection of fifty-two Trading Economics (TE) variables to discover any patterns in the data and find smaller groups of homogeneous items from a larger group of data.
This exercise is a form of clustering and it is unsupervised because it helps find structure in unlabeled data, as opposed to supervised learning, where predictions are based on labeled data or data with a target. So, unlabeled data is data without a target, without labelled responses.
The study will also perform Principal Components Analysis (PCA) analysis, one of the key techniques in unsupervised learning to help with finding patterns in features of data. One way to do this is through dimensionality reduction. Dimensionality reduction is a method to decrease the number of features to describe an observation, while maintaining the maximum information content under the constraints of lower dimensionality.
The main goal of a PCA analysis is to identify patterns in data. PCA aims to detect the correlation between variables. If a strong correlation between variables exists, an attempt to reduce the dimensionality only makes sense. In a nutshell, this is what PCA is all about - finding the directions of maximum variance in high-dimensional data and project it onto a smaller dimensional subspace while retaining most of the information.
Dimensionality Reduction with PCA
Dimensionality reduction has two main goals:
- To find structure within features
- To aid in visualization
Dimensional reduction helps to:
- Find structure in features
- Aid in visualization of high dimensional data
- Pre-processing method used before supervised learning
A popular method of dimensionality reduction is PCA. PCA has three main goals when finding lower dimensional representation of features:
To find linear combination of variables to create principal components.
- This means it takes some fraction of some or all of the features and adds them together. These new features are called principal components.
It maintains and describes the maximum variance in data for a given number of principal components. An important question is “what is the size of k that represents the data ‘well’?”
The new features are uncorrelated, i.e., orthogonal to each other.
Note that data with three or more features is often difficult to develop an effective visualization for and can put high cognitive loads on the viewers of the visualization. PCA can help with this.
In summary, PCA:
- Is a variable reduction technique
- Is used when variables are highly correlated
- Reduces the number of observed variables to a smaller number of principal components which account for most of the variance of the observed variables
- Is a large sample procedure
Some of the challenges of unsupervised learning and PCA include:
- No single goal of analysis
- Requires more creativity
- There is much more unlabeled data available than cleanly labeled data.
Data Understanding
Trading Economics is an online subscription platform that provides historical data, economic forecasts, news, and trading recommendations. Users can view and export up to 100 years of historical data for over 20 million indicators using just a browser or Excel. Some sample variables include:
- Housing market index
- Building permits index
- Index of wages index
- New home sales index
The following United States Trading Economics variables will be used in the model:
| No. | Variable Name | Description | Scale | Notes |
|---|---|---|---|---|
| 1 | seqnum | Sequence Number | index | |
| 2 | te_cpi_yoy | Inflation Rate | % | |
| 3 | te_rstamom | Retail Sales MoM | % | |
| 4 | te_unitedstaavehouear | Average Hourly Earnings | % | |
| 5 | te_unitedstabanlenrat | Bank Lending Rate | % | |
| 6 | te_unitedstacontspe | Construction Spending | % | |
| 7 | te_unitedstaemprat | Employment Rate | % | |
| 8 | te_unitedstahouind | Housing Index | % | |
| 9 | te_unitedstainfratmom | Inflation Rate MoM | % | |
| 10 | te_unitedstalabforparra | Labor Force Participation Rate | % | |
| 11 | te_unitedstalonterunera | Long Term Unemployment Rate | % | |
| 12 | te_unitedstapenhomsal | Pending Home Sales | % | |
| 13 | te_unitedstaperinc | Personal Income | % | |
| 14 | te_unitedstapersav | Personal Savings | % | |
| 15 | te_unitedstaperspe | Personal Spending | % | |
| 16 | te_unitedstaretsalexaut | Retail Sales Excluding Autos | % | |
| 17 | te_unitedstawaggro | Wage Growth | % | |
| 18 | te_unitedstayouunerat | Youth Unemployment Rate | % | |
| 19 | te_usacorecpirate | Core Inflation Rate | % | |
| 20 | te_usaretailsalesyoy | Retail Sales YoY | % | |
| 21 | te_usgg10yr | US Government 10-Yr Bond Rate | % | |
| 22 | te_usurtot | Unemployment Rate | % | |
| 23 | te_ijcusa | Initial Jobless Claims | thousand | |
| 24 | te_nfp_tch | Non-farm Payrolls | thousand | |
| 25 | te_unitedstaadpempcha | ADP Employment Change | thousand | |
| 26 | te_unitedstabuiper | Building Permits | thousand | |
| 27 | te_unitedstaconjobcla | Continuing Jobless Claims | thousand | |
| 28 | te_unitedstaempper | Employed Persons | thousand | |
| 29 | te_unitedstaexihomsal | Existing Home Sales | thousand | |
| 30 | te_unitedstafultimemp | Full-time Employment | thousand | |
| 31 | te_unitedstagovpay | Government Payrolls | thousand | Quarterly series |
| 32 | te_unitedstahousta | Housing Starts | thousand | |
| 33 | te_unitedstajoboff | Job Offers | thousand | |
| 34 | te_unitedstajobvac | Job Vacancies | thousand | |
| 35 | te_unitedstamanpay | Manufacturing Payrolls | thousand | |
| 36 | te_unitedstanewhomsal | New Home Sales | thousand | |
| 37 | te_unitedstanonpay | Nonfarm Payrolls Private | thousand | |
| 38 | te_unitedstauneper | Unemployed Persons | thousand | |
| 39 | te_unitedstachastosal | Chain Store Sales | USD Million | |
| 40 | te_unitedstaconcre | Consumer Credit | USD Million | |
| 41 | te_unitedstapriseccre | Private Sector Credit | USD Million | Annual series |
| 42 | te_unitedstawag | Wages | USD Hour | |
| 43 | te_unitedstawaginman | Wages in Manufacturing | USD Hour | |
| 44 | te_unitedstgaspri | Gasoline Prices | USD Liter | |
| 45 | te_dxy | Exchange Rate | USD Index | |
| 46 | te_indu | Stock Market - DJIA | index points | |
| 47 | te_unitedstaconpriindcp | Consumer Price Index (CPI) | index points | |
| 48 | te_unitedstacorconpri | Core Consumer Prices | index points | |
| 49 | te_unitedstaecooptind | Economic Optimism Index | index 0-100 | |
| 50 | te_unitedstalmci | Labor Market Conditions Index | index points | |
| 51 | te_unitedstanahhoumarin | NAHB Housing Market Index | index 0-100 | Null data until June 2013 |
| 52 | te_unitedstaaveweehou | Average Weekly Hours | hours | |
| 53 | te_unitedstachajobcut | Challenger Job Cuts | persons | |
| 54 | te_unitedstaconconf | United States Consumer Sentiment | index 0-100 | |
| 55 | te_unitedstagdp | United States Gross Domestic Product | USD Billion | Annual series |
Data Preparation
The TE CSV files were manually downloaded from the main Trading Economics site and saved into a subfolder (data/Trading Economics/). With few exceptions, the data in the TE files are collected on a daily basis, but for this study the data were summarized to a annual basis. A metadata file containing the list of variables, variable descriptions, and file names was created (TE_indicators.csv) and saved in a subfolder (data/dictionary/). A separate R script (TEData_fromCSV.R) was executed to aggregate the data from the downloaded files from daily to annually to create a single dataset. The final summarized dataset contains seven years of TE data between January 2010 - July 2017.
Read in summarized dataset containing TE variables:
# read csv flat file containing summarized TE data.
te_vars <-
read_csv("te_all_vars_labels_v1.csv", col_names = TRUE)Exploratory Data Analysis
The first step of any data analysis is to familiarize yourself with the data. The TE data will be explored to answer the following questions:
- How many observations are in this dataset?
- What are the various data types found in the columns of the dataset? Are they integer, decimal, character, dates, etc.
- Note that PCA works on numeric data only. So, if the data has categorical variables they must be converted to numerical.
- How many variables/features in the data are missing?
- Does the data need to be scaled? The mean and standard deviation of the features of the data can be checked to determine if scaling is necessary.
- PCA is not scale invariant, so it is highly recommended to standardize all the variables before applying PCA.
# check number of observations (rows and columns)
dim(te_vars)## [1] 89 53# check if any variable is other than numeric type
str(te_vars)## Classes 'tbl_df', 'tbl' and 'data.frame': 89 obs. of 53 variables:
## $ Seqnum : num 1 2 3 4 5 6 7 8 9 10 ...
## $ InflationRate : int 614 604 636 687 583 536 546 599 594 543 ...
## $ RetailSalesMom : num 0 0.2 2.2 0.7 -0.8 -0.1 0.2 0.6 0.8 1.2 ...
## $ AverageHourlyEarnings : num 0.2 0.2 0 0.2 0.1 0 0.3 0.1 0.2 0.3 ...
## $ BankLendingRate : num 3.25 3.25 3.25 3.25 3.25 3.25 3.25 3.25 3.25 3.25 ...
## $ ConstructionSpending : num -2.9 -0.8 0.8 1.2 -0.4 0.1 -3.1 0.2 0.6 -0.6 ...
## $ EmploymentRate : num 58.5 58.5 58.5 58.7 58.6 58.5 58.5 58.6 58.5 58.3 ...
## $ HousingIndex : int 9 -15 42 51 427 -256 -171 -149 -173 50 ...
## $ InflationRateMoM : num 2.6 2.1 2.3 2.2 2 1.1 1.2 1.1 1.1 1.2 ...
## $ LaborForceParticipationRate: int 2783 2391 2612 3428 2922 2705 3163 2962 2795 3200 ...
## $ LongTermUnemploymentRate : num 10.9 9.9 12.9 12.4 6.2 3.9 1.8 1.3 3.2 5 ...
## $ PendingHomeSales : num 12.1 17.6 21.2 23.9 -15.7 -19.1 -18.9 -19.9 -23.3 -21.9 ...
## $ PersonalIncome : num 0.4 -0.2 0.3 0.7 0.8 0.1 0.3 0.5 0.1 0.5 ...
## $ PersonalSavings : num 5.6 5.2 5 5.6 6 5.9 5.9 5.8 5.6 5.4 ...
## $ PersonalSpending : num 0 0.3 0.6 0.1 0.3 0.3 0.2 0.6 0.2 0.7 ...
## $ RetailSalesExcludingAutos : num 0 0.6 0.9 0.6 -0.9 -0.1 -0.1 0.7 0.8 0.9 ...
## $ WageGrowth : num -0.6 0.03 1.02 1.36 1.92 1.95 2.8 3.09 3.12 3.25 ...
## $ YouthUnemploymentRate : num 18.8 18.7 18.8 19.5 18.1 18.2 18.4 17.7 17.9 18.7 ...
## $ CoreInflationRate : num 1.6 1.3 1.1 0.9 0.9 0.9 0.9 0.9 0.8 0.6 ...
## $ RetailSalesYoY : num 2.8 3.3 7.5 7.8 6 4.2 4.2 2.8 6.2 6.5 ...
## $ USGovernment10yrBondRate : num 3.71 3.68 3.72 3.82 3.4 ...
## $ UnemploymentRate : num 9.8 9.8 9.9 9.9 9.6 9.4 9.4 9.5 9.5 9.4 ...
## $ InitialJoblessClaims : num 0.1 -0.1 0 0 -0.1 0 0.2 0.1 0.2 0.3 ...
## $ NonfarmPayrolls : int 23 -68 164 243 524 -137 -68 -36 -52 262 ...
## $ ADPEmploymentChange : int -22 -116 -20 190 134 119 116 106 143 161 ...
## $ BuildingPermits : int 636 650 687 637 575 587 579 580 563 558 ...
## $ ContinuingJoblessClaims : num 4839 4794 4732 4728 4675 ...
## $ EmployedPersons : int 138438 138581 138751 139297 139241 139141 139179 139438 139396 139119 ...
## $ ExistingHomeSales : num 77.8 80.2 80.7 81.2 85.4 ...
## $ FulltimeEmployment : int 4190 4270 4490 4820 4880 4450 3450 3680 3840 3830 ...
## $ GovernmentPayrolls : num NA NA NA NA NA NA NA NA NA NA ...
## $ HousingStarts : num 0.1 -0.5 0 0.1 0 -1.1 -0.6 -0.1 -1 0.1 ...
## $ JobOffers : num 480 486 470 469 459 ...
## $ JobVacancies : int 2636 2578 2666 3220 2891 2664 2965 2920 2767 3033 ...
## $ ManufacturingPayrolls : num NA NA NA NA NA NA NA NA NA NA ...
## $ NewHomeSales : int 15 17 15 19 22 16 14 13 13 15 ...
## $ NonfarmPayrollsPrivate : int 14 -53 122 192 97 119 103 113 121 212 ...
## $ UnemployedPersons : int 15046 15113 15202 15325 14849 14474 14512 14648 14579 14516 ...
## $ ChainStoreSales : int 3929 4420 4895 5179 5160 4884 4678 4954 4851 5061 ...
## $ ConsumerCredit : num -14 -10.8 6.6 -5.8 -7.8 -3.6 -1.8 0 1 9.1 ...
## $ PrivateSectorCredit : num NA NA NA NA NA NA NA NA NA NA ...
## $ Wages : num 18.9 18.9 18.9 19 19 ...
## $ WagesinManufacturing : num 18.5 18.5 18.5 18.5 18.6 ...
## $ GasolinePrices : int 110613 110778 111162 111854 112539 112608 112248 111847 111926 111723 ...
## $ ExchangeRate : num 77.8 80.2 80.7 81.2 85.4 ...
## $ StockMarket : num 10476 10209 10678 11046 10483 ...
## $ ConsumerPriceIndex : num 217 217 217 217 217 ...
## $ CoreConsumerPrices : num 221 221 221 221 221 ...
## $ EconomicOptimism : num 48.8 46.8 45.4 48.4 48.7 46.2 44.7 43.6 45.3 46.4 ...
## $ LaborMarketConditions : num 64.8 64.9 64.9 65.2 64.9 64.6 64.6 64.7 64.6 64.4 ...
## $ NAHBHousingMarket : int -15 -7 0 36 36 20 16 -8 10 -1 ...
## $ AverageWeeklyHours : num 34 33.8 34 34.1 34.1 34.1 34.2 34.2 34.3 34.3 ...
## $ ChallengerJobCuts : int 71482 42090 67611 38326 38810 39358 41676 34768 37151 37986 ...
## - attr(*, "spec")=List of 2
## ..$ cols :List of 53
## .. ..$ Seqnum : list()
## .. .. ..- attr(*, "class")= chr "collector_double" "collector"
## .. ..$ InflationRate : list()
## .. .. ..- attr(*, "class")= chr "collector_integer" "collector"
## .. ..$ RetailSalesMom : list()
## .. .. ..- attr(*, "class")= chr "collector_double" "collector"
## .. ..$ AverageHourlyEarnings : list()
## .. .. ..- attr(*, "class")= chr "collector_double" "collector"
## .. ..$ BankLendingRate : list()
## .. .. ..- attr(*, "class")= chr "collector_double" "collector"
## .. ..$ ConstructionSpending : list()
## .. .. ..- attr(*, "class")= chr "collector_double" "collector"
## .. ..$ EmploymentRate : list()
## .. .. ..- attr(*, "class")= chr "collector_double" "collector"
## .. ..$ HousingIndex : list()
## .. .. ..- attr(*, "class")= chr "collector_integer" "collector"
## .. ..$ InflationRateMoM : list()
## .. .. ..- attr(*, "class")= chr "collector_double" "collector"
## .. ..$ LaborForceParticipationRate: list()
## .. .. ..- attr(*, "class")= chr "collector_integer" "collector"
## .. ..$ LongTermUnemploymentRate : list()
## .. .. ..- attr(*, "class")= chr "collector_double" "collector"
## .. ..$ PendingHomeSales : list()
## .. .. ..- attr(*, "class")= chr "collector_double" "collector"
## .. ..$ PersonalIncome : list()
## .. .. ..- attr(*, "class")= chr "collector_double" "collector"
## .. ..$ PersonalSavings : list()
## .. .. ..- attr(*, "class")= chr "collector_double" "collector"
## .. ..$ PersonalSpending : list()
## .. .. ..- attr(*, "class")= chr "collector_double" "collector"
## .. ..$ RetailSalesExcludingAutos : list()
## .. .. ..- attr(*, "class")= chr "collector_double" "collector"
## .. ..$ WageGrowth : list()
## .. .. ..- attr(*, "class")= chr "collector_double" "collector"
## .. ..$ YouthUnemploymentRate : list()
## .. .. ..- attr(*, "class")= chr "collector_double" "collector"
## .. ..$ CoreInflationRate : list()
## .. .. ..- attr(*, "class")= chr "collector_double" "collector"
## .. ..$ RetailSalesYoY : list()
## .. .. ..- attr(*, "class")= chr "collector_double" "collector"
## .. ..$ USGovernment10yrBondRate : list()
## .. .. ..- attr(*, "class")= chr "collector_double" "collector"
## .. ..$ UnemploymentRate : list()
## .. .. ..- attr(*, "class")= chr "collector_double" "collector"
## .. ..$ InitialJoblessClaims : list()
## .. .. ..- attr(*, "class")= chr "collector_double" "collector"
## .. ..$ NonfarmPayrolls : list()
## .. .. ..- attr(*, "class")= chr "collector_integer" "collector"
## .. ..$ ADPEmploymentChange : list()
## .. .. ..- attr(*, "class")= chr "collector_integer" "collector"
## .. ..$ BuildingPermits : list()
## .. .. ..- attr(*, "class")= chr "collector_integer" "collector"
## .. ..$ ContinuingJoblessClaims : list()
## .. .. ..- attr(*, "class")= chr "collector_double" "collector"
## .. ..$ EmployedPersons : list()
## .. .. ..- attr(*, "class")= chr "collector_integer" "collector"
## .. ..$ ExistingHomeSales : list()
## .. .. ..- attr(*, "class")= chr "collector_double" "collector"
## .. ..$ FulltimeEmployment : list()
## .. .. ..- attr(*, "class")= chr "collector_integer" "collector"
## .. ..$ GovernmentPayrolls : list()
## .. .. ..- attr(*, "class")= chr "collector_double" "collector"
## .. ..$ HousingStarts : list()
## .. .. ..- attr(*, "class")= chr "collector_double" "collector"
## .. ..$ JobOffers : list()
## .. .. ..- attr(*, "class")= chr "collector_double" "collector"
## .. ..$ JobVacancies : list()
## .. .. ..- attr(*, "class")= chr "collector_integer" "collector"
## .. ..$ ManufacturingPayrolls : list()
## .. .. ..- attr(*, "class")= chr "collector_double" "collector"
## .. ..$ NewHomeSales : list()
## .. .. ..- attr(*, "class")= chr "collector_integer" "collector"
## .. ..$ NonfarmPayrollsPrivate : list()
## .. .. ..- attr(*, "class")= chr "collector_integer" "collector"
## .. ..$ UnemployedPersons : list()
## .. .. ..- attr(*, "class")= chr "collector_integer" "collector"
## .. ..$ ChainStoreSales : list()
## .. .. ..- attr(*, "class")= chr "collector_integer" "collector"
## .. ..$ ConsumerCredit : list()
## .. .. ..- attr(*, "class")= chr "collector_double" "collector"
## .. ..$ PrivateSectorCredit : list()
## .. .. ..- attr(*, "class")= chr "collector_double" "collector"
## .. ..$ Wages : list()
## .. .. ..- attr(*, "class")= chr "collector_double" "collector"
## .. ..$ WagesinManufacturing : list()
## .. .. ..- attr(*, "class")= chr "collector_double" "collector"
## .. ..$ GasolinePrices : list()
## .. .. ..- attr(*, "class")= chr "collector_integer" "collector"
## .. ..$ ExchangeRate : list()
## .. .. ..- attr(*, "class")= chr "collector_double" "collector"
## .. ..$ StockMarket : list()
## .. .. ..- attr(*, "class")= chr "collector_double" "collector"
## .. ..$ ConsumerPriceIndex : list()
## .. .. ..- attr(*, "class")= chr "collector_double" "collector"
## .. ..$ CoreConsumerPrices : list()
## .. .. ..- attr(*, "class")= chr "collector_double" "collector"
## .. ..$ EconomicOptimism : list()
## .. .. ..- attr(*, "class")= chr "collector_double" "collector"
## .. ..$ LaborMarketConditions : list()
## .. .. ..- attr(*, "class")= chr "collector_double" "collector"
## .. ..$ NAHBHousingMarket : list()
## .. .. ..- attr(*, "class")= chr "collector_integer" "collector"
## .. ..$ AverageWeeklyHours : list()
## .. .. ..- attr(*, "class")= chr "collector_double" "collector"
## .. ..$ ChallengerJobCuts : list()
## .. .. ..- attr(*, "class")= chr "collector_integer" "collector"
## ..$ default: list()
## .. ..- attr(*, "class")= chr "collector_guess" "collector"
## ..- attr(*, "class")= chr "col_spec"# another way to look at just the column data types
sapply(te_vars, typeof)## Seqnum InflationRate
## "double" "integer"
## RetailSalesMom AverageHourlyEarnings
## "double" "double"
## BankLendingRate ConstructionSpending
## "double" "double"
## EmploymentRate HousingIndex
## "double" "integer"
## InflationRateMoM LaborForceParticipationRate
## "double" "integer"
## LongTermUnemploymentRate PendingHomeSales
## "double" "double"
## PersonalIncome PersonalSavings
## "double" "double"
## PersonalSpending RetailSalesExcludingAutos
## "double" "double"
## WageGrowth YouthUnemploymentRate
## "double" "double"
## CoreInflationRate RetailSalesYoY
## "double" "double"
## USGovernment10yrBondRate UnemploymentRate
## "double" "double"
## InitialJoblessClaims NonfarmPayrolls
## "double" "integer"
## ADPEmploymentChange BuildingPermits
## "integer" "integer"
## ContinuingJoblessClaims EmployedPersons
## "double" "integer"
## ExistingHomeSales FulltimeEmployment
## "double" "integer"
## GovernmentPayrolls HousingStarts
## "double" "double"
## JobOffers JobVacancies
## "double" "integer"
## ManufacturingPayrolls NewHomeSales
## "double" "integer"
## NonfarmPayrollsPrivate UnemployedPersons
## "integer" "integer"
## ChainStoreSales ConsumerCredit
## "integer" "double"
## PrivateSectorCredit Wages
## "double" "double"
## WagesinManufacturing GasolinePrices
## "double" "integer"
## ExchangeRate StockMarket
## "double" "double"
## ConsumerPriceIndex CoreConsumerPrices
## "double" "double"
## EconomicOptimism LaborMarketConditions
## "double" "double"
## NAHBHousingMarket AverageWeeklyHours
## "integer" "double"
## ChallengerJobCuts
## "integer"# visualize the data types in a bar plot
res <- lapply(te_vars, class)
res_frame <- data.frame(unlist(res))
barplot(table(res_frame), main="Data Types", col="steelblue", ylab="Number of Features")
# examine stats (measures of central tendency) and look for missing values
summary(te_vars)## Seqnum InflationRate RetailSalesMom AverageHourlyEarnings
## Min. : 1 Min. : 517.0 Min. :-1.0000 Min. :-0.2000
## 1st Qu.:23 1st Qu.: 694.0 1st Qu.: 0.0000 1st Qu.: 0.1000
## Median :45 Median : 921.0 Median : 0.4000 Median : 0.2000
## Mean :45 Mean : 902.2 Mean : 0.3539 Mean : 0.1753
## 3rd Qu.:67 3rd Qu.:1104.0 3rd Qu.: 0.7000 3rd Qu.: 0.3000
## Max. :89 Max. :1328.0 Max. : 2.2000 Max. : 0.4000
##
## BankLendingRate ConstructionSpending EmploymentRate HousingIndex
## Min. :3.250 Min. :-4.1000 Min. :58.20 Min. :-256.000
## 1st Qu.:3.250 1st Qu.:-0.1000 1st Qu.:58.50 1st Qu.: -16.000
## Median :3.250 Median : 0.6000 Median :58.70 Median : 3.000
## Mean :3.322 Mean : 0.4404 Mean :58.93 Mean : -1.843
## 3rd Qu.:3.250 3rd Qu.: 1.2000 3rd Qu.:59.30 3rd Qu.: 16.000
## Max. :4.000 Max. : 2.7000 Max. :60.20 Max. : 427.000
##
## InflationRateMoM LaborForceParticipationRate LongTermUnemploymentRate
## Min. :-0.200 Min. :2391 Min. :-2.600
## 1st Qu.: 1.100 1st Qu.:3416 1st Qu.: 1.300
## Median : 1.600 Median :3954 Median : 3.300
## Mean : 1.666 Mean :4243 Mean : 3.772
## 3rd Qu.: 2.100 3rd Qu.:5135 3rd Qu.: 5.000
## Max. : 3.900 Max. :6371 Max. :12.900
## NA's :1
## PendingHomeSales PersonalIncome PersonalSavings PersonalSpending
## Min. :-24.300 Min. :-5.2000 Min. : 4.500 Min. :-0.2000
## 1st Qu.: -2.000 1st Qu.: 0.2000 1st Qu.: 5.300 1st Qu.: 0.2000
## Median : 3.800 Median : 0.4000 Median : 5.700 Median : 0.3000
## Mean : 2.625 Mean : 0.3438 Mean : 5.866 Mean : 0.3135
## 3rd Qu.: 9.800 3rd Qu.: 0.5000 3rd Qu.: 6.100 3rd Qu.: 0.5000
## Max. : 23.900 Max. : 2.6000 Max. :11.000 Max. : 1.1000
##
## RetailSalesExcludingAutos WageGrowth YouthUnemploymentRate
## Min. :-1.2000 Min. :-0.600 Min. : 8.80
## 1st Qu.: 0.0000 1st Qu.: 2.950 1st Qu.:11.60
## Median : 0.3000 Median : 4.010 Median :15.00
## Mean : 0.3011 Mean : 3.876 Mean :14.41
## 3rd Qu.: 0.7000 3rd Qu.: 4.850 3rd Qu.:16.80
## Max. : 1.2000 Max. : 7.510 Max. :19.50
##
## CoreInflationRate RetailSalesYoY USGovernment10yrBondRate
## Min. :0.600 Min. :1.600 Min. :1.491
## 1st Qu.:1.600 1st Qu.:3.100 1st Qu.:1.934
## Median :1.800 Median :4.100 Median :2.285
## Mean :1.767 Mean :4.469 Mean :2.367
## 3rd Qu.:2.100 3rd Qu.:5.600 3rd Qu.:2.694
## Max. :2.300 Max. :8.300 Max. :3.815
##
## UnemploymentRate InitialJoblessClaims NonfarmPayrolls
## Min. :4.300 Min. :-0.6000 Min. :-137.0
## 1st Qu.:5.200 1st Qu.: 0.0000 1st Qu.: 126.0
## Median :7.200 Median : 0.2000 Median : 206.0
## Mean :7.031 Mean : 0.1326 Mean : 184.3
## 3rd Qu.:8.600 3rd Qu.: 0.3000 3rd Qu.: 246.0
## Max. :9.900 Max. : 0.6000 Max. : 524.0
##
## ADPEmploymentChange BuildingPermits ContinuingJoblessClaims
## Min. :-116.0 Min. : 542.0 Min. :1918
## 1st Qu.: 148.0 1st Qu.: 697.0 1st Qu.:2245
## Median : 199.0 Median :1013.0 Median :2926
## Mean : 186.7 Mean : 943.4 Mean :3024
## 3rd Qu.: 235.0 3rd Qu.:1159.0 3rd Qu.:3652
## Max. : 348.0 Max. :1363.0 Max. :4839
##
## EmployedPersons ExistingHomeSales FulltimeEmployment GovernmentPayrolls
## Min. :138438 Min. : 74.24 Min. :3450 Min. :0.4700
## 1st Qu.:140826 1st Qu.: 79.87 1st Qu.:4480 1st Qu.:0.5900
## Median :144318 Median : 81.87 Median :4960 Median :0.6200
## Mean :145025 Mean : 85.91 Mean :4871 Mean :0.6823
## 3rd Qu.:148890 3rd Qu.: 95.24 3rd Qu.:5270 3rd Qu.:0.7500
## Max. :153156 Max. :102.04 Max. :5700 Max. :0.9900
## NA's :50
## HousingStarts JobOffers JobVacancies ManufacturingPayrolls
## Min. :-1.1000 Min. :239.8 Min. :2578 Min. :1.010
## 1st Qu.: 0.2000 1st Qu.:275.0 1st Qu.:3475 1st Qu.:1.268
## Median : 0.4000 Median :344.8 Median :3959 Median :1.490
## Mean : 0.2943 Mean :342.2 Mean :4263 Mean :1.704
## 3rd Qu.: 0.6000 3rd Qu.:401.2 3rd Qu.:5414 3rd Qu.:2.058
## Max. : 1.1000 Max. :485.8 Max. :5973 Max. :2.800
## NA's :1 NA's :41
## NewHomeSales NonfarmPayrollsPrivate UnemployedPersons ChainStoreSales
## Min. :13.0 Min. :-53.0 Min. : 6861 Min. : 3129
## 1st Qu.:21.0 1st Qu.:135.0 1st Qu.: 8235 1st Qu.: 4264
## Median :47.0 Median :194.0 Median :11264 Median : 4683
## Mean :42.7 Mean :186.2 Mean :10943 Mean : 5074
## 3rd Qu.:58.0 3rd Qu.:237.0 3rd Qu.:13302 3rd Qu.: 5036
## Max. :71.0 Max. :366.0 Max. :15325 Max. :10598
## NA's :1
## ConsumerCredit PrivateSectorCredit Wages WagesinManufacturing
## Min. :-14.00 Min. :196.5 Min. :18.89 Min. :18.47
## 1st Qu.: 9.80 1st Qu.:198.2 1st Qu.:19.56 1st Qu.:18.97
## Median : 14.50 Median :198.7 Median :20.22 Median :19.34
## Mean : 14.46 Mean :199.4 Mean :20.32 Mean :19.48
## 3rd Qu.: 18.60 3rd Qu.:199.6 3rd Qu.:21.05 3rd Qu.:19.92
## Max. :118.60 Max. :204.7 Max. :21.99 Max. :20.84
## NA's :82
## GasolinePrices ExchangeRate StockMarket ConsumerPriceIndex
## Min. :110613 Min. : 74.24 Min. :10159 Min. :217.2
## 1st Qu.:113177 1st Qu.: 79.87 1st Qu.:12516 1st Qu.:227.2
## Median :116453 Median : 81.87 Median :15289 Median :233.6
## Mean :117519 Mean : 85.91 Mean :15025 Mean :232.0
## 3rd Qu.:121589 3rd Qu.: 95.24 3rd Qu.:17658 3rd Qu.:237.5
## Max. :125987 Max. :102.04 Max. :20943 Max. :244.5
##
## CoreConsumerPrices EconomicOptimism LaborMarketConditions
## Min. :220.6 Min. :35.80 Min. :62.40
## 1st Qu.:226.9 1st Qu.:45.10 1st Qu.:62.80
## Median :234.7 Median :46.80 Median :63.30
## Mean :234.9 Mean :46.85 Mean :63.41
## 3rd Qu.:242.5 3rd Qu.:48.60 3rd Qu.:64.00
## Max. :251.3 Max. :56.40 Max. :65.20
##
## NAHBHousingMarket AverageWeeklyHours ChallengerJobCuts
## Min. :-23.00 Min. :33.8 Min. : 23622
## 1st Qu.: 0.00 1st Qu.:34.3 1st Qu.: 36602
## Median : 11.00 Median :34.4 Median : 41186
## Mean : 10.34 Mean :34.4 Mean : 44685
## 3rd Qu.: 21.00 3rd Qu.:34.5 3rd Qu.: 50462
## Max. : 38.00 Max. :34.6 Max. :115730
## # remove any dependent or identifier variables; make a matrix of the given data
my_te_data <- as.matrix(as.data.frame(subset(te_vars,
select = -c(Seqnum, PrivateSectorCredit)), ncol=51))
colnames(my_te_data)## [1] "InflationRate" "RetailSalesMom"
## [3] "AverageHourlyEarnings" "BankLendingRate"
## [5] "ConstructionSpending" "EmploymentRate"
## [7] "HousingIndex" "InflationRateMoM"
## [9] "LaborForceParticipationRate" "LongTermUnemploymentRate"
## [11] "PendingHomeSales" "PersonalIncome"
## [13] "PersonalSavings" "PersonalSpending"
## [15] "RetailSalesExcludingAutos" "WageGrowth"
## [17] "YouthUnemploymentRate" "CoreInflationRate"
## [19] "RetailSalesYoY" "USGovernment10yrBondRate"
## [21] "UnemploymentRate" "InitialJoblessClaims"
## [23] "NonfarmPayrolls" "ADPEmploymentChange"
## [25] "BuildingPermits" "ContinuingJoblessClaims"
## [27] "EmployedPersons" "ExistingHomeSales"
## [29] "FulltimeEmployment" "GovernmentPayrolls"
## [31] "HousingStarts" "JobOffers"
## [33] "JobVacancies" "ManufacturingPayrolls"
## [35] "NewHomeSales" "NonfarmPayrollsPrivate"
## [37] "UnemployedPersons" "ChainStoreSales"
## [39] "ConsumerCredit" "Wages"
## [41] "WagesinManufacturing" "GasolinePrices"
## [43] "ExchangeRate" "StockMarket"
## [45] "ConsumerPriceIndex" "CoreConsumerPrices"
## [47] "EconomicOptimism" "LaborMarketConditions"
## [49] "NAHBHousingMarket" "AverageWeeklyHours"
## [51] "ChallengerJobCuts"Data Preparation
Dealing with Missing Values
Missing values are a problem that plagues any data analysis and this analysis is no exception. Most PCA routines cannot be applied if there are any missing values in the dataset. In the event of missing data, there are typically a series of questions that should be asked:
- Determine the reason for the missing data - what is the process that led to the data being missing?
- Is the proportion of missing values so high as to invalidate any sort of analysis?
- Is there information in the dataset that would allow you to predict/infer the values of the missing data?
Missing values (NA’s) were found in the following TE features:
- unitedstagovpay = 50
- unitedstahousta = 1
- unitedstamanpay = 41
- unitedstachastosal = 1
- unitedstalabforparra = 1
- PrivateSectorCredit = 82
Since we have 89 observations (rows) and 82 missing values in the TE variable named PrivateSectorCredit, so it seems sensible to drop this variable from the analysis.
We will need to impute the remaining five TE variables with missing values in order for PCA to work properly. We will also impute any variables with zero values to improve the accuracy of the analysis.
# impute missing values with mean value
te_vars$GovernmentPayrolls <- with(te_vars, impute(te_vars$GovernmentPayrolls, mean))
te_vars$HousingStarts <- with(te_vars, impute(te_vars$HousingStarts, mean))
te_vars$ManufacturingPayrolls <- with(te_vars, impute(te_vars$ManufacturingPayrolls, mean))
te_vars$ChainStoreSales <- with(te_vars, impute(te_vars$ChainStoreSales, mean))
te_vars$LaborForceParticipationRate <- with(te_vars, impute(te_vars$LaborForceParticipationRate, mean))
# impute 0 values with median value
te_vars$GovernmentPayrolls <- ifelse(te_vars$GovernmentPayrolls == 0,
median(te_vars$GovernmentPayrolls), te_vars$GovernmentPayrolls)
te_vars$HousingStarts <- ifelse(te_vars$HousingStarts == 0,
median(te_vars$HousingStarts), te_vars$HousingStarts)
te_vars$ManufacturingPayrolls <- ifelse(te_vars$ManufacturingPayrolls == 0,
median(te_vars$ManufacturingPayrolls), te_vars$ManufacturingPayrolls)
te_vars$ChainStoreSales <- ifelse(te_vars$ChainStoreSales == 0,
median(te_vars$ChainStoreSales), te_vars$ChainStoreSales)
te_vars$LaborForceParticipationRate <- ifelse(te_vars$LaborForceParticipationRate == 0,
median(te_vars$LaborForceParticipationRate),
te_vars$LaborForceParticipationRate)
# remove identifier variable and any extreme missing values variables
#te_vars_cln <- te_vars
te_vars_cln <- as.data.frame(subset(te_vars, select = -c(Seqnum, PrivateSectorCredit)), ncol=51)Scaling the Data
The second important step is to check if the data need to be scaled. Two common reasons for scaling data include:
- The input variables use different units of measurement.
- The input variables have significantly different variances.
It is common to measure lots of different variables that come on different scales. However, performing PCA on un-scaled variables can lead to insanely large loadings for variables with high variance. In turn, this will lead to dependence of a principal component on the variable with high variance. So, this is undesirable.
# check column means and standard deviations
colMeans(te_vars_cln)## InflationRate RetailSalesMom
## 9.022360e+02 3.539326e-01
## AverageHourlyEarnings BankLendingRate
## 1.752809e-01 3.321798e+00
## ConstructionSpending EmploymentRate
## 4.404494e-01 5.893146e+01
## HousingIndex InflationRateMoM
## -1.842697e+00 1.666292e+00
## LaborForceParticipationRate LongTermUnemploymentRate
## 4.243159e+03 3.771910e+00
## PendingHomeSales PersonalIncome
## 2.624719e+00 3.438202e-01
## PersonalSavings PersonalSpending
## 5.866292e+00 3.134831e-01
## RetailSalesExcludingAutos WageGrowth
## 3.011236e-01 3.875618e+00
## YouthUnemploymentRate CoreInflationRate
## 1.440674e+01 1.767416e+00
## RetailSalesYoY USGovernment10yrBondRate
## 4.468539e+00 2.367464e+00
## UnemploymentRate InitialJoblessClaims
## 7.031461e+00 1.325843e-01
## NonfarmPayrolls ADPEmploymentChange
## 1.843146e+02 1.867416e+02
## BuildingPermits ContinuingJoblessClaims
## 9.434382e+02 3.023880e+03
## EmployedPersons ExistingHomeSales
## 1.450250e+05 8.590894e+01
## FulltimeEmployment GovernmentPayrolls
## 4.871124e+03 6.823077e-01
## HousingStarts JobOffers
## 3.078013e-01 3.421809e+02
## JobVacancies ManufacturingPayrolls
## 4.263191e+03 1.703958e+00
## NewHomeSales NonfarmPayrollsPrivate
## 4.269663e+01 1.861573e+02
## UnemployedPersons ChainStoreSales
## 1.094307e+04 5.073818e+03
## ConsumerCredit Wages
## 1.445562e+01 2.031719e+01
## WagesinManufacturing GasolinePrices
## 1.947573e+01 1.175191e+05
## ExchangeRate StockMarket
## 8.590894e+01 1.502452e+04
## ConsumerPriceIndex CoreConsumerPrices
## 2.320358e+02 2.349155e+02
## EconomicOptimism LaborMarketConditions
## 4.685056e+01 6.341011e+01
## NAHBHousingMarket AverageWeeklyHours
## 1.033708e+01 3.439551e+01
## ChallengerJobCuts
## 4.468493e+04# observe the standard deviation of each variable
apply(te_vars_cln, 2, sd)## InflationRate RetailSalesMom
## 2.331850e+02 5.931534e-01
## AverageHourlyEarnings BankLendingRate
## 1.325286e-01 1.657323e-01
## ConstructionSpending EmploymentRate
## 1.146707e+00 5.426357e-01
## HousingIndex InflationRateMoM
## 6.608395e+01 9.538134e-01
## LaborForceParticipationRate LongTermUnemploymentRate
## 1.047223e+03 3.610905e+00
## PendingHomeSales PersonalIncome
## 9.974391e+00 7.066173e-01
## PersonalSavings PersonalSpending
## 9.134032e-01 2.408277e-01
## RetailSalesExcludingAutos WageGrowth
## 5.373534e-01 1.448449e+00
## YouthUnemploymentRate CoreInflationRate
## 3.011448e+00 4.266080e-01
## RetailSalesYoY USGovernment10yrBondRate
## 1.767612e+00 5.807941e-01
## UnemploymentRate InitialJoblessClaims
## 1.776334e+00 2.098567e-01
## NonfarmPayrolls ADPEmploymentChange
## 1.048681e+02 7.295493e+01
## BuildingPermits ContinuingJoblessClaims
## 2.412555e+02 8.239206e+02
## EmployedPersons ExistingHomeSales
## 4.580797e+03 8.380687e+00
## FulltimeEmployment GovernmentPayrolls
## 5.074299e+02 9.825132e-02
## HousingStarts JobOffers
## 4.495530e-01 6.987069e+01
## JobVacancies ManufacturingPayrolls
## 1.042006e+03 4.011878e-01
## NewHomeSales NonfarmPayrollsPrivate
## 1.900682e+01 7.540118e+01
## UnemployedPersons ChainStoreSales
## 2.640933e+03 1.547356e+03
## ConsumerCredit Wages
## 1.391188e+01 9.066948e-01
## WagesinManufacturing GasolinePrices
## 6.496046e-01 4.536619e+03
## ExchangeRate StockMarket
## 8.380687e+00 2.994473e+03
## ConsumerPriceIndex CoreConsumerPrices
## 7.693804e+00 9.446230e+00
## EconomicOptimism LaborMarketConditions
## 3.695271e+00 7.231630e-01
## NAHBHousingMarket AverageWeeklyHours
## 1.415414e+01 1.421508e-01
## ChallengerJobCuts
## 1.436366e+04Since some of the TE variables have different and incomparable scales, it is advisable to scale them to unit variance before finding the principal components. By default, prcomp centers the variable to have mean equals to zero. With parameter scale set to equal TRUE, the variables are normalized to have standard deviation equal to 1.
Modeling
We will use the prcomp function in base R to perform the PCA model. prcomp has the following important options:
- The scale parameter indicates that the data should be scaled to a sd of 1 before performing PCA.
- The center parameter indicates that the data should be centered around 0 before performing PCA.
# clear format of any previous plots
#dev.off()
# perform PCA using prcomp function
te_vars.prc <- prcomp(te_vars_cln, center=TRUE, scale=TRUE)
te_vars.prc## Standard deviations (1, .., p=51):
## [1] 4.898388e+00 2.034390e+00 1.811548e+00 1.497006e+00 1.432282e+00
## [6] 1.408990e+00 1.293312e+00 1.244932e+00 1.067712e+00 1.023595e+00
## [11] 9.600120e-01 9.392304e-01 8.960365e-01 8.860921e-01 7.720803e-01
## [16] 7.421731e-01 7.000046e-01 6.366015e-01 5.931009e-01 5.532642e-01
## [21] 5.393719e-01 5.093687e-01 4.603099e-01 4.080971e-01 3.816194e-01
## [26] 3.487762e-01 3.441621e-01 2.981229e-01 2.768252e-01 2.725065e-01
## [31] 2.531506e-01 2.134222e-01 1.809297e-01 1.744951e-01 1.375670e-01
## [36] 1.226195e-01 1.098159e-01 9.404680e-02 8.415947e-02 6.737896e-02
## [41] 6.522681e-02 4.921031e-02 4.669059e-02 3.505696e-02 2.921692e-02
## [46] 2.191440e-02 1.715842e-02 1.124805e-02 5.923902e-03 4.560184e-16
## [51] 4.560184e-16
##
## Rotation (n x k) = (51 x 51):
## PC1 PC2 PC3
## InflationRate 0.194499219 -0.008983769 0.0044082349
## RetailSalesMom -0.039604781 -0.098187206 -0.3959033877
## AverageHourlyEarnings 0.031336528 0.034796203 0.0025821342
## BankLendingRate 0.124160342 0.187547017 -0.1720988503
## ConstructionSpending 0.047878986 -0.169388477 0.0910606099
## EmploymentRate 0.186806830 0.115593735 -0.0892505364
## HousingIndex 0.044958142 -0.095295981 0.0341742628
## InflationRateMoM -0.098363802 -0.081358792 -0.0722319523
## LaborForceParticipationRate 0.191933563 -0.001761173 -0.0339297743
## LongTermUnemploymentRate -0.109019234 -0.120817482 -0.1657967348
## PendingHomeSales 0.016100922 -0.034926876 0.2194398671
## PersonalIncome -0.009896372 -0.071030365 -0.0083044405
## PersonalSavings -0.032717062 -0.082558445 0.2013036581
## PersonalSpending -0.004639575 -0.102774948 -0.3961681943
## RetailSalesExcludingAutos -0.034766500 -0.097056451 -0.4369915239
## WageGrowth 0.076405115 -0.235057743 0.0455249805
## YouthUnemploymentRate -0.199816217 -0.031227191 0.0454383476
## CoreInflationRate 0.125616095 -0.097599755 0.1405577981
## RetailSalesYoY -0.129536739 -0.113474794 -0.1499631845
## USGovernment10yrBondRate -0.113989958 0.077032009 -0.1948937518
## UnemploymentRate -0.202216917 0.004149693 0.0196131419
## InitialJoblessClaims -0.028494191 -0.095296651 -0.3202889809
## NonfarmPayrolls 0.060054709 -0.341553026 -0.0432234581
## ADPEmploymentChange 0.072765962 -0.340460486 -0.0418847260
## BuildingPermits 0.196623623 -0.020625793 0.0062714003
## ContinuingJoblessClaims -0.198137113 0.084903518 -0.0128727761
## EmployedPersons 0.202204492 0.037121364 -0.0437548914
## ExistingHomeSales 0.176905291 0.171402044 -0.0619795769
## FulltimeEmployment 0.184010754 -0.002454845 0.0137145145
## GovernmentPayrolls -0.067106448 -0.196210535 0.0782717364
## HousingStarts 0.118675919 -0.114363631 0.2064693655
## JobOffers -0.199943704 0.061045991 0.0132418459
## JobVacancies 0.200262149 0.006710509 -0.0342109121
## ManufacturingPayrolls -0.082113141 -0.220268088 0.1001237625
## NewHomeSales 0.192319468 -0.031805209 0.0218502774
## NonfarmPayrollsPrivate 0.044121440 -0.391511974 -0.0900666264
## UnemployedPersons -0.201900081 0.007363974 0.0191672343
## ChainStoreSales -0.031565388 -0.077843900 0.0678155797
## ConsumerCredit 0.059123772 -0.087509157 -0.0060588255
## Wages 0.201228525 0.023671631 -0.0574115853
## WagesinManufacturing 0.196764819 0.046410664 -0.0857584767
## GasolinePrices 0.201783705 0.040427728 -0.0475632730
## ExchangeRate 0.176905291 0.171402044 -0.0619795769
## StockMarket 0.196050414 -0.026541364 -0.0551922011
## ConsumerPriceIndex 0.195840214 -0.079811250 0.0002913054
## CoreConsumerPrices 0.202359519 0.016268727 -0.0369083512
## EconomicOptimism 0.088855954 0.137658055 -0.1038054859
## LaborMarketConditions -0.187584039 0.107801765 -0.0403779113
## NAHBHousingMarket -0.051691595 -0.215354803 -0.0762396099
## AverageWeeklyHours 0.116498165 -0.306087246 0.1139073713
## ChallengerJobCuts -0.012829348 -0.014422316 0.0499398759
## PC4 PC5 PC6
## InflationRate 0.0474622860 -0.030397325 0.032574432
## RetailSalesMom -0.0058407471 0.191986044 0.189700257
## AverageHourlyEarnings -0.0689695329 0.110607730 -0.211080535
## BankLendingRate -0.1935441320 -0.073547677 -0.079075929
## ConstructionSpending 0.0062379534 -0.068747480 0.232461862
## EmploymentRate -0.0745813880 -0.036949998 -0.079048883
## HousingIndex -0.1863612051 -0.352865197 -0.109679759
## InflationRateMoM -0.3582236425 -0.052472302 0.093184605
## LaborForceParticipationRate -0.0377337532 0.011126806 0.018939278
## LongTermUnemploymentRate -0.0979762892 -0.247904564 -0.099957208
## PendingHomeSales -0.3177121733 0.057629346 0.252854456
## PersonalIncome -0.0165289301 0.176933134 -0.395609293
## PersonalSavings -0.2197946620 0.411044403 -0.283184087
## PersonalSpending 0.0619994497 0.132168948 0.069410366
## RetailSalesExcludingAutos 0.0106752152 0.187459499 0.185053421
## WageGrowth -0.0039269835 0.294056887 -0.287776959
## YouthUnemploymentRate -0.0098505809 0.001203922 0.026793654
## CoreInflationRate -0.2839967505 0.144750528 0.226421923
## RetailSalesYoY -0.2717822523 -0.042854872 -0.072060804
## USGovernment10yrBondRate 0.1482018471 -0.305731480 -0.142463706
## UnemploymentRate -0.0495982322 0.002414268 0.013537844
## InitialJoblessClaims -0.0132904147 0.201390208 0.135310303
## NonfarmPayrolls -0.1695376626 -0.266743778 -0.099490624
## ADPEmploymentChange -0.0439388159 -0.073892020 -0.092949099
## BuildingPermits 0.0789888874 -0.021622349 0.059686614
## ContinuingJoblessClaims -0.0351840195 -0.034917561 -0.028685841
## EmployedPersons 0.0001789244 -0.006553109 -0.013951853
## ExistingHomeSales -0.0349647011 -0.020794748 -0.090371730
## FulltimeEmployment -0.0269600915 -0.113992008 0.056737290
## GovernmentPayrolls 0.2849268572 -0.045220418 -0.033732015
## HousingStarts -0.1009612478 -0.083839004 0.187332694
## JobOffers -0.0262060273 0.002995958 -0.003455851
## JobVacancies -0.0306079937 0.000439853 -0.011932759
## ManufacturingPayrolls 0.3460071278 -0.129743167 0.151202196
## NewHomeSales 0.1179445996 -0.065014043 0.026634493
## NonfarmPayrollsPrivate -0.0724604023 -0.061725486 -0.042244913
## UnemployedPersons -0.0527693703 0.002132902 0.014514611
## ChainStoreSales 0.1429809183 0.081995621 -0.218158608
## ConsumerCredit 0.1743819636 0.178736270 -0.073494973
## Wages 0.0082407266 -0.009061802 -0.006369826
## WagesinManufacturing -0.0280728733 0.005211773 -0.028395165
## GasolinePrices 0.0021808788 -0.014183527 -0.030410943
## ExchangeRate -0.0349647011 -0.020794748 -0.090371730
## StockMarket 0.0636626696 -0.073823473 -0.019163499
## ConsumerPriceIndex 0.0126406755 0.003921556 0.057606648
## CoreConsumerPrices 0.0032857856 -0.008166877 0.014097971
## EconomicOptimism -0.1669209468 -0.050316811 -0.225376218
## LaborMarketConditions -0.1397233153 -0.032578069 -0.055755164
## NAHBHousingMarket -0.0752872314 -0.232449357 -0.233982505
## AverageWeeklyHours 0.0508240216 0.141480251 -0.008649133
## ChallengerJobCuts -0.2356700163 0.042800145 0.174111956
## PC7 PC8 PC9
## InflationRate -0.018945652 0.067201915 -0.019531687
## RetailSalesMom -0.049575006 0.244408039 0.129905243
## AverageHourlyEarnings 0.356490929 0.201789185 -0.481952717
## BankLendingRate -0.202046455 -0.170110004 -0.187174083
## ConstructionSpending -0.130697660 -0.077054461 0.185648731
## EmploymentRate -0.024115992 -0.024023321 0.033884870
## HousingIndex -0.035278624 0.427089212 0.136069799
## InflationRateMoM -0.261408149 -0.219130970 -0.360399143
## LaborForceParticipationRate 0.148996506 0.001635950 0.001870099
## LongTermUnemploymentRate -0.091646606 0.106196545 0.061737252
## PendingHomeSales -0.134366624 0.295229998 0.003995903
## PersonalIncome -0.104848184 0.259297556 -0.183447786
## PersonalSavings -0.068500825 0.112713547 0.058866350
## PersonalSpending 0.150669469 0.217315702 -0.125790931
## RetailSalesExcludingAutos -0.096695393 0.133594497 0.093964064
## WageGrowth 0.147497472 -0.050448348 0.127218421
## YouthUnemploymentRate 0.007117785 0.043816773 0.014255622
## CoreInflationRate -0.119562642 -0.036310699 -0.178938860
## RetailSalesYoY -0.148792136 -0.181083001 -0.053006263
## USGovernment10yrBondRate -0.007821352 -0.006692486 -0.100499406
## UnemploymentRate -0.004225766 0.013947321 0.008486132
## InitialJoblessClaims -0.051575152 -0.037522622 -0.113416223
## NonfarmPayrolls 0.094202300 0.180830199 0.198821037
## ADPEmploymentChange -0.020039385 -0.321683600 0.050792549
## BuildingPermits -0.029288470 0.090716817 -0.010291675
## ContinuingJoblessClaims 0.014592880 0.039618164 0.053142287
## EmployedPersons -0.023423285 -0.008969354 -0.010210493
## ExistingHomeSales 0.031177864 0.014087903 0.177253808
## FulltimeEmployment -0.080643646 0.135029243 -0.030500720
## GovernmentPayrolls -0.230632184 0.168596329 -0.267079364
## HousingStarts -0.096824324 0.214226675 -0.080707285
## JobOffers 0.025191460 0.037974283 0.025857480
## JobVacancies 0.067851373 -0.014601233 0.025915264
## ManufacturingPayrolls 0.043261603 0.103279257 -0.275061201
## NewHomeSales -0.025872421 0.075512959 -0.070530964
## NonfarmPayrollsPrivate 0.161935956 -0.122815463 0.157264711
## UnemployedPersons -0.005659495 0.017632259 0.009595110
## ChainStoreSales -0.463603077 0.028929666 0.131290747
## ConsumerCredit -0.226555277 -0.008523357 0.129496715
## Wages -0.029145787 -0.041539056 -0.050109030
## WagesinManufacturing -0.062776514 -0.061250376 -0.059325677
## GasolinePrices -0.012968383 -0.022043995 -0.005020592
## ExchangeRate 0.031177864 0.014087903 0.177253808
## StockMarket -0.048315187 -0.015893707 -0.090279252
## ConsumerPriceIndex -0.070260466 -0.032556492 -0.141424078
## CoreConsumerPrices -0.039035881 -0.022026032 -0.053753402
## EconomicOptimism -0.109918746 0.159480668 -0.032457960
## LaborMarketConditions -0.027003473 0.011509755 0.053718840
## NAHBHousingMarket 0.117014069 -0.156403493 -0.146630693
## AverageWeeklyHours 0.084253513 -0.118741599 -0.034453506
## ChallengerJobCuts 0.393838560 0.015938303 0.016614156
## PC10 PC11 PC12
## InflationRate 0.0048573920 -0.0615434283 0.028631032
## RetailSalesMom -0.0034167442 -0.0552864469 0.153832488
## AverageHourlyEarnings -0.2350111986 0.1589573774 0.111580369
## BankLendingRate 0.0304547294 0.0326735306 -0.072162448
## ConstructionSpending 0.4906390317 -0.1196148570 -0.195237476
## EmploymentRate 0.0499618398 -0.0294646846 -0.017484067
## HousingIndex -0.0372628675 0.2741532623 -0.308575313
## InflationRateMoM -0.0164544930 0.0014322266 -0.174197034
## LaborForceParticipationRate -0.0055917542 -0.0366591978 -0.035700144
## LongTermUnemploymentRate -0.1139119803 -0.2343401327 0.245144001
## PendingHomeSales -0.0626314697 -0.3088629450 0.254342769
## PersonalIncome 0.2744819827 -0.2334357342 -0.408458374
## PersonalSavings 0.1833080029 0.0836037298 0.062538568
## PersonalSpending 0.0217829957 0.0389211079 -0.096115851
## RetailSalesExcludingAutos 0.0717811621 0.0036716397 0.112962247
## WageGrowth 0.1685668285 -0.1329759278 0.009216951
## YouthUnemploymentRate 0.0006996837 0.0338306484 0.070050970
## CoreInflationRate -0.0416142049 0.0924728685 0.035485021
## RetailSalesYoY -0.0537064762 -0.1514180100 -0.040454826
## USGovernment10yrBondRate 0.0257405452 -0.2951434931 -0.132696075
## UnemploymentRate -0.0026040487 0.0384442422 0.018442017
## InitialJoblessClaims -0.0172738918 0.2237321780 -0.199797458
## NonfarmPayrolls -0.0789042018 0.2051185976 -0.125781111
## ADPEmploymentChange 0.0999053101 0.0070204676 0.075345767
## BuildingPermits 0.0260342740 -0.0368383618 0.051399706
## ContinuingJoblessClaims 0.0156945238 0.0148716191 -0.015343967
## EmployedPersons 0.0159795527 -0.0060833437 -0.011557079
## ExistingHomeSales -0.0055879471 -0.0422917684 0.032406910
## FulltimeEmployment 0.0218146362 -0.0809616087 0.021576163
## GovernmentPayrolls 0.0808404944 -0.1317917431 0.146573977
## HousingStarts -0.0727207854 0.1240965895 -0.022405522
## JobOffers 0.0253118824 0.0233035402 -0.024453767
## JobVacancies -0.0252302098 -0.0452665975 -0.027900971
## ManufacturingPayrolls 0.1268227965 -0.0778651461 -0.022602006
## NewHomeSales 0.0253668358 -0.0165979728 0.067759903
## NonfarmPayrollsPrivate -0.0770817158 0.0450027094 0.095508519
## UnemployedPersons -0.0037275925 0.0423620972 0.019965022
## ChainStoreSales -0.3461884110 -0.0036680176 -0.038096351
## ConsumerCredit -0.5199388375 -0.1921559994 -0.169789660
## Wages -0.0105897141 -0.0020631190 -0.043250707
## WagesinManufacturing -0.0365687023 0.0207590479 -0.054631683
## GasolinePrices 0.0101617844 -0.0020282472 -0.012599842
## ExchangeRate -0.0055879471 -0.0422917684 0.032406910
## StockMarket 0.0171042758 -0.0742933723 0.026716469
## ConsumerPriceIndex 0.0006319265 -0.0060748732 0.011756478
## CoreConsumerPrices -0.0012588278 0.0001277947 -0.017081602
## EconomicOptimism 0.2022902137 -0.0932511889 0.405908556
## LaborMarketConditions 0.0342679698 0.0269742519 0.007799026
## NAHBHousingMarket -0.0325558825 -0.0543248707 0.249342717
## AverageWeeklyHours -0.0951147331 0.0186229336 0.070906916
## ChallengerJobCuts -0.1883767368 -0.5637585003 -0.275296667
## PC13 PC14 PC15
## InflationRate -0.033686687 0.068234896 -0.055447799
## RetailSalesMom -0.210432145 0.051827576 0.104729749
## AverageHourlyEarnings -0.175816439 -0.285720065 -0.192026531
## BankLendingRate -0.021158889 0.073734386 0.105504003
## ConstructionSpending -0.194599086 -0.499319618 -0.099793395
## EmploymentRate -0.039368424 0.054010546 -0.001603777
## HousingIndex 0.107856120 -0.080195411 0.262927947
## InflationRateMoM -0.009496675 0.087483493 0.081409360
## LaborForceParticipationRate 0.020564458 -0.012797276 0.028602235
## LongTermUnemploymentRate 0.079178010 0.246954235 -0.216679295
## PendingHomeSales -0.036964523 -0.228893479 -0.122562277
## PersonalIncome -0.090262026 0.068598782 -0.367261542
## PersonalSavings 0.059464319 0.126113757 0.138631291
## PersonalSpending -0.224340108 -0.058382788 0.302021405
## RetailSalesExcludingAutos -0.021078545 0.043232714 -0.011936240
## WageGrowth 0.112855731 0.025656347 0.331745395
## YouthUnemploymentRate 0.057243757 -0.010603698 -0.013727060
## CoreInflationRate -0.046833238 0.055237053 -0.071389648
## RetailSalesYoY -0.046442777 -0.134002408 0.245415232
## USGovernment10yrBondRate 0.063401581 -0.028427046 -0.012288180
## UnemploymentRate 0.018572835 -0.029110756 -0.006776759
## InitialJoblessClaims 0.446372417 -0.163629060 -0.379014195
## NonfarmPayrolls 0.035695385 0.053920636 -0.063585452
## ADPEmploymentChange 0.075374343 0.079290352 -0.074595211
## BuildingPermits 0.003050214 0.019740314 -0.052868637
## ContinuingJoblessClaims -0.003397247 0.015055961 -0.041355640
## EmployedPersons -0.023798533 0.028439315 0.013135742
## ExistingHomeSales -0.066276471 -0.060319620 -0.104078750
## FulltimeEmployment 0.047008420 -0.030250724 0.058091701
## GovernmentPayrolls 0.085758543 0.252546754 0.144835731
## HousingStarts 0.024120435 0.118878645 0.044648961
## JobOffers -0.007771553 0.008366625 -0.014462272
## JobVacancies -0.033199162 -0.008883558 0.009496178
## ManufacturingPayrolls 0.065119942 -0.072286095 0.019385304
## NewHomeSales 0.037836297 0.048047165 0.007330126
## NonfarmPayrollsPrivate -0.044883287 0.145278645 -0.318872979
## UnemployedPersons 0.018676661 -0.029868536 -0.006896956
## ChainStoreSales -0.455455987 -0.017048130 -0.147611056
## ConsumerCredit 0.368271446 -0.348152618 0.137499381
## Wages -0.033242769 0.026223759 0.023455263
## WagesinManufacturing -0.035919318 0.019600951 0.022811693
## GasolinePrices -0.026033664 0.025794388 0.013265513
## ExchangeRate -0.066276471 -0.060319620 -0.104078750
## StockMarket 0.023246496 0.002392774 0.002839644
## ConsumerPriceIndex 0.020054743 0.040567016 0.039992912
## CoreConsumerPrices -0.022657355 0.033719939 0.012258432
## EconomicOptimism 0.388921449 -0.194747441 -0.022759551
## LaborMarketConditions -0.018478808 -0.004007111 -0.012093280
## NAHBHousingMarket -0.203625516 -0.353519716 0.121992804
## AverageWeeklyHours 0.013199890 -0.064863600 0.010016823
## ChallengerJobCuts 0.032978459 0.173399506 0.021628825
## PC16 PC17 PC18
## InflationRate 0.025038228 -0.052823011 0.011136442
## RetailSalesMom -0.083725738 0.162550180 0.057003792
## AverageHourlyEarnings -0.268224408 0.224349866 0.045398528
## BankLendingRate -0.007480783 0.196817534 -0.241888960
## ConstructionSpending -0.112691649 0.081012960 -0.246427080
## EmploymentRate -0.056322515 0.042556382 -0.055322072
## HousingIndex 0.056714902 -0.089918839 0.095382813
## InflationRateMoM -0.137561963 -0.051878511 0.023675282
## LaborForceParticipationRate 0.010418451 -0.046521598 0.056773345
## LongTermUnemploymentRate -0.248138149 -0.346605935 -0.215451563
## PendingHomeSales -0.057562354 -0.090968305 0.284551652
## PersonalIncome -0.026697781 -0.011396149 -0.049633611
## PersonalSavings 0.133045687 -0.138635432 -0.010314009
## PersonalSpending 0.112365707 -0.089511840 -0.283470613
## RetailSalesExcludingAutos 0.003840572 0.030820945 0.184207759
## WageGrowth -0.189408215 -0.049606958 0.240813936
## YouthUnemploymentRate 0.032490490 0.057328612 -0.009890739
## CoreInflationRate 0.092325036 -0.315708693 -0.116739462
## RetailSalesYoY -0.183044872 0.138323993 0.162219998
## USGovernment10yrBondRate -0.038204782 -0.077369932 0.432399595
## UnemploymentRate 0.028823876 0.069195119 -0.046624239
## InitialJoblessClaims 0.332858515 -0.096204298 0.178102001
## NonfarmPayrolls -0.066041847 0.020530860 -0.077125027
## ADPEmploymentChange 0.127028052 0.341788901 0.209787954
## BuildingPermits -0.015022094 -0.051787310 0.021860654
## ContinuingJoblessClaims 0.030989873 0.066895579 -0.044843769
## EmployedPersons -0.016769717 -0.002919546 -0.017440789
## ExistingHomeSales 0.142602520 0.063717868 0.021107700
## FulltimeEmployment 0.095840230 -0.037327556 0.029968683
## GovernmentPayrolls 0.238902848 0.151020264 -0.171580773
## HousingStarts 0.082520584 0.444609711 0.120707245
## JobOffers 0.012810574 0.074896213 -0.007799104
## JobVacancies -0.004351188 -0.011584202 0.040189915
## ManufacturingPayrolls -0.082979228 0.014281566 -0.017146089
## NewHomeSales 0.035779715 -0.001684207 0.046098650
## NonfarmPayrollsPrivate -0.141557788 0.107362021 -0.190862051
## UnemployedPersons 0.031305006 0.080098762 -0.051292287
## ChainStoreSales 0.209875606 0.085632513 0.097317314
## ConsumerCredit -0.150288761 0.032988674 -0.263028994
## Wages -0.032597759 -0.009341866 0.017089763
## WagesinManufacturing -0.012211602 -0.015802210 -0.009234338
## GasolinePrices -0.005212350 -0.010339697 0.001516420
## ExchangeRate 0.142602520 0.063717868 0.021107700
## StockMarket -0.071403701 0.023790710 0.060926632
## ConsumerPriceIndex -0.037298501 -0.054736756 -0.003258696
## CoreConsumerPrices -0.014925182 -0.020079276 -0.004579812
## EconomicOptimism 0.070348053 0.237643451 -0.210677433
## LaborMarketConditions 0.019025406 0.146555050 -0.094671850
## NAHBHousingMarket 0.489350788 -0.241711669 -0.022705195
## AverageWeeklyHours -0.012124373 0.001514669 -0.018187984
## ChallengerJobCuts 0.331173280 0.177733504 -0.132534409
## PC19 PC20 PC21
## InflationRate 8.831330e-02 -0.0647906543 -0.013025436
## RetailSalesMom 1.268188e-01 0.1429336574 -0.178956828
## AverageHourlyEarnings -1.832501e-02 -0.2330202956 -0.117023537
## BankLendingRate 3.615886e-02 0.0075858689 0.050244075
## ConstructionSpending -3.120979e-02 -0.2681234619 0.005747547
## EmploymentRate -4.377209e-02 -0.1191005453 0.026236527
## HousingIndex -3.010140e-02 -0.0045319077 -0.261666043
## InflationRateMoM 2.930032e-02 -0.1370748922 0.050636556
## LaborForceParticipationRate -1.383791e-01 0.0535042149 0.013085713
## LongTermUnemploymentRate 5.529626e-02 -0.0755358721 0.074735147
## PendingHomeSales -2.488136e-01 0.0209926009 0.191377812
## PersonalIncome -1.211767e-02 0.4764483277 0.029343731
## PersonalSavings 1.852448e-01 -0.2256702730 0.041615457
## PersonalSpending -7.650357e-02 -0.0938946836 0.406810125
## RetailSalesExcludingAutos 1.054525e-01 0.0966207332 -0.175056014
## WageGrowth 3.283373e-02 -0.2283662328 0.145823458
## YouthUnemploymentRate -5.941953e-03 0.0334224411 0.051162032
## CoreInflationRate 1.571885e-01 0.0341129682 -0.120655241
## RetailSalesYoY -1.986401e-01 0.1169731073 -0.235227716
## USGovernment10yrBondRate 6.275486e-02 -0.2291560188 0.208594570
## UnemploymentRate 1.398443e-03 0.0072203510 0.028440064
## InitialJoblessClaims -1.239536e-01 -0.2174737374 0.037658155
## NonfarmPayrolls -4.815654e-02 -0.0779438302 -0.047633256
## ADPEmploymentChange 1.159556e-01 0.1021507504 0.003900532
## BuildingPermits 4.589111e-02 -0.0531625526 0.056314880
## ContinuingJoblessClaims 2.078988e-02 -0.0305241621 0.047197773
## EmployedPersons 6.668727e-03 -0.0362493762 -0.004475975
## ExistingHomeSales -1.150027e-01 0.0086708878 -0.005014231
## FulltimeEmployment 4.663904e-02 -0.0094085408 -0.008004751
## GovernmentPayrolls -5.603090e-01 -0.1710248383 -0.102290698
## HousingStarts 1.737611e-01 0.1548625929 0.494643808
## JobOffers 2.879744e-02 -0.0347019777 0.052054615
## JobVacancies -9.681857e-02 0.0080090474 -0.014862706
## ManufacturingPayrolls 3.900981e-01 -0.0194801978 -0.213918033
## NewHomeSales 1.403436e-01 -0.0693623149 -0.013078565
## NonfarmPayrollsPrivate -4.059437e-02 -0.1044324645 0.163083898
## UnemployedPersons 4.182370e-03 0.0030112868 0.026989253
## ChainStoreSales 1.496946e-01 -0.3191516243 -0.116107008
## ConsumerCredit 9.326876e-02 0.1329783052 0.141366695
## Wages -8.731419e-03 -0.0194198130 -0.015934063
## WagesinManufacturing -4.852881e-03 -0.0416418346 -0.016818675
## GasolinePrices 5.680361e-05 -0.0228425337 0.000711360
## ExchangeRate -1.150027e-01 0.0086708878 -0.005014231
## StockMarket -3.306334e-02 0.0134296328 0.044420935
## ConsumerPriceIndex 5.916851e-02 0.0006734763 -0.027104678
## CoreConsumerPrices 2.182697e-02 -0.0216189239 -0.015549912
## EconomicOptimism 1.842187e-01 -0.0681376015 -0.148130787
## LaborMarketConditions -4.319718e-02 -0.0805064375 0.067062707
## NAHBHousingMarket 1.220496e-01 0.2376832102 0.118111402
## AverageWeeklyHours -2.636039e-01 0.1674287567 -0.188779508
## ChallengerJobCuts 1.355276e-01 -0.1793519739 -0.216991991
## PC22 PC23 PC24
## InflationRate 0.046249632 0.0272447746 0.065690591
## RetailSalesMom -0.053073830 0.1186044629 0.180234587
## AverageHourlyEarnings -0.195955115 0.1158162988 0.047061283
## BankLendingRate -0.165027412 -0.2761872213 0.268754792
## ConstructionSpending -0.103304552 0.2016105989 0.139173273
## EmploymentRate -0.068203664 0.0881902079 0.027175546
## HousingIndex -0.009101624 0.0934629611 -0.030039113
## InflationRateMoM 0.046180879 0.0868567882 -0.074723631
## LaborForceParticipationRate -0.102413512 0.1372199312 -0.018222211
## LongTermUnemploymentRate 0.064137387 0.2253703831 0.298270483
## PendingHomeSales -0.024335163 -0.2786849329 -0.179148353
## PersonalIncome 0.035638620 -0.0159000639 -0.033127492
## PersonalSavings -0.253064651 -0.1500946202 0.074125065
## PersonalSpending 0.322660974 -0.1338603417 -0.285153508
## RetailSalesExcludingAutos -0.324218453 0.0713652098 -0.039431713
## WageGrowth 0.097597234 0.0883401794 0.151528934
## YouthUnemploymentRate -0.015016191 -0.0968817423 0.117384683
## CoreInflationRate -0.040267384 0.2161305745 -0.237258285
## RetailSalesYoY 0.198502208 -0.1884689411 0.069348209
## USGovernment10yrBondRate -0.074868889 0.0889231660 -0.066829986
## UnemploymentRate -0.035077853 -0.0635333470 0.040075382
## InitialJoblessClaims 0.126231284 -0.0328712696 0.219329580
## NonfarmPayrolls -0.054144192 -0.1275192218 -0.097905146
## ADPEmploymentChange -0.121159470 0.0403932867 -0.341603159
## BuildingPermits -0.015306790 -0.0367102722 0.028397436
## ContinuingJoblessClaims -0.038207445 -0.0022539308 0.027004961
## EmployedPersons -0.025243951 -0.0134843896 -0.011758908
## ExistingHomeSales 0.013043992 -0.0422345476 0.009111726
## FulltimeEmployment -0.111852117 -0.3757028420 0.244814039
## GovernmentPayrolls -0.246985648 0.1331461908 -0.058690571
## HousingStarts 0.104961507 0.2861806202 0.263798694
## JobOffers -0.024802319 0.0078489640 0.023019528
## JobVacancies -0.082725144 0.0897239154 -0.039695296
## ManufacturingPayrolls 0.061851609 -0.2782522406 -0.025023368
## NewHomeSales 0.011965754 -0.2700770278 0.114492132
## NonfarmPayrollsPrivate -0.067326858 -0.2592677382 -0.109839374
## UnemployedPersons -0.037329094 -0.0709585631 0.040579581
## ChainStoreSales 0.239196093 0.0062036334 -0.028131750
## ConsumerCredit -0.262178303 0.0394113313 -0.087860201
## Wages -0.024523637 -0.0154906632 -0.020890405
## WagesinManufacturing -0.030301144 -0.0141054098 0.004231883
## GasolinePrices -0.038847432 -0.0046116204 0.019333693
## ExchangeRate 0.013043992 -0.0422345476 0.009111726
## StockMarket 0.046778674 -0.0044247624 -0.037224552
## ConsumerPriceIndex 0.014877733 0.0001978001 0.020396691
## CoreConsumerPrices -0.028238648 -0.0323729436 -0.012487815
## EconomicOptimism 0.360436985 0.1172161511 -0.225141302
## LaborMarketConditions -0.101932474 -0.0657031674 0.066876757
## NAHBHousingMarket -0.179228230 0.0831507236 0.161699240
## AverageWeeklyHours 0.347874661 0.0316728489 0.338213517
## ChallengerJobCuts 0.028185221 0.0408262038 0.040774496
## PC25 PC26 PC27
## InflationRate -0.1584724661 0.2406793995 0.1660964861
## RetailSalesMom -0.0659948240 -0.0510958153 0.1597004169
## AverageHourlyEarnings 0.0811726426 0.1460027326 -0.0528552126
## BankLendingRate 0.1425106992 -0.2820625456 -0.2602486959
## ConstructionSpending 0.0243777291 0.0335590115 -0.0445137028
## EmploymentRate 0.1633247533 -0.0526931630 0.1674541574
## HousingIndex 0.1188318411 0.0108334423 -0.1259008827
## InflationRateMoM -0.0025196559 0.0292374496 0.0941629871
## LaborForceParticipationRate -0.1700342297 0.0116743399 0.3550127580
## LongTermUnemploymentRate 0.2234794020 0.2214058953 -0.0578524685
## PendingHomeSales 0.0901891223 -0.1037949484 -0.1772911454
## PersonalIncome -0.0601339855 0.0398772132 -0.0275319642
## PersonalSavings 0.1505586394 -0.1598801666 0.2195708423
## PersonalSpending 0.2094504443 0.1203401502 0.0076640596
## RetailSalesExcludingAutos 0.0572810913 -0.1240736067 -0.1987807050
## WageGrowth -0.1497918486 0.0991020608 -0.1698020379
## YouthUnemploymentRate -0.0650379467 0.1186715100 0.0156768090
## CoreInflationRate 0.0740388813 -0.0353011753 0.2102296042
## RetailSalesYoY -0.2153213014 0.1261637025 0.3000260487
## USGovernment10yrBondRate 0.1593562300 -0.3182892446 0.1902391273
## UnemploymentRate -0.0168099027 0.0392716210 0.0075757605
## InitialJoblessClaims -0.0788892805 0.0490702089 0.0007326947
## NonfarmPayrolls -0.1045241284 -0.1475642690 0.0038072126
## ADPEmploymentChange 0.4056146338 0.3797785636 -0.0977288682
## BuildingPermits -0.0773340488 0.0525944075 0.0381415132
## ContinuingJoblessClaims 0.0638946566 0.0219938640 0.1354777495
## EmployedPersons 0.0006798963 -0.0035499383 0.0431201235
## ExistingHomeSales 0.1664354673 0.0016398755 0.2251772370
## FulltimeEmployment 0.0526054393 0.4530201445 0.0731637991
## GovernmentPayrolls -0.0752444277 0.0058219020 0.0461738128
## HousingStarts -0.0535288419 -0.0720100583 0.1075109248
## JobOffers 0.0545841213 0.0013083729 0.0580603767
## JobVacancies -0.0152533474 -0.0480013911 0.0990243909
## ManufacturingPayrolls 0.1241024139 -0.1381537443 0.2759126634
## NewHomeSales 0.0050641690 0.0335096705 -0.0383948873
## NonfarmPayrollsPrivate -0.2495202137 -0.2147273242 0.1156385385
## UnemployedPersons -0.0111422388 0.0531146664 0.0190583428
## ChainStoreSales -0.0641905592 -0.0030959256 -0.0670507214
## ConsumerCredit 0.0527758611 0.0190937906 0.0879832996
## Wages -0.0645874008 -0.0020064495 -0.0355023780
## WagesinManufacturing -0.0455504840 0.0354469759 -0.0810495311
## GasolinePrices 0.0177488853 0.0072041547 0.0355169862
## ExchangeRate 0.1664354673 0.0016398755 0.2251772370
## StockMarket -0.0794165684 -0.1394314839 -0.1781880772
## ConsumerPriceIndex -0.1144316095 -0.0007036543 -0.1196600929
## CoreConsumerPrices -0.0623161296 -0.0011639694 -0.0127978154
## EconomicOptimism -0.1238256211 -0.1052019484 -0.0033421232
## LaborMarketConditions 0.0615770248 0.0747263239 0.1581129403
## NAHBHousingMarket -0.1730400903 -0.0752683256 -0.0397622341
## AverageWeeklyHours 0.4601725343 -0.2742675040 0.0421153569
## ChallengerJobCuts -0.0242911074 -0.0243522051 -0.1436045799
## PC28 PC29 PC30
## InflationRate 0.3155490194 -0.073916986 -0.351821658
## RetailSalesMom -0.1483574214 0.407390770 -0.221692699
## AverageHourlyEarnings 0.0454226448 0.055407663 0.011888379
## BankLendingRate -0.0940714859 -0.029234208 -0.168832057
## ConstructionSpending 0.0926849465 0.033514795 -0.028657809
## EmploymentRate -0.2383592596 -0.261110408 0.012863711
## HousingIndex -0.0478265666 0.001328925 -0.065689333
## InflationRateMoM 0.0687277729 0.038883390 0.260702785
## LaborForceParticipationRate 0.0007246506 -0.495082241 -0.097999256
## LongTermUnemploymentRate -0.1171659069 -0.077763737 -0.111796135
## PendingHomeSales -0.1596347619 -0.072490590 -0.161604262
## PersonalIncome -0.0042034430 -0.003278201 0.052656917
## PersonalSavings 0.1993878123 0.035765125 -0.263453100
## PersonalSpending 0.0620752995 -0.058406634 -0.040536899
## RetailSalesExcludingAutos 0.2073256289 -0.369711539 0.350393016
## WageGrowth -0.3227335670 0.018187566 0.257251646
## YouthUnemploymentRate -0.0338748301 0.009139408 0.060088058
## CoreInflationRate -0.1071247981 0.169462584 0.191999665
## RetailSalesYoY 0.0604614810 -0.016603627 -0.124025866
## USGovernment10yrBondRate 0.2517745795 0.191141085 0.023251023
## UnemploymentRate -0.0085853606 -0.009231402 0.031876360
## InitialJoblessClaims -0.1714457493 0.013009700 -0.145126074
## NonfarmPayrolls 0.0137494972 0.047116457 0.041494628
## ADPEmploymentChange -0.0390262386 0.021320379 -0.223176562
## BuildingPermits 0.1468691914 0.111648281 -0.008462547
## ContinuingJoblessClaims -0.0002140546 -0.046022036 -0.007901849
## EmployedPersons -0.0650423105 -0.048899189 -0.031277564
## ExistingHomeSales -0.1512976632 0.201242456 0.210598661
## FulltimeEmployment 0.1770099454 0.031992097 0.358734516
## GovernmentPayrolls -0.0546815043 0.094765653 0.037663998
## HousingStarts -0.0098422547 -0.040988235 0.062951528
## JobOffers 0.0917243683 -0.030969430 0.009072857
## JobVacancies -0.0285847476 -0.193886939 -0.073229749
## ManufacturingPayrolls -0.4052530871 -0.155018005 -0.018168084
## NewHomeSales 0.1606587772 -0.022457413 0.075463211
## NonfarmPayrollsPrivate 0.0610395123 0.064364912 0.115282959
## UnemployedPersons -0.0161304421 -0.012533148 0.037459717
## ChainStoreSales -0.0429226252 -0.187850938 0.035871719
## ConsumerCredit 0.0270170650 0.045637234 0.009643879
## Wages -0.0716505600 0.046299809 -0.045387279
## WagesinManufacturing -0.0701794447 0.081959704 -0.052069253
## GasolinePrices -0.0834569887 0.007329112 -0.042584712
## ExchangeRate -0.1512976632 0.201242456 0.210598661
## StockMarket -0.0882510503 -0.017548537 -0.072973884
## ConsumerPriceIndex 0.0384966876 0.059042328 0.010918304
## CoreConsumerPrices -0.0414775395 0.044305861 -0.034553778
## EconomicOptimism 0.1246996806 -0.040775140 0.083580193
## LaborMarketConditions -0.1688698954 -0.226538426 0.073552684
## NAHBHousingMarket -0.0705705207 -0.008708897 0.001387577
## AverageWeeklyHours 0.2547788829 -0.070691053 0.084295789
## ChallengerJobCuts 0.0213361140 -0.004635394 -0.001814011
## PC31 PC32 PC33
## InflationRate -0.5202155790 0.14205165 0.279532586
## RetailSalesMom -0.0925357854 -0.30291688 -0.178711660
## AverageHourlyEarnings 0.0519719406 0.04613762 0.032416726
## BankLendingRate -0.0887394679 -0.14831382 0.316360813
## ConstructionSpending 0.0873016610 0.02497629 0.035529309
## EmploymentRate -0.1784549369 -0.01564240 -0.461008201
## HousingIndex -0.0831641643 0.01533710 0.022918608
## InflationRateMoM -0.2009721875 0.02080264 -0.246059491
## LaborForceParticipationRate 0.0849513634 -0.33259529 0.182829639
## LongTermUnemploymentRate 0.1999952893 0.12798738 0.088883066
## PendingHomeSales -0.1111974764 -0.01787191 -0.032805748
## PersonalIncome -0.0479636422 -0.02130645 -0.024110997
## PersonalSavings 0.2410651066 0.11331513 -0.068746037
## PersonalSpending 0.0585404470 0.02970420 0.054785415
## RetailSalesExcludingAutos 0.0060608555 0.26445946 0.133592613
## WageGrowth -0.2113372293 -0.04535802 0.170730015
## YouthUnemploymentRate -0.1487860352 -0.15577058 0.108219908
## CoreInflationRate -0.0353702133 -0.11514732 0.289853450
## RetailSalesYoY 0.2431836335 0.31339515 0.071131225
## USGovernment10yrBondRate 0.0007983696 -0.18704211 0.126831826
## UnemploymentRate -0.0902852719 -0.05175304 0.178802777
## InitialJoblessClaims 0.0413305540 0.01792504 -0.047270053
## NonfarmPayrolls -0.0479728163 -0.02483072 0.006336534
## ADPEmploymentChange -0.0167391055 -0.06702657 0.009782106
## BuildingPermits -0.2726408449 0.10669973 -0.080989500
## ContinuingJoblessClaims -0.1378657226 0.01467354 -0.110500693
## EmployedPersons -0.0241728520 0.03641923 -0.150239163
## ExistingHomeSales -0.0153337815 0.22671851 0.164951036
## FulltimeEmployment 0.2119489017 -0.41500863 -0.074675213
## GovernmentPayrolls -0.0021061361 0.04430186 0.023704231
## HousingStarts 0.1818626866 0.15573792 0.050832040
## JobOffers -0.1012540559 -0.05638816 -0.056258959
## JobVacancies 0.1460864735 -0.10951526 -0.003880738
## ManufacturingPayrolls 0.0117937800 0.09259464 0.094842816
## NewHomeSales -0.0640505954 0.25425028 -0.161031943
## NonfarmPayrollsPrivate 0.0061670202 -0.04797650 -0.011273724
## UnemployedPersons -0.1057466627 -0.02980252 0.174839251
## ChainStoreSales 0.0381512874 -0.12673229 0.032903193
## ConsumerCredit -0.0707195375 0.03415182 -0.029580092
## Wages 0.0126490891 0.03154025 -0.003990587
## WagesinManufacturing 0.0654934756 -0.02894238 0.221768338
## GasolinePrices 0.0217076519 0.04596166 -0.134865284
## ExchangeRate -0.0153337815 0.22671851 0.164951036
## StockMarket 0.0961270600 -0.02734364 0.023908538
## ConsumerPriceIndex -0.0289767113 0.03982748 -0.069038001
## CoreConsumerPrices -0.0109839202 0.04281880 -0.024035727
## EconomicOptimism 0.0114642486 -0.13083212 -0.014785190
## LaborMarketConditions -0.2833470536 0.01343645 -0.006666579
## NAHBHousingMarket -0.0817756716 0.06548758 -0.121708144
## AverageWeeklyHours -0.1780555077 -0.12405357 -0.015683469
## ChallengerJobCuts 0.0231932230 0.04473268 -0.033789209
## PC34 PC35 PC36
## InflationRate 1.410073e-01 -0.3486008757 -0.1304010902
## RetailSalesMom -9.034019e-02 -0.0742360049 0.0845571165
## AverageHourlyEarnings 5.416276e-05 0.0204678717 0.0001136121
## BankLendingRate -8.646298e-02 0.0854858358 -0.1414253804
## ConstructionSpending -1.472654e-02 -0.0050949392 0.0424008164
## EmploymentRate 3.224096e-01 0.0244430505 -0.0487373680
## HousingIndex -5.159277e-02 0.0146477461 0.0252971301
## InflationRateMoM -3.539708e-01 -0.2663116613 0.0920441764
## LaborForceParticipationRate -3.844389e-01 0.0717802033 0.0870706360
## LongTermUnemploymentRate -8.727725e-02 0.0299257434 0.0152397689
## PendingHomeSales -8.473739e-03 -0.0693269550 -0.0693305646
## PersonalIncome 2.558211e-02 0.0102772989 0.0214150547
## PersonalSavings -5.926468e-02 -0.0340494791 0.0993166441
## PersonalSpending 7.210329e-03 0.0300302444 0.0261501471
## RetailSalesExcludingAutos 6.349718e-02 -0.0104488387 -0.0627642549
## WageGrowth -7.348553e-03 0.0078842207 -0.1235958847
## YouthUnemploymentRate -7.795468e-03 0.0233549303 0.3354431924
## CoreInflationRate 2.272866e-01 0.2074972032 -0.0496574652
## RetailSalesYoY 1.550818e-01 0.2062709297 -0.0759964304
## USGovernment10yrBondRate 1.801848e-01 0.0505907656 0.0831572817
## UnemploymentRate -6.354436e-03 0.0525053937 0.1726935143
## InitialJoblessClaims 1.555175e-02 0.0191048395 -0.0436762672
## NonfarmPayrolls 1.340362e-02 -0.0261160554 -0.0306336956
## ADPEmploymentChange -6.818665e-02 0.0618630844 0.0287023204
## BuildingPermits -1.914263e-01 0.7822080285 -0.0113077330
## ContinuingJoblessClaims -7.921698e-02 0.0777129451 -0.2218071353
## EmployedPersons 1.273043e-01 0.0214623966 0.0095771370
## ExistingHomeSales -2.104081e-01 -0.1070426716 0.0142976839
## FulltimeEmployment 1.226865e-01 -0.0499956535 -0.1436807005
## GovernmentPayrolls 2.031551e-02 -0.0158745815 -0.0579104657
## HousingStarts 9.779457e-02 -0.0015141302 -0.0385281800
## JobOffers -2.716417e-01 0.0439284170 -0.6157158760
## JobVacancies -6.112557e-02 -0.0681380920 0.0747882211
## ManufacturingPayrolls -5.090468e-02 -0.0600404748 -0.0951818586
## NewHomeSales -1.387602e-01 -0.0274541372 0.2922427894
## NonfarmPayrollsPrivate 6.385928e-02 -0.0491599821 -0.0647765739
## UnemployedPersons 3.403557e-03 0.0511990565 0.1729588876
## ChainStoreSales -5.108543e-02 0.0479922541 0.0024246507
## ConsumerCredit -1.747373e-02 -0.0294041006 0.0035721351
## Wages 5.817825e-02 -0.0036757470 0.0157812530
## WagesinManufacturing 2.329901e-01 -0.0581407847 -0.0298256705
## GasolinePrices 9.023047e-02 0.0354920710 -0.0973061736
## ExchangeRate -2.104081e-01 -0.1070426716 0.0142976839
## StockMarket -1.835926e-01 0.0237399763 0.1642581044
## ConsumerPriceIndex -1.620489e-01 -0.0298037153 0.1068274685
## CoreConsumerPrices 3.584595e-02 -0.0008570194 0.0085432804
## EconomicOptimism 1.610386e-02 -0.0200902305 -0.0121689012
## LaborMarketConditions 2.136215e-01 0.1096703915 0.3106748364
## NAHBHousingMarket 7.619947e-03 -0.0261759343 -0.0460480228
## AverageWeeklyHours 5.551126e-02 0.0032488578 0.0287364361
## ChallengerJobCuts 7.391113e-03 0.0394889913 0.0354498795
## PC37 PC38 PC39
## InflationRate 0.187963892 3.627517e-02 0.080517194
## RetailSalesMom -0.055554604 6.748041e-02 -0.034116460
## AverageHourlyEarnings 0.025095712 4.420358e-02 0.019881979
## BankLendingRate -0.037869428 2.024999e-01 0.214906766
## ConstructionSpending -0.024761235 2.565212e-02 -0.034042604
## EmploymentRate 0.176709219 7.615551e-02 -0.014374567
## HousingIndex 0.003233624 3.676296e-05 -0.004709762
## InflationRateMoM -0.038045969 -1.893174e-01 0.101080188
## LaborForceParticipationRate -0.155349923 2.883988e-02 -0.080187816
## LongTermUnemploymentRate -0.040674529 -2.296358e-02 -0.010156723
## PendingHomeSales -0.087614235 -3.528356e-02 0.019859153
## PersonalIncome -0.040809662 2.552010e-02 -0.012416162
## PersonalSavings 0.109734396 -1.508195e-01 -0.016981259
## PersonalSpending 0.017703099 1.152895e-02 0.006794680
## RetailSalesExcludingAutos 0.056740236 -9.249322e-02 0.027970597
## WageGrowth -0.059150082 1.176856e-01 0.027326547
## YouthUnemploymentRate 0.057266857 -1.510801e-01 0.027650296
## CoreInflationRate 0.034848831 3.205329e-01 -0.141766674
## RetailSalesYoY 0.055312409 1.056666e-01 -0.033519390
## USGovernment10yrBondRate -0.030008227 4.321711e-02 -0.018302579
## UnemploymentRate 0.018498839 -9.654885e-02 -0.142042812
## InitialJoblessClaims 0.010475037 5.448473e-02 0.018795134
## NonfarmPayrolls 0.005736055 3.837364e-03 0.006816416
## ADPEmploymentChange -0.030234894 2.936433e-02 -0.009106397
## BuildingPermits 0.024982053 -2.174648e-01 0.193786792
## ContinuingJoblessClaims -0.138726994 -3.702658e-03 -0.192865670
## EmployedPersons -0.079106080 -6.805086e-02 -0.075117616
## ExistingHomeSales 0.060855910 -3.462924e-02 0.037336826
## FulltimeEmployment 0.161758784 -2.932968e-02 0.035688410
## GovernmentPayrolls -0.033707200 4.100558e-02 0.006676598
## HousingStarts -0.004305277 -1.831576e-02 0.025932787
## JobOffers 0.018815276 1.059854e-02 -0.420424391
## JobVacancies -0.104184522 1.950394e-02 0.215976996
## ManufacturingPayrolls 0.025159779 -1.491815e-01 0.089489711
## NewHomeSales -0.419503896 5.082785e-01 -0.326874668
## NonfarmPayrollsPrivate 0.005143664 5.304791e-03 0.013608062
## UnemployedPersons -0.012839510 -1.388043e-01 -0.139504965
## ChainStoreSales -0.007494216 1.299708e-02 0.029795387
## ConsumerCredit 0.004231674 2.523068e-02 -0.027481819
## Wages -0.116431512 -2.809367e-01 -0.082931769
## WagesinManufacturing -0.342809020 -4.528870e-01 -0.410963545
## GasolinePrices -0.046098065 -1.084970e-01 -0.011257525
## ExchangeRate 0.060855910 -3.462924e-02 0.037336826
## StockMarket 0.667756787 4.751946e-02 -0.465800297
## ConsumerPriceIndex 0.126552615 -7.890346e-02 -0.035631357
## CoreConsumerPrices -0.093769639 -1.609563e-01 -0.099358745
## EconomicOptimism -0.069140902 -4.881925e-02 0.027413045
## LaborMarketConditions 0.057914012 8.778490e-03 -0.157828814
## NAHBHousingMarket -0.007697051 -3.419793e-02 -0.009971186
## AverageWeeklyHours -0.019435452 -1.223467e-01 -0.086375988
## ChallengerJobCuts 0.027261584 -3.695634e-02 -0.017148944
## PC40 PC41 PC42
## InflationRate 0.0091733257 0.009257981 0.0126091470
## RetailSalesMom -0.0369777085 0.065302350 -0.0482760648
## AverageHourlyEarnings -0.0173883963 -0.027796081 0.0196467033
## BankLendingRate -0.0201552587 -0.058226767 0.0251195760
## ConstructionSpending 0.0002323835 -0.035847809 0.0223383825
## EmploymentRate -0.0331853961 -0.288649441 -0.1760192994
## HousingIndex 0.0006341283 -0.015897380 0.0019341075
## InflationRateMoM -0.0431564080 0.012903749 -0.0585740794
## LaborForceParticipationRate -0.3101093613 -0.090415938 -0.0041589589
## LongTermUnemploymentRate 0.0203327357 0.009991426 -0.0076124132
## PendingHomeSales -0.0402239968 0.002990245 0.0067892585
## PersonalIncome 0.0121604729 0.003091253 -0.0109107239
## PersonalSavings -0.0177073194 -0.018932032 -0.0174303015
## PersonalSpending 0.0221810538 -0.029215384 0.0189315522
## RetailSalesExcludingAutos 0.0007238496 -0.055974923 0.0453232550
## WageGrowth 0.0145115236 0.032838044 0.0451898388
## YouthUnemploymentRate 0.3478292248 -0.701075003 0.1076899011
## CoreInflationRate 0.0896157263 -0.039479716 0.0728550394
## RetailSalesYoY 0.0516893539 -0.072529651 0.0147063666
## USGovernment10yrBondRate 0.0179639759 0.001498483 0.0283667706
## UnemploymentRate 0.0494271275 0.198471739 0.0149846559
## InitialJoblessClaims 0.0109894401 0.027411181 -0.0088901077
## NonfarmPayrolls -0.0116556400 -0.009005942 -0.0015237929
## ADPEmploymentChange -0.0134840086 -0.012480489 -0.0001164001
## BuildingPermits -0.0030167677 0.016667070 -0.1330303322
## ContinuingJoblessClaims -0.0310859309 -0.061295477 0.7441699026
## EmployedPersons 0.1382765091 -0.020051667 0.0338240238
## ExistingHomeSales -0.0109394226 -0.038051502 -0.0209324772
## FulltimeEmployment -0.0369060185 0.097015690 0.0241462746
## GovernmentPayrolls -0.0075968901 -0.003635186 -0.0022673232
## HousingStarts -0.0090674053 0.001209650 -0.0036795917
## JobOffers 0.3306175542 -0.013476217 -0.2932478796
## JobVacancies 0.7480426382 0.240452497 0.0022958437
## ManufacturingPayrolls 0.0055357309 0.015012364 -0.0320274674
## NewHomeSales 0.0602599877 -0.054582075 -0.0632764531
## NonfarmPayrollsPrivate -0.0167664475 0.001407485 -0.0038144051
## UnemployedPersons 0.0627080423 0.217553624 -0.0089490794
## ChainStoreSales 0.0011517425 -0.010403569 0.0223414193
## ConsumerCredit 0.0014433113 -0.010294481 -0.0034621899
## Wages 0.1474074144 0.093120272 0.1530604663
## WagesinManufacturing -0.1128263002 -0.150647776 -0.2507178456
## GasolinePrices -0.0764642629 0.111167493 0.3363745257
## ExchangeRate -0.0109394226 -0.038051502 -0.0209324772
## StockMarket 0.0349763295 0.003489999 0.0895567073
## ConsumerPriceIndex 0.0287095169 0.102362045 0.1307686966
## CoreConsumerPrices 0.1377028891 0.024512477 0.1930161275
## EconomicOptimism 0.0108419906 0.013574283 0.0050924701
## LaborMarketConditions -0.0181260409 0.412750968 -0.0897371221
## NAHBHousingMarket -0.0042464915 0.008571021 -0.0293918470
## AverageWeeklyHours 0.0060642840 0.049149150 0.0429952227
## ChallengerJobCuts -0.0145643528 -0.026064721 0.0010826022
## PC43 PC44 PC45
## InflationRate 0.067434069 0.006792346 0.0290181582
## RetailSalesMom 0.032588460 0.009187244 0.0003187991
## AverageHourlyEarnings -0.002364292 -0.003651108 -0.0135634466
## BankLendingRate -0.005096527 0.001829502 -0.0169262872
## ConstructionSpending -0.009532157 -0.012844689 -0.0134856350
## EmploymentRate -0.126047763 0.230905635 0.2566467067
## HousingIndex -0.005957373 -0.007916221 -0.0048082489
## InflationRateMoM 0.206300414 -0.035213456 0.0254758599
## LaborForceParticipationRate -0.070895873 -0.042153394 -0.0681271367
## LongTermUnemploymentRate -0.018240413 -0.011345936 -0.0298329948
## PendingHomeSales -0.023994153 -0.000629528 0.0007615252
## PersonalIncome -0.010660236 -0.007298384 0.0122458211
## PersonalSavings 0.007532658 0.016432216 -0.0171119532
## PersonalSpending -0.036263124 -0.003785650 0.0039644972
## RetailSalesExcludingAutos 0.009972916 -0.015612832 -0.0058061800
## WageGrowth 0.032040098 0.016702843 0.0113762488
## YouthUnemploymentRate -0.087009522 -0.212819965 -0.0731516458
## CoreInflationRate -0.018759503 -0.018110190 -0.0121536445
## RetailSalesYoY -0.061586771 0.011293747 0.0161226874
## USGovernment10yrBondRate -0.107298258 -0.031138268 -0.0175524968
## UnemploymentRate -0.004651277 0.130118850 0.3415159849
## InitialJoblessClaims 0.001105546 0.012010507 -0.0051174228
## NonfarmPayrolls -0.006457230 -0.003493731 0.0019079475
## ADPEmploymentChange -0.025520641 0.008005040 -0.0121174837
## BuildingPermits 0.136179273 0.060373471 0.0646372374
## ContinuingJoblessClaims 0.121783516 0.339030848 0.1011327681
## EmployedPersons 0.132696099 -0.197291333 -0.2537675737
## ExistingHomeSales -0.032622617 0.015744470 0.0069000971
## FulltimeEmployment 0.048179093 0.064668402 -0.0152457371
## GovernmentPayrolls 0.011645259 -0.002939559 0.0006422443
## HousingStarts 0.002897818 0.014397637 -0.0112796501
## JobOffers -0.181418333 -0.178967331 -0.0943895265
## JobVacancies 0.136712644 0.111187858 0.2508746595
## ManufacturingPayrolls 0.026964590 0.012871008 0.0123087932
## NewHomeSales 0.053746993 -0.010195458 0.0658743950
## NonfarmPayrollsPrivate -0.003759495 0.002078938 0.0068676750
## UnemployedPersons -0.027282598 -0.036899114 0.2963423358
## ChainStoreSales -0.018840731 -0.014086579 -0.0003564059
## ConsumerCredit 0.006174110 -0.006693848 -0.0079792661
## Wages -0.042611478 0.163086880 -0.4265753358
## WagesinManufacturing 0.092607477 0.120834704 0.2458914531
## GasolinePrices -0.021109189 -0.772766012 0.3165729760
## ExchangeRate -0.032622617 0.015744470 0.0069000971
## StockMarket 0.298627942 0.011463520 0.0207315989
## ConsumerPriceIndex -0.822291493 0.115017753 0.1359529750
## CoreConsumerPrices -0.074453905 0.052896494 -0.2912734142
## EconomicOptimism 0.004453046 -0.013252822 -0.0051090806
## LaborMarketConditions -0.091023408 -0.120328700 -0.3342232297
## NAHBHousingMarket 0.033874420 0.014375231 -0.0078011484
## AverageWeeklyHours 0.012640216 -0.009177435 -0.0341634788
## ChallengerJobCuts 0.003313355 -0.016955816 -0.0069486044
## PC46 PC47 PC48
## InflationRate 0.0028404293 1.128712e-02 1.857335e-02
## RetailSalesMom -0.0140592214 7.426725e-03 -1.056149e-03
## AverageHourlyEarnings 0.0060856023 -1.243370e-02 -6.821241e-04
## BankLendingRate 0.0062070164 -2.011299e-02 1.682453e-02
## ConstructionSpending 0.0092092387 5.542114e-03 2.731776e-03
## EmploymentRate 0.2291979769 5.650646e-02 -5.864882e-03
## HousingIndex 0.0039905987 -4.867378e-03 -3.044383e-03
## InflationRateMoM -0.0083710403 1.166963e-02 -1.194411e-02
## LaborForceParticipationRate 0.0819407326 -2.724437e-02 6.023504e-03
## LongTermUnemploymentRate 0.0308689162 -1.337563e-02 -4.516728e-04
## PendingHomeSales -0.0010804941 -6.687501e-03 2.706495e-03
## PersonalIncome 0.0091361045 3.421179e-04 -2.100638e-03
## PersonalSavings -0.0080399121 1.406780e-02 1.835656e-03
## PersonalSpending 0.0062088872 7.195417e-03 -2.594073e-03
## RetailSalesExcludingAutos 0.0078015115 -1.036817e-03 1.003636e-03
## WageGrowth 0.0035883740 -1.873410e-02 -2.560646e-03
## YouthUnemploymentRate -0.0429132789 7.307641e-02 1.598656e-02
## CoreInflationRate 0.0133461425 2.630706e-02 3.610519e-02
## RetailSalesYoY 0.0106209352 -8.016881e-03 2.859478e-03
## USGovernment10yrBondRate 0.0192464902 -2.479863e-02 1.549729e-03
## UnemploymentRate 0.4804380107 -2.092164e-01 -4.508812e-02
## InitialJoblessClaims 0.0005586965 -5.952668e-03 6.975617e-05
## NonfarmPayrolls -0.0047120682 6.851872e-04 2.871768e-03
## ADPEmploymentChange 0.0069254085 -7.007564e-03 -4.766344e-03
## BuildingPermits -0.0597121687 2.917606e-02 -1.129751e-02
## ContinuingJoblessClaims -0.2299581496 -3.124695e-02 5.076950e-02
## EmployedPersons -0.0303486805 -8.219024e-01 2.531054e-01
## ExistingHomeSales -0.0014637164 -2.178190e-02 6.835943e-03
## FulltimeEmployment -0.0127352061 9.673801e-04 3.364875e-03
## GovernmentPayrolls 0.0059273722 -2.826891e-03 3.603638e-03
## HousingStarts 0.0041459685 5.189278e-04 6.597381e-04
## JobOffers 0.0549771038 3.478602e-02 -7.121931e-04
## JobVacancies -0.1983764917 8.045858e-02 -2.479415e-02
## ManufacturingPayrolls -0.0372135802 5.954568e-03 -3.065446e-03
## NewHomeSales 0.0613556612 9.448205e-02 3.468160e-02
## NonfarmPayrollsPrivate -0.0100510351 5.218881e-03 6.662250e-03
## UnemployedPersons 0.2591515517 -1.287037e-01 1.186311e-01
## ChainStoreSales 0.0152366108 -5.839605e-03 1.386625e-04
## ConsumerCredit 0.0001488614 1.050767e-02 -5.073365e-04
## Wages 0.4067747182 3.721902e-01 5.049237e-01
## WagesinManufacturing -0.3343371710 4.937854e-02 -3.178991e-02
## GasolinePrices 0.0739212661 2.073138e-01 4.222455e-03
## ExchangeRate -0.0014637164 -2.178190e-02 6.835943e-03
## StockMarket -0.0683277219 4.516588e-02 2.978455e-03
## ConsumerPriceIndex -0.2193820617 -1.337871e-01 8.799713e-02
## CoreConsumerPrices 0.2714857974 -6.061687e-02 -8.024771e-01
## EconomicOptimism 0.0065644829 2.111198e-02 7.754935e-05
## LaborMarketConditions -0.3518314614 1.549115e-01 -6.934889e-02
## NAHBHousingMarket -0.0008253694 8.225284e-04 -4.564658e-04
## AverageWeeklyHours -0.0041974571 6.881658e-03 -1.175267e-02
## ChallengerJobCuts 0.0054411248 -8.759441e-06 9.294993e-05
## PC49 PC50 PC51
## InflationRate 0.0027976194 0.000000e+00 0.000000e+00
## RetailSalesMom -0.0016905302 -4.098284e-16 -5.637851e-17
## AverageHourlyEarnings -0.0001442473 7.295130e-17 -2.943257e-16
## BankLendingRate -0.0090142121 2.644447e-16 -2.162063e-17
## ConstructionSpending -0.0009612439 -3.223451e-17 7.716831e-17
## EmploymentRate -0.0670294754 -7.906434e-16 -1.928264e-16
## HousingIndex 0.0006610187 -4.554103e-01 -8.165401e-03
## InflationRateMoM 0.0100386724 1.049508e-16 1.279359e-17
## LaborForceParticipationRate -0.0153642670 -5.248623e-16 1.094587e-16
## LongTermUnemploymentRate -0.0025804839 6.288373e-17 -4.445229e-17
## PendingHomeSales 0.0026926303 -1.431147e-16 -1.034600e-16
## PersonalIncome -0.0025719680 2.255141e-17 3.759471e-17
## PersonalSavings 0.0021849593 -1.235990e-16 -5.402376e-17
## PersonalSpending 0.0005489337 -8.933826e-17 -6.900947e-17
## RetailSalesExcludingAutos 0.0039163754 3.174544e-16 1.662217e-16
## WageGrowth -0.0075768340 -1.613293e-16 -2.200930e-17
## YouthUnemploymentRate 0.0395218501 1.299742e-15 -3.808938e-17
## CoreInflationRate -0.0055282191 4.618972e-16 -9.128262e-17
## RetailSalesYoY -0.0030165901 -2.775558e-17 -4.128100e-17
## USGovernment10yrBondRate -0.0072189292 1.058181e-16 -8.527250e-17
## UnemploymentRate 0.6136725875 2.238877e-15 -1.080790e-15
## InitialJoblessClaims -0.0023766172 2.265983e-17 -1.164331e-17
## NonfarmPayrolls 0.0013759048 7.226867e-01 1.295761e-02
## ADPEmploymentChange 0.0001864769 1.387779e-17 -2.320193e-17
## BuildingPermits 0.0095732424 8.430756e-16 9.413585e-17
## ContinuingJoblessClaims -0.0121018813 4.857226e-16 3.352895e-17
## EmployedPersons 0.0244529565 1.145106e-15 -4.296151e-18
## ExistingHomeSales 0.0046066569 -1.267622e-02 7.069931e-01
## FulltimeEmployment -0.0073344205 -2.149183e-16 -2.775558e-17
## GovernmentPayrolls 0.0003138645 -1.189728e-16 -5.551115e-17
## HousingStarts 0.0020724932 2.218743e-16 -1.267432e-16
## JobOffers 0.0272374046 -3.216188e-16 1.908196e-16
## JobVacancies 0.0184007359 1.564932e-15 1.743397e-16
## ManufacturingPayrolls 0.0142474854 3.295975e-17 3.816392e-17
## NewHomeSales -0.0021425704 3.747003e-16 -3.642919e-17
## NonfarmPayrollsPrivate 0.0013342727 -5.196190e-01 -9.316648e-03
## UnemployedPersons -0.7581253716 -2.663668e-15 1.722580e-15
## ChainStoreSales -0.0025577510 7.285839e-17 1.431147e-16
## ConsumerCredit 0.0030114982 1.097213e-16 -2.656295e-17
## Wages 0.0227985719 1.903425e-15 -4.280430e-16
## WagesinManufacturing 0.0504790701 -5.212844e-16 1.582935e-16
## GasolinePrices 0.0863546948 1.891282e-15 2.957704e-16
## ExchangeRate 0.0046066569 1.267622e-02 -7.069931e-01
## StockMarket -0.0183903905 8.326673e-17 3.469447e-17
## ConsumerPriceIndex 0.0296354822 -1.131907e-15 9.324139e-17
## CoreConsumerPrices -0.1326079591 -3.405262e-15 -2.602085e-17
## EconomicOptimism 0.0047904238 2.723516e-16 7.372575e-17
## LaborMarketConditions 0.1018140860 3.295975e-17 -2.966377e-16
## NAHBHousingMarket -0.0015666157 3.816392e-17 -7.979728e-17
## AverageWeeklyHours 0.0023356143 4.163336e-17 -2.428613e-17
## ChallengerJobCuts 0.0019051757 -1.734723e-17 9.020562e-17# Principal components scores result if the loadings of the variables
# (listed under rotation) are applied to the values on the
# original variables. We achieve this with the predict function:
tepc <- predict(te_vars.prc)
tepc## PC1 PC2 PC3 PC4 PC5
## [1,] -7.99093397 5.433911640 0.9940121098 -1.25570220 -1.605403812
## [2,] -8.44080214 6.633875136 -0.0005955268 0.11261463 -1.393532665
## [3,] -7.98395167 2.070547323 -2.3078722976 -1.48540652 -1.731640075
## [4,] -7.17176410 0.498676857 -0.8589208224 -2.13610962 -3.434985661
## [5,] -6.10001210 0.842769913 1.2031202522 -1.92744314 -6.298400500
## [6,] -7.28472501 4.362366163 -0.3459038583 2.50476069 0.406199742
## [7,] -7.34055319 3.955858831 -0.3416739751 1.93292917 1.307037271
## [8,] -6.82376424 3.143892255 -0.9299586103 2.49912301 1.836695683
## [9,] -7.22316354 2.564564434 -1.6015813225 2.12408186 1.508575126
## [10,] -6.65760570 0.020720774 -2.6778771023 0.87829718 0.143726088
## [11,] -7.23796181 1.073776765 -2.1291861740 0.79030730 0.009122793
## [12,] -6.54659129 0.629947853 -1.4461726837 3.08407528 2.060618637
## [13,] -6.45056956 2.113644642 -2.1074184829 -0.14750933 0.862766460
## [14,] -6.48892654 -0.873652609 -2.8007706345 -0.46356584 -0.125967996
## [15,] -6.26819348 -1.865145471 -3.5116962591 0.17672150 0.085752555
## [16,] -5.74923095 -3.396002751 -2.6274032183 -0.23460892 -1.028977015
## [17,] -5.78254413 0.102611451 0.3707848357 -0.79706471 0.536253294
## [18,] -5.38484470 -1.444146481 0.6404020449 -1.95598760 -0.341320005
## [19,] -5.30467872 -0.573574079 0.5103153183 -1.40979041 1.886591128
## [20,] -5.73629905 -0.835429424 0.7854279411 -1.52244191 0.651969604
## [21,] -4.88879382 -1.914059461 0.3892963264 -3.28782971 0.830840479
## [22,] -4.98974018 -0.511930883 -0.4386534163 -1.64398190 0.305283423
## [23,] -4.50910716 -0.209554950 1.1938169613 -1.01822846 0.167488496
## [24,] -3.99035866 -1.738955402 2.1836573685 -1.50530696 -0.538921363
## [25,] -3.33139808 -3.717382346 -2.1101382189 -2.66148696 0.210605638
## [26,] -3.25932546 -2.722953429 -1.4661545559 -2.45411816 0.934999219
## [27,] -2.50482652 -2.435570381 0.8830669581 -2.33039031 0.458138844
## [28,] -2.40617373 0.261830748 2.8555152411 -1.45843944 1.415791093
## [29,] -2.24460350 0.639865040 3.9432688427 -1.38057720 0.811930811
## [30,] -2.07330081 2.079605530 4.9539212905 -0.73308160 1.149608928
## [31,] -2.20517054 0.291786962 1.4227683744 -0.73092314 1.075777760
## [32,] -2.27023824 -0.604206491 -0.2454892706 -0.51360074 2.075761359
## [33,] -1.71252029 -0.762681367 -0.5911471757 -1.23395387 2.522713863
## [34,] -1.24340011 -0.910929637 1.6887404685 -1.56729720 1.754542117
## [35,] -1.45606504 -0.094359590 2.6917144410 -1.06062651 3.328788037
## [36,] -1.62212287 -3.395560747 2.8991855046 -0.41979309 3.942257247
## [37,] -1.28411983 -0.315440126 -0.1363885917 -0.12259930 -1.749462957
## [38,] -0.83918669 -2.301980277 -1.7233951830 -0.71046697 -0.137893305
## [39,] -0.62511104 0.409409835 4.1651333995 1.02634257 -1.734462116
## [40,] -0.43827522 0.931314938 3.0888941649 0.33192189 -1.289274907
## [41,] -0.32766103 -0.719377051 1.2238504140 0.30056510 -0.125822779
## [42,] -0.80860920 -1.740345703 1.0480651213 1.60223269 -0.248371752
## [43,] -0.54000502 -0.114108147 0.4427248209 1.52630666 -0.156817739
## [44,] -0.39390380 -1.990420005 1.1721364212 1.03730878 -1.688125309
## [45,] -0.44237219 -1.684124242 0.5558717333 2.76101270 -1.325690359
## [46,] -0.62401829 -1.404357038 0.6559198632 2.51022437 -1.479031988
## [47,] -0.08459974 -1.976383559 -0.3663189885 2.26375260 -1.442247914
## [48,] -0.71921887 0.080725403 0.1664890987 4.25372740 -0.634148383
## [49,] 0.64524954 -0.642591620 2.1204618908 2.04415388 -1.136284088
## [50,] 0.08340920 -0.786335757 -1.4728938785 2.37534557 0.250416990
## [51,] 0.18778653 -3.535325003 -1.5104983562 2.71019859 -0.393552127
## [52,] 1.06038678 -3.515581052 -1.7701740890 0.69466856 -1.028853856
## [53,] 0.91829282 -1.878412292 -0.1192827020 0.51849732 -0.459376425
## [54,] 1.09110952 -2.694993141 -0.3311663714 1.27508876 -1.248903201
## [55,] 1.66273264 -1.723058654 0.3458135918 0.72775194 -0.162649069
## [56,] 1.55561913 -1.729907515 -1.2851139821 0.73496354 0.196994691
## [57,] 1.87787519 -1.984428610 0.7268677146 0.89906485 -1.372879463
## [58,] 2.56296460 -2.053147231 -0.2501091357 0.28314375 -0.383918554
## [59,] 2.67226862 -2.346960356 0.2361348058 0.48002071 -0.574875776
## [60,] 3.24413458 -2.007753407 3.3468711425 1.77499765 -1.532373242
## [61,] 3.86786050 0.980979679 3.7538016133 -0.15269463 -1.616750560
## [62,] 3.97233037 -0.492707882 1.2289620548 0.54715876 0.095476205
## [63,] 3.81695834 0.602416306 -1.7001836091 1.32175930 1.590051661
## [64,] 4.95980037 -0.236616788 1.2608091353 0.33199455 0.078449368
## [65,] 4.50617195 -2.302308931 -1.5558960967 0.52038088 0.236979227
## [66,] 4.77147943 -0.495107721 0.5969862340 1.14254044 0.006821861
## [67,] 4.84664863 -0.846134753 -1.0663910391 -0.46401967 0.684954290
## [68,] 4.97584041 1.333660693 1.3012163841 0.97589669 0.577805132
## [69,] 4.95114501 1.345274081 1.6681936253 1.15774182 0.804599528
## [70,] 5.25294223 -0.157138324 0.8909066466 -0.05264652 0.130988128
## [71,] 5.15793956 -0.006559292 -0.1993994563 0.70886010 0.679772125
## [72,] 5.33066668 -0.089246052 0.0244650777 0.22541356 0.206355363
## [73,] 6.15599904 1.660778736 2.3931317706 -1.52555274 -0.138273214
## [74,] 5.89700500 0.767226901 -0.1180037744 -1.22640384 0.530605987
## [75,] 6.21173269 1.448567615 0.7800214298 -0.58437712 0.663918723
## [76,] 5.66943030 1.535785387 -2.2741566457 -0.35460827 2.257670626
## [77,] 5.81602997 3.386884798 -0.1893841007 0.59054126 1.944725114
## [78,] 6.04379076 -0.606081618 -1.7370729087 -0.65458566 0.262020697
## [79,] 6.64646115 0.256478076 0.2887125221 -0.90181707 -0.169742502
## [80,] 6.34676332 2.303329412 0.6824808321 -0.20619808 0.649344650
## [81,] 6.48753279 0.167672188 -2.0777322338 -0.85238899 1.390902363
## [82,] 6.44186077 2.433118209 -1.7718549559 -0.19627768 0.883456072
## [83,] 6.63533948 1.718209618 -1.0627410978 -0.02971969 -0.939526922
## [84,] 6.66254211 1.918036137 -2.8865984603 -0.31878374 -0.624154978
## [85,] 6.99311161 1.104406496 -3.6399911615 -2.17597598 -0.688848228
## [86,] 7.37938828 1.270689382 -1.1779551954 -2.37640954 -1.780712581
## [87,] 7.46201054 3.151259342 -1.4663077951 -1.41471432 -1.243084837
## [88,] 7.75802689 2.014712578 -2.7404216375 -1.49146158 -0.907936355
## [89,] 7.22270448 2.841839517 -0.6058907367 -0.60952102 -1.488949920
## PC6 PC7 PC8 PC9 PC10
## [1,] 1.007270260 0.548610204 2.1843832472 -1.237706815 -0.776934637
## [2,] 1.484446858 -0.377567190 2.4468990648 -0.518202649 0.315821246
## [3,] 2.322535621 -0.780135393 3.3187671855 1.560789178 -0.267125646
## [4,] -0.846728675 -0.589213269 1.4626577723 0.815114931 0.832927369
## [5,] -3.717986160 0.576387196 2.9974836279 1.531101368 0.436256760
## [6,] -0.924900393 0.824919967 -3.0848781788 0.702659524 1.622240786
## [7,] -1.826999153 2.198034166 -1.9015272165 -0.783996327 -0.490073148
## [8,] 0.092853804 0.936532791 -0.7725523928 0.899360554 1.415515714
## [9,] -0.664717123 1.016708942 -2.4650272322 0.681832716 1.185286087
## [10,] -1.069954220 1.444279287 0.9923954132 0.947129187 -0.305629063
## [11,] -0.372292064 0.008679128 -0.0357133875 1.391349673 0.427232276
## [12,] -2.841814068 -3.758111405 0.1513996951 1.677459759 -5.225167612
## [13,] -3.467090625 1.096427328 1.1864921260 -1.855647834 -0.517660382
## [14,] -2.464663592 0.760268227 -1.4064837624 0.824127663 0.966693619
## [15,] -0.651359208 0.172992284 -1.4871196554 1.275502035 1.153478228
## [16,] -1.259994336 0.852517361 -2.3124569461 0.145950047 -0.253891659
## [17,] 0.232177942 -0.374925178 -1.6016524671 -1.495366514 0.196218203
## [18,] 0.867496796 -1.223048734 -0.4924018987 0.349645277 1.235509894
## [19,] 0.137861535 1.133803960 -1.5253038276 -2.480984532 -1.481046184
## [20,] 2.221483553 -1.080129810 -2.6087389521 0.480045673 1.508448975
## [21,] 2.446465891 1.901096013 -0.9057434159 -0.243971873 -1.967171823
## [22,] 0.562596647 0.428090508 -0.4640874549 -1.156220683 -1.465593782
## [23,] 2.041726568 -2.496091970 -1.2760751111 0.847573742 -1.024105271
## [24,] -1.216235893 -2.180071468 -1.1039039884 0.024133359 -1.511095231
## [25,] 0.263160436 0.533747737 -0.1388001169 0.423936428 0.259405917
## [26,] 0.387712099 -0.481165728 0.4402123171 0.020224084 0.353759362
## [27,] -0.457183181 -0.322530329 -0.2170745430 -0.833738840 -0.272313581
## [28,] -0.013621025 -0.397378650 -0.0622572838 -1.226232156 0.415562032
## [29,] 0.783556845 -0.549162214 -0.6229325532 0.910690320 0.614380325
## [30,] -0.306670371 0.076389602 0.1530703238 -0.741772897 0.460729068
## [31,] 0.582833332 0.389494048 0.2119819191 -0.087313472 -0.004270613
## [32,] 1.828896403 -1.442030499 0.8813322165 0.997438451 0.143421069
## [33,] 0.488755309 -0.387009850 2.2512968780 -0.673817921 0.924110843
## [34,] -0.343141404 -1.162736897 0.1609203436 0.474336737 1.587852707
## [35,] -1.617669580 -0.168250588 1.7559650337 0.084943930 0.200796953
## [36,] -4.217895614 -2.318655340 2.3331072951 -0.516860860 0.976993575
## [37,] 4.378027541 0.904913005 -1.6679812484 1.311511034 -2.871367526
## [38,] 2.343364813 0.144549614 1.0576037735 -0.153775865 -0.458466176
## [39,] 1.289061957 -0.636490533 -0.4925871232 -0.315483194 -0.931056613
## [40,] 0.829566335 0.512824937 0.4098073126 -0.005838476 -0.225647986
## [41,] 1.004319806 -0.449168842 0.5804377743 0.593094722 0.022856352
## [42,] 0.919208713 -0.334374189 0.9830071550 -2.147170165 1.032092259
## [43,] 2.682513661 -1.142784770 0.3488729130 -0.055969246 1.398646470
## [44,] 0.131809018 0.970512612 -0.0636362483 -1.204707059 -0.116506693
## [45,] 0.721110718 -0.240378283 0.9973735950 -1.841018979 0.755189123
## [46,] 2.122180913 0.779889132 -0.0823552775 -0.267984130 -0.246391455
## [47,] 1.040917756 0.802143291 0.6064924742 -1.397992241 -0.706250673
## [48,] 1.568867375 -3.264348590 0.4806222530 -1.686339882 -0.115965995
## [49,] 0.079501995 0.666305400 -1.3144106166 -0.690581707 1.157981313
## [50,] 0.376002484 1.365551584 1.3968150790 -1.435060782 0.503235065
## [51,] 0.186436615 -0.619825832 0.8713907319 -0.474710745 0.997604781
## [52,] -0.148362932 -0.023883176 -0.4962770993 0.228945831 0.262698013
## [53,] -0.044617253 0.440738295 -0.6157240305 -0.918882320 -0.543672742
## [54,] -0.918319249 -0.384842100 0.4853297141 -1.756084894 0.173970177
## [55,] 0.019110747 0.274251077 -1.1460476306 -0.623283146 -0.058933047
## [56,] -0.276380533 1.083063063 0.6568981246 -0.991533713 -0.543194775
## [57,] -0.650620054 -0.466084773 -0.8942963432 0.249945463 0.506322044
## [58,] 0.378695034 0.551862873 -0.0734095806 0.133650528 1.158724437
## [59,] -1.428439377 0.617888114 0.0485604031 0.024439413 -0.793338296
## [60,] -1.855973500 -2.246838709 -0.7786439352 1.412063752 -0.708483957
## [61,] -1.826896040 2.659011563 -0.6119089693 0.771501488 -0.185338009
## [62,] 0.245070719 1.440924522 0.3855385171 0.870851637 0.148146435
## [63,] 1.305077773 0.129982472 1.4181779949 0.989879276 0.584267377
## [64,] -0.127401325 1.273771891 1.1463846569 1.520842388 0.767398607
## [65,] -0.406543001 0.984470443 1.7319285383 1.069996501 0.517709440
## [66,] 0.022855760 -0.160256274 0.0710553801 1.739972163 0.924104591
## [67,] 0.254804364 2.876220281 0.9351102150 0.842627935 -1.242415990
## [68,] -0.450892239 1.105626683 0.8602698033 0.423959543 0.077148936
## [69,] 0.433436384 1.208318364 -0.0312381312 0.943570544 -1.102828107
## [70,] -1.426003348 2.400806718 0.5078245065 0.408905852 -1.466492243
## [71,] -0.285820870 0.143954246 0.2198355565 1.938204139 -0.841162723
## [72,] -1.297472810 -1.363031655 -0.0971015170 1.853622917 -0.345631900
## [73,] -0.557919746 2.459178146 -1.6543042235 -0.882491227 -0.156284218
## [74,] 1.302347222 0.483792838 -1.1048707810 2.248570258 0.769477857
## [75,] 0.445390388 0.261116815 -0.0334344649 0.702357594 -0.003807405
## [76,] 0.676118982 1.591823653 1.2415397037 -0.928470508 -0.754510837
## [77,] 0.818378943 -1.088522136 0.8985716457 -0.183331397 0.603412394
## [78,] 0.174210415 0.045103126 -0.0808804776 0.945215662 0.577823098
## [79,] -1.017122210 1.644421255 0.0305727168 -0.158010259 -0.607843793
## [80,] 0.288448136 -0.809782689 0.0169003681 0.499856770 -0.077277538
## [81,] 0.922342986 0.313292867 0.8485907623 -0.066012306 -0.056756185
## [82,] 0.831832937 -0.557763974 0.5503873653 -0.688855984 -0.101182122
## [83,] 0.774858770 -2.184727166 -1.1256706262 0.849093532 0.153365262
## [84,] -0.660240705 -2.004492340 0.0005639822 -1.337060465 -1.372533999
## [85,] 0.001336719 -1.034387103 -1.0068929258 -1.260170098 0.690993851
## [86,] -1.381343721 -0.912711594 -1.0993664057 -1.414628260 0.762429851
## [87,] -0.893646393 -1.810785231 -1.1822887986 -0.962200961 0.700541171
## [88,] -0.895701203 -1.066343179 -0.7131899364 -1.331155053 -0.361747370
## [89,] -1.488402687 -2.145249980 -1.8635807006 -0.607387141 0.078387096
## PC11 PC12 PC13 PC14 PC15
## [1,] -0.86780044 0.0275972953 1.19460218 2.367585058 -1.498420359
## [2,] -0.75879050 0.8193180510 -0.55379182 0.303530618 -0.229640185
## [3,] -2.43404201 -0.3042478875 -1.00329565 0.305300198 0.403689450
## [4,] -1.28640239 1.2595719989 -0.77436735 -1.197229176 -1.180896583
## [5,] 2.88486753 -2.7655586848 0.53238926 -0.317101300 1.784192219
## [6,] -0.90829393 1.1625282988 -0.74505273 0.168946956 -0.643375603
## [7,] 0.77811854 0.7986452884 -0.03839225 0.962557976 -0.577222272
## [8,] 0.98039886 -0.3984217271 -0.86402240 0.834042427 -0.143806335
## [9,] 0.30545754 0.6673104979 -0.56932431 -0.375788042 -0.384552757
## [10,] 2.16078068 -0.8626585886 0.11362815 0.260998618 0.332609158
## [11,] 0.14441494 -0.1973642148 -0.56938440 -0.291761454 0.309895692
## [12,] -1.49726812 -1.5322679512 2.24925610 -2.588090246 0.477971460
## [13,] -0.29370584 0.9636006440 1.43876201 0.202351392 0.602043690
## [14,] -1.41466184 0.7108645694 0.98676261 -0.426217295 0.417876877
## [15,] -0.18978208 -0.7453358267 0.26974499 0.012784125 0.315732222
## [16,] 1.82890505 -0.9240170267 0.23308335 0.865914487 -0.644970031
## [17,] 0.09521508 -0.6094507251 -0.21528277 -1.275383484 -0.203689227
## [18,] -0.22497410 -0.9559042456 -0.78718706 -1.422229421 0.832242221
## [19,] -1.03466658 -1.0215649000 -0.16416319 -0.804627077 0.330391238
## [20,] -0.03370581 -1.7160665397 -0.92839856 0.342599867 0.808462276
## [21,] -1.75926102 -1.8564096908 -0.59545220 1.342332660 0.293289671
## [22,] 0.89866700 -0.0673186356 -1.13767274 -0.061455741 -0.124761833
## [23,] 0.26290756 -0.7573237129 0.08043067 1.256018260 -0.596385806
## [24,] 0.09408376 0.1900541026 -2.31376565 -0.108817588 -1.603870457
## [25,] -0.26132298 0.9410780616 -0.06830378 0.050380899 -0.887630080
## [26,] -0.38481758 1.1338531341 0.11479181 0.481690258 0.058731776
## [27,] 0.66515671 1.3077930087 0.06993742 -0.748291783 -0.397355962
## [28,] 0.44487951 0.2595053797 0.60521339 -1.391355256 0.634682927
## [29,] -0.65425998 0.0272485179 0.28291221 -0.511579761 1.034178088
## [30,] 1.36816931 -0.1282869389 0.01463338 -1.340939366 0.545919323
## [31,] 1.38023474 1.6759861887 -0.52475101 -0.603867394 0.468269075
## [32,] 1.58572466 0.2338163461 1.40876148 0.290432173 -0.543496871
## [33,] 1.26714125 0.4627509748 0.80711170 -0.601567853 -0.490039037
## [34,] -0.36253620 0.7173018362 2.13002250 0.106858755 -0.214765533
## [35,] -0.62617314 0.3129995194 -0.10264025 1.207135557 0.899189676
## [36,] 0.22610577 0.3874578888 -1.21040935 1.896523905 0.009054472
## [37,] 2.42353298 4.2424804846 1.39915025 0.010852384 2.454717223
## [38,] 0.27900334 -0.0226699713 0.67162472 -1.053679612 -1.511458054
## [39,] -0.34181276 0.4846563201 0.16408469 2.166265804 -0.106680389
## [40,] 0.59834146 0.5700478571 -0.39070669 -0.644467992 -0.901569746
## [41,] 0.46056533 0.0178099424 0.14872289 0.453727127 -0.884145795
## [42,] -0.35013228 0.6834880128 0.76347877 -0.465127211 0.142717248
## [43,] -0.40907152 -0.4778136761 1.20611111 -0.027522231 0.213066801
## [44,] -0.33660772 -0.1325378219 0.42132029 -0.951426491 -0.406501844
## [45,] -0.33358957 0.2619968728 0.46749519 0.003728229 0.318572509
## [46,] 0.01908820 -0.7974794925 -1.02371079 -0.375372188 0.299031222
## [47,] 0.36709316 -0.8537071933 -1.48684766 -0.675414510 -0.970021635
## [48,] 0.10215509 -0.6202076111 -1.21221708 0.098853075 -0.647887903
## [49,] 0.13880322 -1.6711624847 1.94183559 -0.018857243 -0.968002832
## [50,] 0.23782050 -0.1211186511 -0.61593915 -0.606787696 0.629820209
## [51,] -0.57942093 0.2459071240 0.07771880 1.409136670 1.193655332
## [52,] -0.17049999 0.0005818528 0.76753908 0.427215626 -0.035557989
## [53,] -0.45546385 -0.5698305051 0.53991213 0.761213999 -0.421244922
## [54,] 0.40217236 0.3799307445 0.54504169 1.104545297 0.718907338
## [55,] -0.42707194 -0.2656966053 1.09624320 0.736339715 0.072942799
## [56,] -0.18289514 0.4253568410 -1.28786271 0.246197848 1.335499074
## [57,] -0.03137729 0.0224236049 0.98775466 0.605482131 0.547629129
## [58,] -1.09405994 -0.4309240529 -1.04325987 -0.625045010 1.090980195
## [59,] -0.10884743 0.9543912383 -1.37714942 0.013807083 0.430858107
## [60,] -0.95416136 0.9646732755 -0.72411091 0.271992341 0.243891389
## [61,] -0.99650484 1.3507564627 -0.18385845 -0.753475413 -0.034407357
## [62,] -0.13214747 -0.0521851063 0.68597683 -0.896137895 -0.702503534
## [63,] -0.47927518 1.1659352182 -0.63545041 -1.779568412 0.430193971
## [64,] -1.35047574 -0.1414060573 0.46705480 -1.035205597 -1.054295227
## [65,] -0.23539567 0.8497161958 -0.21511065 -0.876482679 -1.023548379
## [66,] -1.13723720 0.0299115450 1.15024347 -0.066516489 -0.378258137
## [67,] -2.69669782 -0.1628048798 0.62084566 1.265275559 0.137910334
## [68,] 0.01868130 -0.0426962108 -0.12397362 -0.355715430 0.166425073
## [69,] -0.94682480 -0.7327831072 -0.18022159 0.800673950 0.875097176
## [70,] 0.50855583 0.5122627769 0.34323171 0.772589509 -1.370727219
## [71,] 1.00214271 -0.0229350552 -0.12699521 0.923662959 -1.029289477
## [72,] 0.69117833 0.7592799008 -2.21422657 0.257763572 -0.258727994
## [73,] -0.56473902 -1.2807739702 -0.10067727 -1.978581442 0.210583990
## [74,] -0.21586587 -0.8692183842 -0.55423332 0.501355153 0.369300978
## [75,] 0.81912260 -1.3074242908 0.28839142 0.040282157 -0.805201447
## [76,] -0.01933687 -1.4009468837 -0.88526845 0.092667945 0.845052587
## [77,] 0.87125916 -0.9363877727 0.19616799 -0.283476059 0.756005231
## [78,] 0.82139023 -0.0287925750 -0.30959476 0.404435245 -0.049757187
## [79,] 0.85665516 -0.4256474646 -0.88652431 0.123952674 -0.310599352
## [80,] 1.23552525 -0.8031286195 1.14795439 0.803356368 0.037859037
## [81,] 1.02254126 -0.8795331422 -0.37377789 0.441865118 0.114008249
## [82,] 0.62713017 0.1146080066 -0.01042701 -0.948044295 0.046326081
## [83,] 0.33197774 -0.1798037374 0.08549550 -0.528867677 -0.514998432
## [84,] 0.40600188 0.6458559694 -1.50597456 -0.739645655 -0.627065584
## [85,] -0.16204027 0.1262785036 1.50687192 -0.147713171 -1.166333795
## [86,] -0.27453699 0.6376072023 0.53898568 -0.512657327 -0.442563220
## [87,] -1.62355769 0.6585916497 -0.23841792 0.398597430 1.978476717
## [88,] 0.28615109 0.0335223167 0.06718307 0.960584749 0.503221443
## [89,] -0.29600515 -0.1255389027 0.04371502 1.057686054 0.265055427
## PC16 PC17 PC18 PC19 PC20
## [1,] 0.18517560 -0.638322427 0.344238917 0.488020826 -0.442845978
## [2,] -0.74596077 -1.674756086 0.171614217 0.069131899 -0.951143987
## [3,] -0.78049276 0.323326857 -0.146573894 0.037813822 -0.101437475
## [4,] -0.25626959 0.550190820 1.077859258 -0.335399128 0.130711458
## [5,] 1.56738644 -0.016348824 -0.195087682 0.006503842 0.017368976
## [6,] 1.05899193 -0.377503086 -1.089306977 0.135843494 -0.106875870
## [7,] 0.68650537 0.464872691 0.330939848 -0.133296510 0.419954505
## [8,] -0.04233544 1.116403232 -0.796262551 0.045414447 0.057067450
## [9,] -0.30219591 0.792852005 -0.456527359 -0.314281547 0.029022404
## [10,] -0.78942920 2.011187728 -0.581628029 -0.340892815 0.212923138
## [11,] 1.29000680 0.555570315 0.103169236 -0.031052582 0.493341611
## [12,] -0.42355492 -0.082058163 -0.952397254 0.084251549 0.320035924
## [13,] -0.01932221 -0.318367827 1.514855958 0.255933834 1.310458663
## [14,] -0.34935871 -0.949030874 -0.125629947 0.124605993 -0.750379454
## [15,] -0.38362572 -1.440216040 -0.092945459 -0.198202631 -1.301260086
## [16,] -0.11444918 0.738209604 0.774819181 0.233171520 -0.429699429
## [17,] -0.90453840 0.113012473 1.311978069 -1.079777940 -0.573605508
## [18,] -0.90826973 1.168450358 1.222993774 -0.323971085 0.186666145
## [19,] -0.12570921 0.620976757 0.946217498 -0.593452273 -0.130912823
## [20,] 0.81071767 -1.354537225 1.175914763 -0.827075225 0.305384995
## [21,] 0.50360553 1.651231637 -0.637548908 -0.100974157 -0.721610143
## [22,] -1.48597788 -0.977265983 -0.047899302 -0.973755991 0.140081620
## [23,] -0.78950071 -0.581279613 -0.272184529 -0.236352207 0.339825404
## [24,] 0.34556318 -0.415335840 -0.608974160 0.580460871 -0.247980256
## [25,] 0.21055040 -1.547345130 -1.854756736 -0.187790388 0.485140904
## [26,] 0.16287359 -0.230713833 -0.537819809 0.438302791 0.993583681
## [27,] 0.24984208 0.236975876 0.483402418 0.436935957 0.792364666
## [28,] 0.65689582 -0.597308821 -0.162317545 -0.280430396 0.068024794
## [29,] 0.68398550 0.388565776 -0.774914368 0.757140721 0.207648035
## [30,] 0.09627335 -0.585925078 -0.707148711 -0.207625183 -0.219196020
## [31,] 1.09338604 -0.403530591 -0.521527305 0.146588491 0.953518505
## [32,] 0.63659750 -0.006679476 0.678414647 -0.017556135 0.722792239
## [33,] -0.33877635 0.109519733 -0.362380075 -0.049633739 -0.502470667
## [34,] 0.68667683 0.245499597 -0.226139216 0.991494221 0.004372872
## [35,] -0.94368721 0.402161048 -0.158390993 1.313227081 -0.793359543
## [36,] 0.70900194 0.220953362 0.097718636 -0.032618591 -1.147724902
## [37,] 0.55411103 0.270485679 0.339505497 0.292227396 -1.921045952
## [38,] 0.72538286 0.122140156 -0.816474862 -0.017744727 0.456734731
## [39,] 0.44994832 0.520718319 -0.276935869 -1.056361422 1.486941471
## [40,] -1.20504375 0.323570796 -0.866901646 -0.571222622 0.463529450
## [41,] -0.51445520 0.788551953 0.033922820 -0.127255741 0.932211699
## [42,] -0.23337944 1.723435009 0.002671088 -0.765423288 -0.461672754
## [43,] -1.39729102 0.678695984 0.660876750 0.789279331 -0.141203621
## [44,] 0.65095225 -0.684372873 0.154820456 1.021407745 0.145392176
## [45,] 0.81154366 0.428572467 -0.574700981 -0.319920631 -0.227570172
## [46,] -0.02910882 -0.663560783 -0.178157754 0.926522480 0.642632277
## [47,] 0.19748957 -0.331623255 -0.494000760 0.775301332 -0.120785849
## [48,] 1.53861458 0.438802087 0.657863370 0.225421341 -0.656522861
## [49,] -0.16459016 -0.082201946 0.186529163 0.811184610 -0.350486912
## [50,] -0.49726578 -0.008208427 0.202277412 1.871078465 0.195033216
## [51,] -0.18884634 -0.360039978 -0.979898283 -0.891686349 0.072985907
## [52,] -0.89097630 -0.178071117 -0.039137799 1.120545989 0.184525219
## [53,] -0.75571312 -0.758323225 -0.233064708 0.631471162 -0.269136485
## [54,] 0.13420529 -0.350645432 -0.680707614 -1.273989343 -0.058891861
## [55,] -0.24121927 -0.542969297 0.516322787 0.618673722 0.276227350
## [56,] -1.01545554 -0.515443948 0.193967315 0.218536893 0.646499134
## [57,] -0.41157723 -1.035536987 0.723910151 -0.683465481 0.286332287
## [58,] -0.03300439 -0.714701163 -0.151865436 0.034240777 -0.082531909
## [59,] -0.99223266 0.233583339 0.478298399 -0.100532482 0.176541005
## [60,] 0.24238714 0.651486942 0.169058188 -0.551262581 -0.614227419
## [61,] -2.03360932 0.535222889 -0.416753591 0.294365706 0.256027572
## [62,] 0.30648367 0.168392665 0.268027088 -0.687362040 0.183082851
## [63,] 0.50687023 0.275711855 0.830388695 -0.213374377 1.080051452
## [64,] 0.22802405 0.255183669 -0.600723881 -0.694929555 -0.574635130
## [65,] 0.08673831 0.384110941 -0.088882866 -0.802672402 -0.654357642
## [66,] 0.41682372 -0.556313853 0.353093978 -1.020033237 -0.268784333
## [67,] 2.19331889 0.601998059 0.711592897 -0.293142091 -0.218530332
## [68,] -0.45651578 -0.748014092 0.703475146 -1.136009613 -0.231575480
## [69,] 0.42210719 -0.593704743 0.714622268 -0.131098790 0.866515902
## [70,] -0.30772838 -0.407736179 0.582477894 -0.136470877 -0.157973000
## [71,] -0.32374042 -0.767238211 1.113884728 0.179946310 -0.064998770
## [72,] 0.31046789 -0.446012200 1.574417498 0.823024345 -0.077774317
## [73,] 1.21326502 -0.306931830 -0.122870765 0.364507527 -0.598843075
## [74,] -0.32296688 0.550569401 -0.182871083 0.834426029 0.238010009
## [75,] -0.72073259 1.003848621 0.253712743 0.398218444 -0.391362379
## [76,] 0.92111859 -0.547762276 -0.565209874 -0.182267588 -0.102967315
## [77,] -0.52854148 -0.680013071 0.168785903 -0.353288234 -0.061929245
## [78,] 0.43653457 -0.298008663 -0.376954301 0.097939536 0.393013874
## [79,] -0.38644775 -0.272939850 -0.736090363 -0.202086110 -0.034695041
## [80,] -0.52015606 -0.542266603 -0.124209468 0.114983895 0.028382525
## [81,] -0.56434705 0.617704747 -0.051802504 -0.097532162 -0.318149746
## [82,] -0.73368548 -0.690186661 -0.533809916 -0.479553885 -0.430500853
## [83,] 0.15058907 0.006116750 -0.245261825 0.266819865 0.001032460
## [84,] 1.37474479 0.402905896 0.191969632 0.370663368 -0.522084388
## [85,] 0.77869272 0.309545219 0.464846701 -0.158818904 0.193367999
## [86,] -0.24201211 0.729003328 -0.177069864 0.307669173 -0.203620523
## [87,] -0.12303259 0.734538296 -0.323540496 0.679505503 0.500505875
## [88,] -0.09860973 0.337583882 -0.560132676 -0.792463664 0.206631021
## [89,] -0.64871042 0.476912768 -0.779064999 -0.208695582 0.134864052
## PC21 PC22 PC23 PC24 PC25
## [1,] -0.14700453 0.233227874 0.579706649 -0.339852173 0.411139311
## [2,] 0.34158461 -0.590410059 0.139162561 -0.543165901 -0.009183724
## [3,] 0.76456579 0.112216522 0.017837513 0.690670237 -0.289691431
## [4,] 0.59676807 -1.157140171 -0.470755955 0.401537357 0.155339252
## [5,] -0.43074973 0.380853807 -0.107192212 -0.153665521 0.316787774
## [6,] 0.56981847 -0.143755899 -0.813157670 -0.190756537 0.040547823
## [7,] 0.10903854 0.043900538 0.112130859 -0.140381115 -0.179698314
## [8,] 1.07673916 0.175722370 0.550804502 -0.193950158 0.400349855
## [9,] -1.22160336 -0.181249566 0.242299885 0.659055079 -0.048952157
## [10,] -0.26370546 0.672956685 0.257259373 0.151576876 -0.150960748
## [11,] -0.57079213 0.664484174 -0.306132878 0.060183506 -0.717611754
## [12,] 0.19245023 -0.904746301 -0.335936266 -0.295230865 0.194641519
## [13,] -0.37068544 0.569169790 -0.102018263 0.149434972 -0.343670567
## [14,] -0.69556023 0.354913919 0.002638748 0.165420852 -0.351341340
## [15,] 0.26928588 -0.010761184 0.027969871 0.308153375 0.235774919
## [16,] 0.73993339 -0.617872767 0.619405660 0.291600263 0.213080122
## [17,] 0.27007083 0.204230553 -0.080514341 -0.450335036 0.669660630
## [18,] 0.16234365 -0.430010022 0.196134995 -0.832082948 -0.394882617
## [19,] 1.15500027 0.142943320 -0.135997396 -0.447323029 -0.200926735
## [20,] -0.20739549 -0.328878479 -0.669811203 -0.138608460 -0.413480728
## [21,] -0.46029660 0.311188047 0.053484285 0.142538099 -0.414220411
## [22,] -1.52142801 -0.609425636 -0.086552487 -0.394437453 0.328446316
## [23,] -0.31540003 0.553632791 0.379093978 0.403809488 0.328499137
## [24,] -0.14448190 0.378374562 0.760832530 0.321436420 0.173409382
## [25,] -0.09974608 -0.065145577 -0.202387634 -0.515720610 0.462662217
## [26,] -0.20809784 0.536293261 0.300102878 -0.337255408 0.925566441
## [27,] 0.40779354 -0.563560304 0.861907338 0.422217993 0.184324966
## [28,] 0.67572190 1.096553177 0.417077101 0.053788851 0.158911802
## [29,] -0.23120368 -0.334615132 0.272588786 -0.531710809 -0.768467414
## [30,] -0.28313196 -0.217088138 0.377172981 0.184553865 -0.223745429
## [31,] -0.22922109 -0.654908631 -0.375076999 -0.569085471 0.077431893
## [32,] -0.14951151 -0.208213547 -0.509593881 0.519300583 -0.053079674
## [33,] 0.31078484 1.000429377 -0.460035993 -0.401739915 -0.020817312
## [34,] 0.39092126 0.466455703 0.175628439 -0.120169750 -0.092131486
## [35,] -0.13911940 0.047127805 -0.398123186 0.393489129 0.052563606
## [36,] -0.02926397 -1.098089832 -0.773435746 0.662030083 -0.022809589
## [37,] 0.29270867 -0.294657412 -0.215807921 0.274698380 0.073155118
## [38,] 0.01354390 0.073014003 0.259238307 0.715546272 -0.210431628
## [39,] 0.25590086 -0.024940222 0.315014270 0.303087582 0.080401923
## [40,] -0.24337066 -0.202963807 -0.104770383 0.356819588 -0.695721805
## [41,] 0.89451941 0.254353659 -0.524533987 0.145377436 -0.435805757
## [42,] -0.47516547 0.467453080 -0.766712004 -0.263018201 0.585721849
## [43,] -0.55527125 0.769144257 -0.910991334 0.318179657 0.178851778
## [44,] -0.22060387 -0.017178735 -1.273255564 0.510307559 -0.021609801
## [45,] 0.10389270 -0.834933955 -0.302477129 -0.607581981 0.526994951
## [46,] 0.12442902 -0.579838899 -0.799701608 0.339998366 -0.212783024
## [47,] 0.18850847 0.618983597 -0.741706179 0.028437064 0.399184659
## [48,] -1.11313952 -0.280781419 0.612669440 -0.119150795 -0.430542230
## [49,] 0.83501101 0.066138836 0.183968953 0.033785815 -0.166548017
## [50,] 0.02688902 -0.965926967 0.947698196 0.084615061 0.058793484
## [51,] 0.02385107 0.071040988 0.565949637 0.200691750 0.192693331
## [52,] -0.40563816 0.059943804 0.676713745 -0.366421401 0.110497064
## [53,] -0.61250846 0.485940733 0.240902371 -0.065241113 -0.082823380
## [54,] -0.14257190 -0.296555349 0.636061594 -0.720850282 -0.283196855
## [55,] 0.10049312 0.035669525 0.168610210 -0.064856804 -0.191933892
## [56,] 0.38654195 0.076029040 0.130673976 0.171035184 -0.563510453
## [57,] -0.14783245 -0.032395572 -0.310229387 -0.044252502 -0.387573818
## [58,] 0.47897435 0.364281496 0.424836073 0.380228731 -0.388573370
## [59,] 0.14590587 0.522864434 -0.285498251 -0.203295296 -0.096674128
## [60,] 0.05478848 0.844519733 0.337027705 -0.129855569 -0.121882788
## [61,] -0.89530323 0.420569366 -0.274297687 -0.465553309 -0.140586584
## [62,] 0.50580057 0.392833000 -0.248243440 -0.027462371 0.743475826
## [63,] -0.46422985 -0.169791606 0.691019645 0.258412462 0.601176595
## [64,] -0.28398437 0.183418770 0.346106220 -0.151978651 -0.330549726
## [65,] 0.70629915 -0.310596199 0.243969554 0.004755505 -0.153052969
## [66,] 0.31401555 -0.131782305 0.283735973 -0.073920253 0.177539458
## [67,] -0.92166855 -0.518634986 0.153949020 -0.242992188 -0.146422376
## [68,] -0.72191179 0.271954617 0.344054463 0.388867278 0.573590287
## [69,] 0.71532100 0.073198619 0.188204111 0.287497492 0.397113741
## [70,] 0.08699795 -0.255457947 -0.395774026 -0.086569662 -0.483578760
## [71,] 0.69338230 0.654085074 0.029592514 -0.390454608 0.165561962
## [72,] -0.13326755 0.457786122 -0.186859983 -0.128108934 0.152341029
## [73,] 0.01787523 0.030733777 0.439405575 1.190794888 0.712641572
## [74,] -1.13961629 -0.068022685 -0.347451533 -0.535503869 0.513262853
## [75,] -0.54087469 -1.378394346 0.163396119 -0.317397427 0.570349191
## [76,] 0.66623004 0.397218108 -0.779921873 -0.570408180 -0.076444599
## [77,] -0.62346814 -0.068308335 0.679815447 0.441672854 -0.009824381
## [78,] 0.14368148 -0.633641600 -0.306856726 -0.453542718 -0.347087618
## [79,] 0.46061997 -0.753903372 0.002807249 -0.212194904 -0.122405153
## [80,] 0.23969969 -0.256313630 0.123390405 -0.163057707 -0.632956179
## [81,] 0.50284517 -0.271524235 -0.134108404 -0.221453364 0.160603643
## [82,] -0.28948417 0.380292939 0.051296961 0.464768030 -0.740065391
## [83,] 0.64184050 0.331166002 0.176251198 -0.583012117 -0.318058249
## [84,] -0.37537229 1.362159359 0.055863984 -0.520577634 -0.144138889
## [85,] -1.08196720 -0.327738217 -0.335962420 0.077285873 -0.212276098
## [86,] 0.48598575 -0.502138238 0.420714129 -0.090153659 -0.751774257
## [87,] 0.59152978 -0.123748065 0.325053679 -0.207897094 0.611813749
## [88,] 0.32655943 -0.002692069 -0.401174309 1.336104772 -0.004968871
## [89,] 0.04818648 -0.190735793 -1.088173401 0.328469127 0.684571453
## PC26 PC27 PC28 PC29 PC30
## [1,] -0.57200777 -0.079863564 1.438724e-01 -0.0877496248 -1.104781e-01
## [2,] -0.07628045 -0.507046641 -6.442972e-02 -0.2709631175 2.517903e-01
## [3,] -0.38848661 0.561209562 -1.327841e-01 0.6422773824 -3.588641e-01
## [4,] 0.40472224 0.584847965 1.458504e-01 -0.6604757548 -4.586177e-01
## [5,] 0.37088241 0.336633898 1.082828e-02 0.1822906856 3.019448e-01
## [6,] 0.51730358 0.731312784 2.670325e-01 0.3538926519 4.837257e-01
## [7,] -0.37590457 0.499946854 -5.168062e-01 0.0956502345 -3.395161e-01
## [8,] -0.25480905 0.115092336 2.837820e-02 -0.3936853831 2.577331e-01
## [9,] 0.22778113 0.106278205 2.336858e-01 -0.0755460377 -6.186830e-02
## [10,] 0.01382765 -0.224615950 2.139175e-01 -0.2594777745 -3.220098e-01
## [11,] -0.16052003 -0.838754144 -2.376413e-01 -0.4519334905 -2.567948e-01
## [12,] -0.09856651 0.110863167 2.936828e-02 -0.1445798763 1.986011e-01
## [13,] 0.63860234 -0.224209916 -2.484987e-01 0.1850212894 -6.161175e-02
## [14,] -0.27772395 -0.335452877 -1.160961e-01 0.4231424581 3.930596e-02
## [15,] 0.25331493 -0.436083350 -5.697996e-02 0.0629039671 5.571020e-02
## [16,] -0.04067270 0.168171838 3.245448e-01 0.0657484274 3.225194e-01
## [17,] -0.01002367 -0.367966615 3.209375e-01 -0.0521738569 -2.163765e-01
## [18,] -0.58275688 -0.226168407 4.840825e-01 0.3443920421 -7.948923e-02
## [19,] -0.09023128 0.235491863 1.903859e-01 0.0779092756 -5.091282e-02
## [20,] 0.21170822 -0.350861909 -5.040311e-01 -0.1517208188 -1.802838e-02
## [21,] 0.09037912 0.013076903 -4.059845e-01 0.3120102905 1.698054e-01
## [22,] 0.40011496 -0.014131134 -1.848394e-01 0.0726022712 2.801391e-01
## [23,] 0.22620193 0.263372854 -3.993981e-01 -0.1086179246 1.907017e-01
## [24,] 0.46411027 -0.037934405 -7.117771e-01 -0.1942927585 -2.277508e-01
## [25,] -0.28741320 0.068025181 1.503344e-01 -0.0301588658 1.739617e-01
## [26,] 0.21517411 -0.256515028 7.046047e-02 -0.0507787129 1.487224e-01
## [27,] -0.37125439 0.343978073 -1.627818e-01 0.0141330811 1.742047e-01
## [28,] -0.24195692 0.255764172 -7.406522e-02 -0.0865866284 -2.014648e-01
## [29,] -0.53739409 0.166171915 -5.262038e-03 0.2857995958 7.546876e-02
## [30,] -0.54714489 0.142623676 4.454977e-04 0.0295094615 2.347837e-02
## [31,] -0.45980759 -0.012439851 1.451138e-01 -0.0360492233 -1.789199e-01
## [32,] -0.26128072 0.104234166 -2.841563e-02 0.0129599495 -2.939675e-01
## [33,] 0.09557618 -0.214059231 2.794803e-01 0.0886650635 -2.167988e-01
## [34,] 0.35155083 -0.156295122 -2.154330e-03 -0.4837581963 -2.522922e-01
## [35,] 0.30453145 -0.380341562 2.553124e-01 0.0659223322 5.756585e-01
## [36,] 0.15729008 0.227893506 3.841071e-01 0.2942033991 -4.066880e-01
## [37,] 0.24043658 -0.147015119 6.730839e-02 -0.0187505329 -2.223528e-01
## [38,] 0.27921918 -0.603815170 5.797671e-01 0.1059240695 2.235466e-01
## [39,] 0.48543778 -0.290125309 4.393090e-01 0.0474709065 2.821310e-01
## [40,] 0.25517000 -0.574724919 3.238683e-01 0.0699247452 3.358964e-01
## [41,] 0.50130044 -0.316663960 1.047931e-01 0.2765272660 2.114872e-01
## [42,] 0.48252335 0.461800737 -7.346763e-01 0.1081165132 -1.122289e-02
## [43,] 0.07235165 0.817425497 -1.901705e-01 -0.2487496824 5.266455e-01
## [44,] -0.41393431 0.236160860 -4.559852e-01 -0.2001886811 3.660053e-01
## [45,] 0.12191611 -0.012485218 -1.037734e-01 0.0139246677 3.192683e-01
## [46,] -0.20448016 0.032263332 1.571600e-01 0.2666142349 -3.220031e-01
## [47,] -0.04097170 -0.113360195 3.930118e-01 0.0094184317 -5.025622e-01
## [48,] -0.07165919 -0.216024753 2.356986e-01 -0.1800332057 -2.050593e-01
## [49,] -0.20525714 -0.610331788 -1.644985e-01 0.1642338728 1.541410e-01
## [50,] 0.17350020 -0.729484409 -4.522666e-01 0.3822232493 -2.626608e-02
## [51,] -0.61739511 0.367838727 -3.917120e-01 0.3580245056 3.963529e-02
## [52,] 0.04155306 0.305686982 2.734577e-01 -0.1931967994 -5.025157e-01
## [53,] 0.01789325 0.298400513 1.273913e-01 -0.1376824136 -3.398184e-01
## [54,] -0.22643116 0.433819373 3.181668e-01 0.1065003032 2.701653e-02
## [55,] 0.20922816 0.299986340 2.079994e-01 -0.3412588719 -4.566761e-01
## [56,] -0.22776302 0.191787257 4.132160e-01 -0.5305964470 2.898513e-01
## [57,] -0.02228293 -0.003465054 -1.019263e-01 0.0012361431 -1.176136e-01
## [58,] 0.20440666 0.017493096 1.094071e-01 -0.4637100226 1.047933e-01
## [59,] -0.34660495 -0.121294498 8.819311e-03 -0.0087822703 4.651743e-01
## [60,] -0.13268683 -0.275824566 -2.229669e-01 -0.3477514774 -8.817031e-05
## [61,] 0.14185397 -0.049915313 -2.896644e-02 0.0251222836 -4.531778e-01
## [62,] -0.60012096 -0.496487564 -3.242896e-01 -0.3352123939 1.872253e-01
## [63,] -0.04814611 0.023587886 -5.929777e-01 0.5573375594 1.326656e-03
## [64,] -0.19128530 0.005490568 -1.602890e-01 -0.2459298216 -1.631148e-01
## [65,] -0.12755302 -0.031440390 -3.742226e-01 -0.0185479607 1.430215e-01
## [66,] 0.51757374 -0.137341598 2.913469e-01 0.3285538362 -3.039704e-01
## [67,] 0.03531302 -0.237909251 1.583169e-01 -0.3424302684 2.039630e-01
## [68,] 0.06903849 0.033302932 3.445788e-01 0.0327815749 7.501414e-02
## [69,] 0.58004157 0.308316040 3.263330e-01 0.3028847366 -3.436666e-01
## [70,] -0.32384525 0.015699685 7.709272e-02 0.1641335862 3.851537e-01
## [71,] -0.53656295 0.245559029 -8.050533e-05 0.5375663701 -4.885754e-02
## [72,] -0.23793472 -0.013467276 -2.095061e-02 -0.2245115752 6.254963e-01
## [73,] 0.08058573 0.105446383 2.802976e-01 0.2340183395 3.323178e-02
## [74,] -0.45813972 0.437649545 3.835281e-01 0.0909700397 -2.333328e-01
## [75,] -0.11051002 0.232887356 -1.307423e-03 -0.4521540204 3.562118e-03
## [76,] 0.15935306 0.004532340 2.007124e-02 -0.5399490512 -1.095856e-01
## [77,] 0.01613794 -0.089434293 1.333217e-01 -0.0931793632 1.407204e-01
## [78,] 0.22499571 -0.168321350 -7.803190e-02 -0.0785439617 -6.228098e-02
## [79,] 0.68350848 0.035950898 -7.405116e-01 -0.0694668557 -5.684864e-01
## [80,] 0.16048848 0.116092255 -6.994234e-01 0.0002124098 -1.494349e-01
## [81,] 0.45609616 -0.108893699 -4.096488e-01 0.1738658946 2.197857e-01
## [82,] 0.38385385 0.580167749 3.356019e-01 -0.0880463355 3.551442e-03
## [83,] 0.47967479 0.347580184 1.744887e-01 0.2304464685 9.908554e-02
## [84,] 0.06671332 0.130660003 1.659064e-01 0.5361028692 1.009925e-02
## [85,] 0.01426410 0.147611566 1.784608e-01 0.1513059979 2.725294e-01
## [86,] -0.05258838 0.149282776 -1.597806e-01 0.1681115415 2.568866e-01
## [87,] 0.39495324 -0.286645465 6.852020e-02 -0.3330043156 1.198049e-01
## [88,] -1.12280087 0.094882532 -1.274782e-01 -0.5293741604 -8.498474e-02
## [89,] -0.53926442 -1.215140496 1.250270e-02 0.5070122578 -5.100004e-01
## PC31 PC32 PC33 PC34 PC35
## [1,] -0.099748694 -0.344391038 -0.110930268 0.025927318 0.083362834
## [2,] -0.074567324 0.309327893 0.095906509 -0.040411317 -0.046987814
## [3,] 0.094530139 0.108388078 -0.171511860 0.115703076 0.081109517
## [4,] -0.090899022 -0.208613587 -0.202625163 0.121845080 -0.035837538
## [5,] -0.077977743 0.105541965 -0.117695291 -0.136277215 -0.028155048
## [6,] 0.377614477 -0.274689772 -0.269183071 -0.053088582 -0.040127734
## [7,] -0.444528599 0.156922338 -0.222059005 -0.242720300 0.103919299
## [8,] -0.329131961 0.116547517 0.121630182 -0.069513520 -0.163905863
## [9,] -0.155660845 0.245964056 -0.069003554 0.145623298 0.072740276
## [10,] 0.290708253 0.158094364 0.133049596 -0.127353811 0.070858085
## [11,] 0.057458703 0.094827284 0.114946765 0.210068568 0.118977769
## [12,] -0.324174414 0.006106660 -0.073925942 -0.053936575 0.027110517
## [13,] 0.186793856 0.001983479 0.304062242 0.358264012 -0.006431778
## [14,] 0.371507497 -0.185881851 0.163634797 0.275320761 -0.013579472
## [15,] 0.125096769 -0.181619908 0.303435438 -0.045283771 -0.174527916
## [16,] 0.257309999 -0.306953669 0.124344818 0.014852607 -0.026536358
## [17,] -0.033207935 -0.042170811 -0.145029789 -0.044022518 -0.019623090
## [18,] -0.087329419 -0.158912472 0.101916786 -0.063813568 -0.211217675
## [19,] 0.051322140 0.029924197 0.224370615 0.061671619 0.069824774
## [20,] -0.263062207 0.053037730 -0.184262192 -0.100778667 0.094162450
## [21,] -0.034603372 0.103838022 -0.155191067 -0.102762977 -0.152061509
## [22,] -0.259910206 -0.072934434 0.058707437 -0.306448175 0.082119180
## [23,] -0.039670171 0.302421549 0.026037061 0.212821091 0.009213379
## [24,] 0.358773435 -0.017636473 -0.006933020 -0.078053854 0.098803397
## [25,] 0.152011030 0.217817103 0.181473411 -0.121466896 -0.250942125
## [26,] 0.199127621 0.190981805 0.054844658 0.080255487 -0.046769216
## [27,] 0.169050038 0.158082002 -0.012196473 0.192726918 0.179518198
## [28,] 0.108702390 -0.422388146 0.315415232 0.177899596 0.026252598
## [29,] 0.352188030 -0.113243924 0.156230159 0.176055038 0.245972998
## [30,] -0.036876076 0.059540546 0.117364453 0.020817535 -0.057114471
## [31,] -0.037790297 0.124320315 -0.098934433 -0.187592845 0.013028298
## [32,] 0.411394843 0.123636833 -0.076891496 -0.338658703 -0.156828079
## [33,] -0.026414739 0.058225214 -0.148419385 -0.064880718 0.078901418
## [34,] -0.416009911 -0.161682223 -0.306412099 -0.118833026 -0.118784637
## [35,] -0.090414166 0.017243224 -0.099102404 -0.245140182 0.106013967
## [36,] -0.085738459 0.211651430 0.080325735 0.035856602 -0.055264667
## [37,] -0.112941755 0.010489047 0.028310528 0.108400921 -0.010868724
## [38,] -0.453720876 -0.056313544 -0.189832961 0.191746144 -0.105321906
## [39,] -0.283381107 -0.049477599 0.316136532 0.118971315 -0.157194867
## [40,] 0.105730246 -0.503647480 -0.048857750 -0.150877530 0.251972248
## [41,] -0.164240158 -0.332718021 0.005022295 -0.021161045 0.066783047
## [42,] 0.018147591 -0.078501419 0.056636493 -0.065280166 -0.067272590
## [43,] 0.256614540 0.004505366 0.246711779 -0.204305664 -0.062501138
## [44,] -0.272384382 -0.148570942 -0.082456511 0.201082353 -0.109462074
## [45,] 0.222923403 0.190395667 0.247036015 0.233416578 0.134971657
## [46,] 0.267720941 -0.127831020 0.406326697 -0.467051192 0.131293351
## [47,] -0.221139180 0.269444168 0.238728845 0.286688341 -0.012696475
## [48,] 0.375731752 0.335297360 -0.031400347 0.066767609 -0.015732597
## [49,] -0.141663612 0.081301713 -0.163023131 0.005011514 -0.092459117
## [50,] -0.242496025 -0.026825647 -0.064833594 -0.106688729 -0.105873066
## [51,] -0.675622090 -0.298003831 -0.035672325 0.233452076 0.105024686
## [52,] -0.015453692 -0.141011655 -0.040689211 -0.279031972 0.157474415
## [53,] 0.417463667 -0.160004933 -0.114273280 -0.018970137 0.117005906
## [54,] 0.371114976 -0.100047348 -0.370316608 0.012515106 -0.002100834
## [55,] -0.196681816 -0.262454452 -0.225525534 -0.068594880 -0.032111079
## [56,] 0.196786419 0.161272431 -0.238067619 -0.025005768 -0.006660907
## [57,] -0.140366624 0.284486953 -0.256188426 -0.100446620 -0.036552191
## [58,] -0.115018185 0.335601897 -0.217931479 0.004655110 0.019268739
## [59,] -0.251929315 0.151888627 -0.283291594 0.151440069 -0.017344243
## [60,] -0.002885195 -0.097421456 -0.076149999 0.042388908 -0.179171626
## [61,] 0.030177128 0.234332433 0.007825739 0.093663491 -0.037815859
## [62,] 0.469836041 0.426108312 -0.219364592 0.017867683 0.200022383
## [63,] 0.173918804 -0.279336206 -0.282756533 -0.175309882 -0.278785917
## [64,] -0.131406881 -0.099047337 0.409937186 -0.312759215 -0.179128207
## [65,] 0.115379794 -0.018125732 0.102775392 0.036274844 0.234903824
## [66,] -0.325186527 0.057870552 0.161783615 -0.111484580 0.516802806
## [67,] 0.198577147 -0.042936298 0.067684867 -0.222549881 -0.083696253
## [68,] -0.054647285 -0.107151475 -0.041683179 -0.076050862 -0.139279913
## [69,] 0.316551207 0.190698608 0.199647626 0.199303881 -0.219483163
## [70,] 0.383258366 0.020847059 -0.146149097 0.004991210 -0.084299863
## [71,] -0.292910530 0.444147908 0.210468940 -0.213985642 0.009035006
## [72,] -0.172676764 -0.365439984 0.254195777 0.094620916 0.123477501
## [73,] -0.075809490 0.161425674 -0.184968820 -0.077265419 0.148209405
## [74,] -0.489557470 -0.072382837 -0.035392552 0.419354608 0.106405482
## [75,] 0.173258226 -0.184760803 0.029259504 0.371247941 -0.148538386
## [76,] 0.013079484 -0.327709360 -0.097521060 0.172600008 -0.023148898
## [77,] 0.239372073 -0.328521644 -0.055171441 0.159659471 -0.023147205
## [78,] -0.111944833 -0.180751440 0.046061737 0.169897432 -0.068638273
## [79,] -0.131949666 -0.198395171 0.152086103 -0.025294649 -0.066943063
## [80,] 0.085378475 0.014658850 0.071168608 0.147768175 -0.013212682
## [81,] 0.160917604 0.091591994 -0.257724298 0.185808312 0.305919508
## [82,] -0.147051811 0.306026056 0.043494041 0.063991804 -0.171082748
## [83,] 0.624544142 0.145415632 -0.214102892 -0.011221797 -0.033828136
## [84,] -0.301357054 -0.379864729 -0.043250645 -0.203931507 -0.049295739
## [85,] -0.310365274 0.292857380 0.155544375 0.029679937 0.237230494
## [86,] -0.317136555 0.441943722 0.223166267 -0.013147329 -0.285468462
## [87,] 0.005618820 0.159003752 0.192884402 -0.379349856 0.191404010
## [88,] 0.040105411 -0.249455700 0.194211666 -0.264299155 0.065712225
## [89,] 0.333844233 -0.058208397 -0.257898931 0.086126332 -0.160994454
## PC36 PC37 PC38 PC39 PC40
## [1,] 0.104624044 -0.009268660 -9.869291e-02 -0.0424432209 -0.0240601420
## [2,] -0.064419132 -0.006641011 4.813075e-02 0.0910036089 0.0868250519
## [3,] -0.010524869 -0.009884060 1.162150e-01 -0.0328037148 -0.0710715180
## [4,] -0.073660217 -0.103182810 -8.052394e-02 -0.1187572570 0.0948796942
## [5,] -0.034792002 0.143115679 2.335969e-02 -0.0122600564 -0.0472157954
## [6,] -0.125175684 0.150872795 -9.568225e-03 0.0636255586 -0.0736093852
## [7,] 0.037866907 -0.139974987 1.020212e-01 0.1371312259 0.0309504129
## [8,] -0.078257303 0.021372822 3.395123e-02 -0.0129792735 0.0571400385
## [9,] -0.103420027 0.043420128 -1.504179e-02 -0.1905922168 -0.0481439635
## [10,] 0.014488670 -0.130500484 2.070762e-05 0.0379492136 0.0560380294
## [11,] 0.250286261 -0.125137183 -1.163399e-01 -0.0195916592 0.0640417105
## [12,] -0.004893526 -0.004437912 -2.740954e-02 0.0237221985 -0.0122989936
## [13,] -0.236324413 0.027070285 -6.699373e-02 0.1193380018 -0.0116455675
## [14,] 0.146443063 -0.117079255 3.312759e-02 -0.0532445851 -0.0045111221
## [15,] 0.187408720 0.033904160 1.114943e-01 0.0425050018 -0.0058624297
## [16,] 0.088781115 0.134315251 1.820656e-02 -0.0906570635 0.0057118945
## [17,] 0.061740071 0.038393041 -1.212074e-01 -0.1034208564 -0.0285346757
## [18,] -0.106760194 -0.143541880 -1.470074e-01 0.0837662653 -0.0977820110
## [19,] -0.043109462 0.076009061 1.002561e-01 -0.0217734538 0.0670608502
## [20,] -0.091050257 0.031133849 -7.399814e-03 0.0554565331 -0.0017417640
## [21,] -0.081651689 0.034628803 -1.275586e-01 -0.0016457909 -0.0042091183
## [22,] -0.014128651 -0.051532038 8.902710e-02 0.0065846109 -0.0603286410
## [23,] 0.018820081 0.151222155 -2.566734e-03 0.0612947914 0.0238028250
## [24,] 0.110055509 -0.086091002 2.717791e-02 0.0626841561 -0.0660800806
## [25,] -0.175859193 -0.110993663 -3.481554e-02 -0.0820107698 0.0148144299
## [26,] 0.153419764 -0.051166799 -2.779244e-02 -0.0004412691 -0.0367003462
## [27,] -0.037539404 -0.037252031 9.514598e-02 0.0491791078 0.0085157178
## [28,] -0.083084192 0.021114292 -3.386988e-02 -0.1182294769 0.0152973526
## [29,] 0.043158016 0.065001645 1.355622e-01 -0.0101970337 0.0007133899
## [30,] 0.165461919 0.132465076 -4.381854e-02 0.0212924182 0.1188356626
## [31,] 0.159216995 0.008915305 -1.165783e-01 -0.0539431745 -0.0741306219
## [32,] 0.036582567 0.081156097 2.477559e-02 0.0970707557 0.0179573021
## [33,] -0.232106027 -0.014982114 2.687433e-01 0.0386565134 -0.0819319919
## [34,] 0.052140542 -0.124214004 -1.169154e-02 0.0099233484 -0.0505272943
## [35,] -0.256576299 -0.030984997 -2.200216e-02 -0.1485745437 0.0392722492
## [36,] 0.193040439 0.080566749 -1.001615e-01 -0.0125073479 0.0158311286
## [37,] -0.050325967 0.048544303 -1.311565e-02 0.0090614826 -0.0150971191
## [38,] 0.021892557 -0.087491203 1.794350e-01 0.0045534895 0.1437899229
## [39,] -0.044602322 0.022223660 -3.947844e-02 -0.0046757052 0.0265259238
## [40,] 0.170744623 -0.007341320 -1.412062e-01 0.0518434203 -0.1130329119
## [41,] 0.062455739 0.173002704 5.628487e-02 0.0707627825 -0.0065173562
## [42,] -0.022125907 -0.163514923 -5.542384e-02 0.1439103822 0.0686260661
## [43,] 0.107573283 0.020230660 4.736305e-02 0.0096570125 0.0234078020
## [44,] 0.028584411 -0.037819998 4.358758e-02 -0.0645884397 0.0270476080
## [45,] 0.251544109 -0.216252646 2.121686e-02 0.1060804724 -0.0868493528
## [46,] -0.125964414 -0.143657089 4.260072e-03 -0.1337991658 0.1153319591
## [47,] -0.033443194 0.235713052 -9.691935e-03 0.0026631615 -0.0974017032
## [48,] -0.177318369 -0.013019168 1.510281e-01 -0.1301550327 -0.1047859493
## [49,] 0.081926882 -0.170058912 1.697225e-02 -0.1157011195 -0.0728846819
## [50,] -0.054363465 0.120994521 -1.778030e-01 0.1366112776 -0.0156704838
## [51,] -0.052670511 -0.058938355 -2.156068e-01 -0.1168331076 0.0363477047
## [52,] -0.267623177 -0.032650523 -9.853016e-02 0.0908479594 -0.0585045850
## [53,] -0.055543948 0.218786699 -7.070430e-02 0.0554177322 0.0648757064
## [54,] -0.117917526 0.041796112 8.878502e-02 0.0450711770 0.1050725567
## [55,] 0.223340701 0.135298023 4.617965e-03 -0.0484942332 -0.0325563517
## [56,] 0.181422869 -0.008661229 6.617555e-02 0.0530949581 0.1422545943
## [57,] 0.075381271 -0.078310960 1.026545e-01 -0.0461577405 0.0845274185
## [58,] 0.065469748 0.050741773 3.611287e-02 0.1969101250 -0.0573746747
## [59,] 0.003097148 0.105757111 1.174546e-01 -0.1073355393 -0.0478006244
## [60,] -0.198019053 -0.076526319 4.036573e-02 0.0330540610 -0.0416482847
## [61,] 0.104122744 0.098123235 4.312784e-02 -0.0936353554 -0.0033411838
## [62,] -0.227779816 0.183575216 -1.799663e-01 -0.1280834454 0.0411357732
## [63,] 0.103691269 0.220759951 8.441682e-02 -0.1002804198 -0.0607660005
## [64,] -0.034764580 0.068074392 7.856742e-02 0.0115337945 0.0495295114
## [65,] -0.016906834 0.152958441 -5.821335e-02 0.1531239856 0.0022836127
## [66,] 0.013986339 0.094690971 7.062000e-02 0.0491576445 -0.0076035988
## [67,] 0.064123686 0.090438570 -4.736632e-02 -0.0216244511 0.0438473282
## [68,] 0.101539041 -0.108684350 1.821521e-02 -0.0113796946 -0.0849523579
## [69,] -0.017601156 -0.198306450 4.772440e-02 -0.0198181973 -0.0695260416
## [70,] -0.034872314 -0.238568511 4.312455e-02 0.0530311683 -0.1176942246
## [71,] 0.152226175 -0.029836920 -7.076909e-02 -0.0032410437 0.0216946782
## [72,] -0.150952785 -0.019232788 -8.120878e-02 0.0835592055 0.0700507284
## [73,] -0.135747706 -0.229366872 -4.588694e-02 0.1300260775 0.0501859598
## [74,] 0.018225076 -0.090573998 6.125173e-02 0.1522646716 0.0154587875
## [75,] 0.078961029 0.053931980 9.284712e-02 0.0656350343 0.0160078828
## [76,] 0.083995983 0.125658110 -3.508494e-02 0.0244686336 -0.1064812445
## [77,] -0.103898134 -0.045877105 7.621962e-02 -0.0104845141 0.0823166533
## [78,] -0.302133474 -0.132294164 1.930449e-02 -0.0348385436 -0.0320826424
## [79,] -0.048486298 -0.014109215 1.024421e-01 -0.0849751489 0.0927586344
## [80,] -0.101063083 0.023147787 -2.558080e-02 -0.1499648236 -0.0544684233
## [81,] 0.158343827 -0.185328212 -7.900029e-03 -0.1735495829 0.0062427055
## [82,] 0.026511202 -0.027745301 -2.928900e-01 0.0050627936 -0.0350256604
## [83,] 0.081657059 -0.125165387 -2.650579e-01 0.0069228724 0.0967632525
## [84,] 0.158099753 -0.052539296 6.495795e-02 -0.0649644532 0.1124486360
## [85,] -0.059223566 0.144128618 -1.880425e-02 -0.0520836108 0.0036961812
## [86,] 0.008172452 0.127514918 1.435632e-02 0.0254906330 -0.0510595321
## [87,] 0.126376629 0.010245980 1.052264e-01 -0.1196723140 -0.0164231377
## [88,] -0.027903405 -0.063617469 2.762869e-02 0.0452125470 -0.1973394658
## [89,] -0.062416749 0.111339590 7.769243e-03 0.1461965817 0.1273577263
## PC41 PC42 PC43 PC44
## [1,] -0.0117791382 2.732181e-02 0.1029800743 0.0243674957
## [2,] -0.0584678265 -4.169103e-03 -0.0216112952 -0.0006684653
## [3,] 0.0147564590 -1.285902e-02 0.0145578713 -0.0016829325
## [4,] -0.0044999372 2.733842e-02 -0.0572151362 -0.0082245776
## [5,] 0.0359498457 1.446277e-02 0.0160558170 0.0059549086
## [6,] 0.0133151030 7.228842e-02 -0.0103485283 -0.0295109195
## [7,] -0.0346841991 -1.916986e-02 0.0094893235 -0.0536757507
## [8,] 0.1028503316 -6.744731e-02 -0.0369237563 0.0134214313
## [9,] 0.0045435279 6.219503e-02 0.0657637261 0.0760463282
## [10,] -0.1893700953 1.353447e-01 0.0032544017 -0.0098146054
## [11,] 0.1544573260 2.992896e-02 -0.0247597163 0.0332819017
## [12,] 0.0026308172 9.994198e-03 0.0097426933 -0.0150801936
## [13,] -0.0065443534 -1.643414e-01 -0.0074891119 0.0126577904
## [14,] -0.0476608678 -5.518806e-03 -0.0102482274 -0.0186010641
## [15,] -0.0068003034 -4.938604e-02 0.0108818317 -0.0564345663
## [16,] 0.0420997879 -1.415999e-01 -0.0053747233 0.0024568281
## [17,] -0.0583224384 -3.165080e-02 -0.0302484440 -0.0168079620
## [18,] 0.0081657842 -3.986398e-03 0.0461479011 0.0110199521
## [19,] 0.0069247607 9.923139e-02 0.0535237705 0.0239434031
## [20,] 0.0515591069 4.114805e-02 0.0085216810 -0.0195248166
## [21,] 0.0337621244 -5.894371e-02 -0.0293946999 -0.0039369808
## [22,] 0.0931521136 2.627231e-02 -0.0466616853 0.0168156934
## [23,] -0.0195321685 -6.810930e-03 -0.0435579934 0.0441216063
## [24,] -0.0712763136 -2.197587e-02 -0.0637844325 0.0413015410
## [25,] -0.0662333114 -3.140107e-02 0.0794486360 0.0378703030
## [26,] -0.0069719519 6.569833e-02 0.0987594745 -0.0492468592
## [27,] 0.0623874092 4.053796e-02 0.0864405265 -0.0660590014
## [28,] -0.0164295647 5.052590e-02 -0.0967303339 -0.0142305031
## [29,] -0.0276756917 -6.890791e-03 -0.0454027902 0.0490055264
## [30,] 0.0123709556 -1.777972e-02 -0.0145863886 -0.0248919328
## [31,] -0.0320568244 -1.050801e-01 0.0213608063 0.0501655414
## [32,] -0.0163226092 3.258706e-02 -0.0339356987 0.0101821152
## [33,] 0.0994873630 1.972095e-02 -0.0794010616 -0.0014446850
## [34,] 0.0841759676 -3.525688e-02 -0.0470602467 0.0022550658
## [35,] -0.0491552716 -3.127826e-02 -0.0750689964 -0.0239712800
## [36,] -0.0242159781 3.851871e-02 0.0525219623 0.0028875881
## [37,] -0.0031093862 1.714228e-02 0.0093633283 0.0118219342
## [38,] -0.0229086334 -6.322563e-03 -0.0214065643 -0.0264930876
## [39,] 0.0057742253 1.198630e-02 -0.0022535919 -0.0451122809
## [40,] -0.0545260043 -2.729216e-02 -0.0055481734 -0.0143021161
## [41,] -0.0006634143 -2.330478e-02 0.0672204075 -0.0368232937
## [42,] -0.0451456323 -4.194013e-02 0.0303640715 0.0138044098
## [43,] -0.0439590330 -3.098862e-03 0.0684764356 0.0427319672
## [44,] 0.0304868628 -1.964091e-02 -0.0035332504 0.0416367290
## [45,] 0.0838726063 5.319045e-02 -0.0284237516 -0.0245032351
## [46,] 0.0284863261 -2.998719e-03 -0.0015160989 -0.0286897652
## [47,] 0.0028776405 4.058136e-03 -0.0166120453 0.0082336784
## [48,] -0.0003764842 -1.048271e-01 0.0188096465 -0.0630684244
## [49,] -0.1126082366 7.330301e-02 -0.0004664073 0.0336209835
## [50,] -0.0607150644 5.660201e-02 -0.0452644455 0.0285906946
## [51,] 0.0263368385 7.991777e-03 -0.0592055929 -0.0094030931
## [52,] 0.1065606318 6.372429e-03 0.0098535314 0.0156552174
## [53,] 0.0674603643 1.175033e-01 -0.0329127532 -0.0735727278
## [54,] -0.0557318642 -3.395721e-02 -0.0153738433 0.0363099260
## [55,] 0.0151980596 -2.319826e-02 0.0434709705 0.0421252382
## [56,] 0.1207208561 3.567883e-03 0.0113626544 0.0606179238
## [57,] -0.1358684889 -2.887566e-02 -0.0333303806 -0.0872986005
## [58,] -0.0594687570 -9.029625e-03 0.0218156453 -0.0091407004
## [59,] -0.0241164353 -8.973578e-03 -0.0175166664 0.0327555131
## [60,] -0.0290772508 3.411902e-02 0.0934956906 0.0308154477
## [61,] 0.0019868892 -4.116600e-02 0.0072731021 -0.0664455209
## [62,] -0.0263391484 -2.077396e-02 -0.0111128569 -0.0260929553
## [63,] -0.0410803638 2.204289e-02 0.0127616321 -0.0153194347
## [64,] 0.1132307905 -5.675111e-02 0.0421297956 0.0230808900
## [65,] 0.0646267548 -2.579981e-02 -0.0280177798 -0.0163713633
## [66,] -0.0558261572 -3.337998e-02 0.0456566290 0.0832437833
## [67,] -0.0344591158 4.947093e-02 -0.0308262454 -0.0195897540
## [68,] 0.1473299959 7.606853e-02 -0.0860329494 0.0103702242
## [69,] 0.0106454288 1.190081e-02 -0.0904500149 0.0526101089
## [70,] 0.0606146820 1.110995e-02 -0.0058454813 0.0276979140
## [71,] -0.0176264371 -1.526109e-02 -0.0113139704 -0.0128718041
## [72,] -0.0242051637 5.359337e-02 0.0209582779 -0.0044965905
## [73,] 0.0217962950 -3.965363e-02 0.0893427553 -0.0128785944
## [74,] -0.0891257726 -4.165025e-02 -0.0409645022 -0.0426228199
## [75,] 0.0032284719 -2.979149e-02 -0.0392539225 -0.0195074220
## [76,] -0.1233681575 -1.400350e-02 0.0357038912 -0.0095058420
## [77,] -0.0773933245 -6.398997e-04 0.0730357266 0.0384294020
## [78,] -0.0372090701 2.838023e-02 0.0291588698 -0.0250422935
## [79,] -0.0902581222 1.032431e-02 0.0099336603 0.0443662703
## [80,] 0.1483880210 3.498082e-02 0.0839863001 -0.0562482935
## [81,] 0.0780330047 -1.661429e-02 0.0481960212 0.0022757235
## [82,] 0.0400277489 -3.831811e-02 0.0597992093 -0.0438909557
## [83,] -0.0407333961 -3.496993e-02 -0.0848956648 0.0215270674
## [84,] 0.0272180770 4.329219e-03 0.0076673718 -0.0148052844
## [85,] -0.0424704373 3.193497e-02 -0.0619027572 0.0474252478
## [86,] -0.0156119732 9.865465e-02 -0.0319387884 0.0107203188
## [87,] 0.0333061128 -2.862057e-02 -0.0034678814 -0.0273901116
## [88,] -0.0857711906 -2.527899e-02 -0.0552555514 -0.0159424107
## [89,] 0.1369558923 -6.411497e-05 0.0251590760 0.0430142380
## PC45 PC46 PC47 PC48
## [1,] -0.0345840753 0.0017133329 8.588080e-03 -2.005993e-03
## [2,] -0.0026807708 -0.0099252450 -6.194912e-03 1.432903e-03
## [3,] 0.0042540215 0.0338067487 -2.335612e-02 -8.337511e-03
## [4,] -0.0157794264 -0.0353315135 7.536885e-05 3.705835e-03
## [5,] 0.0399770306 -0.0106261154 1.416257e-02 -8.230032e-04
## [6,] 0.0181933112 0.0099164598 3.177086e-02 1.709143e-02
## [7,] 0.0417108028 -0.0296180251 -1.769157e-02 -2.337611e-02
## [8,] -0.0008022954 0.0395241914 8.990583e-03 6.591276e-04
## [9,] 0.0056158471 -0.0270487589 -3.125124e-02 -4.065525e-03
## [10,] -0.0660849951 -0.0018257175 2.373707e-02 1.272223e-02
## [11,] -0.0250380739 0.0283755311 -2.400907e-02 1.106729e-02
## [12,] -0.0018726508 0.0022601834 -6.547717e-03 -6.278403e-04
## [13,] 0.0120501206 -0.0042701414 1.018343e-03 3.070424e-03
## [14,] -0.0058618722 0.0091036336 2.972630e-02 -1.274763e-02
## [15,] -0.0063162862 0.0019185240 -1.300819e-02 -1.089181e-02
## [16,] -0.0216438014 -0.0009972931 -2.468723e-02 -4.652920e-03
## [17,] 0.0396872805 0.0110818093 -9.719308e-04 9.478766e-03
## [18,] 0.0306164862 0.0259696350 1.265672e-02 -1.024257e-03
## [19,] -0.0155304992 -0.0081627787 2.947455e-02 4.741304e-05
## [20,] -0.0030981429 0.0290095508 6.827289e-05 -3.684688e-06
## [21,] -0.0238561836 -0.0324655715 2.026960e-03 -5.900618e-04
## [22,] -0.0344694633 -0.0202515381 -1.414620e-02 3.233997e-03
## [23,] -0.0261905987 -0.0067504908 3.989151e-02 2.066150e-02
## [24,] 0.0161728529 0.0289235090 2.981025e-02 -1.727127e-02
## [25,] 0.0581028750 -0.0144337572 -2.734119e-02 7.190664e-03
## [26,] -0.0171710967 0.0012743467 -2.813763e-02 3.539298e-03
## [27,] 0.0497347870 -0.0072515241 7.777586e-03 1.992348e-02
## [28,] -0.0221323989 -0.0305614394 -5.167491e-03 -6.649907e-04
## [29,] 0.0097531346 -0.0084075982 -2.263042e-02 2.551665e-02
## [30,] 0.0195857271 -0.0101102231 2.551810e-03 -7.245565e-03
## [31,] -0.0126954325 0.0053953154 2.904208e-02 -1.145446e-02
## [32,] -0.0042164866 0.0275770298 -3.356396e-02 -1.487411e-02
## [33,] 0.0018910122 -0.0167135530 -2.726504e-02 3.340942e-03
## [34,] -0.0235476796 -0.0157376786 1.406571e-02 -1.408197e-03
## [35,] -0.0051916256 0.0029869050 -5.901557e-03 -3.682726e-03
## [36,] -0.0044270046 0.0072826768 1.128336e-02 1.054976e-03
## [37,] 0.0142841739 0.0010096852 8.890165e-03 -1.506119e-03
## [38,] 0.0572605393 0.0369055034 3.795264e-02 1.113377e-03
## [39,] -0.0156840288 0.0113238152 -1.046795e-02 -1.561778e-03
## [40,] 0.0134293341 -0.0156752139 -1.962382e-02 -1.731390e-02
## [41,] 0.0022101442 -0.0177143234 -7.784606e-03 -6.625995e-03
## [42,] 0.0200504449 0.0329050762 3.731561e-03 -1.847095e-03
## [43,] 0.0264775128 0.0007868600 -7.226386e-03 1.067224e-02
## [44,] -0.0362505581 0.0241889973 -3.388617e-03 -2.930432e-03
## [45,] -0.0339320159 -0.0360488323 -3.865888e-03 -9.336728e-03
## [46,] -0.0294219432 -0.0028128782 1.495363e-02 -1.113663e-02
## [47,] -0.0160066648 -0.0054300979 -6.182360e-03 -1.485392e-03
## [48,] 0.0207655307 -0.0103572897 5.251221e-03 8.768910e-03
## [49,] 0.0470326786 -0.0322238281 6.685335e-03 2.922130e-03
## [50,] 0.0163383868 0.0162450683 -6.453958e-03 2.522720e-02
## [51,] 0.0215499106 0.0087551670 -6.767927e-03 1.388836e-03
## [52,] 0.0114985038 -0.0215455814 1.751798e-02 -1.337323e-02
## [53,] 0.0304728344 0.0270167762 -3.823981e-03 -1.387116e-02
## [54,] -0.0553458882 0.0245405789 -5.086902e-03 1.126810e-03
## [55,] -0.0435969622 0.0007341986 -8.495237e-03 1.232983e-02
## [56,] 0.0679293296 0.0036514974 9.814549e-03 -8.062001e-03
## [57,] -0.0223209156 -0.0244777767 -3.021316e-03 1.518914e-02
## [58,] -0.0220605404 -0.0456490757 -5.077284e-03 5.222511e-04
## [59,] -0.0455028496 -0.0127262109 -8.917297e-03 -1.794007e-03
## [60,] 0.0232562803 0.0138550332 -1.446592e-02 -1.812775e-02
## [61,] 0.0232495801 0.0403752349 2.735951e-03 8.144217e-03
## [62,] -0.0094673135 0.0160050107 3.832929e-04 -1.276661e-02
## [63,] -0.0424954239 -0.0263146578 2.035568e-02 -2.531299e-03
## [64,] -0.0127423772 -0.0414212521 2.950192e-02 2.592055e-03
## [65,] -0.0202418673 -0.0068365594 6.472344e-03 -2.138027e-04
## [66,] 0.0116766737 0.0261751486 -7.284169e-03 1.558950e-02
## [67,] 0.0514055485 -0.0166397705 -9.928352e-03 6.370529e-03
## [68,] 0.0302856481 0.0367344526 2.627702e-04 5.385756e-03
## [69,] 0.0284279845 -0.0261983446 -1.538329e-03 -1.423433e-02
## [70,] -0.0473567140 0.0304215711 2.076731e-02 1.621293e-02
## [71,] -0.0056632868 0.0027744017 8.241815e-03 2.655355e-03
## [72,] 0.0100590073 -0.0311329845 -7.079570e-03 2.156560e-03
## [73,] -0.0615277595 0.0277645116 -3.251253e-02 5.386184e-03
## [74,] -0.0044292359 -0.0144237326 1.044088e-02 -1.180348e-02
## [75,] 0.0023892411 0.0150853291 -2.034228e-02 6.252839e-03
## [76,] 0.0232291404 0.0222341336 -1.175968e-03 -1.061668e-02
## [77,] -0.0243814232 -0.0025273171 1.107906e-02 -2.676475e-02
## [78,] -0.0533412743 0.0277703688 1.466504e-02 3.949525e-03
## [79,] 0.0349164673 0.0030726289 -1.992383e-02 1.075512e-02
## [80,] 0.0062492966 0.0101401074 -7.685111e-03 2.915253e-03
## [81,] 0.0425222365 -0.0288192556 1.978323e-02 -8.145663e-03
## [82,] -0.0128439245 -0.0199731855 1.558031e-02 1.316010e-02
## [83,] -0.0013536146 -0.0068371436 -2.817240e-02 3.714237e-03
## [84,] -0.0034312630 0.0022765372 -1.999431e-02 1.504115e-02
## [85,] 0.0215830333 0.0077601392 1.898473e-02 -2.056425e-02
## [86,] -0.0322436746 0.0208295849 -2.106238e-02 -2.944445e-02
## [87,] -0.0265849746 0.0323999652 1.810311e-02 -2.626808e-03
## [88,] 0.0387524407 -0.0024531519 8.480837e-04 2.515736e-02
## [89,] 0.0067701618 -0.0511293396 9.499562e-03 5.995677e-03
## PC49 PC50 PC51
## [1,] 1.534825e-03 1.223678e-15 9.900763e-16
## [2,] -4.767644e-03 9.195703e-16 1.262875e-15
## [3,] 3.201858e-03 -1.260977e-15 1.050190e-15
## [4,] -1.304594e-04 2.122380e-17 4.212684e-16
## [5,] 2.923447e-03 -1.350323e-15 6.628255e-17
## [6,] -1.065997e-03 -2.128291e-16 1.497435e-15
## [7,] -1.445365e-03 6.298383e-16 8.401775e-16
## [8,] -2.729038e-03 -6.026250e-16 1.623753e-15
## [9,] 7.010432e-03 -6.462826e-16 1.601591e-15
## [10,] 1.374102e-03 -8.572535e-16 1.231858e-15
## [11,] -6.743624e-03 -5.553208e-16 1.778696e-15
## [12,] -1.549349e-03 2.364685e-16 1.459812e-15
## [13,] 3.311805e-04 -2.716688e-16 3.248151e-16
## [14,] -1.281639e-02 -1.311321e-15 1.114085e-15
## [15,] 1.312876e-02 -1.504314e-15 1.311450e-15
## [16,] -1.602536e-03 -1.161623e-15 7.457119e-16
## [17,] -6.650679e-03 -3.646643e-16 7.303710e-16
## [18,] 7.174908e-04 -1.156334e-15 1.260316e-15
## [19,] 1.206345e-02 -8.309537e-18 4.304606e-16
## [20,] -3.583559e-03 -1.604726e-15 2.084795e-15
## [21,] 3.893028e-03 -1.869620e-15 7.037275e-16
## [22,] -4.802495e-03 -1.294904e-15 2.653301e-16
## [23,] -6.091058e-06 -3.437119e-16 1.172123e-15
## [24,] -1.706768e-03 -4.308277e-16 4.636196e-16
## [25,] -3.370513e-03 -1.412159e-15 3.925806e-16
## [26,] 9.972263e-03 -2.365531e-16 5.780423e-16
## [27,] -1.819272e-03 8.909928e-16 -2.964276e-16
## [28,] 4.850989e-03 6.904231e-17 1.350089e-16
## [29,] -7.769151e-03 -1.378208e-16 1.093419e-15
## [30,] 1.056349e-03 4.109483e-16 3.529473e-16
## [31,] 8.432584e-03 -3.274575e-16 2.914955e-16
## [32,] 6.751701e-03 -4.011629e-16 1.015642e-15
## [33,] -1.546581e-03 -8.078562e-16 2.465365e-16
## [34,] 7.130359e-03 2.626484e-17 1.005621e-15
## [35,] -2.496377e-03 -1.028790e-15 5.741297e-16
## [36,] -7.552841e-03 -1.119302e-15 1.752546e-16
## [37,] -2.472726e-03 -4.414071e-18 1.575089e-16
## [38,] -3.197697e-03 3.672966e-16 5.073968e-16
## [39,] -6.819888e-03 2.961474e-16 6.229368e-16
## [40,] 4.590680e-03 1.314509e-16 2.113557e-16
## [41,] -3.376616e-03 -7.445282e-17 3.244564e-16
## [42,] -7.000489e-03 -2.636255e-16 -4.843078e-17
## [43,] 5.902417e-04 1.535753e-16 1.098909e-15
## [44,] 7.425420e-03 1.165747e-17 -4.125868e-17
## [45,] -6.224753e-05 3.692552e-16 2.624574e-16
## [46,] 1.544713e-03 -2.359286e-16 9.390428e-16
## [47,] -1.784017e-02 -2.694869e-16 1.804236e-16
## [48,] 1.182224e-02 4.976728e-16 1.400851e-15
## [49,] -5.891561e-04 4.817396e-16 4.822822e-16
## [50,] 8.377036e-03 1.479558e-16 4.335375e-17
## [51,] 6.288593e-03 -9.143432e-16 3.574362e-16
## [52,] -1.362983e-02 -4.688077e-16 6.980409e-16
## [53,] 8.406218e-03 1.978327e-16 3.452723e-16
## [54,] -2.410302e-03 -7.134210e-17 -3.200324e-16
## [55,] -5.647487e-03 4.051650e-17 2.998231e-16
## [56,] -1.006984e-03 -2.216053e-16 -4.985103e-16
## [57,] 4.854553e-04 -3.000729e-16 -1.252940e-16
## [58,] -3.439352e-03 -4.179268e-16 -1.918299e-16
## [59,] 2.722396e-03 -6.498423e-16 -1.102412e-15
## [60,] 4.069038e-03 -2.362175e-16 -2.493613e-16
## [61,] 2.159555e-03 5.023420e-17 -1.265112e-15
## [62,] -2.043131e-03 1.987022e-16 -8.350337e-16
## [63,] -1.041641e-02 -5.822632e-16 -7.923069e-16
## [64,] -7.720633e-04 -2.787853e-16 -6.802094e-16
## [65,] 8.107914e-03 -1.582986e-16 -1.395664e-15
## [66,] 5.248109e-03 4.097855e-16 -6.379199e-16
## [67,] -7.097931e-03 -3.473626e-16 -7.704193e-16
## [68,] 1.653124e-03 -1.875228e-16 -1.119467e-15
## [69,] 4.598807e-03 -3.240121e-16 -7.103810e-16
## [70,] 5.905265e-03 2.819973e-16 -1.508096e-15
## [71,] -2.610023e-03 1.069971e-16 -1.134508e-15
## [72,] 2.216366e-03 4.990520e-16 -6.820938e-16
## [73,] -5.596686e-03 1.401165e-15 -1.137050e-15
## [74,] 4.328203e-03 -6.677401e-17 -3.912463e-16
## [75,] 8.911402e-04 5.726742e-16 -1.236716e-15
## [76,] 4.627222e-04 -3.213183e-16 -1.167997e-15
## [77,] -6.486701e-03 3.736043e-16 -5.699238e-16
## [78,] -9.646436e-05 -3.297022e-16 -7.488964e-16
## [79,] 6.711975e-03 -2.843325e-16 -1.682594e-15
## [80,] -8.428585e-03 2.725091e-16 -8.547008e-16
## [81,] -5.424846e-03 -7.011946e-17 -1.449663e-15
## [82,] 8.113809e-03 4.234403e-16 -1.324322e-15
## [83,] -3.454382e-03 7.733795e-16 -5.983762e-16
## [84,] -2.221235e-05 1.014066e-15 -1.296432e-15
## [85,] 5.596060e-03 8.977113e-16 -1.679249e-15
## [86,] -6.716547e-03 6.988433e-16 -2.457748e-15
## [87,] 4.592048e-03 1.350235e-15 -1.270724e-15
## [88,] -6.197425e-03 5.386260e-16 -1.807938e-15
## [89,] 5.731149e-03 8.889545e-16 -1.455876e-15# get information stored within PCA object
ls(te_vars.prc) ## [1] "center" "rotation" "scale" "sdev" "x"plot(te_vars.prc$x[,1:2])
As shown, the prcomp function results in five useful measures:
- center and scale refers to respective mean and standard deviation of the variables that are used for normalization prior to implementing PCA.
# outputs the mean of variables
te_vars.prc$center## InflationRate RetailSalesMom
## 9.022360e+02 3.539326e-01
## AverageHourlyEarnings BankLendingRate
## 1.752809e-01 3.321798e+00
## ConstructionSpending EmploymentRate
## 4.404494e-01 5.893146e+01
## HousingIndex InflationRateMoM
## -1.842697e+00 1.666292e+00
## LaborForceParticipationRate LongTermUnemploymentRate
## 4.243159e+03 3.771910e+00
## PendingHomeSales PersonalIncome
## 2.624719e+00 3.438202e-01
## PersonalSavings PersonalSpending
## 5.866292e+00 3.134831e-01
## RetailSalesExcludingAutos WageGrowth
## 3.011236e-01 3.875618e+00
## YouthUnemploymentRate CoreInflationRate
## 1.440674e+01 1.767416e+00
## RetailSalesYoY USGovernment10yrBondRate
## 4.468539e+00 2.367464e+00
## UnemploymentRate InitialJoblessClaims
## 7.031461e+00 1.325843e-01
## NonfarmPayrolls ADPEmploymentChange
## 1.843146e+02 1.867416e+02
## BuildingPermits ContinuingJoblessClaims
## 9.434382e+02 3.023880e+03
## EmployedPersons ExistingHomeSales
## 1.450250e+05 8.590894e+01
## FulltimeEmployment GovernmentPayrolls
## 4.871124e+03 6.823077e-01
## HousingStarts JobOffers
## 3.078013e-01 3.421809e+02
## JobVacancies ManufacturingPayrolls
## 4.263191e+03 1.703958e+00
## NewHomeSales NonfarmPayrollsPrivate
## 4.269663e+01 1.861573e+02
## UnemployedPersons ChainStoreSales
## 1.094307e+04 5.073818e+03
## ConsumerCredit Wages
## 1.445562e+01 2.031719e+01
## WagesinManufacturing GasolinePrices
## 1.947573e+01 1.175191e+05
## ExchangeRate StockMarket
## 8.590894e+01 1.502452e+04
## ConsumerPriceIndex CoreConsumerPrices
## 2.320358e+02 2.349155e+02
## EconomicOptimism LaborMarketConditions
## 4.685056e+01 6.341011e+01
## NAHBHousingMarket AverageWeeklyHours
## 1.033708e+01 3.439551e+01
## ChallengerJobCuts
## 4.468493e+04# outputs the standard deviation of variables
te_vars.prc$scale## InflationRate RetailSalesMom
## 2.331850e+02 5.931534e-01
## AverageHourlyEarnings BankLendingRate
## 1.325286e-01 1.657323e-01
## ConstructionSpending EmploymentRate
## 1.146707e+00 5.426357e-01
## HousingIndex InflationRateMoM
## 6.608395e+01 9.538134e-01
## LaborForceParticipationRate LongTermUnemploymentRate
## 1.047223e+03 3.610905e+00
## PendingHomeSales PersonalIncome
## 9.974391e+00 7.066173e-01
## PersonalSavings PersonalSpending
## 9.134032e-01 2.408277e-01
## RetailSalesExcludingAutos WageGrowth
## 5.373534e-01 1.448449e+00
## YouthUnemploymentRate CoreInflationRate
## 3.011448e+00 4.266080e-01
## RetailSalesYoY USGovernment10yrBondRate
## 1.767612e+00 5.807941e-01
## UnemploymentRate InitialJoblessClaims
## 1.776334e+00 2.098567e-01
## NonfarmPayrolls ADPEmploymentChange
## 1.048681e+02 7.295493e+01
## BuildingPermits ContinuingJoblessClaims
## 2.412555e+02 8.239206e+02
## EmployedPersons ExistingHomeSales
## 4.580797e+03 8.380687e+00
## FulltimeEmployment GovernmentPayrolls
## 5.074299e+02 9.825132e-02
## HousingStarts JobOffers
## 4.495530e-01 6.987069e+01
## JobVacancies ManufacturingPayrolls
## 1.042006e+03 4.011878e-01
## NewHomeSales NonfarmPayrollsPrivate
## 1.900682e+01 7.540118e+01
## UnemployedPersons ChainStoreSales
## 2.640933e+03 1.547356e+03
## ConsumerCredit Wages
## 1.391188e+01 9.066948e-01
## WagesinManufacturing GasolinePrices
## 6.496046e-01 4.536619e+03
## ExchangeRate StockMarket
## 8.380687e+00 2.994473e+03
## ConsumerPriceIndex CoreConsumerPrices
## 7.693804e+00 9.446230e+00
## EconomicOptimism LaborMarketConditions
## 3.695271e+00 7.231630e-01
## NAHBHousingMarket AverageWeeklyHours
## 1.415414e+01 1.421508e-01
## ChallengerJobCuts
## 1.436366e+04- The rotation measure provides the principal component loading. Each column of the rotation matrix contains the principal component loading vector. This is the most important measure to look at. The following code picks out the matrix of projections of the variables on to the components. This is called rotation because it can be thought of as rotating the coordinate axes in feature-vector space.
te_vars.prc$rotation[1:10]## [1] 0.19449922 -0.03960478 0.03133653 0.12416034 0.04787899
## [6] 0.18680683 0.04495814 -0.09836380 0.19193356 -0.10901923te_vars.prc$rotation[1:20]## [1] 0.194499219 -0.039604781 0.031336528 0.124160342 0.047878986
## [6] 0.186806830 0.044958142 -0.098363802 0.191933563 -0.109019234
## [11] 0.016100922 -0.009896372 -0.032717062 -0.004639575 -0.034766500
## [16] 0.076405115 -0.199816217 0.125616095 -0.129536739 -0.113989958te.latent.sem <- te_vars.prc$rotation
round(te_vars.prc$rotation[, 1:2], 2)## PC1 PC2
## InflationRate 0.19 -0.01
## RetailSalesMom -0.04 -0.10
## AverageHourlyEarnings 0.03 0.03
## BankLendingRate 0.12 0.19
## ConstructionSpending 0.05 -0.17
## EmploymentRate 0.19 0.12
## HousingIndex 0.04 -0.10
## InflationRateMoM -0.10 -0.08
## LaborForceParticipationRate 0.19 0.00
## LongTermUnemploymentRate -0.11 -0.12
## PendingHomeSales 0.02 -0.03
## PersonalIncome -0.01 -0.07
## PersonalSavings -0.03 -0.08
## PersonalSpending 0.00 -0.10
## RetailSalesExcludingAutos -0.03 -0.10
## WageGrowth 0.08 -0.24
## YouthUnemploymentRate -0.20 -0.03
## CoreInflationRate 0.13 -0.10
## RetailSalesYoY -0.13 -0.11
## USGovernment10yrBondRate -0.11 0.08
## UnemploymentRate -0.20 0.00
## InitialJoblessClaims -0.03 -0.10
## NonfarmPayrolls 0.06 -0.34
## ADPEmploymentChange 0.07 -0.34
## BuildingPermits 0.20 -0.02
## ContinuingJoblessClaims -0.20 0.08
## EmployedPersons 0.20 0.04
## ExistingHomeSales 0.18 0.17
## FulltimeEmployment 0.18 0.00
## GovernmentPayrolls -0.07 -0.20
## HousingStarts 0.12 -0.11
## JobOffers -0.20 0.06
## JobVacancies 0.20 0.01
## ManufacturingPayrolls -0.08 -0.22
## NewHomeSales 0.19 -0.03
## NonfarmPayrollsPrivate 0.04 -0.39
## UnemployedPersons -0.20 0.01
## ChainStoreSales -0.03 -0.08
## ConsumerCredit 0.06 -0.09
## Wages 0.20 0.02
## WagesinManufacturing 0.20 0.05
## GasolinePrices 0.20 0.04
## ExchangeRate 0.18 0.17
## StockMarket 0.20 -0.03
## ConsumerPriceIndex 0.20 -0.08
## CoreConsumerPrices 0.20 0.02
## EconomicOptimism 0.09 0.14
## LaborMarketConditions -0.19 0.11
## NAHBHousingMarket -0.05 -0.22
## AverageWeeklyHours 0.12 -0.31
## ChallengerJobCuts -0.01 -0.01# sort the rotations in decreasing order of importance
signif(sort(te.latent.sem[, 1], decreasing = TRUE)[1:30], 2)## CoreConsumerPrices EmployedPersons
## 0.200 0.200
## GasolinePrices Wages
## 0.200 0.200
## JobVacancies WagesinManufacturing
## 0.200 0.200
## BuildingPermits StockMarket
## 0.200 0.200
## ConsumerPriceIndex InflationRate
## 0.200 0.190
## NewHomeSales LaborForceParticipationRate
## 0.190 0.190
## EmploymentRate FulltimeEmployment
## 0.190 0.180
## ExistingHomeSales ExchangeRate
## 0.180 0.180
## CoreInflationRate BankLendingRate
## 0.130 0.120
## HousingStarts AverageWeeklyHours
## 0.120 0.120
## EconomicOptimism WageGrowth
## 0.089 0.076
## ADPEmploymentChange NonfarmPayrolls
## 0.073 0.060
## ConsumerCredit ConstructionSpending
## 0.059 0.048
## HousingIndex NonfarmPayrollsPrivate
## 0.045 0.044
## AverageHourlyEarnings PendingHomeSales
## 0.031 0.016We see that there are 51 distinct components (PC1-PC51) and start to identify what is being represented in each one. For example, the first component (PC1) seems to explain the divergence of Employment to Unemployment-related features. This also makes a lot of intuitive sense that this is the first component, since it is the largest divergence in variance. The second component (PC2) shows the correlation between Bank Lending Rate and Existing Home Sales and a divergence with Payrolls (Manufacturing and Non-farm) and ADP Employment Change.
- The x value is the value (scores) of each observation in the original dataset projected to the principal components.
The summary method describes the importance of the principle components as it outputs the proportion of variance explained by the components.
- The first row describes the standard deviation associated with each principle component.
- The second row shows the proportion of the variance in the data explained by each component.
- The third row describes the cumulative proportion of explained variance.
# view summary of PCA results
summary(te_vars.prc)## Importance of components%s:
## PC1 PC2 PC3 PC4 PC5 PC6
## Standard deviation 4.8984 2.03439 1.81155 1.49701 1.43228 1.40899
## Proportion of Variance 0.4705 0.08115 0.06435 0.04394 0.04022 0.03893
## Cumulative Proportion 0.4705 0.55163 0.61597 0.65992 0.70014 0.73907
## PC7 PC8 PC9 PC10 PC11 PC12
## Standard deviation 1.2933 1.24493 1.06771 1.02360 0.96001 0.9392
## Proportion of Variance 0.0328 0.03039 0.02235 0.02054 0.01807 0.0173
## Cumulative Proportion 0.7719 0.80225 0.82461 0.84515 0.86322 0.8805
## PC13 PC14 PC15 PC16 PC17 PC18
## Standard deviation 0.89604 0.8861 0.77208 0.7422 0.70000 0.63660
## Proportion of Variance 0.01574 0.0154 0.01169 0.0108 0.00961 0.00795
## Cumulative Proportion 0.89626 0.9117 0.92334 0.9341 0.94375 0.95170
## PC19 PC20 PC21 PC22 PC23 PC24
## Standard deviation 0.5931 0.5533 0.5394 0.50937 0.46031 0.40810
## Proportion of Variance 0.0069 0.0060 0.0057 0.00509 0.00415 0.00327
## Cumulative Proportion 0.9586 0.9646 0.9703 0.97539 0.97954 0.98281
## PC25 PC26 PC27 PC28 PC29 PC30
## Standard deviation 0.38162 0.34878 0.34416 0.29812 0.2768 0.27251
## Proportion of Variance 0.00286 0.00239 0.00232 0.00174 0.0015 0.00146
## Cumulative Proportion 0.98567 0.98805 0.99037 0.99212 0.9936 0.99507
## PC31 PC32 PC33 PC34 PC35 PC36
## Standard deviation 0.25315 0.21342 0.18093 0.1745 0.13757 0.12262
## Proportion of Variance 0.00126 0.00089 0.00064 0.0006 0.00037 0.00029
## Cumulative Proportion 0.99633 0.99722 0.99787 0.9985 0.99883 0.99913
## PC37 PC38 PC39 PC40 PC41 PC42
## Standard deviation 0.10982 0.09405 0.08416 0.06738 0.06523 0.04921
## Proportion of Variance 0.00024 0.00017 0.00014 0.00009 0.00008 0.00005
## Cumulative Proportion 0.99937 0.99954 0.99968 0.99977 0.99985 0.99990
## PC43 PC44 PC45 PC46 PC47 PC48
## Standard deviation 0.04669 0.03506 0.02922 0.02191 0.01716 0.01125
## Proportion of Variance 0.00004 0.00002 0.00002 0.00001 0.00001 0.00000
## Cumulative Proportion 0.99994 0.99996 0.99998 0.99999 1.00000 1.00000
## PC49 PC50 PC51
## Standard deviation 0.005924 4.56e-16 4.56e-16
## Proportion of Variance 0.000000 0.00e+00 0.00e+00
## Cumulative Proportion 1.000000 1.00e+00 1.00e+00- We can see that the first component accounts for 46% of the variance, while the second component accounts for 8% of the variance.
- It appears that 80% of the (cumulative) variance can be explained in the first nine components (PC1 - PC9).
- It appears that 90% of the variance can be explained in the first fourteen components (PC1 - PC14).
Using PCA we have reduced 92 predictors to 14 without compromising on explained variance. This is the power of PCA!
Calculating and Visualizing the Variance Explained
Next, we calculate the variance explained of each principal component by squaring the standard deviation in the components of te_vars.prc. We save the result in object prc.var. This is done in preparation for visualizing the results in a plot.
- Calculate the variance explained by each principal component by dividing by the total variance explained of all principal components. We assign this to a variable called pve.
# calculating the variance
# 2 figures arranged in 2 rows and 2 columns
par(mfrow = c(2, 2))
# calculate variability of each component: pr.var
prc.var <- te_vars.prc$sdev^2
# this gives the eigenvalues of each prc (must be > 1 to keep)
prc.var## [1] 2.399420e+01 4.138743e+00 3.281706e+00 2.241028e+00 2.051431e+00
## [6] 1.985252e+00 1.672656e+00 1.549857e+00 1.140009e+00 1.047747e+00
## [11] 9.216231e-01 8.821537e-01 8.028814e-01 7.851592e-01 5.961081e-01
## [16] 5.508209e-01 4.900064e-01 4.052615e-01 3.517687e-01 3.061013e-01
## [21] 2.909221e-01 2.594565e-01 2.118852e-01 1.665433e-01 1.456334e-01
## [26] 1.216449e-01 1.184475e-01 8.887729e-02 7.663220e-02 7.425980e-02
## [31] 6.408524e-02 4.554905e-02 3.273555e-02 3.044853e-02 1.892467e-02
## [36] 1.503554e-02 1.205952e-02 8.844800e-03 7.082816e-03 4.539924e-03
## [41] 4.254537e-03 2.421654e-03 2.180012e-03 1.228990e-03 8.536283e-04
## [46] 4.802408e-04 2.944113e-04 1.265186e-04 3.509261e-05 2.079528e-31
## [51] 2.079528e-31The goal here is to find the components which explain the maximum variance. We want to retain as much information as possible using the fewest components. So, the higher the explained variance, the higher the information contained will be in those components.
To compute the proportion of variance explained by each component, we divide the variance by sum of total variance. This results in:
# calculate proportion of variance explained by each component: pve
# this is done in preparation for visualizing the results in a scree plot (below)
prc.ve <- prc.var/sum(prc.var)
prc.ve[1:20]## [1] 0.470474599 0.081151825 0.064347180 0.043941723 0.040224138
## [6] 0.038926507 0.032797167 0.030389348 0.022353124 0.020544054
## [11] 0.018071040 0.017297131 0.015742773 0.015395278 0.011688393
## [16] 0.010800409 0.009607968 0.007946303 0.006897426 0.006001987This shows that first principal component explains 47.0% variance of the TE data. The second component explains 8.1% variance. Third component explains 6.4% variance and so on. So, how do we decide how many components to select for the modeling stage? Which of the fifty-one principal components do we keep?
There are several ways to answer this question.
- The first is using the Kaiser criterion, which states that if the eigenvalue of each component is greater than 1, then we retain that component. If it is not greater than 1, then we reject it. To calculate the eigenvalues we look at the standard deviations of the components and square them.
# calculate variability of each component: pr.var
# (done in preparation for visualizing the results in scree plot below)
prc.var <- te_vars.prc$sdev^2
# this gives the eigenvalues of each prc (must be > 1 to keep)
prc.var## [1] 2.399420e+01 4.138743e+00 3.281706e+00 2.241028e+00 2.051431e+00
## [6] 1.985252e+00 1.672656e+00 1.549857e+00 1.140009e+00 1.047747e+00
## [11] 9.216231e-01 8.821537e-01 8.028814e-01 7.851592e-01 5.961081e-01
## [16] 5.508209e-01 4.900064e-01 4.052615e-01 3.517687e-01 3.061013e-01
## [21] 2.909221e-01 2.594565e-01 2.118852e-01 1.665433e-01 1.456334e-01
## [26] 1.216449e-01 1.184475e-01 8.887729e-02 7.663220e-02 7.425980e-02
## [31] 6.408524e-02 4.554905e-02 3.273555e-02 3.044853e-02 1.892467e-02
## [36] 1.503554e-02 1.205952e-02 8.844800e-03 7.082816e-03 4.539924e-03
## [41] 4.254537e-03 2.421654e-03 2.180012e-03 1.228990e-03 8.536283e-04
## [46] 4.802408e-04 2.944113e-04 1.265186e-04 3.509261e-05 2.079528e-31
## [51] 2.079528e-31# eigenvalues of each prc can be plotted
source('evplot.R')
par(mar = rep(2, 4))
evplot(prc.var)
The sum of the eigenvalues is also equal to the sum total of the variance in the data. The standard deviation is the square root of this. We can see that the first ten are greater than 1, while the eleventh is not. So, according to the Kaiser criterion we would retain principle components 1 through 10. The first barplot tells us (according to Kaiser criterion) that only the first ten principal components should be chosen (because the beige bar is above the red line). The first three components should be chosen according to the second barplot (broken stick model), because it has a tan bar above the red bar.
- The second way to answer this question is by using a scree plot. A scree plot is used to access components or factors which explains the most variability in the data. It represents values in descending order.
Visualizing and Interpreting the PCA Results
- Create a scree plot of variance explained for each principal component.
- Using the cumsum() function, create a plot of cumulative proportion of variance explained.
# scree plot of variance explained for each principal component
plot(prc.ve, xlab = "Principal Component",
ylab = "Proportion of Variance Explained",
ylim = c(0, 1), type = "b")
# scree plot cumulative proportion of variance explained
plot(cumsum(prc.ve), xlab = "Principal Component",
ylab = "Cumulative Proportion of Variance Explained",
ylim = c(0, 1), type = "b")
By looking at the elbow (bend) in the second scree plot, we can see that 80% of the variance is explained by a minimum of 8 principle components.
- Also note when the number of principle components and number of original features are the same, the cumulative proportion of variance explained is 1.
- After PC20 or so we see that the rate of change is approaching zero.
So, using this visual criteria we might only keep principal components 1 through 20.
To examine these results further, We can drill down and look at the principal components. One useful technique is to look at a dot plot of the component loadings or rotations.
# dotplot for PC1
load <- te_vars.prc$rotation
sorted.loadings <- load[order(load[, 1]), 1]
myTitle <- "Loadings Plot for PC1"
myXlab <- "Variable Loadings"
dotplot(sorted.loadings, main = myTitle, xlab = myXlab, cex = 1.5, col = "red")
In the loadings plot above we can see that Consumer Price Index (CPI), Employed Persons, Gasoline Prices, Wages, Job Vacancies, Wages in Manufacturing, Full-time Employment, Employment Rate, Labor Force Participation Rate, New Home Sales, Inflation Rate, Consumer Price Index, Stock Market, Building Permits, Wages in Manufacturing are all contributing to component PC1. Job and wage-related variables seem to be the dominant contributors here. As we move to the left we see other variables that are not contributors.
# dotplot PC2
sorted.loadings <- load[order(load[, 2]), 2]
myTitle <- "Loadings Plot for PC2"
myXlab <- "Variable Loadings"
dotplot(sorted.loadings, main=myTitle, xlab=myXlab, cex=1.5, col="red")
In the second dot plot, Existing Home Sales, Exchange Rate, Bank Lending Rate are all contributing to component PC2.
Using the Biplot to Visualize the PCA Results
The premier visualization tool for PCA is the biplot. The bi in the biplot refers to the plotting the variables as vectors onto the same graph containing the observations plotted as points.
# draw the biplot
biplot(te_vars.prc, cex = c(0.1, 0.7))
#biplot(te_vars.prc, col = c("white","red"),
# cex = c(0.1, 0.7))
#points(te_vars.prc$x[,1],te_vars.prc$x[,2],
# pch = as.character(1:5),
# col = c("green","green","black","blue","blue"))
# create a biplot of te_vars.prc
#layout(matrix(c(1,1,2,3), 2, 2, byrow = TRUE),
# widths=c(3,1), heights=c(1,2))
#biplot(te_vars.prc, cex = c(0.1, .65))
#autoplot(te_vars.prc, label = TRUE, label.size = 3, shape = TRUE)In the biplot we can see a lot of information. First, by looking at the axis, PC1 on the x-axis and PC2 on the y-axis. The arrows show how they are moving across the two dimensions. We see that the Employment-related features are moving along the first PC while the Unemployment-related features move mostly across the second PC.
An alternative and perhaps better way to plot the PCA results is by using ggbiplot:
# create a biplot using ggbiplot pkg
g <- ggbiplot(te_vars.prc,
choices=c(1,2),
obs.scale = 1,
var.scale = 1,
ellipse = TRUE,
circle = TRUE)
g <- g + scale_color_discrete(name = '')
g <- g + theme(legend.direction = 'horizontal',
legend.position = 'top')
print(g)
For the most part, the points are clustered in the circle. However, there does appear to be quite a few points outside that may be considered outliers, though they do not appear to be too far off from the cluster. Note that the first two principle components of the matrix only represent 55.1% of the total variance and thus may not be fully representative of the variance in the dataset.
Evaluation
The biplot above shows that Consumer Price Index (CPI), Employed Persons, Gasoline Prices, Wages, and Job Vacancies have the highest variance in the PC1 grouping.
Here is a more complete grouping of the variables:
| PC | Rank | Variable |
|---|---|---|
| PC1 | 1 | Consumer Price Index (CPI) |
| PC1 | 2 | Employed Persons |
| PC1 | 3 | Gasoline Prices |
| PC1 | 4 | Wages |
| PC1 | 5 | Job Vacancies |
| PC1 | 6 | Wages in Manufacturing |
| PC1 | 7 | Building Permits |
| PC1 | 8 | Stock Market |
| PC1 | 9 | Consumer Price Index |
| PC1 | 10 | Inflation Rate |
| PC2 | 11 | Bank Lending Rate |
| PC2 | 12 | Exchange Rate |
| PC2 | 13 | Existing Home Sales |
| PC2 | 14 | Economic Optimism |
| PC2 | 15 | Employment Rate |
signif(te_vars.prc$rotation[, 1:2], 2)## PC1 PC2
## InflationRate 0.1900 -0.0090
## RetailSalesMom -0.0400 -0.0980
## AverageHourlyEarnings 0.0310 0.0350
## BankLendingRate 0.1200 0.1900
## ConstructionSpending 0.0480 -0.1700
## EmploymentRate 0.1900 0.1200
## HousingIndex 0.0450 -0.0950
## InflationRateMoM -0.0980 -0.0810
## LaborForceParticipationRate 0.1900 -0.0018
## LongTermUnemploymentRate -0.1100 -0.1200
## PendingHomeSales 0.0160 -0.0350
## PersonalIncome -0.0099 -0.0710
## PersonalSavings -0.0330 -0.0830
## PersonalSpending -0.0046 -0.1000
## RetailSalesExcludingAutos -0.0350 -0.0970
## WageGrowth 0.0760 -0.2400
## YouthUnemploymentRate -0.2000 -0.0310
## CoreInflationRate 0.1300 -0.0980
## RetailSalesYoY -0.1300 -0.1100
## USGovernment10yrBondRate -0.1100 0.0770
## UnemploymentRate -0.2000 0.0041
## InitialJoblessClaims -0.0280 -0.0950
## NonfarmPayrolls 0.0600 -0.3400
## ADPEmploymentChange 0.0730 -0.3400
## BuildingPermits 0.2000 -0.0210
## ContinuingJoblessClaims -0.2000 0.0850
## EmployedPersons 0.2000 0.0370
## ExistingHomeSales 0.1800 0.1700
## FulltimeEmployment 0.1800 -0.0025
## GovernmentPayrolls -0.0670 -0.2000
## HousingStarts 0.1200 -0.1100
## JobOffers -0.2000 0.0610
## JobVacancies 0.2000 0.0067
## ManufacturingPayrolls -0.0820 -0.2200
## NewHomeSales 0.1900 -0.0320
## NonfarmPayrollsPrivate 0.0440 -0.3900
## UnemployedPersons -0.2000 0.0074
## ChainStoreSales -0.0320 -0.0780
## ConsumerCredit 0.0590 -0.0880
## Wages 0.2000 0.0240
## WagesinManufacturing 0.2000 0.0460
## GasolinePrices 0.2000 0.0400
## ExchangeRate 0.1800 0.1700
## StockMarket 0.2000 -0.0270
## ConsumerPriceIndex 0.2000 -0.0800
## CoreConsumerPrices 0.2000 0.0160
## EconomicOptimism 0.0890 0.1400
## LaborMarketConditions -0.1900 0.1100
## NAHBHousingMarket -0.0520 -0.2200
## AverageWeeklyHours 0.1200 -0.3100
## ChallengerJobCuts -0.0130 -0.0140Applying the Varimax Rotation
Another way to improve our understanding of the biplot and the principal components is to apply a technique called the varimax rotation. This tweak involves a change of coordinates to maximize the sum of the variances of the squared loadings. The goal here is to clean up the rotations that were found when prcomp was first called.
# apply varimax rotation to PC1 & PC2
my.var = varimax(te_vars.prc$rotation[, 1:2])
my.var## $loadings
##
## Loadings:
## PC1 PC2
## InflationRate 0.194
## RetailSalesMom -0.103
## AverageHourlyEarnings
## BankLendingRate 0.204
## ConstructionSpending -0.160
## EmploymentRate 0.167 0.143
## HousingIndex
## InflationRateMoM
## LaborForceParticipationRate 0.190
## LongTermUnemploymentRate -0.136
## PendingHomeSales
## PersonalIncome
## PersonalSavings
## PersonalSpending -0.102
## RetailSalesExcludingAutos -0.101
## WageGrowth 0.112 -0.221
## YouthUnemploymentRate -0.193
## CoreInflationRate 0.139
## RetailSalesYoY -0.111 -0.132
## USGovernment10yrBondRate -0.124
## UnemploymentRate -0.200
## InitialJoblessClaims
## NonfarmPayrolls 0.112 -0.328
## ADPEmploymentChange 0.124 -0.325
## BuildingPermits 0.197
## ContinuingJoblessClaims -0.209
## EmployedPersons 0.194
## ExistingHomeSales 0.148 0.197
## FulltimeEmployment 0.182
## GovernmentPayrolls -0.204
## HousingStarts 0.135
## JobOffers -0.207
## JobVacancies 0.197
## ManufacturingPayrolls -0.230
## NewHomeSales 0.195
## NonfarmPayrollsPrivate 0.104 -0.380
## UnemployedPersons -0.201
## ChainStoreSales
## ConsumerCredit
## Wages 0.195
## WagesinManufacturing 0.187
## GasolinePrices 0.193
## ExchangeRate 0.148 0.197
## StockMarket 0.198
## ConsumerPriceIndex 0.206
## CoreConsumerPrices 0.197
## EconomicOptimism 0.150
## LaborMarketConditions -0.202
## NAHBHousingMarket -0.221
## AverageWeeklyHours 0.162 -0.285
## ChallengerJobCuts
##
## PC1 PC2
## SS loadings 1.00 1.000
## Proportion Var 0.02 0.020
## Cumulative Var 0.02 0.039
##
## $rotmat
## [,1] [,2]
## [1,] 0.9880906 0.1538732
## [2,] -0.1538732 0.9880906#my.var <- varimax(te_vars.prc$rotation)
#my.varAdditional Analysis Using pcaMethods
We can use the pcaMethods suite of tools to conduct further study on the TE variables.
pcaMethods has the following advantages:
- It provides several different algorithms, e.g., Bayesian PCA, probabilistic PCA, robust PCA, etc.
- Some of these algorithms tolerate and impute missing values.
- The object structure and plotting capabilities are user-friendly.
## try http:// if https:// URLs are not supported
#source("http://bioconductor.org/biocLite.R")
#biocLite()
#biocLite("pcaMethods")
library(pcaMethods, quietly=TRUE)
te_pca_methods <- pca(te_vars_cln, scale = "uv", center = T, nPcs = 2, method = "svd")
#par(mfrow=c(2,1))
slplot(te_pca_methods, scoresLoadings = c(T,T))
# slots are marked with @
str(te_pca_methods) ## Formal class 'pcaRes' [package "pcaMethods"] with 19 slots
## ..@ completeObs: num [1:89, 1:51] 614 604 636 687 583 536 546 599 594 543 ...
## .. ..- attr(*, "dimnames")=List of 2
## .. .. ..$ : NULL
## .. .. ..$ : chr [1:51] "InflationRate" "RetailSalesMom" "AverageHourlyEarnings" "BankLendingRate" ...
## ..@ scores : num [1:89, 1:2] -7.99 -8.44 -7.98 -7.17 -6.1 ...
## .. ..- attr(*, "dimnames")=List of 2
## .. .. ..$ : NULL
## .. .. ..$ : chr [1:2] "PC1" "PC2"
## ..@ loadings : num [1:51, 1:2] 0.1945 -0.0396 0.0313 0.1242 0.0479 ...
## .. ..- attr(*, "dimnames")=List of 2
## .. .. ..$ : chr [1:51] "InflationRate" "RetailSalesMom" "AverageHourlyEarnings" "BankLendingRate" ...
## .. .. ..$ : chr [1:2] "PC1" "PC2"
## ..@ R2cum : Named num [1:2] 0.47 0.552
## .. ..- attr(*, "names")= chr [1:2] "PC1" "PC2"
## ..@ R2 : Named num [1:2] 0.4705 0.0812
## .. ..- attr(*, "names")= chr [1:2] "PC1" "PC2"
## ..@ cvstat : NULL
## ..@ sDev : Named num [1:2] 4.9 2.03
## .. ..- attr(*, "names")= chr [1:2] "PC1" "PC2"
## ..@ nObs : int 89
## ..@ nVar : int 51
## ..@ centered : logi TRUE
## ..@ center : Named num [1:51] 902.236 0.354 0.175 3.322 0.44 ...
## .. ..- attr(*, "names")= chr [1:51] "InflationRate" "RetailSalesMom" "AverageHourlyEarnings" "BankLendingRate" ...
## ..@ subset : NULL
## ..@ scaled : chr "uv"
## ..@ scale : Named num [1:51] 233.185 0.593 0.133 0.166 1.147 ...
## .. ..- attr(*, "names")= chr [1:51] "InflationRate" "RetailSalesMom" "AverageHourlyEarnings" "BankLendingRate" ...
## ..@ varLimit : num 1
## ..@ nPcs : int 2
## ..@ method : chr "svd"
## ..@ missing : logi [1:89, 1:51] FALSE FALSE FALSE FALSE FALSE FALSE ...
## .. ..- attr(*, "dimnames")=List of 2
## .. .. ..$ : NULL
## .. .. ..$ : chr [1:51] "InflationRate" "RetailSalesMom" "AverageHourlyEarnings" "BankLendingRate" ...
## ..@ network : NULLte_pca_methods@R2## PC1 PC2
## 0.47047 0.08116So, according to pcaMethods, it seems that PC1 and PC2 explain 47.0% and 8.1% of the variance in the TE data set, respectively. The Scores dot plot displays a total of 55.16% of variance explained by PC1 + PC2.
Summary
This study used Principle Components Analysis (PCA) to examine over fifty Trading Economics macroeconomic variables. PCA is one of the best known dimensionality reduction techniques and a staple procedure in many scientific fields. PCA is used because:
- It can find important latent structure and relations.
- It can reduce overfitting.
- It can ease the curse of dimensionality, thus removing irrelevant or redundant variables that can produce noise and decrease the predictive power of any algorithm.
- It is often used as a precursor before conducting a regression on the original variables.
During the introductory PCA modeling we were able to quickly identify a subset of TE variables that have the highest variance of the group. Using PCA techniques, we were able to reduce the number of dimensions from 52 variables to under 20 variables. While many manual methods apply dimension reduction through trial-and-error, PCA accomplishes this by systematic computation.
Using PCA-related visualizations, we were also able to uncover and differentiate clusters of data by identifying categories of variables that are highly correlated, e.g., employment-related and price-related TE variables. By examining generated scree plots, PCA helped choose the right number of components, ordering the components by their effectiveness in differentiating data points. A rule-of-thumb is to use the number of principle components corresponding to the location of a kink or bend in the line. By looking at the bend in the scree plot, we saw that 80% of the variance is explained by a minimum of 8 principle components. So we were successful in using PCA as a dimension-reduction technique to compress the total number of variables from 52 to 14 variables representing 90% variation.
Adding additional components beyond this may better differentiate data points, but this extra information may not justify the resulting complexity of the solution. Using fewer components to explain the current data sample ensures the same components can be generalized to another data sample.
For further study, a next step might involve the inclusion of our response variables to conduct a supervised learning by using a PCA regression model. Typically, one uses only a subset of the most important principal components in the regression.
Caveat: As noted, a major drawback of PCA is that the directions that best represent each predictor are obtained in an unsupervised way. It does not make use of the output response variable even when it is available. So, the response variable is not used to identify each principal component direction. This means there is no guarantee that the directions found will be the optimal directions to use when making predictions on the response variable.
References
Peng, R. (2016). Exploratory Data Analysis with R. Leanpub.
Shalizi, C. (2017). Advanced Data Analysis from an Elementary Point of View. [Retrieved from: http://www.stat.cmu.edu/~cshalizi/ADAfaEPoV/ADAfaEPoV.pdf]
Shlens, J. (2005). A Tutorial on Principal Component Analysis. [Retrieved from: http://www.cs.cmu.edu/~elaw/papers/pca.pdf