Libraries

library(kableExtra)
library(tidyverse)

Computational Mathematics

  1. Using R, generate a random variable X that has 10,000 random uniform numbers from 1 to N, where N can be any number of your choosing greater than or equal to 6. Then generate a random variable Y that has 10,000 random normal numbers with a mean of \(\mu =\sigma =\frac { (N+1) }{ 2 }\)

Ans:

Random Variable X

N<- round(runif(1, 6, 100))
n <- 10000

X <- runif(n,1,N)
hist(X)

Random Variable Y

mu <- (N+1)/2
sigma <- (N+1)/2

Y <- rnorm(n,mu,sigma)
hist(Y)
abline(v=(N+1)/2, col = "red")

Probability

Calculate as a minimum the below probabilities a through c. Assume the small letter “x” is estimated as the median of the X variable, and the small letter “y” is estimated as the 1st quartile of the Y variable. Interpret the meaning of all probabilities.

x <- median(X)
round(x,2)
## [1] 49.35
y <- quantile(Y,0.25)[[1]]
round(y,2)
## [1] 17.01

1.a. P(X>x | X>y) - Probability that X is greater than its median given that X is greater than the first quartile of Y.

\(P(X>x|X>y)=\frac { P(X>x,X>y) }{ P(X>y) }\)

# Probability expression for X greater than mean(x) given that X greater than the 1st quartile of Y(y) 
# Numerator: Sum of all random numbers X greater than the mean and Q1 of Y divided by all possible X 
prob_num <- sum(X>x & X>y)/n

# Denominator: Sum of all X greater than y divided by all X
prob_den <- sum(X>y)/n

prob = round(prob_num/prob_den, 2)

cat('Probability: ',prob)
## Probability:  0.6

1.b. P(X>x, Y>y) - Probability that X is greater than its median as well as greater than the first quartile of Y.

# Probablity of X greater than the mean and Y greater than 1st quartile

prob = round(sum(X>x & Y>y)/n, 2)

cat('Probability: ',prob)
## Probability:  0.37

1.c. \(P(X<x|X>y)\) - Probability that X is lesser than its median given that X is greater than the first quartile of Y.

\(P(X<x|X>y)=\frac { P(X<x,X>y) }{ P(X>y) }\)

# Probability expression for X lesser than mean(x) given that X greater than the 1st quartile of Y(y) 
# Numerator: Sum of all random numbers X lesser than the mean but greater than Q1 of Y divided by all possible X 
prob_num <- sum(X<x & X>y)/n

# Denominator: Sum of all X greater than y divided by all X
prob_den <- sum(X>y)/n

prob = round(prob_num/prob_den, 2)

cat('Probability: ',prob)
## Probability:  0.4

Independence Test

Investigate whether P(X>x and Y>y)=P(X>x)P(Y>y) by building a table and evaluating the marginal and joint probabilities.

matrix<-matrix( c(sum(X>x & Y<y),sum(X>x & Y>y), sum(X<x & Y<y),sum(X<x & Y>y)), nrow = 2,ncol = 2)
matrix<-cbind(matrix,c(matrix[1,1]+matrix[1,2],matrix[2,1]+matrix[2,2]))
matrix<-rbind(matrix,c(matrix[1,1]+matrix[2,1],matrix[1,2]+matrix[2,2],matrix[1,3]+matrix[2,3]))
contingency<-as.data.frame(matrix)
names(contingency) <- c("X>x","X<x", "Total")
row.names(contingency) <- c("Y<y","Y>y", "Total")

contingency %>% kable() %>% kable_styling(bootstrap_options = c("striped", "hover", "condensed", "responsive")) %>% scroll_box(width="100%",height="300px")
X>x X<x Total
Y<y 1252 1248 2500
Y>y 3748 3752 7500
Total 5000 5000 10000 
prob_matrix<-matrix/matrix[3,3]
contingency_p<-as.data.frame(prob_matrix)
names(contingency_p) <- c("X>x","X<x", "Total")
row.names(contingency_p) <- c("Y<y","Y>y", "Total")

round(contingency_p,2) %>% kable() %>% kable_styling(bootstrap_options = c("striped", "hover", "condensed", "responsive")) %>% scroll_box(width="100%",height="300px")
X>x X<x Total
Y<y 0.13 0.12 0.25
Y>y 0.37 0.38 0.75
Total 0.50 0.50 1.00

Compute P(X>x)P(Y>y)

round(prob_matrix[3,1]*prob_matrix[2,3],3)
## [1] 0.375

Compute P(X>x and Y>y)

round(prob_matrix[2,1],digits = 3)
## [1] 0.375

Euality Check

round(prob_matrix[3,1]*prob_matrix[2,3],3)==round(prob_matrix[2,1],digits = 3)
## [1] TRUE

Since the results are so similar we would conclude that X and Y are indeed independent.

Check to see if independence holds by using Fisher’s Exact Test and the Chi Square Test. What is the difference between the two? Which is most appropriate?

Ans:

fisher.test(matrix,simulate.p.value=TRUE)
## 
##  Fisher's Exact Test for Count Data with simulated p-value (based
##  on 2000 replicates)
## 
## data:  matrix
## p-value = 1
## alternative hypothesis: two.sided
chisq.test(matrix, correct=TRUE)
## 
##  Pearson's Chi-squared test
## 
## data:  matrix
## X-squared = 0.0085333, df = 4, p-value = 1

Fisher’s Exact Test for Independence is known to be more accurate than Chi Square Test and recommended to be used when we have small sample sizes (typically less than 1000). The Chi Square Test is used when the cell sizes are large. Since, in our case we are dealing with a distribution of 10,000 random numbers, it would be appropriate to use Chi Square test.

  1. You are to register for Kaggle.com (free) and compete in the House Prices: Advanced Regression Techniques competition. https://www.kaggle.com/c/house-prices-advanced-regression-techniques . I want you to do the following.

5 points. Descriptive and Inferential Statistics. Provide univariate descriptive statistics and appropriate plots for the training data set. Provide a scatterplot matrix for at least two of the independent variables and the dependent variable. Derive a correlation matrix for any three quantitative variables in the dataset. Test the hypotheses that the correlations between each pairwise set of variables is 0 and provide an 80% confidence interval. Discuss the meaning of your analysis. Would you be worried about familywise error? Why or why not?

5 points. Linear Algebra and Correlation. Invert your correlation matrix from above. (This is known as the precision matrix and contains variance inflation factors on the diagonal.) Multiply the correlation matrix by the precision matrix, and then multiply the precision matrix by the correlation matrix. Conduct LU decomposition on the matrix.

5 points. Calculus-Based Probability & Statistics. Many times, it makes sense to fit a closed form distribution to data. Select a variable in the Kaggle.com training dataset that is skewed to the right, shift it so that the minimum value is absolutely above zero if necessary. Then load the MASS package and run fitdistr to fit an exponential probability density function. (See https://stat.ethz.ch/R-manual/R-devel/library/MASS/html/fitdistr.html ). Find the optimal value of \(\lambda\) for this distribution, and then take 1000 samples from this exponential distribution using this value (e.g., rexp(1000, \(\lambda\)???)). Plot a histogram and compare it with a histogram of your original variable. Using the exponential pdf, find the 5th and 95th percentiles using the cumulative distribution function (CDF). Also generate a 95% confidence interval from the empirical data, assuming normality. Finally, provide the empirical 5th percentile and 95th percentile of the data. Discuss.

10 points. Modeling. Build some type of multiple regression model and submit your model to the competition board. Provide your complete model summary and results with analysis. Report your Kaggle.com user name and score.

Kaggle House Prices Competition

Libraries Used

library(ggplot2)
library(dplyr)
library(plotly)
library(MASS)
library(corrplot)
library(RColorBrewer)
library(GGally)
library(ggResidpanel)

Data Sources

The following data source and metadata dictionary files were downloaded from the Kaggle Competition site to my GitHub project directory -

Load Train Data into R

The train.csv file is loaded into a R data frame ‘housing_pr_train’ -

housing_pr_train <- read.csv('https://raw.githubusercontent.com/soumya2g/R-CUNY-MSDS/master/DATA-605/House%20Price%20Analysis/Data%20Source/house-prices-advanced-regression-techniques/train.csv',sep = ',')

## Sample Data
head(housing_pr_train) %>% kable() %>% kable_styling(bootstrap_options = c("striped", "hover", "condensed", "responsive")) %>% scroll_box(width="100%",height="300px")
Id MSSubClass MSZoning LotFrontage LotArea Street Alley LotShape LandContour Utilities LotConfig LandSlope Neighborhood Condition1 Condition2 BldgType HouseStyle OverallQual OverallCond YearBuilt YearRemodAdd RoofStyle RoofMatl Exterior1st Exterior2nd MasVnrType MasVnrArea ExterQual ExterCond Foundation BsmtQual BsmtCond BsmtExposure BsmtFinType1 BsmtFinSF1 BsmtFinType2 BsmtFinSF2 BsmtUnfSF TotalBsmtSF Heating HeatingQC CentralAir Electrical X1stFlrSF X2ndFlrSF LowQualFinSF GrLivArea BsmtFullBath BsmtHalfBath FullBath HalfBath BedroomAbvGr KitchenAbvGr KitchenQual TotRmsAbvGrd Functional Fireplaces FireplaceQu GarageType GarageYrBlt GarageFinish GarageCars GarageArea GarageQual GarageCond PavedDrive WoodDeckSF OpenPorchSF EnclosedPorch X3SsnPorch ScreenPorch PoolArea PoolQC Fence MiscFeature MiscVal MoSold YrSold SaleType SaleCondition SalePrice
1 60 RL 65 8450 Pave NA Reg Lvl AllPub Inside Gtl CollgCr Norm Norm 1Fam 2Story 7 5 2003 2003 Gable CompShg VinylSd VinylSd BrkFace 196 Gd TA PConc Gd TA No GLQ 706 Unf 0 150 856 GasA Ex Y SBrkr 856 854 0 1710 1 0 2 1 3 1 Gd 8 Typ 0 NA Attchd 2003 RFn 2 548 TA TA Y 0 61 0 0 0 0 NA NA NA 0 2 2008 WD Normal 208500
2 20 RL 80 9600 Pave NA Reg Lvl AllPub FR2 Gtl Veenker Feedr Norm 1Fam 1Story 6 8 1976 1976 Gable CompShg MetalSd MetalSd None 0 TA TA CBlock Gd TA Gd ALQ 978 Unf 0 284 1262 GasA Ex Y SBrkr 1262 0 0 1262 0 1 2 0 3 1 TA 6 Typ 1 TA Attchd 1976 RFn 2 460 TA TA Y 298 0 0 0 0 0 NA NA NA 0 5 2007 WD Normal 181500
3 60 RL 68 11250 Pave NA IR1 Lvl AllPub Inside Gtl CollgCr Norm Norm 1Fam 2Story 7 5 2001 2002 Gable CompShg VinylSd VinylSd BrkFace 162 Gd TA PConc Gd TA Mn GLQ 486 Unf 0 434 920 GasA Ex Y SBrkr 920 866 0 1786 1 0 2 1 3 1 Gd 6 Typ 1 TA Attchd 2001 RFn 2 608 TA TA Y 0 42 0 0 0 0 NA NA NA 0 9 2008 WD Normal 223500
4 70 RL 60 9550 Pave NA IR1 Lvl AllPub Corner Gtl Crawfor Norm Norm 1Fam 2Story 7 5 1915 1970 Gable CompShg Wd Sdng Wd Shng None 0 TA TA BrkTil TA Gd No ALQ 216 Unf 0 540 756 GasA Gd Y SBrkr 961 756 0 1717 1 0 1 0 3 1 Gd 7 Typ 1 Gd Detchd 1998 Unf 3 642 TA TA Y 0 35 272 0 0 0 NA NA NA 0 2 2006 WD Abnorml 140000
5 60 RL 84 14260 Pave NA IR1 Lvl AllPub FR2 Gtl NoRidge Norm Norm 1Fam 2Story 8 5 2000 2000 Gable CompShg VinylSd VinylSd BrkFace 350 Gd TA PConc Gd TA Av GLQ 655 Unf 0 490 1145 GasA Ex Y SBrkr 1145 1053 0 2198 1 0 2 1 4 1 Gd 9 Typ 1 TA Attchd 2000 RFn 3 836 TA TA Y 192 84 0 0 0 0 NA NA NA 0 12 2008 WD Normal 250000
6 50 RL 85 14115 Pave NA IR1 Lvl AllPub Inside Gtl Mitchel Norm Norm 1Fam 1.5Fin 5 5 1993 1995 Gable CompShg VinylSd VinylSd None 0 TA TA Wood Gd TA No GLQ 732 Unf 0 64 796 GasA Ex Y SBrkr 796 566 0 1362 1 0 1 1 1 1 TA 5 Typ 0 NA Attchd 1993 Unf 2 480 TA TA Y 40 30 0 320 0 0 NA MnPrv Shed 700 10 2009 WD Normal 143000 
glimpse(housing_pr_train)
## Observations: 1,460
## Variables: 81
## $ Id            <int> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 1...
## $ MSSubClass    <int> 60, 20, 60, 70, 60, 50, 20, 60, 50, 190, 20, 60,...
## $ MSZoning      <fct> RL, RL, RL, RL, RL, RL, RL, RL, RM, RL, RL, RL, ...
## $ LotFrontage   <int> 65, 80, 68, 60, 84, 85, 75, NA, 51, 50, 70, 85, ...
## $ LotArea       <int> 8450, 9600, 11250, 9550, 14260, 14115, 10084, 10...
## $ Street        <fct> Pave, Pave, Pave, Pave, Pave, Pave, Pave, Pave, ...
## $ Alley         <fct> NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, ...
## $ LotShape      <fct> Reg, Reg, IR1, IR1, IR1, IR1, Reg, IR1, Reg, Reg...
## $ LandContour   <fct> Lvl, Lvl, Lvl, Lvl, Lvl, Lvl, Lvl, Lvl, Lvl, Lvl...
## $ Utilities     <fct> AllPub, AllPub, AllPub, AllPub, AllPub, AllPub, ...
## $ LotConfig     <fct> Inside, FR2, Inside, Corner, FR2, Inside, Inside...
## $ LandSlope     <fct> Gtl, Gtl, Gtl, Gtl, Gtl, Gtl, Gtl, Gtl, Gtl, Gtl...
## $ Neighborhood  <fct> CollgCr, Veenker, CollgCr, Crawfor, NoRidge, Mit...
## $ Condition1    <fct> Norm, Feedr, Norm, Norm, Norm, Norm, Norm, PosN,...
## $ Condition2    <fct> Norm, Norm, Norm, Norm, Norm, Norm, Norm, Norm, ...
## $ BldgType      <fct> 1Fam, 1Fam, 1Fam, 1Fam, 1Fam, 1Fam, 1Fam, 1Fam, ...
## $ HouseStyle    <fct> 2Story, 1Story, 2Story, 2Story, 2Story, 1.5Fin, ...
## $ OverallQual   <int> 7, 6, 7, 7, 8, 5, 8, 7, 7, 5, 5, 9, 5, 7, 6, 7, ...
## $ OverallCond   <int> 5, 8, 5, 5, 5, 5, 5, 6, 5, 6, 5, 5, 6, 5, 5, 8, ...
## $ YearBuilt     <int> 2003, 1976, 2001, 1915, 2000, 1993, 2004, 1973, ...
## $ YearRemodAdd  <int> 2003, 1976, 2002, 1970, 2000, 1995, 2005, 1973, ...
## $ RoofStyle     <fct> Gable, Gable, Gable, Gable, Gable, Gable, Gable,...
## $ RoofMatl      <fct> CompShg, CompShg, CompShg, CompShg, CompShg, Com...
## $ Exterior1st   <fct> VinylSd, MetalSd, VinylSd, Wd Sdng, VinylSd, Vin...
## $ Exterior2nd   <fct> VinylSd, MetalSd, VinylSd, Wd Shng, VinylSd, Vin...
## $ MasVnrType    <fct> BrkFace, None, BrkFace, None, BrkFace, None, Sto...
## $ MasVnrArea    <int> 196, 0, 162, 0, 350, 0, 186, 240, 0, 0, 0, 286, ...
## $ ExterQual     <fct> Gd, TA, Gd, TA, Gd, TA, Gd, TA, TA, TA, TA, Ex, ...
## $ ExterCond     <fct> TA, TA, TA, TA, TA, TA, TA, TA, TA, TA, TA, TA, ...
## $ Foundation    <fct> PConc, CBlock, PConc, BrkTil, PConc, Wood, PConc...
## $ BsmtQual      <fct> Gd, Gd, Gd, TA, Gd, Gd, Ex, Gd, TA, TA, TA, Ex, ...
## $ BsmtCond      <fct> TA, TA, TA, Gd, TA, TA, TA, TA, TA, TA, TA, TA, ...
## $ BsmtExposure  <fct> No, Gd, Mn, No, Av, No, Av, Mn, No, No, No, No, ...
## $ BsmtFinType1  <fct> GLQ, ALQ, GLQ, ALQ, GLQ, GLQ, GLQ, ALQ, Unf, GLQ...
## $ BsmtFinSF1    <int> 706, 978, 486, 216, 655, 732, 1369, 859, 0, 851,...
## $ BsmtFinType2  <fct> Unf, Unf, Unf, Unf, Unf, Unf, Unf, BLQ, Unf, Unf...
## $ BsmtFinSF2    <int> 0, 0, 0, 0, 0, 0, 0, 32, 0, 0, 0, 0, 0, 0, 0, 0,...
## $ BsmtUnfSF     <int> 150, 284, 434, 540, 490, 64, 317, 216, 952, 140,...
## $ TotalBsmtSF   <int> 856, 1262, 920, 756, 1145, 796, 1686, 1107, 952,...
## $ Heating       <fct> GasA, GasA, GasA, GasA, GasA, GasA, GasA, GasA, ...
## $ HeatingQC     <fct> Ex, Ex, Ex, Gd, Ex, Ex, Ex, Ex, Gd, Ex, Ex, Ex, ...
## $ CentralAir    <fct> Y, Y, Y, Y, Y, Y, Y, Y, Y, Y, Y, Y, Y, Y, Y, Y, ...
## $ Electrical    <fct> SBrkr, SBrkr, SBrkr, SBrkr, SBrkr, SBrkr, SBrkr,...
## $ X1stFlrSF     <int> 856, 1262, 920, 961, 1145, 796, 1694, 1107, 1022...
## $ X2ndFlrSF     <int> 854, 0, 866, 756, 1053, 566, 0, 983, 752, 0, 0, ...
## $ LowQualFinSF  <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
## $ GrLivArea     <int> 1710, 1262, 1786, 1717, 2198, 1362, 1694, 2090, ...
## $ BsmtFullBath  <int> 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 0, ...
## $ BsmtHalfBath  <int> 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
## $ FullBath      <int> 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 1, 3, 1, 2, 1, 1, ...
## $ HalfBath      <int> 1, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, ...
## $ BedroomAbvGr  <int> 3, 3, 3, 3, 4, 1, 3, 3, 2, 2, 3, 4, 2, 3, 2, 2, ...
## $ KitchenAbvGr  <int> 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, ...
## $ KitchenQual   <fct> Gd, TA, Gd, Gd, Gd, TA, Gd, TA, TA, TA, TA, Ex, ...
## $ TotRmsAbvGrd  <int> 8, 6, 6, 7, 9, 5, 7, 7, 8, 5, 5, 11, 4, 7, 5, 5,...
## $ Functional    <fct> Typ, Typ, Typ, Typ, Typ, Typ, Typ, Typ, Min1, Ty...
## $ Fireplaces    <int> 0, 1, 1, 1, 1, 0, 1, 2, 2, 2, 0, 2, 0, 1, 1, 0, ...
## $ FireplaceQu   <fct> NA, TA, TA, Gd, TA, NA, Gd, TA, TA, TA, NA, Gd, ...
## $ GarageType    <fct> Attchd, Attchd, Attchd, Detchd, Attchd, Attchd, ...
## $ GarageYrBlt   <int> 2003, 1976, 2001, 1998, 2000, 1993, 2004, 1973, ...
## $ GarageFinish  <fct> RFn, RFn, RFn, Unf, RFn, Unf, RFn, RFn, Unf, RFn...
## $ GarageCars    <int> 2, 2, 2, 3, 3, 2, 2, 2, 2, 1, 1, 3, 1, 3, 1, 2, ...
## $ GarageArea    <int> 548, 460, 608, 642, 836, 480, 636, 484, 468, 205...
## $ GarageQual    <fct> TA, TA, TA, TA, TA, TA, TA, TA, Fa, Gd, TA, TA, ...
## $ GarageCond    <fct> TA, TA, TA, TA, TA, TA, TA, TA, TA, TA, TA, TA, ...
## $ PavedDrive    <fct> Y, Y, Y, Y, Y, Y, Y, Y, Y, Y, Y, Y, Y, Y, Y, Y, ...
## $ WoodDeckSF    <int> 0, 298, 0, 0, 192, 40, 255, 235, 90, 0, 0, 147, ...
## $ OpenPorchSF   <int> 61, 0, 42, 35, 84, 30, 57, 204, 0, 4, 0, 21, 0, ...
## $ EnclosedPorch <int> 0, 0, 0, 272, 0, 0, 0, 228, 205, 0, 0, 0, 0, 0, ...
## $ X3SsnPorch    <int> 0, 0, 0, 0, 0, 320, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0...
## $ ScreenPorch   <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 176, 0, 0, 0...
## $ PoolArea      <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
## $ PoolQC        <fct> NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, ...
## $ Fence         <fct> NA, NA, NA, NA, NA, MnPrv, NA, NA, NA, NA, NA, N...
## $ MiscFeature   <fct> NA, NA, NA, NA, NA, Shed, NA, Shed, NA, NA, NA, ...
## $ MiscVal       <int> 0, 0, 0, 0, 0, 700, 0, 350, 0, 0, 0, 0, 0, 0, 0,...
## $ MoSold        <int> 2, 5, 9, 2, 12, 10, 8, 11, 4, 1, 2, 7, 9, 8, 5, ...
## $ YrSold        <int> 2008, 2007, 2008, 2006, 2008, 2009, 2007, 2009, ...
## $ SaleType      <fct> WD, WD, WD, WD, WD, WD, WD, WD, WD, WD, WD, New,...
## $ SaleCondition <fct> Normal, Normal, Normal, Abnorml, Normal, Normal,...
## $ SalePrice     <int> 208500, 181500, 223500, 140000, 250000, 143000, ...

Descriptive and Inferential Statistics

Provide univariate descriptive statistics and appropriate plots for the training data set. Provide a scatterplot matrix for at least two of the independent variables and the dependent variable. Derive a correlation matrix for any three quantitative variables in the dataset. Test the hypotheses that the correlations between each pairwise set of variables is 0 and provide an 80% confidence interval. Discuss the meaning of your analysis. Would you be worried about familywise error? Why or why not?

** Ans:**

Statistical Summary

summary(housing_pr_train)
##        Id           MSSubClass       MSZoning     LotFrontage    
##  Min.   :   1.0   Min.   : 20.0   C (all):  10   Min.   : 21.00  
##  1st Qu.: 365.8   1st Qu.: 20.0   FV     :  65   1st Qu.: 59.00  
##  Median : 730.5   Median : 50.0   RH     :  16   Median : 69.00  
##  Mean   : 730.5   Mean   : 56.9   RL     :1151   Mean   : 70.05  
##  3rd Qu.:1095.2   3rd Qu.: 70.0   RM     : 218   3rd Qu.: 80.00  
##  Max.   :1460.0   Max.   :190.0                  Max.   :313.00  
##                                                  NA's   :259     
##     LotArea        Street      Alley      LotShape  LandContour
##  Min.   :  1300   Grvl:   6   Grvl:  50   IR1:484   Bnk:  63   
##  1st Qu.:  7554   Pave:1454   Pave:  41   IR2: 41   HLS:  50   
##  Median :  9478               NA's:1369   IR3: 10   Low:  36   
##  Mean   : 10517                           Reg:925   Lvl:1311   
##  3rd Qu.: 11602                                                
##  Max.   :215245                                                
##                                                                
##   Utilities      LotConfig    LandSlope   Neighborhood   Condition1  
##  AllPub:1459   Corner : 263   Gtl:1382   NAmes  :225   Norm   :1260  
##  NoSeWa:   1   CulDSac:  94   Mod:  65   CollgCr:150   Feedr  :  81  
##                FR2    :  47   Sev:  13   OldTown:113   Artery :  48  
##                FR3    :   4              Edwards:100   RRAn   :  26  
##                Inside :1052              Somerst: 86   PosN   :  19  
##                                          Gilbert: 79   RRAe   :  11  
##                                          (Other):707   (Other):  15  
##    Condition2     BldgType      HouseStyle   OverallQual    
##  Norm   :1445   1Fam  :1220   1Story :726   Min.   : 1.000  
##  Feedr  :   6   2fmCon:  31   2Story :445   1st Qu.: 5.000  
##  Artery :   2   Duplex:  52   1.5Fin :154   Median : 6.000  
##  PosN   :   2   Twnhs :  43   SLvl   : 65   Mean   : 6.099  
##  RRNn   :   2   TwnhsE: 114   SFoyer : 37   3rd Qu.: 7.000  
##  PosA   :   1                 1.5Unf : 14   Max.   :10.000  
##  (Other):   2                 (Other): 19                   
##   OverallCond      YearBuilt     YearRemodAdd    RoofStyle   
##  Min.   :1.000   Min.   :1872   Min.   :1950   Flat   :  13  
##  1st Qu.:5.000   1st Qu.:1954   1st Qu.:1967   Gable  :1141  
##  Median :5.000   Median :1973   Median :1994   Gambrel:  11  
##  Mean   :5.575   Mean   :1971   Mean   :1985   Hip    : 286  
##  3rd Qu.:6.000   3rd Qu.:2000   3rd Qu.:2004   Mansard:   7  
##  Max.   :9.000   Max.   :2010   Max.   :2010   Shed   :   2  
##                                                              
##     RoofMatl     Exterior1st   Exterior2nd    MasVnrType    MasVnrArea    
##  CompShg:1434   VinylSd:515   VinylSd:504   BrkCmn : 15   Min.   :   0.0  
##  Tar&Grv:  11   HdBoard:222   MetalSd:214   BrkFace:445   1st Qu.:   0.0  
##  WdShngl:   6   MetalSd:220   HdBoard:207   None   :864   Median :   0.0  
##  WdShake:   5   Wd Sdng:206   Wd Sdng:197   Stone  :128   Mean   : 103.7  
##  ClyTile:   1   Plywood:108   Plywood:142   NA's   :  8   3rd Qu.: 166.0  
##  Membran:   1   CemntBd: 61   CmentBd: 60                 Max.   :1600.0  
##  (Other):   2   (Other):128   (Other):136                 NA's   :8       
##  ExterQual ExterCond  Foundation  BsmtQual   BsmtCond    BsmtExposure
##  Ex: 52    Ex:   3   BrkTil:146   Ex  :121   Fa  :  45   Av  :221    
##  Fa: 14    Fa:  28   CBlock:634   Fa  : 35   Gd  :  65   Gd  :134    
##  Gd:488    Gd: 146   PConc :647   Gd  :618   Po  :   2   Mn  :114    
##  TA:906    Po:   1   Slab  : 24   TA  :649   TA  :1311   No  :953    
##            TA:1282   Stone :  6   NA's: 37   NA's:  37   NA's: 38    
##                      Wood  :  3                                      
##                                                                      
##  BsmtFinType1   BsmtFinSF1     BsmtFinType2   BsmtFinSF2     
##  ALQ :220     Min.   :   0.0   ALQ :  19    Min.   :   0.00  
##  BLQ :148     1st Qu.:   0.0   BLQ :  33    1st Qu.:   0.00  
##  GLQ :418     Median : 383.5   GLQ :  14    Median :   0.00  
##  LwQ : 74     Mean   : 443.6   LwQ :  46    Mean   :  46.55  
##  Rec :133     3rd Qu.: 712.2   Rec :  54    3rd Qu.:   0.00  
##  Unf :430     Max.   :5644.0   Unf :1256    Max.   :1474.00  
##  NA's: 37                      NA's:  38                     
##    BsmtUnfSF       TotalBsmtSF      Heating     HeatingQC CentralAir
##  Min.   :   0.0   Min.   :   0.0   Floor:   1   Ex:741    N:  95    
##  1st Qu.: 223.0   1st Qu.: 795.8   GasA :1428   Fa: 49    Y:1365    
##  Median : 477.5   Median : 991.5   GasW :  18   Gd:241              
##  Mean   : 567.2   Mean   :1057.4   Grav :   7   Po:  1              
##  3rd Qu.: 808.0   3rd Qu.:1298.2   OthW :   2   TA:428              
##  Max.   :2336.0   Max.   :6110.0   Wall :   4                       
##                                                                     
##  Electrical     X1stFlrSF      X2ndFlrSF     LowQualFinSF    
##  FuseA:  94   Min.   : 334   Min.   :   0   Min.   :  0.000  
##  FuseF:  27   1st Qu.: 882   1st Qu.:   0   1st Qu.:  0.000  
##  FuseP:   3   Median :1087   Median :   0   Median :  0.000  
##  Mix  :   1   Mean   :1163   Mean   : 347   Mean   :  5.845  
##  SBrkr:1334   3rd Qu.:1391   3rd Qu.: 728   3rd Qu.:  0.000  
##  NA's :   1   Max.   :4692   Max.   :2065   Max.   :572.000  
##                                                              
##    GrLivArea     BsmtFullBath     BsmtHalfBath        FullBath    
##  Min.   : 334   Min.   :0.0000   Min.   :0.00000   Min.   :0.000  
##  1st Qu.:1130   1st Qu.:0.0000   1st Qu.:0.00000   1st Qu.:1.000  
##  Median :1464   Median :0.0000   Median :0.00000   Median :2.000  
##  Mean   :1515   Mean   :0.4253   Mean   :0.05753   Mean   :1.565  
##  3rd Qu.:1777   3rd Qu.:1.0000   3rd Qu.:0.00000   3rd Qu.:2.000  
##  Max.   :5642   Max.   :3.0000   Max.   :2.00000   Max.   :3.000  
##                                                                   
##     HalfBath       BedroomAbvGr    KitchenAbvGr   KitchenQual
##  Min.   :0.0000   Min.   :0.000   Min.   :0.000   Ex:100     
##  1st Qu.:0.0000   1st Qu.:2.000   1st Qu.:1.000   Fa: 39     
##  Median :0.0000   Median :3.000   Median :1.000   Gd:586     
##  Mean   :0.3829   Mean   :2.866   Mean   :1.047   TA:735     
##  3rd Qu.:1.0000   3rd Qu.:3.000   3rd Qu.:1.000              
##  Max.   :2.0000   Max.   :8.000   Max.   :3.000              
##                                                              
##   TotRmsAbvGrd    Functional    Fireplaces    FireplaceQu   GarageType 
##  Min.   : 2.000   Maj1:  14   Min.   :0.000   Ex  : 24    2Types :  6  
##  1st Qu.: 5.000   Maj2:   5   1st Qu.:0.000   Fa  : 33    Attchd :870  
##  Median : 6.000   Min1:  31   Median :1.000   Gd  :380    Basment: 19  
##  Mean   : 6.518   Min2:  34   Mean   :0.613   Po  : 20    BuiltIn: 88  
##  3rd Qu.: 7.000   Mod :  15   3rd Qu.:1.000   TA  :313    CarPort:  9  
##  Max.   :14.000   Sev :   1   Max.   :3.000   NA's:690    Detchd :387  
##                   Typ :1360                               NA's   : 81  
##   GarageYrBlt   GarageFinish   GarageCars      GarageArea     GarageQual 
##  Min.   :1900   Fin :352     Min.   :0.000   Min.   :   0.0   Ex  :   3  
##  1st Qu.:1961   RFn :422     1st Qu.:1.000   1st Qu.: 334.5   Fa  :  48  
##  Median :1980   Unf :605     Median :2.000   Median : 480.0   Gd  :  14  
##  Mean   :1979   NA's: 81     Mean   :1.767   Mean   : 473.0   Po  :   3  
##  3rd Qu.:2002                3rd Qu.:2.000   3rd Qu.: 576.0   TA  :1311  
##  Max.   :2010                Max.   :4.000   Max.   :1418.0   NA's:  81  
##  NA's   :81                                                              
##  GarageCond  PavedDrive   WoodDeckSF      OpenPorchSF     EnclosedPorch   
##  Ex  :   2   N:  90     Min.   :  0.00   Min.   :  0.00   Min.   :  0.00  
##  Fa  :  35   P:  30     1st Qu.:  0.00   1st Qu.:  0.00   1st Qu.:  0.00  
##  Gd  :   9   Y:1340     Median :  0.00   Median : 25.00   Median :  0.00  
##  Po  :   7              Mean   : 94.24   Mean   : 46.66   Mean   : 21.95  
##  TA  :1326              3rd Qu.:168.00   3rd Qu.: 68.00   3rd Qu.:  0.00  
##  NA's:  81              Max.   :857.00   Max.   :547.00   Max.   :552.00  
##                                                                           
##    X3SsnPorch      ScreenPorch        PoolArea        PoolQC    
##  Min.   :  0.00   Min.   :  0.00   Min.   :  0.000   Ex  :   2  
##  1st Qu.:  0.00   1st Qu.:  0.00   1st Qu.:  0.000   Fa  :   2  
##  Median :  0.00   Median :  0.00   Median :  0.000   Gd  :   3  
##  Mean   :  3.41   Mean   : 15.06   Mean   :  2.759   NA's:1453  
##  3rd Qu.:  0.00   3rd Qu.:  0.00   3rd Qu.:  0.000              
##  Max.   :508.00   Max.   :480.00   Max.   :738.000              
##                                                                 
##    Fence      MiscFeature    MiscVal             MoSold      
##  GdPrv:  59   Gar2:   2   Min.   :    0.00   Min.   : 1.000  
##  GdWo :  54   Othr:   2   1st Qu.:    0.00   1st Qu.: 5.000  
##  MnPrv: 157   Shed:  49   Median :    0.00   Median : 6.000  
##  MnWw :  11   TenC:   1   Mean   :   43.49   Mean   : 6.322  
##  NA's :1179   NA's:1406   3rd Qu.:    0.00   3rd Qu.: 8.000  
##                           Max.   :15500.00   Max.   :12.000  
##                                                              
##      YrSold        SaleType    SaleCondition    SalePrice     
##  Min.   :2006   WD     :1267   Abnorml: 101   Min.   : 34900  
##  1st Qu.:2007   New    : 122   AdjLand:   4   1st Qu.:129975  
##  Median :2008   COD    :  43   Alloca :  12   Median :163000  
##  Mean   :2008   ConLD  :   9   Family :  20   Mean   :180921  
##  3rd Qu.:2009   ConLI  :   5   Normal :1198   3rd Qu.:214000  
##  Max.   :2010   ConLw  :   5   Partial: 125   Max.   :755000  
##                 (Other):   9

Categorizing Attributes and Feature engineering

I have categorized all the Numeric variables present in the data set into ‘Discrete’ and ‘Continuous’. Similarly Categorical variables into ‘Ordinal’ and ‘Nominal’.

num_discrete <- c('BsmtFullBath','BsmtHalfBath','FullBath','HalfBath','BedroomAbvGr','KitchenAbvGr','TotRmsAbvGrd',
                'Fireplaces','GarageCars','GarageYrBlt','YearBuilt','YearRemodAdd','YrSold','MoSold')

num_continuous <- c('LotFrontage','LotArea','MasVnrArea','BsmtFinSF1','BsmtFinSF2','BsmtUnfSF','TotalBsmtSF','1stFlrSF',
                  '2ndFlrSF','LowQualFinSF','GrLivArea','GarageArea','WoodDeckSF','OpenPorchSF','EnclosedPorch',
                  '3SsnPorch','ScreenPorch','PoolArea','MiscVal','SalePrice')

cat_ordinal <- c('LotShape','Utilities','LandSlope','OverallQual','OverallCond','ExterQual','ExterCond',
               'BsmtQual','BsmtCond','BsmtExposure','BsmtFinType1','BsmtFinType2','HeatingQC','CentralAir',
               'Electrical','KitchenQual','Functional','FireplaceQu','GarageFinish','GarageQual','GarageCond',
               'PavedDrive','PoolQC','Fence')

cat_nominal <- c('MSSubClass','MSZoning','Street','Alley','LandContour','LotConfig','Neighborhood',
               'Condition1','Condition2','BldgType','HouseStyle','RoofStyle','RoofMatl','Exterior1st',
               'Exterior2nd','MasVnrType','Foundation','Heating','GarageType','MiscFeature',
               'SaleType','SaleCondition')

Feature engineering

Consolidate similar variables like No. of Bathrooms - BsmtFullBath, BsmtHalfBath, FullBath, HalfBath etc. into a single variable Baths.

# Those variables tell almost the same information, so let's add them.
housing_pr_train$Baths <- housing_pr_train$BsmtFullBath + 0.5*housing_pr_train$BsmtHalfBath + housing_pr_train$FullBath + 0.5*housing_pr_train$HalfBath

# Derive CentralAirScore as discrete variable
housing_pr_train$CentralAirScore <- ifelse(housing_pr_train$CentralAir == "N", 0, 1)

# Remove redundant variables
housing_pr_train <- subset(housing_pr_train, select = -c(Id,BsmtFullBath,BsmtHalfBath,FullBath,HalfBath))

# Remove Variable with 'NA' values
housing_pr_train <- housing_pr_train %>% subset(select = -c(LotFrontage,MasVnrArea,GarageYrBlt))

Visualizations

A. Scatterplot Matrix

For building the scatterplot matrix with the dependant variable (SalePrice), I have chosen following independent variables - LotArea, TotRmsAbvGrd, Total No. of Bathrooms(Baths), GrLivArea, GarageCars etc.

train_subset <- subset(housing_pr_train,select = c(LotArea,TotRmsAbvGrd,Baths,GrLivArea,OverallQual,SalePrice))

p <- ggpairs(data=train_subset, # data.frame with variables
             columns=1:6, # columns to plot, default to all.
             title="House Prices data") # title of the plot

ggplotly(p)
A.1. Box Plot - Overall Quality

From the Scatterplot matrix above, Overall Quality can be inferred as an influential variable for the SalePrice of a house.

train_subset$OverallQual_factor <- factor(train_subset$OverallQual)
p <- ggplot(train_subset, aes(x=OverallQual, y=SalePrice, fill=OverallQual_factor, group=OverallQual_factor)) + 
     geom_boxplot() +
     ggtitle("Sale Price Vs. Overall Quality")

ggplotly(p)
A.2. Box Plot - No. of Bathrooms

From the Scatterplot matrix above, No. of Bathrooms is also identified as an influential variable for the SalePrice of a house.

train_subset$OverallQual_factor <- factor(train_subset$OverallQual)
p <- ggplot(train_subset, aes(x=Baths, y=SalePrice, fill=Baths, group=Baths)) + 
     geom_boxplot() +
     ggtitle("Sale Price Vs. No. of Bathrooms")

ggplotly(p)
B. Correlation Matrix
housing_pr_train_num <- housing_pr_train %>% select_if(is.numeric) 
corrMatrix <- round(cor(housing_pr_train_num),4)
corrMatrix %>% kable() %>% kable_styling(bootstrap_options = c("striped", "hover", "condensed", "responsive")) %>% scroll_box(width="100%",height="300px")
MSSubClass LotArea OverallQual OverallCond YearBuilt YearRemodAdd BsmtFinSF1 BsmtFinSF2 BsmtUnfSF TotalBsmtSF X1stFlrSF X2ndFlrSF LowQualFinSF GrLivArea BedroomAbvGr KitchenAbvGr TotRmsAbvGrd Fireplaces GarageCars GarageArea WoodDeckSF OpenPorchSF EnclosedPorch X3SsnPorch ScreenPorch PoolArea MiscVal MoSold YrSold SalePrice Baths CentralAirScore
MSSubClass 1.0000 -0.1398 0.0326 -0.0593 0.0279 0.0406 -0.0698 -0.0656 -0.1408 -0.2385 -0.2518 0.3079 0.0465 0.0749 -0.0234 0.2817 0.0404 -0.0456 -0.0401 -0.0987 -0.0126 -0.0061 -0.0120 -0.0438 -0.0260 0.0083 -0.0077 -0.0136 -0.0214 -0.0843 0.1510 -0.1018
LotArea -0.1398 1.0000 0.1058 -0.0056 0.0142 0.0138 0.2141 0.1112 -0.0026 0.2608 0.2995 0.0510 0.0048 0.2631 0.1197 -0.0178 0.1900 0.2714 0.1549 0.1804 0.1717 0.0848 -0.0183 0.0204 0.0432 0.0777 0.0381 0.0012 -0.0143 0.2638 0.2048 0.0498
OverallQual 0.0326 0.1058 1.0000 -0.0919 0.5723 0.5507 0.2397 -0.0591 0.3082 0.5378 0.4762 0.2955 -0.0304 0.5930 0.1017 -0.1839 0.4275 0.3968 0.6007 0.5620 0.2389 0.3088 -0.1139 0.0304 0.0649 0.0652 -0.0314 0.0708 -0.0273 0.7910 0.5411 0.2720
OverallCond -0.0593 -0.0056 -0.0919 1.0000 -0.3760 0.0737 -0.0462 0.0402 -0.1368 -0.1711 -0.1442 0.0289 0.0255 -0.0797 0.0130 -0.0870 -0.0576 -0.0238 -0.1858 -0.1515 -0.0033 -0.0326 0.0704 0.0255 0.0548 -0.0020 0.0688 -0.0035 0.0439 -0.0779 -0.1740 0.1190
YearBuilt 0.0279 0.0142 0.5723 -0.3760 1.0000 0.5929 0.2495 -0.0491 0.1490 0.3915 0.2820 0.0103 -0.1838 0.1990 -0.0707 -0.1748 0.0956 0.1477 0.5379 0.4790 0.2249 0.1887 -0.3873 0.0314 -0.0504 0.0049 -0.0344 0.0124 -0.0136 0.5229 0.5243 0.3818
YearRemodAdd 0.0406 0.0138 0.5507 0.0737 0.5929 1.0000 0.1285 -0.0678 0.1811 0.2911 0.2404 0.1400 -0.0624 0.2874 -0.0406 -0.1496 0.1917 0.1126 0.4206 0.3716 0.2057 0.2263 -0.1939 0.0453 -0.0387 0.0058 -0.0103 0.0215 0.0357 0.5071 0.4437 0.2989
BsmtFinSF1 -0.0698 0.2141 0.2397 -0.0462 0.2495 0.1285 1.0000 -0.0501 -0.4953 0.5224 0.4459 -0.1371 -0.0645 0.2082 -0.1074 -0.0810 0.0443 0.2600 0.2241 0.2970 0.2043 0.1118 -0.1023 0.0265 0.0620 0.1405 0.0036 -0.0157 0.0144 0.3864 0.4816 0.1665
BsmtFinSF2 -0.0656 0.1112 -0.0591 0.0402 -0.0491 -0.0678 -0.0501 1.0000 -0.2093 0.1048 0.0971 -0.0993 0.0148 -0.0096 -0.0157 -0.0408 -0.0352 0.0469 -0.0383 -0.0182 0.0679 0.0031 0.0365 -0.0300 0.0889 0.0417 0.0049 -0.0152 0.0317 -0.0114 0.0517 0.0399
BsmtUnfSF -0.1408 -0.0026 0.3082 -0.1368 0.1490 0.1811 -0.4953 -0.2093 1.0000 0.4154 0.3180 0.0045 0.0282 0.2403 0.1666 0.0301 0.2506 0.0516 0.2142 0.1833 -0.0053 0.1290 -0.0025 0.0208 -0.0126 -0.0351 -0.0238 0.0349 -0.0413 0.2145 -0.1045 0.0201
TotalBsmtSF -0.2385 0.2608 0.5378 -0.1711 0.3915 0.2911 0.5224 0.1048 0.4154 1.0000 0.8195 -0.1745 -0.0332 0.4549 0.0504 -0.0689 0.2856 0.3395 0.4346 0.4867 0.2320 0.2473 -0.0955 0.0374 0.0845 0.1261 -0.0185 0.0132 -0.0150 0.6136 0.4145 0.2080
X1stFlrSF -0.2518 0.2995 0.4762 -0.1442 0.2820 0.2404 0.4459 0.0971 0.3180 0.8195 1.0000 -0.2026 -0.0142 0.5660 0.1274 0.0681 0.4095 0.4105 0.4393 0.4898 0.2355 0.2117 -0.0653 0.0561 0.0888 0.1315 -0.0211 0.0314 -0.0136 0.6059 0.3906 0.1470
X2ndFlrSF 0.3079 0.0510 0.2955 0.0289 0.0103 0.1400 -0.1371 -0.0993 0.0045 -0.1745 -0.2026 1.0000 0.0634 0.6875 0.5029 0.0593 0.6164 0.1946 0.1839 0.1383 0.0922 0.2080 0.0620 -0.0244 0.0406 0.0815 0.0162 0.0352 -0.0287 0.3193 0.3752 -0.0118
LowQualFinSF 0.0465 0.0048 -0.0304 0.0255 -0.1838 -0.0624 -0.0645 0.0148 0.0282 -0.0332 -0.0142 0.0634 1.0000 0.1347 0.1056 0.0075 0.1312 -0.0213 -0.0945 -0.0676 -0.0254 0.0183 0.0611 -0.0043 0.0268 0.0622 -0.0038 -0.0222 -0.0289 -0.0256 -0.0412 -0.0501
GrLivArea 0.0749 0.2631 0.5930 -0.0797 0.1990 0.2874 0.2082 -0.0096 0.2403 0.4549 0.5660 0.6875 0.1347 1.0000 0.5213 0.1001 0.8255 0.4617 0.4672 0.4690 0.2474 0.3302 0.0091 0.0206 0.1015 0.1702 -0.0024 0.0502 -0.0365 0.7086 0.5952 0.0937
BedroomAbvGr -0.0234 0.1197 0.1017 0.0130 -0.0707 -0.0406 -0.1074 -0.0157 0.1666 0.0504 0.1274 0.5029 0.1056 0.5213 1.0000 0.1986 0.6766 0.1076 0.0861 0.0653 0.0469 0.0938 0.0416 -0.0245 0.0443 0.0707 0.0078 0.0465 -0.0360 0.1682 0.2349 0.0079
KitchenAbvGr 0.2817 -0.0178 -0.1839 -0.0870 -0.1748 -0.1496 -0.0810 -0.0408 0.0301 -0.0689 0.0681 0.0593 0.0075 0.1001 0.1986 1.0000 0.2560 -0.1239 -0.0506 -0.0644 -0.0901 -0.0701 0.0373 -0.0246 -0.0516 -0.0145 0.0623 0.0266 0.0317 -0.1359 0.0383 -0.2468
TotRmsAbvGrd 0.0404 0.1900 0.4275 -0.0576 0.0956 0.1917 0.0443 -0.0352 0.2506 0.2856 0.4095 0.6164 0.1312 0.8255 0.6766 0.2560 1.0000 0.3261 0.3623 0.3378 0.1660 0.2342 0.0042 -0.0067 0.0594 0.0838 0.0248 0.0369 -0.0345 0.5337 0.4603 0.0345
Fireplaces -0.0456 0.2714 0.3968 -0.0238 0.1477 0.1126 0.2600 0.0469 0.0516 0.3395 0.4105 0.1946 -0.0213 0.4617 0.1076 -0.1239 0.3261 1.0000 0.3008 0.2691 0.2000 0.1694 -0.0248 0.0113 0.1845 0.0951 0.0014 0.0464 -0.0241 0.4669 0.3317 0.1863
GarageCars -0.0401 0.1549 0.6007 -0.1858 0.5379 0.4206 0.2241 -0.0383 0.2142 0.4346 0.4393 0.1839 -0.0945 0.4672 0.0861 -0.0506 0.3623 0.3008 1.0000 0.8825 0.2263 0.2136 -0.1514 0.0358 0.0505 0.0209 -0.0431 0.0405 -0.0391 0.6404 0.4836 0.2337
GarageArea -0.0987 0.1804 0.5620 -0.1515 0.4790 0.3716 0.2970 -0.0182 0.1833 0.4867 0.4898 0.1383 -0.0676 0.4690 0.0653 -0.0644 0.3378 0.2691 0.8825 1.0000 0.2247 0.2414 -0.1218 0.0351 0.0514 0.0610 -0.0274 0.0280 -0.0274 0.6234 0.4516 0.2307
WoodDeckSF -0.0126 0.1717 0.2389 -0.0033 0.2249 0.2057 0.2043 0.0679 -0.0053 0.2320 0.2355 0.0922 -0.0254 0.2474 0.0469 -0.0901 0.1660 0.2000 0.2263 0.2247 1.0000 0.0587 -0.1260 -0.0328 -0.0742 0.0734 -0.0096 0.0210 0.0223 0.3244 0.2882 0.1460
OpenPorchSF -0.0061 0.0848 0.3088 -0.0326 0.1887 0.2263 0.1118 0.0031 0.1290 0.2473 0.2117 0.2080 0.0183 0.3302 0.0938 -0.0701 0.2342 0.1694 0.2136 0.2414 0.0587 1.0000 -0.0931 -0.0058 0.0743 0.0608 -0.0186 0.0713 -0.0576 0.3159 0.2869 0.0259
EnclosedPorch -0.0120 -0.0183 -0.1139 0.0704 -0.3873 -0.1939 -0.1023 0.0365 -0.0025 -0.0955 -0.0653 0.0620 0.0611 0.0091 0.0416 0.0373 0.0042 -0.0248 -0.1514 -0.1218 -0.1260 -0.0931 1.0000 -0.0373 -0.0829 0.0542 0.0184 -0.0289 -0.0099 -0.1286 -0.1455 -0.1569
X3SsnPorch -0.0438 0.0204 0.0304 0.0255 0.0314 0.0453 0.0265 -0.0300 0.0208 0.0374 0.0561 -0.0244 -0.0043 0.0206 -0.0245 -0.0246 -0.0067 0.0113 0.0358 0.0351 -0.0328 -0.0058 -0.0373 1.0000 -0.0314 -0.0080 0.0004 0.0295 0.0186 0.0446 0.0285 0.0307
ScreenPorch -0.0260 0.0432 0.0649 0.0548 -0.0504 -0.0387 0.0620 0.0889 -0.0126 0.0845 0.0888 0.0406 0.0268 0.1015 0.0443 -0.0516 0.0594 0.1845 0.0505 0.0514 -0.0742 0.0743 -0.0829 -0.0314 1.0000 0.0513 0.0319 0.0232 0.0107 0.1114 0.0377 0.0512
PoolArea 0.0083 0.0777 0.0652 -0.0020 0.0049 0.0058 0.1405 0.0417 -0.0351 0.1261 0.1315 0.0815 0.0622 0.1702 0.0707 -0.0145 0.0838 0.0951 0.0209 0.0610 0.0734 0.0608 0.0542 -0.0080 0.0513 1.0000 0.0297 -0.0337 -0.0597 0.0924 0.0897 0.0181
MiscVal -0.0077 0.0381 -0.0314 0.0688 -0.0344 -0.0103 0.0036 0.0049 -0.0238 -0.0185 -0.0211 0.0162 -0.0038 -0.0024 0.0078 0.0623 0.0248 0.0014 -0.0431 -0.0274 -0.0096 -0.0186 0.0184 0.0004 0.0319 0.0297 1.0000 -0.0065 0.0049 -0.0212 -0.0260 -0.0025
MoSold -0.0136 0.0012 0.0708 -0.0035 0.0124 0.0215 -0.0157 -0.0152 0.0349 0.0132 0.0314 0.0352 -0.0222 0.0502 0.0465 0.0266 0.0369 0.0464 0.0405 0.0280 0.0210 0.0713 -0.0289 0.0295 0.0232 -0.0337 -0.0065 1.0000 -0.1457 0.0464 0.0245 0.0098
YrSold -0.0214 -0.0143 -0.0273 0.0439 -0.0136 0.0357 0.0144 0.0317 -0.0413 -0.0150 -0.0136 -0.0287 -0.0289 -0.0365 -0.0360 0.0317 -0.0345 -0.0241 -0.0391 -0.0274 0.0223 -0.0576 -0.0099 0.0186 0.0107 -0.0597 0.0049 -0.1457 1.0000 -0.0289 0.0201 -0.0094
SalePrice -0.0843 0.2638 0.7910 -0.0779 0.5229 0.5071 0.3864 -0.0114 0.2145 0.6136 0.6059 0.3193 -0.0256 0.7086 0.1682 -0.1359 0.5337 0.4669 0.6404 0.6234 0.3244 0.3159 -0.1286 0.0446 0.1114 0.0924 -0.0212 0.0464 -0.0289 1.0000 0.6317 0.2513
Baths 0.1510 0.2048 0.5411 -0.1740 0.5243 0.4437 0.4816 0.0517 -0.1045 0.4145 0.3906 0.3752 -0.0412 0.5952 0.2349 0.0383 0.4603 0.3317 0.4836 0.4516 0.2882 0.2869 -0.1455 0.0285 0.0377 0.0897 -0.0260 0.0245 0.0201 0.6317 1.0000 0.2016
CentralAirScore -0.1018 0.0498 0.2720 0.1190 0.3818 0.2989 0.1665 0.0399 0.0201 0.2080 0.1470 -0.0118 -0.0501 0.0937 0.0079 -0.2468 0.0345 0.1863 0.2337 0.2307 0.1460 0.0259 -0.1569 0.0307 0.0512 0.0181 -0.0025 0.0098 -0.0094 0.2513 0.2016 1.0000
C. Correlation Plots
C.1. Correlation Plot1 of Numeric Variables
corrplot(corrMatrix,method ="color")

C.2. Correlation Plot2 of Numeric Variables
corrplot(corrMatrix, method = "color", outline = T, addgrid.col = "darkgray", order="hclust", addrect = 4, rect.col = "black", rect.lwd = 5,cl.pos = "b", tl.col = "indianred4", tl.cex = 1.5, cl.cex = 1.5, addCoef.col = "white", number.digits = 2, number.cex = 0.75, col = colorRampPalette(c("darkred","white","midnightblue"))(100))

Based on the above Corrleation plot, 3 variables with highest correlation scores are - OverallQual, GrLivArea and GarageCars.

Hypothesis Testing

Test the hypotheses that the correlations between each pairwise set of variables is 0 and provide an 80% confidence interval. Discuss the meaning of your analysis.

corr_train_data <- subset(housing_pr_train_num, select=c(OverallQual,GrLivArea,GarageCars,SalePrice))
1. SalePrice Vs. Overall Quality

Null Hypothesis \({ H }_{ 0 }\): Correlation between House Sale Price and Overll Quality of houses is zero.

Alt. Hypothesis \({ H }_{ A }\): Correlation between House Sale Price and Overll Quality of houses is not zero.

cor.test(corr_train_data$SalePrice,corr_train_data$OverallQual, conf.level = 0.8)
## 
##  Pearson's product-moment correlation
## 
## data:  corr_train_data$SalePrice and corr_train_data$OverallQual
## t = 49.364, df = 1458, p-value < 2.2e-16
## alternative hypothesis: true correlation is not equal to 0
## 80 percent confidence interval:
##  0.7780752 0.8032204
## sample estimates:
##       cor 
## 0.7909816

With a low P value, we are confident the correlation between these two variables is not zero, and we are 80% confident it is between 0.778 and 0.803. Hence we accept alternate hypothesis.

2. SalePrice Vs. Above Ground Living Area Sq. Ft.(GrLivArea)

Null Hypothesis \({ H }_{ 0 }\): Correlation between House Sale Price and GrLivAreay of houses is zero.

Alt. Hypothesis \({ H }_{ A }\): Correlation between House Sale Price and GrLivArea of houses is not zero.

cor.test(corr_train_data$SalePrice,corr_train_data$GrLivArea, conf.level = 0.8)
## 
##  Pearson's product-moment correlation
## 
## data:  corr_train_data$SalePrice and corr_train_data$GrLivArea
## t = 38.348, df = 1458, p-value < 2.2e-16
## alternative hypothesis: true correlation is not equal to 0
## 80 percent confidence interval:
##  0.6915087 0.7249450
## sample estimates:
##       cor 
## 0.7086245

With a low P value, we are confident the correlation between these two variables is not zero, and we are 80% confident it is between 0.692 and 0.725. Hence we accept alternate hypothesis.

#### 3. SalePrice Vs. Size of garage in car capacity(GarageCars)q Null Hypothesis \({ H }_{ 0 }\): Correlation between House Sale Price and GarageCars of houses is zero.

Alt. Hypothesis \({ H }_{ A }\): Correlation between House Sale Price and GarageCars of houses is not zero.

cor.test(corr_train_data$SalePrice,corr_train_data$GarageCars, conf.level = 0.8)
## 
##  Pearson's product-moment correlation
## 
## data:  corr_train_data$SalePrice and corr_train_data$GarageCars
## t = 31.839, df = 1458, p-value < 2.2e-16
## alternative hypothesis: true correlation is not equal to 0
## 80 percent confidence interval:
##  0.6201771 0.6597899
## sample estimates:
##       cor 
## 0.6404092

With a low P value, we are confident the correlation between these two variables is not zero, and we are 80% confident it is between 0.620 and 0.659. Hence we accept alternate hypothesis.

I will not be worried about Family Wise Error in this case, since for all 3 variables I have suffcient statistical evidence to reject the Null Hypothesis.

Linear Algebra and Correlation

Invert your correlation matrix from above. (This is known as the precision matrix and contains variance inflation factors on the diagonal.) Multiply the correlation matrix by the precision matrix, and then multiply the precision matrix by the correlation matrix. Conduct LU decomposition on the matrix.

Ans:

A. Precision Matrix

Invert the correlation matrix to create the precision matrix:

precMatrix <- solve(corrMatrix)

precMatrix %>% kable() %>% kable_styling(bootstrap_options = c("striped", "hover", "condensed", "responsive")) %>% scroll_box(width="100%",height="300px")
MSSubClass LotArea OverallQual OverallCond YearBuilt YearRemodAdd BsmtFinSF1 BsmtFinSF2 BsmtUnfSF TotalBsmtSF X1stFlrSF X2ndFlrSF LowQualFinSF GrLivArea BedroomAbvGr KitchenAbvGr TotRmsAbvGrd Fireplaces GarageCars GarageArea WoodDeckSF OpenPorchSF EnclosedPorch X3SsnPorch ScreenPorch PoolArea MiscVal MoSold YrSold SalePrice Baths CentralAirScore
MSSubClass 1.5187929 0.0896258 -0.3728027 0.0192462 -0.1294784 -0.0281652 -0.3266891 -0.0973287 -0.2304666 0.3868408 0.8815329 0.3119878 -0.0060351 -1.1792054 0.2878876 -0.5186781 0.2580809 -0.0355909 -0.0854236 0.2122713 -0.0452697 0.0418729 0.0251561 0.0322569 -0.0314368 -0.0136819 0.0349678 0.0487569 0.0693108 0.4541448 -0.2475740 0.0586666
LotArea 0.0896258 1.2286247 0.2526113 0.0205515 0.1772295 0.0356887 0.3605135 0.0630309 0.3971923 -0.4585826 -0.3882111 -0.3354522 -0.0255314 0.2992615 -0.0687503 0.0137128 0.0919937 -0.1716822 -0.0112965 -0.0478966 -0.0780148 0.0002499 0.0489740 -0.0101098 0.0423613 -0.0119318 -0.0509599 0.0142185 0.0165527 -0.2649300 -0.1016003 -0.0020999
OverallQual -0.3728027 0.2526113 3.7195170 -0.2302517 -0.7600601 -0.2615928 2.6379228 0.9683630 2.1290467 -2.8918479 0.5926402 0.4332855 0.0331779 -1.0559318 0.2318922 0.3819201 -0.1609482 -0.2208109 -0.3062727 0.0254338 0.1161789 -0.0689868 -0.1975677 0.0298428 0.0157615 0.0154577 0.0108863 -0.1415124 -0.0370148 -1.6280933 0.0057152 -0.0013015
OverallCond 0.0192462 0.0205515 -0.2302517 1.6852943 1.3850804 -0.7131503 -0.0539844 -0.0460802 0.1313585 0.1212273 0.1207346 0.0412350 0.1009629 0.2312606 -0.1981359 0.0430524 0.1235128 0.0073703 0.1515479 -0.1011573 -0.0621166 -0.0372609 0.1728914 -0.0398407 -0.0037972 -0.0160637 -0.0845752 0.0216522 -0.0251138 -0.3694899 0.1235527 -0.4029192
YearBuilt -0.1294784 0.1772295 -0.7600601 1.3850804 4.1640910 -1.1376843 -1.0965069 -0.3970827 -1.2223569 1.0193900 -0.1815921 -0.2136660 0.2633852 1.3964835 -0.2620557 0.1508261 0.3691451 0.1294159 -0.4088882 -0.1070433 -0.0455798 -0.0142679 0.7970473 0.0085539 0.2241807 -0.0776674 -0.0933470 0.0776147 0.0449552 -0.7129222 -0.9057226 -0.7067990
YearRemodAdd -0.0281652 0.0356887 -0.2615928 -0.7131503 -1.1376843 2.2162347 -0.0118517 0.0162748 -0.3516506 0.3278749 0.5336157 0.7533361 0.0182839 -0.9759247 0.3499482 0.0704274 -0.1690508 0.2457118 -0.1364954 0.1683705 -0.0311433 -0.0937114 -0.0722429 -0.0103045 0.0585208 0.0035889 -0.0039778 -0.0150763 -0.0827315 -0.1958077 -0.3945433 -0.0249592
BsmtFinSF1 -0.3266891 0.3605135 2.6379228 -0.0539844 -1.0965069 -0.0118517 -6081.2006422 -2151.1712098 -5893.1072242 5849.7033555 -674.0564312 -762.0888549 -84.9733601 920.5059851 0.5334566 0.4507445 -3.0073581 -0.4749160 2.3964999 -1.9906259 0.3296336 0.1404175 -0.0185164 0.3488403 -0.3231962 -0.5720369 0.4849930 0.1715235 0.1605179 -2.2513881 -1.2301068 0.2845166
BsmtFinSF2 -0.0973287 0.0630309 0.9683630 -0.0460802 -0.3970827 0.0162748 -2151.1712098 -759.8151773 -2084.2773993 2068.9496512 -237.7561951 -268.7410383 -29.9887063 324.6211598 0.1596079 0.2008021 -1.0975963 -0.1462996 0.7757920 -0.6017772 0.0578102 0.0244388 -0.0693203 0.1516215 -0.2011882 -0.1927933 0.1776360 0.0594454 0.0306899 -0.6378003 -0.3979701 0.0752365
BsmtUnfSF -0.2304666 0.3971923 2.1290467 0.1313585 -1.2223569 -0.3516506 -5893.1072242 -2084.2773993 -5708.8051564 5667.5405647 -653.5589414 -739.2352494 -82.4358202 892.5066898 0.2709481 0.3142328 -3.1648903 -0.3627352 1.8969195 -1.5501798 0.3568447 0.0561981 -0.0808533 0.3122064 -0.2990549 -0.4158191 0.5054999 0.1559984 0.1625537 -1.7899723 0.0170691 0.3508237
TotalBsmtSF 0.3868408 -0.4585826 -2.8918479 0.1212273 1.0193900 0.3278749 5849.7033555 2068.9496512 5667.5405647 -5622.4778577 648.5758435 736.5698612 82.0899364 -888.9931141 -0.4772984 -0.2319624 3.3647617 0.5702412 -1.7031882 1.4144370 -0.3510774 -0.2597476 0.0289856 -0.2782575 0.2644169 0.3880750 -0.5281458 -0.0888259 -0.1508482 1.4617095 0.2156941 -0.4257052
X1stFlrSF 0.8815329 -0.3882111 0.5926402 0.1207346 -0.1815921 0.5336157 -674.0564312 -237.7561951 -653.5589414 648.5758435 5180.4586671 5848.1465220 650.8760929 -7039.7518114 0.4650636 -0.1848814 0.3671321 0.2777617 -0.8720624 1.1639323 -0.8370026 -0.2443897 -0.3634361 -0.0011621 -0.4788594 -0.1459882 0.1647503 -0.5817615 0.0729080 -1.2012921 -0.9304826 -0.0761503
X2ndFlrSF 0.3119878 -0.3354522 0.4332855 0.0412350 -0.2136660 0.7533361 -762.0888549 -268.7410383 -739.2352494 736.5698612 5848.1465220 6607.4187144 735.2965696 -7952.1470869 0.1344684 0.5408196 0.4240139 0.7205234 -0.9922997 1.5718057 -0.8767633 -0.4402663 -0.4725391 0.1029198 -0.5514440 -0.1661124 0.0800826 -0.6282969 0.0674675 -1.2661755 -1.5808144 -0.1206293
LowQualFinSF -0.0060351 -0.0255314 0.0331779 0.1009629 0.2633852 0.0182839 -84.9733601 -29.9887063 -82.4358202 82.0899364 650.8760929 735.2965696 82.9336728 -885.2454652 0.0684629 0.1488813 -0.0425811 0.1715691 0.0098023 0.1376780 -0.0884707 -0.0425301 -0.0039633 0.0056909 -0.0704553 -0.0547437 0.0179818 -0.0424173 0.0348689 -0.0817947 -0.1244076 -0.0634499
GrLivArea -1.1792054 0.2992615 -1.0559318 0.2312606 1.3964835 -0.9759247 920.5059851 324.6211598 892.5066898 -888.9931141 -7039.7518114 -7952.1470869 -885.2454652 9577.2989883 -0.8557446 -0.2501504 -3.3203721 -1.2727456 1.5698862 -2.4966084 0.9654451 0.1784581 0.4436743 -0.1826584 0.6564571 -0.1658447 -0.1131617 0.7291888 -0.1116095 -0.0661081 0.8306440 0.1477788
BedroomAbvGr 0.2878876 -0.0687503 0.2318922 -0.1981359 -0.2620557 0.3499482 0.5334566 0.1596079 0.2709481 -0.4772984 0.4650636 0.1344684 0.0684629 -0.8557446 2.3328897 0.0346439 -1.5547523 0.1886041 -0.0118737 0.1961902 0.0199195 0.0734214 -0.0102789 0.0285296 -0.0587883 -0.0470828 0.0521659 -0.0706347 0.0262787 0.5526497 -0.2918891 -0.1121002
KitchenAbvGr -0.5186781 0.0137128 0.3819201 0.0430524 0.1508261 0.0704274 0.4507445 0.2008021 0.3142328 -0.2319624 -0.1848814 0.5408196 0.1488813 -0.2501504 0.0346439 1.5978617 -0.7645964 0.2072343 -0.1087378 0.0122804 0.0985822 0.0786318 0.0426879 0.0211170 0.0438416 0.0064231 -0.0892323 -0.0774241 -0.0682586 0.2115985 -0.2625170 0.1735325
TotRmsAbvGrd 0.2580809 0.0919937 -0.1609482 0.1235128 0.3691451 -0.1690508 -3.0073581 -1.0975963 -3.1648903 3.3647617 0.3671321 0.4240139 -0.0425811 -3.3203721 -1.5547523 -0.7645964 4.9258676 -0.0790649 -0.2613985 0.1402299 0.0015691 0.0178050 0.1394060 0.0655276 0.0925771 0.1619020 -0.1007079 0.1137357 0.0426725 -0.5506856 -0.0605385 -0.0131231
Fireplaces -0.0355909 -0.1716822 -0.2208109 0.0073703 0.1294159 0.2457118 -0.4749160 -0.1462996 -0.3627352 0.5702412 0.2777617 0.7205234 0.1715691 -1.2727456 0.1886041 0.2072343 -0.0790649 1.5773935 -0.2944998 0.3579616 -0.0575346 -0.0133843 -0.0026849 0.0141650 -0.1623785 -0.0077151 -0.0166039 -0.0295346 -0.0027785 -0.1669234 -0.0586763 -0.1745289
GarageCars -0.0854236 -0.0112965 -0.3062727 0.1515479 -0.4088882 -0.1364954 2.3964999 0.7757920 1.8969195 -1.7031882 -0.8720624 -0.9922997 0.0098023 1.5698862 -0.0118737 -0.1087378 -0.2613985 -0.2944998 5.5159885 -4.2608057 -0.0120595 0.1908130 0.0320692 -0.0222845 -0.0599161 0.1226352 0.0841633 -0.0218457 0.0871742 -0.4987449 -0.2919750 0.0299363
GarageArea 0.2122713 -0.0478966 0.0254338 -0.1011573 -0.1070433 0.1683705 -1.9906259 -0.6017772 -1.5501798 1.4144370 1.1639323 1.5718057 0.1376780 -2.4966084 0.1961902 0.0122804 0.1402299 0.3579616 -4.2608057 5.1586993 -0.0143448 -0.2004188 -0.0748826 0.0149721 0.0163981 -0.0758021 -0.0421272 0.0271031 -0.0447443 -0.0540746 0.2279026 -0.1016034
WoodDeckSF -0.0452697 -0.0780148 0.1161789 -0.0621166 -0.0455798 -0.0311433 0.3296336 0.0578102 0.3568447 -0.3510774 -0.8370026 -0.8767633 -0.0884707 0.9654451 0.0199195 0.0985822 0.0015691 -0.0575346 -0.0120595 -0.0143448 1.2098882 0.0822247 0.1070574 0.0732364 0.1618373 -0.0436141 0.0002592 -0.0306097 -0.0371833 -0.2233329 -0.1055616 -0.0251613
OpenPorchSF 0.0418729 0.0002499 -0.0689868 -0.0372609 -0.0142679 -0.0937114 0.1404175 0.0244388 0.0561981 -0.2597476 -0.2443897 -0.4402663 -0.0425301 0.1784581 0.0734214 0.0786318 0.0178050 -0.0133843 0.1908130 -0.2004188 0.0822247 1.2203300 0.0921435 0.0300788 -0.0395150 -0.0105990 0.0138818 -0.0601017 0.0567198 0.0457699 -0.1659391 0.1189225
EnclosedPorch 0.0251561 0.0489740 -0.1975677 0.1728914 0.7970473 -0.0722429 -0.0185164 -0.0693203 -0.0808533 0.0289856 -0.3634361 -0.4725391 -0.0039633 0.4436743 -0.0102789 0.0426879 0.1394060 -0.0026849 0.0320692 -0.0748826 0.1070574 0.0921435 1.2791244 0.0425749 0.1682078 -0.0648520 -0.0274937 0.0361818 0.0161678 -0.0438255 -0.1028568 -0.0204113
X3SsnPorch 0.0322569 -0.0101098 0.0298428 -0.0398407 0.0085539 -0.0103045 0.3488403 0.1516215 0.3122064 -0.2782575 -0.0011621 0.1029198 0.0056909 -0.1826584 0.0285296 0.0211170 0.0655276 0.0141650 -0.0222845 0.0149721 0.0732364 0.0300788 0.0425749 1.0197154 0.0478761 0.0045333 -0.0048382 -0.0337145 -0.0222234 -0.0341076 -0.0510731 -0.0113314
ScreenPorch -0.0314368 0.0423613 0.0157615 -0.0037972 0.2241807 0.0585208 -0.3231962 -0.2011882 -0.2990549 0.2644169 -0.4788594 -0.5514440 -0.0704553 0.6564571 -0.0587883 0.0438416 0.0925771 -0.1623785 -0.0599161 0.0163981 0.1618373 -0.0395150 0.1682078 0.0478761 1.1142253 -0.0389950 -0.0402725 -0.0181218 -0.0310499 -0.2131914 0.0088053 -0.0544112
PoolArea -0.0136819 -0.0119318 0.0154577 -0.0160637 -0.0776674 0.0035889 -0.5720369 -0.1927933 -0.4158191 0.3880750 -0.1459882 -0.1661124 -0.0547437 -0.1658447 -0.0470828 0.0064231 0.1619020 -0.0077151 0.1226352 -0.0758021 -0.0436141 -0.0105990 -0.0648520 0.0045333 -0.0389950 1.0807211 -0.0281238 0.0476788 0.0676284 0.1094033 0.0793226 0.0103748
MiscVal 0.0349678 -0.0509599 0.0108863 -0.0845752 -0.0933470 -0.0039778 0.4849930 0.1776360 0.5054999 -0.5281458 0.1647503 0.0800826 0.0179818 -0.1131617 0.0521659 -0.0892323 -0.1007079 -0.0166039 0.0841633 -0.0421272 0.0002592 0.0138818 -0.0274937 -0.0048382 -0.0402725 -0.0281238 1.0220520 0.0028627 0.0016487 0.0317613 0.0608899 0.0045922
MoSold 0.0487569 0.0142185 -0.1415124 0.0216522 0.0776147 -0.0150763 0.1715235 0.0594454 0.1559984 -0.0888259 -0.5817615 -0.6282969 -0.0424173 0.7291888 -0.0706347 -0.0774241 0.1137357 -0.0295346 -0.0218457 0.0271031 -0.0306097 -0.0601017 0.0361818 -0.0337145 -0.0181218 0.0476788 0.0028627 1.0459557 0.1552631 0.0158545 -0.0018102 -0.0106120
YrSold 0.0693108 0.0165527 -0.0370148 -0.0251138 0.0449552 -0.0827315 0.1605179 0.0306899 0.1625537 -0.1508482 0.0729080 0.0674675 0.0348689 -0.1116095 0.0262787 -0.0682586 0.0426725 -0.0027785 0.0871742 -0.0447443 -0.0371833 0.0567198 0.0161678 -0.0222234 -0.0310499 0.0676284 0.0016487 0.1552631 1.0486723 0.0652605 -0.0920085 0.0192693
SalePrice 0.4541448 -0.2649300 -1.6280933 -0.3694899 -0.7129222 -0.1958077 -2.2513881 -0.6378003 -1.7899723 1.4617095 -1.2012921 -1.2661755 -0.0817947 -0.0661081 0.5526497 0.2115985 -0.5506856 -0.1669234 -0.4987449 -0.0540746 -0.2233329 0.0457699 -0.0438255 -0.0341076 -0.2131914 0.1094033 0.0317613 0.0158545 0.0652605 5.2397928 -0.2986324 0.1784433
Baths -0.2475740 -0.1016003 0.0057152 0.1235527 -0.9057226 -0.3945433 -1.2301068 -0.3979701 0.0170691 0.2156941 -0.9304826 -1.5808144 -0.1244076 0.8306440 -0.2918891 -0.2625170 -0.0605385 -0.0586763 -0.2919750 0.2279026 -0.1055616 -0.1659391 -0.1028568 -0.0510731 0.0088053 0.0793226 0.0608899 -0.0018102 -0.0920085 -0.2986324 3.1116038 0.0533799
CentralAirScore 0.0586666 -0.0020999 -0.0013015 -0.4029192 -0.7067990 -0.0249592 0.2845166 0.0752365 0.3508237 -0.4257052 -0.0761503 -0.1206293 -0.0634499 0.1477788 -0.1121002 0.1735325 -0.0131231 -0.1745289 0.0299363 -0.1016034 -0.0251613 0.1189225 -0.0204113 -0.0113314 -0.0544112 0.0103748 0.0045922 -0.0106120 0.0192693 0.1784433 0.0533799 1.3937145

B. Correlation Matrix X Precision Matrix

The correlation matrix is now multiplied by the precision matrix (rounding to 15 digits to remove R’s decimilization at the extremes).

round(corrMatrix %*% precMatrix, 15) %>% kable() %>% kable_styling(bootstrap_options = c("striped", "hover", "condensed", "responsive")) %>% scroll_box(width="100%",height="300px")
MSSubClass LotArea OverallQual OverallCond YearBuilt YearRemodAdd BsmtFinSF1 BsmtFinSF2 BsmtUnfSF TotalBsmtSF X1stFlrSF X2ndFlrSF LowQualFinSF GrLivArea BedroomAbvGr KitchenAbvGr TotRmsAbvGrd Fireplaces GarageCars GarageArea WoodDeckSF OpenPorchSF EnclosedPorch X3SsnPorch ScreenPorch PoolArea MiscVal MoSold YrSold SalePrice Baths CentralAirScore
MSSubClass 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
LotArea 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
OverallQual 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
OverallCond 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
YearBuilt 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
YearRemodAdd 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
BsmtFinSF1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
BsmtFinSF2 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
BsmtUnfSF 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
TotalBsmtSF 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
X1stFlrSF 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
X2ndFlrSF 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
LowQualFinSF 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
GrLivArea 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
BedroomAbvGr 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
KitchenAbvGr 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
TotRmsAbvGrd 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Fireplaces 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0
GarageCars 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0
GarageArea 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0
WoodDeckSF 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0
OpenPorchSF 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0
EnclosedPorch 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0
X3SsnPorch 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0
ScreenPorch 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0
PoolArea 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0
MiscVal 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0
MoSold 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0
YrSold 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0
SalePrice 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0
Baths 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0
CentralAirScore 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1

C. Precision Matrix X Correlation Matrix

The precision matrix is now multiplied by the correlation matrix (rounding to 15 digits to remove R’s decimilization at the extremes).

round(precMatrix %*% corrMatrix, 15) %>% kable() %>% kable_styling(bootstrap_options = c("striped", "hover", "condensed", "responsive")) %>% scroll_box(width="100%",height="300px")
MSSubClass LotArea OverallQual OverallCond YearBuilt YearRemodAdd BsmtFinSF1 BsmtFinSF2 BsmtUnfSF TotalBsmtSF X1stFlrSF X2ndFlrSF LowQualFinSF GrLivArea BedroomAbvGr KitchenAbvGr TotRmsAbvGrd Fireplaces GarageCars GarageArea WoodDeckSF OpenPorchSF EnclosedPorch X3SsnPorch ScreenPorch PoolArea MiscVal MoSold YrSold SalePrice Baths CentralAirScore
MSSubClass 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
LotArea 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
OverallQual 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
OverallCond 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
YearBuilt 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
YearRemodAdd 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
BsmtFinSF1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
BsmtFinSF2 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
BsmtUnfSF 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
TotalBsmtSF 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
X1stFlrSF 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
X2ndFlrSF 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
LowQualFinSF 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
GrLivArea 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
BedroomAbvGr 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
KitchenAbvGr 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
TotRmsAbvGrd 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Fireplaces 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0
GarageCars 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0
GarageArea 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0
WoodDeckSF 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0
OpenPorchSF 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0
EnclosedPorch 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0
X3SsnPorch 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0
ScreenPorch 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0
PoolArea 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0
MiscVal 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0
MoSold 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0
YrSold 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0
SalePrice 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0
Baths 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0
CentralAirScore 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1

D. LU Decomposition

## LU Factorization Function

LUDecomposition<-function(A){
  nrows<-nrow(A)
  ncols<-ncol(A)
  
  if(nrows!=ncols){
    print("Please enter a square matrix for LU decomposition")
    return()
  }
  
  ## Initialize Lower Triangular Matrix
  L<- matrix(0, nrow=nrows, ncol=ncols, byrow = T)
  
  ## Upper Triangular Matrix initialized to A
  U <- A
  for(i in 1:nrows){
    pivotElement <- U[i,i]
    L[i,i] <- 1
    if(i!=nrows){
      for(j in (i+1):nrows){
        if(U[j,i]!=0){
          multiplier <- U[j,i]/pivotElement
          newRow <- U[j,] - (multiplier*U[i,])
          U[j,] <- newRow
          L[j,i] <- multiplier
        }
      }
    }
    
  }
  
  result <- list(L,U)

  return(result)
}

result <- LUDecomposition(corrMatrix)

L <- result[[1]]
U <- result[[2]]

print("The LU decomposition of the input matrix : ")
## [1] "The LU decomposition of the input matrix : "
print("A")
## [1] "A"
L%*%U  %>% kable() %>% kable_styling(bootstrap_options = c("striped", "hover", "condensed", "responsive")) %>% scroll_box(width="100%",height="300px")
MSSubClass LotArea OverallQual OverallCond YearBuilt YearRemodAdd BsmtFinSF1 BsmtFinSF2 BsmtUnfSF TotalBsmtSF X1stFlrSF X2ndFlrSF LowQualFinSF GrLivArea BedroomAbvGr KitchenAbvGr TotRmsAbvGrd Fireplaces GarageCars GarageArea WoodDeckSF OpenPorchSF EnclosedPorch X3SsnPorch ScreenPorch PoolArea MiscVal MoSold YrSold SalePrice Baths CentralAirScore
1.0000 -0.1398 0.0326 -0.0593 0.0279 0.0406 -0.0698 -0.0656 -0.1408 -0.2385 -0.2518 0.3079 0.0465 0.0749 -0.0234 0.2817 0.0404 -0.0456 -0.0401 -0.0987 -0.0126 -0.0061 -0.0120 -0.0438 -0.0260 0.0083 -0.0077 -0.0136 -0.0214 -0.0843 0.1510 -0.1018
-0.1398 1.0000 0.1058 -0.0056 0.0142 0.0138 0.2141 0.1112 -0.0026 0.2608 0.2995 0.0510 0.0048 0.2631 0.1197 -0.0178 0.1900 0.2714 0.1549 0.1804 0.1717 0.0848 -0.0183 0.0204 0.0432 0.0777 0.0381 0.0012 -0.0143 0.2638 0.2048 0.0498
0.0326 0.1058 1.0000 -0.0919 0.5723 0.5507 0.2397 -0.0591 0.3082 0.5378 0.4762 0.2955 -0.0304 0.5930 0.1017 -0.1839 0.4275 0.3968 0.6007 0.5620 0.2389 0.3088 -0.1139 0.0304 0.0649 0.0652 -0.0314 0.0708 -0.0273 0.7910 0.5411 0.2720
-0.0593 -0.0056 -0.0919 1.0000 -0.3760 0.0737 -0.0462 0.0402 -0.1368 -0.1711 -0.1442 0.0289 0.0255 -0.0797 0.0130 -0.0870 -0.0576 -0.0238 -0.1858 -0.1515 -0.0033 -0.0326 0.0704 0.0255 0.0548 -0.0020 0.0688 -0.0035 0.0439 -0.0779 -0.1740 0.1190
0.0279 0.0142 0.5723 -0.3760 1.0000 0.5929 0.2495 -0.0491 0.1490 0.3915 0.2820 0.0103 -0.1838 0.1990 -0.0707 -0.1748 0.0956 0.1477 0.5379 0.4790 0.2249 0.1887 -0.3873 0.0314 -0.0504 0.0049 -0.0344 0.0124 -0.0136 0.5229 0.5243 0.3818
0.0406 0.0138 0.5507 0.0737 0.5929 1.0000 0.1285 -0.0678 0.1811 0.2911 0.2404 0.1400 -0.0624 0.2874 -0.0406 -0.1496 0.1917 0.1126 0.4206 0.3716 0.2057 0.2263 -0.1939 0.0453 -0.0387 0.0058 -0.0103 0.0215 0.0357 0.5071 0.4437 0.2989
-0.0698 0.2141 0.2397 -0.0462 0.2495 0.1285 1.0000 -0.0501 -0.4953 0.5224 0.4459 -0.1371 -0.0645 0.2082 -0.1074 -0.0810 0.0443 0.2600 0.2241 0.2970 0.2043 0.1118 -0.1023 0.0265 0.0620 0.1405 0.0036 -0.0157 0.0144 0.3864 0.4816 0.1665
-0.0656 0.1112 -0.0591 0.0402 -0.0491 -0.0678 -0.0501 1.0000 -0.2093 0.1048 0.0971 -0.0993 0.0148 -0.0096 -0.0157 -0.0408 -0.0352 0.0469 -0.0383 -0.0182 0.0679 0.0031 0.0365 -0.0300 0.0889 0.0417 0.0049 -0.0152 0.0317 -0.0114 0.0517 0.0399
-0.1408 -0.0026 0.3082 -0.1368 0.1490 0.1811 -0.4953 -0.2093 1.0000 0.4154 0.3180 0.0045 0.0282 0.2403 0.1666 0.0301 0.2506 0.0516 0.2142 0.1833 -0.0053 0.1290 -0.0025 0.0208 -0.0126 -0.0351 -0.0238 0.0349 -0.0413 0.2145 -0.1045 0.0201
-0.2385 0.2608 0.5378 -0.1711 0.3915 0.2911 0.5224 0.1048 0.4154 1.0000 0.8195 -0.1745 -0.0332 0.4549 0.0504 -0.0689 0.2856 0.3395 0.4346 0.4867 0.2320 0.2473 -0.0955 0.0374 0.0845 0.1261 -0.0185 0.0132 -0.0150 0.6136 0.4145 0.2080
-0.2518 0.2995 0.4762 -0.1442 0.2820 0.2404 0.4459 0.0971 0.3180 0.8195 1.0000 -0.2026 -0.0142 0.5660 0.1274 0.0681 0.4095 0.4105 0.4393 0.4898 0.2355 0.2117 -0.0653 0.0561 0.0888 0.1315 -0.0211 0.0314 -0.0136 0.6059 0.3906 0.1470
0.3079 0.0510 0.2955 0.0289 0.0103 0.1400 -0.1371 -0.0993 0.0045 -0.1745 -0.2026 1.0000 0.0634 0.6875 0.5029 0.0593 0.6164 0.1946 0.1839 0.1383 0.0922 0.2080 0.0620 -0.0244 0.0406 0.0815 0.0162 0.0352 -0.0287 0.3193 0.3752 -0.0118
0.0465 0.0048 -0.0304 0.0255 -0.1838 -0.0624 -0.0645 0.0148 0.0282 -0.0332 -0.0142 0.0634 1.0000 0.1347 0.1056 0.0075 0.1312 -0.0213 -0.0945 -0.0676 -0.0254 0.0183 0.0611 -0.0043 0.0268 0.0622 -0.0038 -0.0222 -0.0289 -0.0256 -0.0412 -0.0501
0.0749 0.2631 0.5930 -0.0797 0.1990 0.2874 0.2082 -0.0096 0.2403 0.4549 0.5660 0.6875 0.1347 1.0000 0.5213 0.1001 0.8255 0.4617 0.4672 0.4690 0.2474 0.3302 0.0091 0.0206 0.1015 0.1702 -0.0024 0.0502 -0.0365 0.7086 0.5952 0.0937
-0.0234 0.1197 0.1017 0.0130 -0.0707 -0.0406 -0.1074 -0.0157 0.1666 0.0504 0.1274 0.5029 0.1056 0.5213 1.0000 0.1986 0.6766 0.1076 0.0861 0.0653 0.0469 0.0938 0.0416 -0.0245 0.0443 0.0707 0.0078 0.0465 -0.0360 0.1682 0.2349 0.0079
0.2817 -0.0178 -0.1839 -0.0870 -0.1748 -0.1496 -0.0810 -0.0408 0.0301 -0.0689 0.0681 0.0593 0.0075 0.1001 0.1986 1.0000 0.2560 -0.1239 -0.0506 -0.0644 -0.0901 -0.0701 0.0373 -0.0246 -0.0516 -0.0145 0.0623 0.0266 0.0317 -0.1359 0.0383 -0.2468
0.0404 0.1900 0.4275 -0.0576 0.0956 0.1917 0.0443 -0.0352 0.2506 0.2856 0.4095 0.6164 0.1312 0.8255 0.6766 0.2560 1.0000 0.3261 0.3623 0.3378 0.1660 0.2342 0.0042 -0.0067 0.0594 0.0838 0.0248 0.0369 -0.0345 0.5337 0.4603 0.0345
-0.0456 0.2714 0.3968 -0.0238 0.1477 0.1126 0.2600 0.0469 0.0516 0.3395 0.4105 0.1946 -0.0213 0.4617 0.1076 -0.1239 0.3261 1.0000 0.3008 0.2691 0.2000 0.1694 -0.0248 0.0113 0.1845 0.0951 0.0014 0.0464 -0.0241 0.4669 0.3317 0.1863
-0.0401 0.1549 0.6007 -0.1858 0.5379 0.4206 0.2241 -0.0383 0.2142 0.4346 0.4393 0.1839 -0.0945 0.4672 0.0861 -0.0506 0.3623 0.3008 1.0000 0.8825 0.2263 0.2136 -0.1514 0.0358 0.0505 0.0209 -0.0431 0.0405 -0.0391 0.6404 0.4836 0.2337
-0.0987 0.1804 0.5620 -0.1515 0.4790 0.3716 0.2970 -0.0182 0.1833 0.4867 0.4898 0.1383 -0.0676 0.4690 0.0653 -0.0644 0.3378 0.2691 0.8825 1.0000 0.2247 0.2414 -0.1218 0.0351 0.0514 0.0610 -0.0274 0.0280 -0.0274 0.6234 0.4516 0.2307
-0.0126 0.1717 0.2389 -0.0033 0.2249 0.2057 0.2043 0.0679 -0.0053 0.2320 0.2355 0.0922 -0.0254 0.2474 0.0469 -0.0901 0.1660 0.2000 0.2263 0.2247 1.0000 0.0587 -0.1260 -0.0328 -0.0742 0.0734 -0.0096 0.0210 0.0223 0.3244 0.2882 0.1460
-0.0061 0.0848 0.3088 -0.0326 0.1887 0.2263 0.1118 0.0031 0.1290 0.2473 0.2117 0.2080 0.0183 0.3302 0.0938 -0.0701 0.2342 0.1694 0.2136 0.2414 0.0587 1.0000 -0.0931 -0.0058 0.0743 0.0608 -0.0186 0.0713 -0.0576 0.3159 0.2869 0.0259
-0.0120 -0.0183 -0.1139 0.0704 -0.3873 -0.1939 -0.1023 0.0365 -0.0025 -0.0955 -0.0653 0.0620 0.0611 0.0091 0.0416 0.0373 0.0042 -0.0248 -0.1514 -0.1218 -0.1260 -0.0931 1.0000 -0.0373 -0.0829 0.0542 0.0184 -0.0289 -0.0099 -0.1286 -0.1455 -0.1569
-0.0438 0.0204 0.0304 0.0255 0.0314 0.0453 0.0265 -0.0300 0.0208 0.0374 0.0561 -0.0244 -0.0043 0.0206 -0.0245 -0.0246 -0.0067 0.0113 0.0358 0.0351 -0.0328 -0.0058 -0.0373 1.0000 -0.0314 -0.0080 0.0004 0.0295 0.0186 0.0446 0.0285 0.0307
-0.0260 0.0432 0.0649 0.0548 -0.0504 -0.0387 0.0620 0.0889 -0.0126 0.0845 0.0888 0.0406 0.0268 0.1015 0.0443 -0.0516 0.0594 0.1845 0.0505 0.0514 -0.0742 0.0743 -0.0829 -0.0314 1.0000 0.0513 0.0319 0.0232 0.0107 0.1114 0.0377 0.0512
0.0083 0.0777 0.0652 -0.0020 0.0049 0.0058 0.1405 0.0417 -0.0351 0.1261 0.1315 0.0815 0.0622 0.1702 0.0707 -0.0145 0.0838 0.0951 0.0209 0.0610 0.0734 0.0608 0.0542 -0.0080 0.0513 1.0000 0.0297 -0.0337 -0.0597 0.0924 0.0897 0.0181
-0.0077 0.0381 -0.0314 0.0688 -0.0344 -0.0103 0.0036 0.0049 -0.0238 -0.0185 -0.0211 0.0162 -0.0038 -0.0024 0.0078 0.0623 0.0248 0.0014 -0.0431 -0.0274 -0.0096 -0.0186 0.0184 0.0004 0.0319 0.0297 1.0000 -0.0065 0.0049 -0.0212 -0.0260 -0.0025
-0.0136 0.0012 0.0708 -0.0035 0.0124 0.0215 -0.0157 -0.0152 0.0349 0.0132 0.0314 0.0352 -0.0222 0.0502 0.0465 0.0266 0.0369 0.0464 0.0405 0.0280 0.0210 0.0713 -0.0289 0.0295 0.0232 -0.0337 -0.0065 1.0000 -0.1457 0.0464 0.0245 0.0098
-0.0214 -0.0143 -0.0273 0.0439 -0.0136 0.0357 0.0144 0.0317 -0.0413 -0.0150 -0.0136 -0.0287 -0.0289 -0.0365 -0.0360 0.0317 -0.0345 -0.0241 -0.0391 -0.0274 0.0223 -0.0576 -0.0099 0.0186 0.0107 -0.0597 0.0049 -0.1457 1.0000 -0.0289 0.0201 -0.0094
-0.0843 0.2638 0.7910 -0.0779 0.5229 0.5071 0.3864 -0.0114 0.2145 0.6136 0.6059 0.3193 -0.0256 0.7086 0.1682 -0.1359 0.5337 0.4669 0.6404 0.6234 0.3244 0.3159 -0.1286 0.0446 0.1114 0.0924 -0.0212 0.0464 -0.0289 1.0000 0.6317 0.2513
0.1510 0.2048 0.5411 -0.1740 0.5243 0.4437 0.4816 0.0517 -0.1045 0.4145 0.3906 0.3752 -0.0412 0.5952 0.2349 0.0383 0.4603 0.3317 0.4836 0.4516 0.2882 0.2869 -0.1455 0.0285 0.0377 0.0897 -0.0260 0.0245 0.0201 0.6317 1.0000 0.2016
-0.1018 0.0498 0.2720 0.1190 0.3818 0.2989 0.1665 0.0399 0.0201 0.2080 0.1470 -0.0118 -0.0501 0.0937 0.0079 -0.2468 0.0345 0.1863 0.2337 0.2307 0.1460 0.0259 -0.1569 0.0307 0.0512 0.0181 -0.0025 0.0098 -0.0094 0.2513 0.2016 1.0000 
print("L")
## [1] "L"
L %>% kable() %>% kable_styling(bootstrap_options = c("striped", "hover", "condensed", "responsive")) %>% scroll_box(width="100%",height="300px")
1.0000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0
-0.1398 1.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0
0.0326 0.1125573 1.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0
-0.0593 -0.0141670 -0.0896117 1.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0
0.0279 0.0184612 0.5771354 -0.3268716 1.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0
0.0406 0.0198641 0.5546635 0.1268941 0.5606666 1.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0
-0.0698 0.2084152 0.2219684 -0.0281490 0.1986970 -0.1068994 1.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0
-0.0656 0.1040629 -0.0693811 0.0319941 0.0012136 -0.0670527 -0.0726611 1.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0
-0.1408 -0.0227280 0.3196080 -0.1185905 -0.1185478 0.1292930 -0.6325254 -0.2328028 1.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0
-0.2385 0.2319918 0.5270804 -0.1370192 0.0878241 -0.0055207 0.3759066 0.1332220 1.0075507 1.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0
-0.2518 0.2695668 0.4608745 -0.1159942 -0.0278532 0.0330431 0.3130825 0.1124265 0.7275876 0.8686173 1.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0
0.3079 0.0959191 0.2786343 0.0739838 -0.2386542 0.0787302 -0.1864521 -0.0835051 -0.3866344 -0.8290493 -0.2866741 1.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0
0.0465 0.0115260 -0.0336415 0.0257431 -0.2795070 0.0781579 -0.0228298 0.0153463 0.0351940 -0.2159394 0.0004768 0.0281176 1.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0
0.0749 0.2790243 0.5674171 -0.0214710 -0.2446245 0.0969687 0.0733663 0.0148611 0.2174705 0.0226581 0.4972974 0.8332271 0.0924648 1.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0
-0.0234 0.1187495 0.0905793 0.0215197 -0.2068666 -0.0573420 -0.1512860 -0.0362348 0.0848586 -0.3835762 0.2176117 0.6971707 0.0642321 0.9695948 1.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0
0.2817 0.0220119 -0.1981850 -0.0885398 -0.1750071 0.0269248 -0.0043681 -0.0355647 0.2207784 0.2316000 0.4664881 0.1267147 -0.0467390 0.6602299 0.1797899 1.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0
0.0404 0.1995479 0.4096857 -0.0164058 -0.2615499 0.0935361 -0.0565254 -0.0253199 0.1548705 -0.7648323 0.4585178 0.7551912 0.0835421 1.1195440 0.3341932 0.1573400 1.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0
-0.0456 0.2703080 0.3734923 0.0103894 -0.1154842 -0.1195969 0.1447066 0.0481544 0.0575950 -0.1331134 0.3105166 0.2058735 -0.0276459 0.4756979 -0.1168356 -0.1450543 0.0685294 1.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0
-0.0401 0.1522700 0.5932020 -0.1351948 0.2742098 0.0495204 0.0324674 -0.0082771 0.0904155 0.2482207 0.2274785 0.1600057 -0.0353759 0.2440601 -0.0578563 0.0513344 0.1103073 0.0179862 1.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0
-0.0987 0.1699227 0.5539348 -0.1072711 0.2278016 0.0210049 0.1218915 0.0078103 0.1549637 0.0608125 0.2425910 0.1642979 -0.0125002 0.5837401 -0.0949269 0.0375050 0.0553527 -0.0537540 0.8231549 1.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0
-0.0126 0.1733260 0.2231925 0.0183043 0.1787744 0.0435338 0.1164523 0.0748956 0.0428974 0.1855104 0.1146058 0.1262064 0.0051600 -0.7748856 -0.0345107 -0.0708464 0.0393545 0.0415520 0.0339049 -0.0014503 1.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0
-0.0061 0.0856206 0.3036444 -0.0049873 0.0227173 0.1041115 0.0345963 0.0211703 0.1176299 -0.0222716 -0.0078490 0.2225467 0.0230129 -0.2079052 -0.0650047 -0.0414202 -0.0227497 -0.0045837 -0.0217443 0.1431861 -0.0642375 1.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0
-0.0120 -0.0203758 -0.1127809 0.0601348 -0.5364990 0.0618948 -0.0103904 0.0308060 -0.0016539 -0.0439411 -0.0462333 0.0244997 -0.0333338 -0.3090294 -0.0246065 -0.0307091 -0.0997648 -0.0203619 0.0187681 0.0512230 -0.0562766 -0.0754322 1.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0
-0.0438 0.0145613 0.0306340 0.0261169 0.0413579 0.0272687 0.0129190 -0.0322679 0.0206528 0.4182648 0.0775129 -0.0127625 0.0042299 0.1229094 -0.0418854 -0.0123838 -0.0563519 -0.0170961 0.0105355 -0.0237351 -0.0574974 -0.0259124 -0.0349034 1.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0
-0.0260 0.0403539 0.0621321 0.0600096 -0.1175737 -0.0852830 0.0555661 0.0886230 0.0461082 -0.3525445 0.0320485 0.0583953 0.0114847 -0.4380803 -0.0076289 -0.0647420 -0.0390480 0.1591972 0.0496885 -0.0077106 -0.1274346 0.0421792 -0.1472531 -0.0411226 1.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0
0.0083 0.0804323 0.0568193 0.0046870 -0.0489178 -0.0245378 0.1331303 0.0467415 0.0742067 -0.2685199 0.0681918 0.1448783 0.0570868 0.0147803 -0.0122645 -0.0471540 -0.1711322 -0.0003597 -0.0655983 0.0794956 0.0227781 0.0043248 0.0554834 -0.0099384 0.0309801 1.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0
-0.0077 0.0377616 -0.0357990 0.0664766 0.0123786 -0.0067672 0.0037520 -0.0041211 -0.0057252 0.3977306 -0.0243030 0.0292725 -0.0051208 0.2309681 -0.0076768 0.0897162 0.0782207 0.0148883 -0.0536951 0.0461768 -0.0087893 -0.0166098 0.0206487 0.0014772 0.0388799 0.0301908 1.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0
-0.0136 -0.0007153 0.0722972 0.0020993 -0.0490392 -0.0044529 -0.0308664 -0.0135904 -0.0285675 0.1240597 0.0457341 0.0091475 -0.0284298 -0.7542500 0.0422253 0.0397457 -0.0956270 0.0348932 0.0082849 -0.0273712 0.0174307 0.0674288 -0.0383953 0.0293894 0.0121978 -0.0364601 -0.0023032 1.0000000 0.0000000 0.0000000 0.0000000 0
-0.0214 -0.0176364 -0.0249931 0.0406495 0.0259942 0.0743659 0.0276306 0.0342817 -0.0271009 0.1812353 -0.0052850 -0.0213214 -0.0290697 0.1616161 -0.0173219 0.0617800 -0.0249049 -0.0023038 -0.0445760 0.0250868 0.0310336 -0.0563914 -0.0160351 0.0173757 0.0226901 -0.0603520 -0.0025398 -0.1484964 1.0000000 0.0000000 0.0000000 0
-0.0843 0.2570384 0.7758438 -0.0108679 0.1334521 0.0800311 0.1681028 0.0242459 0.1452334 -0.3397512 0.2199497 0.3036154 0.0153324 -0.0146475 -0.0852354 -0.0197259 0.1231578 0.0381507 0.1114221 0.0008355 0.0377498 -0.0015405 0.0027067 0.0055633 0.0383071 -0.0218023 -0.0071260 -0.0017551 -0.0103495 1.0000000 0.0000000 0
0.1510 0.2304130 0.5177308 -0.1174425 0.3244299 0.1233467 0.3412330 0.1057817 -0.0640287 -0.2810171 0.1222469 0.4250203 0.0149850 -0.1553362 0.0995736 0.0829466 0.0368740 0.0254777 0.0457997 -0.0675199 0.0320133 0.0501948 0.0307323 0.0177304 -0.0006020 -0.0299292 -0.0203018 -0.0041501 0.0288096 0.0982348 1.0000000 0
-0.1018 0.0362774 0.2750237 0.1394022 0.4824480 -0.0367047 0.0451801 0.0465278 0.0143917 0.2320922 -0.0363669 0.0110753 0.0392450 -0.0207290 0.0525871 -0.1415324 0.0023231 0.1222306 0.0286219 0.0620933 0.0107716 -0.0854061 0.0065374 0.0051862 0.0335793 -0.0030489 -0.0015874 0.0100509 -0.0136042 -0.1317968 -0.0383005 1 
print("U")
## [1] "U"
U  %>% kable() %>% kable_styling(bootstrap_options = c("striped", "hover", "condensed", "responsive")) %>% scroll_box(width="100%",height="300px")
MSSubClass LotArea OverallQual OverallCond YearBuilt YearRemodAdd BsmtFinSF1 BsmtFinSF2 BsmtUnfSF TotalBsmtSF X1stFlrSF X2ndFlrSF LowQualFinSF GrLivArea BedroomAbvGr KitchenAbvGr TotRmsAbvGrd Fireplaces GarageCars GarageArea WoodDeckSF OpenPorchSF EnclosedPorch X3SsnPorch ScreenPorch PoolArea MiscVal MoSold YrSold SalePrice Baths CentralAirScore
MSSubClass 1 -0.139800 0.0326000 -0.0593000 0.0279000 0.0406000 -0.0698000 -0.0656000 -0.1408000 -0.2385000 -0.2518000 0.3079000 0.0465000 0.0749000 -0.0234000 0.2817000 0.0404000 -0.0456000 -0.0401000 -0.0987000 -0.0126000 -0.0061000 -0.0120000 -0.0438000 -0.0260000 0.0083000 -0.0077000 -0.0136000 -0.0214000 -0.0843000 0.1510000 -0.1018000
LotArea 0 0.980456 0.1103575 -0.0138901 0.0181004 0.0194759 0.2043420 0.1020291 -0.0222838 0.2274577 0.2642984 0.0940444 0.0113007 0.2735710 0.1164287 0.0215817 0.1956479 0.2650251 0.1492940 0.1666017 0.1699385 0.0839472 -0.0199776 0.0142768 0.0395652 0.0788603 0.0370235 -0.0007013 -0.0172917 0.2520149 0.2259098 0.0355684
OverallQual 0 0.000000 0.9865157 -0.0884034 0.5693531 0.5471843 0.2189753 -0.0684456 0.3152983 0.5199731 0.4546600 0.2748771 -0.0331879 0.5597658 0.0893579 -0.1955126 0.4041614 0.3684560 0.5852031 0.5464654 0.2201829 0.2995500 -0.1112602 0.0302209 0.0612942 0.0560531 -0.0353162 0.0713223 -0.0246561 0.7653821 0.5107496 0.2713152
OverallCond 0 0.000000 0.0000000 0.9883647 -0.3230684 0.1254176 -0.0278215 0.0316218 -0.1172107 -0.1354250 -0.1146446 0.0731230 0.0254435 -0.0212212 0.0212693 -0.0875097 -0.0162149 0.0102685 -0.1336218 -0.1060229 0.0180913 -0.0049293 0.0594352 0.0258131 0.0593114 0.0046324 0.0657032 0.0020749 0.0401765 -0.0107415 -0.1160761 0.1377802
YearBuilt 0 0.000000 0.0000000 0.0000000 0.5646917 0.3166038 0.1122026 0.0006853 -0.0669429 0.0495936 -0.0157285 -0.1347660 -0.1578353 -0.1381374 -0.1168158 -0.0988250 -0.1476951 -0.0652130 0.1548440 0.1286377 0.1009524 0.0128283 -0.3029565 0.0233544 -0.0663929 -0.0276235 0.0069901 -0.0276920 0.0146787 0.0753593 0.1832029 0.2724344
YearRemodAdd 0 0.000000 0.0000000 0.0000000 0.0000000 0.5010377 -0.0535607 -0.0335959 0.0647807 -0.0027661 0.0165559 0.0394468 0.0391601 0.0485850 -0.0287305 0.0134903 0.0468651 -0.0599225 0.0248116 0.0105243 0.0218121 0.0521638 0.0310116 0.0136627 -0.0427300 -0.0122944 -0.0033906 -0.0022311 0.0372601 0.0400986 0.0618014 -0.0183905
BsmtFinSF1 0 0.000000 0.0000000 0.0000000 0.0000000 0.0000000 0.8751313 -0.0635880 -0.5535428 0.3289676 0.2739883 -0.1631701 -0.0199791 0.0642052 -0.1323951 -0.0038226 -0.0494671 0.1266372 0.0284133 0.1066710 0.1019111 0.0302763 -0.0090930 0.0113059 0.0486276 0.1165065 0.0032835 -0.0270122 0.0241804 0.1471120 0.2986237 0.0395385
BsmtFinSF2 0 0.000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.9724447 -0.2263878 0.1295511 0.1093285 -0.0812041 0.0149234 0.0144516 -0.0352364 -0.0345847 -0.0246222 0.0468275 -0.0080491 0.0075951 0.0728319 0.0205869 0.0299572 -0.0313788 0.0861809 0.0454535 -0.0040075 -0.0132159 0.0333371 0.0235778 0.1028668 0.0452457
BsmtUnfSF 0 0.000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.4458518 0.4492183 0.3243962 -0.1723816 0.0156913 0.0969596 0.0378343 0.0984344 0.0690493 0.0256788 0.0403119 0.0690909 0.0191259 0.0524455 -0.0007374 0.0092081 0.0205574 0.0330852 -0.0025526 -0.0127369 -0.0120830 0.0647526 -0.0285473 0.0064166
TotalBsmtSF 0 0.000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 -0.0001751 -0.0001521 0.0001451 0.0000378 -0.0000040 0.0000672 -0.0000405 0.0001339 0.0000233 -0.0000435 -0.0000106 -0.0000325 0.0000039 0.0000077 -0.0000732 0.0000617 0.0000470 -0.0000696 -0.0000217 -0.0000317 0.0000595 0.0000492 -0.0000406
X1stFlrSF 0 0.000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.3075593 -0.0881693 0.0001466 0.1529484 0.0669285 0.1434728 0.1410214 0.0955023 0.0699631 0.0746111 0.0352481 -0.0024140 -0.0142195 0.0238398 0.0098568 0.0209730 -0.0074746 0.0140659 -0.0016254 0.0676476 0.0375982 -0.0111850
X2ndFlrSF 0 0.000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.6499002 0.0182736 0.5415145 0.4530914 0.0823519 0.4907989 0.1337972 0.1039877 0.1067772 0.0820215 0.1446331 0.0159224 -0.0082943 0.0379511 0.0941564 0.0190242 0.0059449 -0.0138568 0.1973197 0.2762207 0.0071978
LowQualFinSF 0 0.000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.9470162 0.0875657 0.0608288 -0.0442626 0.0791157 -0.0261811 -0.0335016 -0.0118379 0.0048866 0.0217936 -0.0315677 0.0040058 0.0108762 0.0540621 -0.0048495 -0.0269234 -0.0275295 0.0145200 0.0141911 0.0371657
GrLivArea 0 0.000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0001045 0.0001013 0.0000690 0.0001170 0.0000497 0.0000255 0.0000610 -0.0000810 -0.0000217 -0.0000323 0.0000128 -0.0000458 0.0000015 0.0000241 -0.0000788 0.0000169 -0.0000015 -0.0000162 -0.0000022
BedroomAbvGr 0 0.000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.5923192 0.1064930 0.1979490 -0.0692040 -0.0342694 -0.0562270 -0.0204413 -0.0385035 -0.0145749 -0.0248095 -0.0045188 -0.0072645 -0.0045471 0.0250109 -0.0102601 -0.0504865 0.0589793 0.0311484
KitchenAbvGr 0 0.000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.7344223 0.1155540 -0.1065311 0.0377011 0.0275445 -0.0520312 -0.0304199 -0.0225534 -0.0090950 -0.0475480 -0.0346309 0.0658895 0.0291902 0.0453726 -0.0144871 0.0609178 -0.1039446
TotRmsAbvGrd 0 0.000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.2100747 0.0143963 0.0231728 0.0116282 0.0082674 -0.0047791 -0.0209581 -0.0118381 -0.0082030 -0.0359505 0.0164322 -0.0200888 -0.0052319 0.0258723 0.0077463 0.0004880
Fireplaces 0 0.000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.6693337 0.0120388 -0.0359794 0.0278122 -0.0030680 -0.0136289 -0.0114430 0.1065560 -0.0002408 0.0099652 0.0233552 -0.0015420 0.0255355 0.0170531 0.0818131
GarageCars 0 0.000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.5217029 0.4294423 0.0176883 -0.0113440 0.0097914 0.0054964 0.0259226 -0.0342228 -0.0280129 0.0043223 -0.0232554 0.0581292 0.0238938 0.0149321
GarageArea 0 0.000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.1963045 -0.0002847 0.0281081 0.0100553 -0.0046593 -0.0015136 0.0156053 0.0090647 -0.0053731 0.0049247 0.0001640 -0.0132545 0.0121892
WoodDeckSF 0 0.000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.8628258 -0.0554258 -0.0485569 -0.0496102 -0.1099539 0.0196535 -0.0075837 0.0150397 0.0267766 0.0325715 0.0276219 0.0092940
OpenPorchSF 0 0.000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.8439892 -0.0636640 -0.0218698 0.0355988 0.0036501 -0.0140185 0.0569092 -0.0475937 -0.0013002 0.0423639 -0.0720818
EnclosedPorch 0 0.000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.8040901 -0.0280655 -0.1184048 0.0446137 0.0166034 -0.0308733 -0.0128936 0.0021764 0.0247116 0.0052566
X3SsnPorch 0 0.000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.9850631 -0.0405084 -0.0097899 0.0014551 0.0289504 0.0171162 0.0054802 0.0174656 0.0051088
ScreenPorch 0 0.000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.9086558 0.0281502 0.0353284 0.0110836 0.0206175 0.0348080 -0.0005470 0.0305121
PoolArea 0 0.000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.9352252 0.0282352 -0.0340984 -0.0564427 -0.0203900 -0.0279906 -0.0028514
MiscVal 0 0.000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.9798433 -0.0022567 -0.0024886 -0.0069823 -0.0198926 -0.0015554
MoSold 0 0.000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.9777343 -0.1451900 -0.0017160 -0.0040577 0.0098271
YrSold 0 0.000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.9568609 -0.0099031 0.0275667 -0.0130173
SalePrice 0 0.000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.1927920 0.0189389 -0.0254094
Baths 0 0.000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.3215890 -0.0123170
CentralAirScore 0 0.000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.7175071

We can confirm the decomposition by comparing to our original Correlation Matrix -

round(L %*% U ,4)== round(corrMatrix,4)
##       MSSubClass LotArea OverallQual OverallCond YearBuilt YearRemodAdd
##  [1,]       TRUE    TRUE        TRUE        TRUE      TRUE         TRUE
##  [2,]       TRUE    TRUE        TRUE        TRUE      TRUE         TRUE
##  [3,]       TRUE    TRUE        TRUE        TRUE      TRUE         TRUE
##  [4,]       TRUE    TRUE        TRUE        TRUE      TRUE         TRUE
##  [5,]       TRUE    TRUE        TRUE        TRUE      TRUE         TRUE
##  [6,]       TRUE    TRUE        TRUE        TRUE      TRUE         TRUE
##  [7,]       TRUE    TRUE        TRUE        TRUE      TRUE         TRUE
##  [8,]       TRUE    TRUE        TRUE        TRUE      TRUE         TRUE
##  [9,]       TRUE    TRUE        TRUE        TRUE      TRUE         TRUE
## [10,]       TRUE    TRUE        TRUE        TRUE      TRUE         TRUE
## [11,]       TRUE    TRUE        TRUE        TRUE      TRUE         TRUE
## [12,]       TRUE    TRUE        TRUE        TRUE      TRUE         TRUE
## [13,]       TRUE    TRUE        TRUE        TRUE      TRUE         TRUE
## [14,]       TRUE    TRUE        TRUE        TRUE      TRUE         TRUE
## [15,]       TRUE    TRUE        TRUE        TRUE      TRUE         TRUE
## [16,]       TRUE    TRUE        TRUE        TRUE      TRUE         TRUE
## [17,]       TRUE    TRUE        TRUE        TRUE      TRUE         TRUE
## [18,]       TRUE    TRUE        TRUE        TRUE      TRUE         TRUE
## [19,]       TRUE    TRUE        TRUE        TRUE      TRUE         TRUE
## [20,]       TRUE    TRUE        TRUE        TRUE      TRUE         TRUE
## [21,]       TRUE    TRUE        TRUE        TRUE      TRUE         TRUE
## [22,]       TRUE    TRUE        TRUE        TRUE      TRUE         TRUE
## [23,]       TRUE    TRUE        TRUE        TRUE      TRUE         TRUE
## [24,]       TRUE    TRUE        TRUE        TRUE      TRUE         TRUE
## [25,]       TRUE    TRUE        TRUE        TRUE      TRUE         TRUE
## [26,]       TRUE    TRUE        TRUE        TRUE      TRUE         TRUE
## [27,]       TRUE    TRUE        TRUE        TRUE      TRUE         TRUE
## [28,]       TRUE    TRUE        TRUE        TRUE      TRUE         TRUE
## [29,]       TRUE    TRUE        TRUE        TRUE      TRUE         TRUE
## [30,]       TRUE    TRUE        TRUE        TRUE      TRUE         TRUE
## [31,]       TRUE    TRUE        TRUE        TRUE      TRUE         TRUE
## [32,]       TRUE    TRUE        TRUE        TRUE      TRUE         TRUE
##       BsmtFinSF1 BsmtFinSF2 BsmtUnfSF TotalBsmtSF X1stFlrSF X2ndFlrSF
##  [1,]       TRUE       TRUE      TRUE        TRUE      TRUE      TRUE
##  [2,]       TRUE       TRUE      TRUE        TRUE      TRUE      TRUE
##  [3,]       TRUE       TRUE      TRUE        TRUE      TRUE      TRUE
##  [4,]       TRUE       TRUE      TRUE        TRUE      TRUE      TRUE
##  [5,]       TRUE       TRUE      TRUE        TRUE      TRUE      TRUE
##  [6,]       TRUE       TRUE      TRUE        TRUE      TRUE      TRUE
##  [7,]       TRUE       TRUE      TRUE        TRUE      TRUE      TRUE
##  [8,]       TRUE       TRUE      TRUE        TRUE      TRUE      TRUE
##  [9,]       TRUE       TRUE      TRUE        TRUE      TRUE      TRUE
## [10,]       TRUE       TRUE      TRUE        TRUE      TRUE      TRUE
## [11,]       TRUE       TRUE      TRUE        TRUE      TRUE      TRUE
## [12,]       TRUE       TRUE      TRUE        TRUE      TRUE      TRUE
## [13,]       TRUE       TRUE      TRUE        TRUE      TRUE      TRUE
## [14,]       TRUE       TRUE      TRUE        TRUE      TRUE      TRUE
## [15,]       TRUE       TRUE      TRUE        TRUE      TRUE      TRUE
## [16,]       TRUE       TRUE      TRUE        TRUE      TRUE      TRUE
## [17,]       TRUE       TRUE      TRUE        TRUE      TRUE      TRUE
## [18,]       TRUE       TRUE      TRUE        TRUE      TRUE      TRUE
## [19,]       TRUE       TRUE      TRUE        TRUE      TRUE      TRUE
## [20,]       TRUE       TRUE      TRUE        TRUE      TRUE      TRUE
## [21,]       TRUE       TRUE      TRUE        TRUE      TRUE      TRUE
## [22,]       TRUE       TRUE      TRUE        TRUE      TRUE      TRUE
## [23,]       TRUE       TRUE      TRUE        TRUE      TRUE      TRUE
## [24,]       TRUE       TRUE      TRUE        TRUE      TRUE      TRUE
## [25,]       TRUE       TRUE      TRUE        TRUE      TRUE      TRUE
## [26,]       TRUE       TRUE      TRUE        TRUE      TRUE      TRUE
## [27,]       TRUE       TRUE      TRUE        TRUE      TRUE      TRUE
## [28,]       TRUE       TRUE      TRUE        TRUE      TRUE      TRUE
## [29,]       TRUE       TRUE      TRUE        TRUE      TRUE      TRUE
## [30,]       TRUE       TRUE      TRUE        TRUE      TRUE      TRUE
## [31,]       TRUE       TRUE      TRUE        TRUE      TRUE      TRUE
## [32,]       TRUE       TRUE      TRUE        TRUE      TRUE      TRUE
##       LowQualFinSF GrLivArea BedroomAbvGr KitchenAbvGr TotRmsAbvGrd
##  [1,]         TRUE      TRUE         TRUE         TRUE         TRUE
##  [2,]         TRUE      TRUE         TRUE         TRUE         TRUE
##  [3,]         TRUE      TRUE         TRUE         TRUE         TRUE
##  [4,]         TRUE      TRUE         TRUE         TRUE         TRUE
##  [5,]         TRUE      TRUE         TRUE         TRUE         TRUE
##  [6,]         TRUE      TRUE         TRUE         TRUE         TRUE
##  [7,]         TRUE      TRUE         TRUE         TRUE         TRUE
##  [8,]         TRUE      TRUE         TRUE         TRUE         TRUE
##  [9,]         TRUE      TRUE         TRUE         TRUE         TRUE
## [10,]         TRUE      TRUE         TRUE         TRUE         TRUE
## [11,]         TRUE      TRUE         TRUE         TRUE         TRUE
## [12,]         TRUE      TRUE         TRUE         TRUE         TRUE
## [13,]         TRUE      TRUE         TRUE         TRUE         TRUE
## [14,]         TRUE      TRUE         TRUE         TRUE         TRUE
## [15,]         TRUE      TRUE         TRUE         TRUE         TRUE
## [16,]         TRUE      TRUE         TRUE         TRUE         TRUE
## [17,]         TRUE      TRUE         TRUE         TRUE         TRUE
## [18,]         TRUE      TRUE         TRUE         TRUE         TRUE
## [19,]         TRUE      TRUE         TRUE         TRUE         TRUE
## [20,]         TRUE      TRUE         TRUE         TRUE         TRUE
## [21,]         TRUE      TRUE         TRUE         TRUE         TRUE
## [22,]         TRUE      TRUE         TRUE         TRUE         TRUE
## [23,]         TRUE      TRUE         TRUE         TRUE         TRUE
## [24,]         TRUE      TRUE         TRUE         TRUE         TRUE
## [25,]         TRUE      TRUE         TRUE         TRUE         TRUE
## [26,]         TRUE      TRUE         TRUE         TRUE         TRUE
## [27,]         TRUE      TRUE         TRUE         TRUE         TRUE
## [28,]         TRUE      TRUE         TRUE         TRUE         TRUE
## [29,]         TRUE      TRUE         TRUE         TRUE         TRUE
## [30,]         TRUE      TRUE         TRUE         TRUE         TRUE
## [31,]         TRUE      TRUE         TRUE         TRUE         TRUE
## [32,]         TRUE      TRUE         TRUE         TRUE         TRUE
##       Fireplaces GarageCars GarageArea WoodDeckSF OpenPorchSF
##  [1,]       TRUE       TRUE       TRUE       TRUE        TRUE
##  [2,]       TRUE       TRUE       TRUE       TRUE        TRUE
##  [3,]       TRUE       TRUE       TRUE       TRUE        TRUE
##  [4,]       TRUE       TRUE       TRUE       TRUE        TRUE
##  [5,]       TRUE       TRUE       TRUE       TRUE        TRUE
##  [6,]       TRUE       TRUE       TRUE       TRUE        TRUE
##  [7,]       TRUE       TRUE       TRUE       TRUE        TRUE
##  [8,]       TRUE       TRUE       TRUE       TRUE        TRUE
##  [9,]       TRUE       TRUE       TRUE       TRUE        TRUE
## [10,]       TRUE       TRUE       TRUE       TRUE        TRUE
## [11,]       TRUE       TRUE       TRUE       TRUE        TRUE
## [12,]       TRUE       TRUE       TRUE       TRUE        TRUE
## [13,]       TRUE       TRUE       TRUE       TRUE        TRUE
## [14,]       TRUE       TRUE       TRUE       TRUE        TRUE
## [15,]       TRUE       TRUE       TRUE       TRUE        TRUE
## [16,]       TRUE       TRUE       TRUE       TRUE        TRUE
## [17,]       TRUE       TRUE       TRUE       TRUE        TRUE
## [18,]       TRUE       TRUE       TRUE       TRUE        TRUE
## [19,]       TRUE       TRUE       TRUE       TRUE        TRUE
## [20,]       TRUE       TRUE       TRUE       TRUE        TRUE
## [21,]       TRUE       TRUE       TRUE       TRUE        TRUE
## [22,]       TRUE       TRUE       TRUE       TRUE        TRUE
## [23,]       TRUE       TRUE       TRUE       TRUE        TRUE
## [24,]       TRUE       TRUE       TRUE       TRUE        TRUE
## [25,]       TRUE       TRUE       TRUE       TRUE        TRUE
## [26,]       TRUE       TRUE       TRUE       TRUE        TRUE
## [27,]       TRUE       TRUE       TRUE       TRUE        TRUE
## [28,]       TRUE       TRUE       TRUE       TRUE        TRUE
## [29,]       TRUE       TRUE       TRUE       TRUE        TRUE
## [30,]       TRUE       TRUE       TRUE       TRUE        TRUE
## [31,]       TRUE       TRUE       TRUE       TRUE        TRUE
## [32,]       TRUE       TRUE       TRUE       TRUE        TRUE
##       EnclosedPorch X3SsnPorch ScreenPorch PoolArea MiscVal MoSold YrSold
##  [1,]          TRUE       TRUE        TRUE     TRUE    TRUE   TRUE   TRUE
##  [2,]          TRUE       TRUE        TRUE     TRUE    TRUE   TRUE   TRUE
##  [3,]          TRUE       TRUE        TRUE     TRUE    TRUE   TRUE   TRUE
##  [4,]          TRUE       TRUE        TRUE     TRUE    TRUE   TRUE   TRUE
##  [5,]          TRUE       TRUE        TRUE     TRUE    TRUE   TRUE   TRUE
##  [6,]          TRUE       TRUE        TRUE     TRUE    TRUE   TRUE   TRUE
##  [7,]          TRUE       TRUE        TRUE     TRUE    TRUE   TRUE   TRUE
##  [8,]          TRUE       TRUE        TRUE     TRUE    TRUE   TRUE   TRUE
##  [9,]          TRUE       TRUE        TRUE     TRUE    TRUE   TRUE   TRUE
## [10,]          TRUE       TRUE        TRUE     TRUE    TRUE   TRUE   TRUE
## [11,]          TRUE       TRUE        TRUE     TRUE    TRUE   TRUE   TRUE
## [12,]          TRUE       TRUE        TRUE     TRUE    TRUE   TRUE   TRUE
## [13,]          TRUE       TRUE        TRUE     TRUE    TRUE   TRUE   TRUE
## [14,]          TRUE       TRUE        TRUE     TRUE    TRUE   TRUE   TRUE
## [15,]          TRUE       TRUE        TRUE     TRUE    TRUE   TRUE   TRUE
## [16,]          TRUE       TRUE        TRUE     TRUE    TRUE   TRUE   TRUE
## [17,]          TRUE       TRUE        TRUE     TRUE    TRUE   TRUE   TRUE
## [18,]          TRUE       TRUE        TRUE     TRUE    TRUE   TRUE   TRUE
## [19,]          TRUE       TRUE        TRUE     TRUE    TRUE   TRUE   TRUE
## [20,]          TRUE       TRUE        TRUE     TRUE    TRUE   TRUE   TRUE
## [21,]          TRUE       TRUE        TRUE     TRUE    TRUE   TRUE   TRUE
## [22,]          TRUE       TRUE        TRUE     TRUE    TRUE   TRUE   TRUE
## [23,]          TRUE       TRUE        TRUE     TRUE    TRUE   TRUE   TRUE
## [24,]          TRUE       TRUE        TRUE     TRUE    TRUE   TRUE   TRUE
## [25,]          TRUE       TRUE        TRUE     TRUE    TRUE   TRUE   TRUE
## [26,]          TRUE       TRUE        TRUE     TRUE    TRUE   TRUE   TRUE
## [27,]          TRUE       TRUE        TRUE     TRUE    TRUE   TRUE   TRUE
## [28,]          TRUE       TRUE        TRUE     TRUE    TRUE   TRUE   TRUE
## [29,]          TRUE       TRUE        TRUE     TRUE    TRUE   TRUE   TRUE
## [30,]          TRUE       TRUE        TRUE     TRUE    TRUE   TRUE   TRUE
## [31,]          TRUE       TRUE        TRUE     TRUE    TRUE   TRUE   TRUE
## [32,]          TRUE       TRUE        TRUE     TRUE    TRUE   TRUE   TRUE
##       SalePrice Baths CentralAirScore
##  [1,]      TRUE  TRUE            TRUE
##  [2,]      TRUE  TRUE            TRUE
##  [3,]      TRUE  TRUE            TRUE
##  [4,]      TRUE  TRUE            TRUE
##  [5,]      TRUE  TRUE            TRUE
##  [6,]      TRUE  TRUE            TRUE
##  [7,]      TRUE  TRUE            TRUE
##  [8,]      TRUE  TRUE            TRUE
##  [9,]      TRUE  TRUE            TRUE
## [10,]      TRUE  TRUE            TRUE
## [11,]      TRUE  TRUE            TRUE
## [12,]      TRUE  TRUE            TRUE
## [13,]      TRUE  TRUE            TRUE
## [14,]      TRUE  TRUE            TRUE
## [15,]      TRUE  TRUE            TRUE
## [16,]      TRUE  TRUE            TRUE
## [17,]      TRUE  TRUE            TRUE
## [18,]      TRUE  TRUE            TRUE
## [19,]      TRUE  TRUE            TRUE
## [20,]      TRUE  TRUE            TRUE
## [21,]      TRUE  TRUE            TRUE
## [22,]      TRUE  TRUE            TRUE
## [23,]      TRUE  TRUE            TRUE
## [24,]      TRUE  TRUE            TRUE
## [25,]      TRUE  TRUE            TRUE
## [26,]      TRUE  TRUE            TRUE
## [27,]      TRUE  TRUE            TRUE
## [28,]      TRUE  TRUE            TRUE
## [29,]      TRUE  TRUE            TRUE
## [30,]      TRUE  TRUE            TRUE
## [31,]      TRUE  TRUE            TRUE
## [32,]      TRUE  TRUE            TRUE

Calculus-Based Probability & Statistics.

Many times, it makes sense to fit a closed form distribution to data. Select a variable in the Kaggle.com training dataset that is skewed to the right, shift it so that the minimum value is absolutely above zero if necessary. Then load the MASS package and run fitdistr to fit an exponential probability density function. (See https://stat.ethz.ch/R-manual/R-devel/library/MASS/html/fitdistr.html ). Find the optimal value of \(\lambda\) for this distribution, and then take 1000 samples from this exponential distribution using this value (e.g., rexp(1000, \(\lambda\))). Plot a histogram and compare it with a histogram of your original variable. Using the exponential pdf, find the 5th and 95th percentiles using the cumulative distribution function (CDF). Also generate a 95% confidence interval from the empirical data, assuming normality. Finally, provide the empirical 5th percentile and 95th percentile of the data. Discuss.

Ans: For this analysis, I have chosen the GarageArea variable in the Kaggle data set.

Dist <- housing_pr_train$GarageArea

min(Dist)
## [1] 0

Then run fitdistr to fit an exponential probability density function.

fitDist <- fitdistr(Dist, "exponential")
fitDist
##        rate    
##   2.114254e-03 
##  (5.533255e-05)

Find the optimal value of \(\lambda\) for this distribution, and then take 1000 samples from this exponential distribution using this value.

lambda <- fitDist$estimate
sim <- rexp(1000,lambda)
hist(sim,breaks = 100, col="blue", main="Histogram of 1000 samples with Simulation", xlab="Exponential of Lot Area (Square Feet)")

hist(Dist,breaks=100, col="grey", main="Histogram of Original Data", xlab="Garage Area (Square Feet)")

sim.df <- data.frame(length = sim)
Dist.df <- data.frame(length = Dist)

sim.df$from <- 'sim'
Dist.df$from <- 'orig'

both.df <- rbind(sim.df,Dist.df)

ggplotly(ggplot(both.df, aes(length, fill = from)) + geom_density(alpha = 0.2) + scale_fill_brewer(palette="Set1") +
           ggtitle("Density plot for Original Vs. Simulated Distributions"))

Using the exponential pdf, find the 5th and 95th percentiles using the cumulative distribution function (CDF).

quantile(sim, probs=c(0.05, 0.95))
##         5%        95% 
##   24.30699 1437.18642

Also generate a 95% confidence interval from the empirical data, assuming normality.

mean(Dist)
## [1] 472.9801
normal<-rnorm(length(Dist),mean(Dist),sd(Dist))
hist(normal,breaks = 100, col="orange", main="Histogram of Normalized Distribution", xlab="Garage Area (Square Feet)")

quantile(normal, probs=c(0.05, 0.95))
##       5%      95% 
## 120.6115 811.4731
normal.df <- data.frame(length = normal)
normal.df$from <- 'normal'

all.df <- rbind(both.df,normal.df)

ggplotly(ggplot(all.df, aes(length, fill = from)) + geom_density(alpha = 0.2)+ scale_fill_brewer(palette="Set1") +
           ggtitle("Density plot for Original Vs. Simulated Vs. Normal Distributions"))

Finally, provide the empirical 5th percentile and 95th percentile of the data. Discuss.

quantile(Dist, probs=c(0.05, 0.95))
##    5%   95% 
##   0.0 850.1

From this analysis it appears the data select was not very right skew. The exponential simulation does not match our data very well, rather, our selected empirical data matches the normal distribution a lot better. This can be seen in the final density plot, but also on the confidence interval where the limits are much closer than for the exponential approximation.

Modeling

Build some type of multiple regression model and submit your model to the competition board. Provide your complete model summary and results with analysis. Report your Kaggle.com user name and score.

Ans: Based on the Correlation Scores from the previous section, I have implemented a Linear Regresion model with following predictor variables -

  • YearBuilt: Original construction date
  • YearRemodAdd: Remodel date (same as construction date if no remodeling or additions)
  • TotalBsmtSF: Total square feet of basement area
  • OverallQual: Rates the overall material and finish of the house
  • OverallCond: Rates the overall condition of the house
  • GrLivArea: Above grade (ground) living area square feet
  • Baths: Derived attribute from Half and Full Baths at basement and above ground level
  • GarageCars: Size of garage in car capacity
  • GarageArea: Size of garage in square feet
  • BedroomAbvGr: Bedrooms above grade (does NOT include basement bedrooms)
  • KitchenAbvGr: Kitchens above grade
  • TotRmsAbvGrd: Total rooms above grade (does not include bathrooms)
  • CentralAirScore: Does not have A/C = 0, does have A/C = 1
  • LogLotArea: Log transformation of Lot size in square feet

I have used log transformtaion for the Lot Area and SalePrice variables.

housing_pr_train$LogLotArea <- log(housing_pr_train$LotArea)
housing_pr_train$LogSalePrice <- log(housing_pr_train$SalePrice)

model_train_df <- subset(housing_pr_train, select = c(YearBuilt,YearRemodAdd,TotalBsmtSF,OverallQual,OverallCond,GrLivArea,Baths,
                                GarageCars,GarageArea,BedroomAbvGr,KitchenAbvGr,TotRmsAbvGrd,CentralAirScore,LogLotArea, LogSalePrice))

Linear Model Iterations

Next, I created a linear model with all 14 variables and then found the optimized model with the final list of predictors using Stepwise regreesion technique.

Initial Regression Model
## Initial Model
lm_full_model <- lm(LogSalePrice ~ YearBuilt + YearRemodAdd + TotalBsmtSF + OverallQual + OverallCond + GrLivArea + Baths + GarageCars +
                 GarageArea + BedroomAbvGr + KitchenAbvGr + TotRmsAbvGrd + CentralAirScore + LogLotArea, data = model_train_df)

## Model Summary
summary(lm_full_model)
## 
## Call:
## lm(formula = LogSalePrice ~ YearBuilt + YearRemodAdd + TotalBsmtSF + 
##     OverallQual + OverallCond + GrLivArea + Baths + GarageCars + 
##     GarageArea + BedroomAbvGr + KitchenAbvGr + TotRmsAbvGrd + 
##     CentralAirScore + LogLotArea, data = model_train_df)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -2.05965 -0.06474  0.00281  0.07658  0.48775 
## 
## Coefficients:
##                   Estimate Std. Error t value Pr(>|t|)    
## (Intercept)      3.460e+00  5.401e-01   6.407 2.00e-10 ***
## YearBuilt        2.341e-03  2.375e-04   9.859  < 2e-16 ***
## YearRemodAdd     7.635e-04  2.689e-04   2.839  0.00459 ** 
## TotalBsmtSF      9.510e-05  1.165e-05   8.162 7.11e-16 ***
## OverallQual      8.804e-02  4.826e-03  18.245  < 2e-16 ***
## OverallCond      4.669e-02  4.354e-03  10.724  < 2e-16 ***
## GrLivArea        1.647e-04  1.657e-05   9.941  < 2e-16 ***
## Baths            6.306e-02  7.210e-03   8.747  < 2e-16 ***
## GarageCars       7.733e-02  1.181e-02   6.550 7.98e-11 ***
## GarageArea      -2.941e-05  4.020e-05  -0.731  0.46460    
## BedroomAbvGr    -1.937e-02  6.895e-03  -2.810  0.00503 ** 
## KitchenAbvGr    -9.241e-02  1.989e-02  -4.647 3.67e-06 ***
## TotRmsAbvGrd     1.505e-02  5.149e-03   2.922  0.00353 ** 
## CentralAirScore  8.754e-02  1.811e-02   4.833 1.49e-06 ***
## LogLotArea       1.093e-01  8.752e-03  12.487  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.1472 on 1445 degrees of freedom
## Multiple R-squared:  0.8655, Adjusted R-squared:  0.8642 
## F-statistic: 664.4 on 14 and 1445 DF,  p-value: < 2.2e-16
Stepwise Regression Model

I have used the stepAIC() function from MASS package, which choose the best model by AIC. I have used a combined forward and backward elimination process using direcition parameter as ‘both’.

train_model <- stepAIC(lm_full_model, direction = "both", 
                      trace = FALSE)

summary(train_model)
## 
## Call:
## lm(formula = LogSalePrice ~ YearBuilt + YearRemodAdd + TotalBsmtSF + 
##     OverallQual + OverallCond + GrLivArea + Baths + GarageCars + 
##     BedroomAbvGr + KitchenAbvGr + TotRmsAbvGrd + CentralAirScore + 
##     LogLotArea, data = model_train_df)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -2.07392 -0.06451  0.00265  0.07630  0.48977 
## 
## Coefficients:
##                   Estimate Std. Error t value Pr(>|t|)    
## (Intercept)      3.460e+00  5.400e-01   6.407 2.00e-10 ***
## YearBuilt        2.337e-03  2.374e-04   9.845  < 2e-16 ***
## YearRemodAdd     7.713e-04  2.687e-04   2.871  0.00415 ** 
## TotalBsmtSF      9.372e-05  1.150e-05   8.152 7.66e-16 ***
## OverallQual      8.815e-02  4.823e-03  18.277  < 2e-16 ***
## OverallCond      4.654e-02  4.348e-03  10.703  < 2e-16 ***
## GrLivArea        1.635e-04  1.648e-05   9.920  < 2e-16 ***
## Baths            6.318e-02  7.207e-03   8.767  < 2e-16 ***
## GarageCars       7.046e-02  7.161e-03   9.839  < 2e-16 ***
## BedroomAbvGr    -1.900e-02  6.875e-03  -2.764  0.00579 ** 
## KitchenAbvGr    -9.224e-02  1.988e-02  -4.639 3.81e-06 ***
## TotRmsAbvGrd     1.519e-02  5.145e-03   2.953  0.00320 ** 
## CentralAirScore  8.721e-02  1.810e-02   4.817 1.61e-06 ***
## LogLotArea       1.086e-01  8.695e-03  12.486  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.1472 on 1446 degrees of freedom
## Multiple R-squared:  0.8655, Adjusted R-squared:  0.8643 
## F-statistic: 715.7 on 13 and 1446 DF,  p-value: < 2.2e-16

Now all predictors have p vlaues less than 0.05.

train_model$coefficients
##     (Intercept)       YearBuilt    YearRemodAdd     TotalBsmtSF 
##    3.459613e+00    2.336921e-03    7.713103e-04    9.372069e-05 
##     OverallQual     OverallCond       GrLivArea           Baths 
##    8.814626e-02    4.654093e-02    1.634841e-04    6.318413e-02 
##      GarageCars    BedroomAbvGr    KitchenAbvGr    TotRmsAbvGrd 
##    7.046074e-02   -1.900090e-02   -9.223818e-02    1.519203e-02 
## CentralAirScore      LogLotArea 
##    8.720929e-02    1.085608e-01
Model Equation
## Build Function for Regression Model
train_model_equation <- function(YearBuilt,YearRemodAdd,TotalBsmtSF,OverallQual,OverallCond,GrLivArea,Baths,
                                GarageCars,BedroomAbvGr,KitchenAbvGr,TotRmsAbvGrd,CentralAirScore,LogLotArea) { 
  (3.459613 + (0.002336921 * YearBuilt) + (0.0007713103 * YearRemodAdd) + (0.00009372069 * TotalBsmtSF) + (0.08814626 * OverallQual) + (0.04654093 * OverallCond) + (0.0001634841 * GrLivArea) + (0.06318413 * Baths) + (0.07046074 * GarageCars) + (-0.01900090 * BedroomAbvGr) + (-0.09223818 * KitchenAbvGr) + (0.01519203 * TotRmsAbvGrd) + (0.08720929 * CentralAirScore) + (0.01085608 * LogLotArea)) 
  }

The linear regression equation is as follows:

logSalePrice = 3.459613 + (0.002336921 * YearBuilt) + (0.0007713103 * YearRemodAdd) + (0.00009372069 * TotalBsmtSF) + (0.08814626 * OverallQual) + (0.04654093 * OverallCond) + (0.0001634841 * GrLivArea) + (0.06318413 * Baths) + (0.07046074 * GarageCars) + (-0.01900090 * BedroomAbvGr) + (-0.09223818 * KitchenAbvGr) + (0.01519203 * TotRmsAbvGrd) + (0.08720929 * CentralAirScore) + (0.01085608 * LogLotArea)

Model Interpretation

Having constructed a model where all p-values are below 0.05, I looked the residual plot, Q-Q plot and other relevant plots using ggresidpanel package:

resid_panel(train_model, plots='all', smoother = TRUE)

# Residuals plot
resid_interact(train_model, plots='resid', smoother = TRUE)
# Q-Q plot
resid_interact(train_model, plots='qq', smoother = TRUE)

The residual plot is acceptable with few outliers but he Q-Q plot is heavily tailed. This is not necessarily a surprise because during step wise regression model selection process, as the model was evaluated, the residuals, though fairly centered in terms of mean and quartiles, exhibited skew on the minimum end. We can see that in the residual plot in that there are two huge outliers and there appear to be sligtly more plots below zero than above zero.

Test Model

Next, I read in the test data and used the linear regression model to predict logSalePrice, which was then tranformed back to SalePrice:

** Load Test Data**
housing_pr_test <- read.csv('https://raw.githubusercontent.com/soumya2g/R-CUNY-MSDS/master/DATA-605/House%20Price%20Analysis/Data%20Source/house-prices-advanced-regression-techniques/test.csv',sep = ',')
Apply Feature Engineering
# Consolidate no. of baths
housing_pr_test$Baths <- housing_pr_test$BsmtFullBath + 0.5*housing_pr_test$BsmtHalfBath + housing_pr_test$FullBath + 0.5*housing_pr_test$HalfBath

# Derive CentralAirScore as discrete variable
housing_pr_test$CentralAirScore <- ifelse(housing_pr_test$CentralAir == "N", 0, 1)

# Calculate log of LotArea
housing_pr_test$LogLotArea <- log(housing_pr_test$LotArea)
Model Prediction
# predict logSalePrice using the test data and linear regression model
lm_predicted <- train_model_equation(housing_pr_test$YearBuilt,housing_pr_test$YearRemodAdd,housing_pr_test$TotalBsmtSF,
                                     housing_pr_test$OverallQual,housing_pr_test$OverallCond,housing_pr_test$GrLivArea,
                                     housing_pr_test$Baths,housing_pr_test$GarageCars,housing_pr_test$BedroomAbvGr,
                                     housing_pr_test$KitchenAbvGr,housing_pr_test$TotRmsAbvGrd,housing_pr_test$CentralAirScore,
                                     housing_pr_test$LogLotArea)

# compute mean
lm_predicted_mean <- mean(lm_predicted, na.rm = TRUE)

# replace NA values with mean
lm_predicted <- ifelse(is.na(lm_predicted) == TRUE, lm_predicted_mean, lm_predicted)

# transform logSalePrice prediction to SalePrice
lm_predicted <- (exp(1))^lm_predicted

# put transformed SalePrice into data frame
lm_predicted_df <- data.frame(cbind(seq(from = 1461, to = 2919, by = 1), lm_predicted))

# change column names to Id and SalePrice
colnames(lm_predicted_df) <- c("Id","SalePrice")

# export to .csv for submission
write.csv(lm_predicted_df, file = "C:/CUNY/Semester3 (Fall)/DATA 605/Assignments/Final Project/Output/submission.csv",row.names = FALSE)

Kaggle Submission Details

My Kaggle user name is soumya2g. Below is a screenshot of the submission.

Kaggle Score: 0.90655

Kaggle Leaderboard Screenshot