Prologis Analytics Internship
Twinkle Roy
Questions to be addressed in this analysis:
- Please develop a Multi-Variable linear regression model that predicts the average monthly rental price of a property.
Done below in the R markdown:
- What are the key assumptions for Linear Regression?
Assumptions are:
- Linearity in Parameters
- Homoscedasticity - the variance of the residual is the same for any value
- Normality - Errors are normally distributed
- No serial autocorrelation : Independence of observations (E(x’e) = 0) This condition is called exogeneity.
- No Multicollinearity /independence between x’s Full rank = There is no exact linear relationship among any of the independent variables in the model. This assumption will be necessary for estimation of the parameters of the model.
- Do any features require transformation?
Yes, The histogram for rent per sq. feet shows positive skewness, we can adjust it by log transformation, and normalized the distribution. We also removed outliers from the model and check hierarchial clustering the relation between relative important factors. Correlation matrix was used to gain insights about quantitative variables.
- Please share how you determined the accuracy and the model’s goodness of fit.
The following steps were used to identify the model
The model was divided into training and testing set: 80:20 ratio. The model was assessed for accuracy using RMSE (root mean square error) - tells us roughly how much average error the model makes.
We look at two measures that assess how well a model is predicting, the train RMSE and the test RMSE. We achieved as can be seen below under Final model, lower RMSE under test. The best fit model was chosen on the basis of stepwise AIC criterion. R2 was also one of the measures used.
- Are there any violation of the Linear Regression? Which metrics are used to test this?
Yes, There was violation of heteroscedasticity, which meant that The least squares estimator is still a linear and unbiased estimator, but it is no longer best. That is, there is another estimator with a smaller variance.The standard errors computed for the least squares estimators are incorrect. This can affect confidence intervals and hypothesis testing that use those standard errors, which could lead to misleading conclusions.
So we did a Breusch Pagan Test on our model, we then used heteroskedasticity-consistent standard errors or simply robust standard errors for our final model. However, we also found that the linear model is not the correct one for the data, then the slope and intercept estimates and the fitted values from the linear regression is biased, and the fitted slope and intercept estimates will not be meaningful. The model needs to be fitted using a polynomial regression as some of the X variables require higher degrees of polynomial order.
The strict exogeneity is violated when means X and errors are correlated. This causes inconsistency along with biasedness in estimates. In cases where exogeneity is violated, there could be multiple approaches to resolve it: 1. Instrument variable approach: This can help with endogeneity arising due to ommitted regressers or measurement error. 2. Using Newey West cov matrix to address the issue of serial correlation of errors.
- In case of violation, what strategy/strategies do you suggest can improve the model? Show with an example.
As presented under the subheading Improved model, we fit a our final model used uses robust standard errors with polynomial (degree = k) (variable degree for two different variables = Property per Sq and Lease term) for the analysis. We conclude that with lower RMSE and From the output below, the R2 is 0.97, meaning that the observed and the predicted outcome values are highly correlated, which is good. The prediction error RMSE is 22.29, representing an error rate of 1.58% which is good.
Also some of the codes has been inputted as annotations because R-markdown couldn’t knit them. They are working under the R console and produces interesting outputs.
Data Preprocessing
We can observe from the dataset that Monthly rent is our dependent variable for each id associated with Prologis. There are some aspects of the data that would be hard for doing calculations or analysis. Like in Col: Property Office Area has a couple of 0s in place so we need to identify if they are outliers/missing data or not.
Thus, we need to perform several data preprocessing tasks:
- Cleaning the missing values
- Convert string values to numerical values that allows calculations
#load data
library(corrplot)## corrplot 0.90 loadedlibrary(skimr)
library(dplyr); library(tidyr)##
## Attaching package: 'dplyr'## The following objects are masked from 'package:stats':
##
## filter, lag## The following objects are masked from 'package:base':
##
## intersect, setdiff, setequal, unionlibrary(ClustOfVar)
library(readxl)
library(ggplot2)
library(hrbrthemes)## Warning: package 'hrbrthemes' was built under R version 4.1.2## NOTE: Either Arial Narrow or Roboto Condensed fonts are required to use these themes.## Please use hrbrthemes::import_roboto_condensed() to install Roboto Condensed and## if Arial Narrow is not on your system, please see https://bit.ly/arialnarrowlibrary(MASS)##
## Attaching package: 'MASS'## The following object is masked from 'package:dplyr':
##
## selectprologis_data <- read_excel("C:/Users/Twinkle/Downloads/2022 Intern - Advanced Analytics Take Home Assessment (1).xlsx", sheet = "IE_Model_Population", skip = 2)
str(prologis_data) # sanity check## tibble [84 x 10] (S3: tbl_df/tbl/data.frame)
## $ Lease ID : num [1:84] 53884 56140 64052 54897 46043 ...
## $ Monthly Rent ($ per Sq. Ft.): num [1:84] 6.35 6.73 6.67 5.88 6.36 ...
## $ Transaction Type : chr [1:84] "Renewal" "Renewal" "Original Lease" "Renewal" ...
## $ Customer Industry : chr [1:84] "Wholesaler" "Wholesaler" "Retailer" "Wholesaler" ...
## $ Lease - Total Term (Month) : num [1:84] 24 24 24 24 31 33 36 36 36 36 ...
## $ Lease - Free Rent (Month) : num [1:84] 0 0 0 0 2 2 0 1 0 0 ...
## $ Property - Size (Sq. Ft.) : num [1:84] 656040 445200 172998 395954 441970 ...
## $ Property - Speed Bay : num [1:84] 50 60 60 50 58 60 42 60 48 48 ...
## $ Property - Year Built : num [1:84] 2008 1996 1999 1998 2000 ...
## $ Property - Office Area : num [1:84] 33000 4150 12147 5520 8453 ...prologis_data <- na.omit( data.frame(prologis_data)) # remove NA valuesWe can observe that some columns factor, they are actually character like Transaction type and Customer Industry. So we need to change those.
#typecast categorical variables as factors, but also keep the non factor versions for creating interaction terms
prologis_data$Transaction.Type = as.factor(prologis_data$Transaction.Type)
prologis_data$Customer.Industry = as.factor(prologis_data$Customer.Industry)
prologis_data$Property...Year.Built = as.factor(prologis_data$Property...Year.Built)
##### Rent transformation ####
prologis_data$totalrent = prologis_data$Monthly.Rent....per.Sq..Ft.. * prologis_data$Property...Size..Sq..Ft..
hist(prologis_data$totalrent)
plot(log(totalrent) ~ Customer.Industry, data = prologis_data) 
plot(log(totalrent) ~ Transaction.Type, data = prologis_data)
#### There is skewness present in rents so we need to do transformation to correct it. Analysis: The histogram shows positive skewness, we can adjust it by log transformation, and normalized the distribution.
prologis_data$log_totalrent = log(prologis_data$totalrent)
par(mfrow = c(1, 2))
hist(prologis_data$totalrent)
hist(prologis_data$log_totalrent)
### Calculate the average monthly price using subsets
### We checked that the average rent is aggregated based on conditional criterions so we are good to go.
prologis_data %>%
ggplot( aes(x=log_totalrent, y = Property...Year.Built)) +
geom_point(aes(colour = factor(Transaction.Type))) +
labs(fill="") + facet_wrap(~ Customer.Industry)
## renewals are highest in wholesaler and 3 pl in the recent years. and original lease is high under unspecified but they are usually for a earlier time periods.
prologis_data %>%
group_by(Lease.ID, Transaction.Type, Customer.Industry, Property...Year.Built) %>%
mutate(agg_rent = mean(totalrent),
log_aggrent = log(agg_rent))## # A tibble: 84 x 14
## # Groups: Lease.ID, Transaction.Type, Customer.Industry,
## # Property...Year.Built [84]
## Lease.ID Monthly.Rent....~ Transaction.Type Customer.Indust~ Lease...Total.T~
## <dbl> <dbl> <fct> <fct> <dbl>
## 1 53884 6.35 Renewal Wholesaler 24
## 2 56140 6.73 Renewal Wholesaler 24
## 3 64052 6.67 Original Lease Retailer 24
## 4 54897 5.88 Renewal Wholesaler 24
## 5 46043 6.36 Original Lease 3PL 31
## 6 41864 6.25 Original Lease Transportation/~ 33
## 7 44752 7.30 Renewal Manufacturer 36
## 8 49420 6.11 Renewal Manufacturer 36
## 9 48528 6.41 Renewal Retailer 36
## 10 52925 5.89 Renewal 3PL 36
## # ... with 74 more rows, and 9 more variables: Lease...Free.Rent..Month. <dbl>,
## # Property...Size..Sq..Ft.. <dbl>, Property...Speed.Bay <dbl>,
## # Property...Year.Built <fct>, Property...Office.Area <dbl>, totalrent <dbl>,
## # log_totalrent <dbl>, agg_rent <dbl>, log_aggrent <dbl>Exploratory data analysis
First step in analyzing data is by EDA:
- Explore structures of the data - (done by str function) we will use skim to check other descriptive stats
- Makes use of simple descriptive statistics
- Finding errors and outliers
- Correlations between variables
### descriptive Stats
skim(prologis_data)| Name | prologis_data |
| Number of rows | 84 |
| Number of columns | 12 |
| _______________________ | |
| Column type frequency: | |
| factor | 3 |
| numeric | 9 |
| ________________________ | |
| Group variables | None |
Variable type: factor
| skim_variable | n_missing | complete_rate | ordered | n_unique | top_counts |
|---|---|---|---|---|---|
| Transaction.Type | 0 | 1 | FALSE | 2 | Ren: 47, Ori: 37 |
| Customer.Industry | 0 | 1 | FALSE | 7 | Who: 30, 3PL: 21, Uns: 15, Ret: 8 |
| Property…Year.Built | 0 | 1 | FALSE | 26 | 199: 9, 198: 8, 200: 7, 200: 7 |
Variable type: numeric
| skim_variable | n_missing | complete_rate | mean | sd | p0 | p25 | p50 | p75 | p100 | hist |
|---|---|---|---|---|---|---|---|---|---|---|
| Lease.ID | 0 | 1 | 54943.33 | 7483.54 | 40079.00 | 49419.75 | 54896.50 | 62323.25 | 68040.00 | ▅▇▇▆▇ |
| Monthly.Rent….per.Sq..Ft.. | 0 | 1 | 6.88 | 0.98 | 5.62 | 6.21 | 6.50 | 7.40 | 9.94 | ▇▅▂▂▁ |
| Lease…Total.Term..Month. | 0 | 1 | 56.15 | 20.05 | 24.00 | 37.00 | 61.00 | 63.00 | 130.00 | ▅▇▁▁▁ |
| Lease…Free.Rent..Month. | 0 | 1 | 1.55 | 1.57 | 0.00 | 0.00 | 1.00 | 2.00 | 10.00 | ▇▂▁▁▁ |
| Property…Size..Sq..Ft.. | 0 | 1 | 279097.38 | 219862.98 | 9862.00 | 107500.00 | 201473.50 | 419296.50 | 970073.00 | ▇▃▃▁▁ |
| Property…Speed.Bay | 0 | 1 | 54.10 | 9.10 | 18.00 | 48.00 | 58.50 | 60.00 | 73.00 | ▁▁▅▇▁ |
| Property…Office.Area | 0 | 1 | 8454.06 | 13047.11 | 0.00 | 3037.50 | 5712.00 | 8743.50 | 113949.00 | ▇▁▁▁▁ |
| totalrent | 0 | 1 | 1790980.29 | 1366148.84 | 92691.94 | 751808.22 | 1369916.16 | 2577316.50 | 6626790.83 | ▇▅▂▁▁ |
| log_totalrent | 0 | 1 | 14.07 | 0.89 | 11.44 | 13.53 | 14.13 | 14.76 | 15.71 | ▁▂▆▇▅ |
pairs(prologis_data[,c("log_totalrent","Transaction.Type","Customer.Industry",
"Lease...Total.Term..Month.", "Lease...Free.Rent..Month.", "Property...Size..Sq..Ft..", "Property...Speed.Bay", "Property...Office.Area")], col=rainbow(3))
names(prologis_data)## [1] "Lease.ID" "Monthly.Rent....per.Sq..Ft.."
## [3] "Transaction.Type" "Customer.Industry"
## [5] "Lease...Total.Term..Month." "Lease...Free.Rent..Month."
## [7] "Property...Size..Sq..Ft.." "Property...Speed.Bay"
## [9] "Property...Year.Built" "Property...Office.Area"
## [11] "totalrent" "log_totalrent"### Remove outliers by group
boxplot(prologis_data$Property...Office.Area)
outliers <- boxplot(prologis_data$Property...Office.Area,plot=FALSE)$out
prologis_data[which(prologis_data$Property...Office.Area %in% outliers),]## Lease.ID Monthly.Rent....per.Sq..Ft.. Transaction.Type Customer.Industry
## 1 53884 6.350155 Renewal Wholesaler
## 26 45525 6.246059 Renewal Retailer
## 29 41816 6.170980 Renewal Wholesaler
## 31 48903 6.850775 Renewal 3PL
## 32 41732 6.067994 Renewal Wholesaler
## 34 49919 6.734077 Renewal Retailer
## 60 60795 5.706969 Renewal 3PL
## 80 57890 7.268996 Original Lease 3PL
## Lease...Total.Term..Month. Lease...Free.Rent..Month.
## 1 24 0
## 26 38 2
## 29 48 1
## 31 59 1
## 32 60 0
## 34 60 0
## 60 62 2
## 80 86 2
## Property...Size..Sq..Ft.. Property...Speed.Bay Property...Year.Built
## 1 656040 50 2008
## 26 334800 60 2005
## 29 359996 73 2000
## 31 241367 60 2008
## 32 504530 60 2000
## 34 757765 60 2004
## 60 645311 60 2007
## 80 201454 40 1996
## Property...Office.Area totalrent log_totalrent
## 1 33000 4165955 15.24246
## 26 23040 2091180 14.55324
## 29 17590 2221528 14.61371
## 31 113949 1653551 14.31844
## 32 23640 3061485 14.93441
## 34 18138 5102848 15.44531
## 60 17413 3682770 15.11918
## 80 18677 1464368 14.19693prologis_data <-prologis_data[-which(prologis_data$Property...Office.Area %in% outliers),]
## Hierarchial clustering with Qual and Quant variables
xquant = prologis_data[,c(1,5,6,7,8,10,12)]
xqual = prologis_data[,c(3,4,9)]
tree <- hclustvar(xquant, xqual)
plot(tree)
stab <- stability(tree, B=50)
### Correlation Matrix
corrplot(cor(xquant),order = 'hclust',addrect = 2)
We can observe presence of outlier in proper office area and speed bay. We need to correct those outliers for the dataset.
Multiple Linear Regression Model
We will use regression models to estimate the average rental price in the market; and predict the new houses coming on the market based on the current data we have.
We will use Root Mean Squared Error (RMSE) on the test data to evaluate their performance.
First, we split the data into training and testing data on a 80-20% ratio
N <- length(prologis_data$Lease.ID)
set.seed(2022)
all_indices = seq(1, N)
training_indices = sort(sample(1:N, 4*N/5, replace = FALSE)) # ratio is 80/20
training_set = prologis_data[training_indices,]
testing_indices = sort(all_indices[!all_indices %in% training_indices]) # remove training indices from set
testing_set = prologis_data[testing_indices,]
all.equal(sort(c(training_indices, testing_indices)), all_indices) # sanity check to ensure we separated tests correctly## [1] TRUE##Fit the full model
fullmodel <- lm(log_totalrent ~ Lease...Total.Term..Month. + Lease...Free.Rent..Month. + Property...Size..Sq..Ft.. + Property...Speed.Bay +
factor(Property...Year.Built) + Property...Office.Area + factor(Transaction.Type) +
factor(Customer.Industry) , data = training_set)
summary(fullmodel)##
## Call:
## lm(formula = log_totalrent ~ Lease...Total.Term..Month. + Lease...Free.Rent..Month. +
## Property...Size..Sq..Ft.. + Property...Speed.Bay + factor(Property...Year.Built) +
## Property...Office.Area + factor(Transaction.Type) + factor(Customer.Industry),
## data = training_set)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.64855 -0.07872 0.00000 0.10028 0.54965
##
## Coefficients:
## Estimate Std. Error
## (Intercept) 1.430e+01 7.583e-01
## Lease...Total.Term..Month. -7.007e-03 3.649e-03
## Lease...Free.Rent..Month. 5.033e-02 6.696e-02
## Property...Size..Sq..Ft.. 3.507e-06 4.968e-07
## Property...Speed.Bay 1.344e-02 8.741e-03
## factor(Property...Year.Built)1984 -1.593e+00 5.808e-01
## factor(Property...Year.Built)1985 -1.241e+00 6.280e-01
## factor(Property...Year.Built)1986 -1.357e+00 6.198e-01
## factor(Property...Year.Built)1987 -2.233e+00 5.658e-01
## factor(Property...Year.Built)1988 -1.639e+00 5.368e-01
## factor(Property...Year.Built)1989 -2.011e+00 5.315e-01
## factor(Property...Year.Built)1990 -1.485e+00 5.012e-01
## factor(Property...Year.Built)1995 -1.298e+00 5.754e-01
## factor(Property...Year.Built)1996 -1.557e+00 5.626e-01
## factor(Property...Year.Built)1997 -1.385e+00 5.967e-01
## factor(Property...Year.Built)1998 -1.476e+00 5.808e-01
## factor(Property...Year.Built)1999 -1.228e+00 5.916e-01
## factor(Property...Year.Built)2000 -1.535e+00 5.112e-01
## factor(Property...Year.Built)2001 -1.547e+00 5.031e-01
## factor(Property...Year.Built)2002 -1.186e+00 5.125e-01
## factor(Property...Year.Built)2003 -1.504e+00 5.100e-01
## factor(Property...Year.Built)2005 -1.437e+00 5.040e-01
## factor(Property...Year.Built)2006 -1.468e+00 5.241e-01
## factor(Property...Year.Built)2008 -1.611e+00 5.605e-01
## factor(Property...Year.Built)2014 -1.689e+00 5.436e-01
## factor(Property...Year.Built)2015 -1.316e+00 5.787e-01
## factor(Property...Year.Built)2016 -2.079e+00 7.485e-01
## Property...Office.Area -1.959e-05 2.069e-05
## factor(Transaction.Type)Renewal 1.869e-01 1.415e-01
## factor(Customer.Industry)Charity/Prof.Services/Other -1.256e+00 4.393e-01
## factor(Customer.Industry)Manufacturer -2.251e-01 2.616e-01
## factor(Customer.Industry)Retailer -1.535e-01 2.104e-01
## factor(Customer.Industry)Transportation/Freight/Delivery -1.396e-01 3.220e-01
## factor(Customer.Industry)Unspecified 1.287e-02 2.223e-01
## factor(Customer.Industry)Wholesaler -1.332e-01 1.610e-01
## t value Pr(>|t|)
## (Intercept) 18.856 2.70e-16 ***
## Lease...Total.Term..Month. -1.920 0.066305 .
## Lease...Free.Rent..Month. 0.752 0.459312
## Property...Size..Sq..Ft.. 7.059 2.12e-07 ***
## Property...Speed.Bay 1.537 0.136817
## factor(Property...Year.Built)1984 -2.742 0.011109 *
## factor(Property...Year.Built)1985 -1.976 0.059235 .
## factor(Property...Year.Built)1986 -2.189 0.038149 *
## factor(Property...Year.Built)1987 -3.946 0.000569 ***
## factor(Property...Year.Built)1988 -3.054 0.005306 **
## factor(Property...Year.Built)1989 -3.784 0.000861 ***
## factor(Property...Year.Built)1990 -2.963 0.006597 **
## factor(Property...Year.Built)1995 -2.255 0.033108 *
## factor(Property...Year.Built)1996 -2.768 0.010465 *
## factor(Property...Year.Built)1997 -2.321 0.028759 *
## factor(Property...Year.Built)1998 -2.540 0.017661 *
## factor(Property...Year.Built)1999 -2.075 0.048419 *
## factor(Property...Year.Built)2000 -3.003 0.005994 **
## factor(Property...Year.Built)2001 -3.074 0.005054 **
## factor(Property...Year.Built)2002 -2.315 0.029118 *
## factor(Property...Year.Built)2003 -2.950 0.006814 **
## factor(Property...Year.Built)2005 -2.852 0.008585 **
## factor(Property...Year.Built)2006 -2.801 0.009688 **
## factor(Property...Year.Built)2008 -2.874 0.008164 **
## factor(Property...Year.Built)2014 -3.108 0.004653 **
## factor(Property...Year.Built)2015 -2.274 0.031836 *
## factor(Property...Year.Built)2016 -2.778 0.010230 *
## Property...Office.Area -0.947 0.352834
## factor(Transaction.Type)Renewal 1.321 0.198589
## factor(Customer.Industry)Charity/Prof.Services/Other -2.859 0.008447 **
## factor(Customer.Industry)Manufacturer -0.860 0.397744
## factor(Customer.Industry)Retailer -0.729 0.472548
## factor(Customer.Industry)Transportation/Freight/Delivery -0.433 0.668427
## factor(Customer.Industry)Unspecified 0.058 0.954298
## factor(Customer.Industry)Wholesaler -0.827 0.415920
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.2999 on 25 degrees of freedom
## Multiple R-squared: 0.9568, Adjusted R-squared: 0.8981
## F-statistic: 16.3 on 34 and 25 DF, p-value: 2.07e-10#Stepwise regression model
step.model <- stepAIC(fullmodel, direction = "both",
trace = FALSE)
step.model$anova## Stepwise Model Path
## Analysis of Deviance Table
##
## Initial Model:
## log_totalrent ~ Lease...Total.Term..Month. + Lease...Free.Rent..Month. +
## Property...Size..Sq..Ft.. + Property...Speed.Bay + factor(Property...Year.Built) +
## Property...Office.Area + factor(Transaction.Type) + factor(Customer.Industry)
##
## Final Model:
## log_totalrent ~ Lease...Total.Term..Month. + Property...Size..Sq..Ft.. +
## Property...Speed.Bay + factor(Property...Year.Built) + factor(Transaction.Type) +
## factor(Customer.Industry)
##
##
## Step Df Deviance Resid. Df Resid. Dev AIC
## 1 25 2.248309 -127.0500
## 2 - Lease...Free.Rent..Month. 1 0.05080236 26 2.299111 -127.7093
## 3 - Property...Office.Area 1 0.04599654 27 2.345108 -128.5208step.model.backward <- stepAIC(fullmodel, direction = "backward",
trace = FALSE)
summary(step.model.backward)##
## Call:
## lm(formula = log_totalrent ~ Lease...Total.Term..Month. + Property...Size..Sq..Ft.. +
## Property...Speed.Bay + factor(Property...Year.Built) + factor(Transaction.Type) +
## factor(Customer.Industry), data = training_set)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.62376 -0.06495 0.00000 0.09915 0.51053
##
## Coefficients:
## Estimate Std. Error
## (Intercept) 1.423e+01 7.010e-01
## Lease...Total.Term..Month. -5.416e-03 2.932e-03
## Property...Size..Sq..Ft.. 3.361e-06 4.663e-07
## Property...Speed.Bay 1.534e-02 7.823e-03
## factor(Property...Year.Built)1984 -1.611e+00 5.600e-01
## factor(Property...Year.Built)1985 -1.415e+00 5.911e-01
## factor(Property...Year.Built)1986 -1.543e+00 5.594e-01
## factor(Property...Year.Built)1987 -2.339e+00 5.358e-01
## factor(Property...Year.Built)1988 -1.811e+00 4.933e-01
## factor(Property...Year.Built)1989 -2.143e+00 5.033e-01
## factor(Property...Year.Built)1990 -1.557e+00 4.724e-01
## factor(Property...Year.Built)1995 -1.418e+00 5.434e-01
## factor(Property...Year.Built)1996 -1.660e+00 5.421e-01
## factor(Property...Year.Built)1997 -1.550e+00 5.516e-01
## factor(Property...Year.Built)1998 -1.593e+00 5.522e-01
## factor(Property...Year.Built)1999 -1.432e+00 5.330e-01
## factor(Property...Year.Built)2000 -1.578e+00 4.750e-01
## factor(Property...Year.Built)2001 -1.659e+00 4.796e-01
## factor(Property...Year.Built)2002 -1.235e+00 4.988e-01
## factor(Property...Year.Built)2003 -1.559e+00 4.916e-01
## factor(Property...Year.Built)2005 -1.559e+00 4.697e-01
## factor(Property...Year.Built)2006 -1.604e+00 4.884e-01
## factor(Property...Year.Built)2008 -1.704e+00 5.436e-01
## factor(Property...Year.Built)2014 -1.729e+00 5.317e-01
## factor(Property...Year.Built)2015 -1.393e+00 5.168e-01
## factor(Property...Year.Built)2016 -2.004e+00 6.172e-01
## factor(Transaction.Type)Renewal 1.371e-01 1.286e-01
## factor(Customer.Industry)Charity/Prof.Services/Other -1.233e+00 3.985e-01
## factor(Customer.Industry)Manufacturer -1.694e-01 2.454e-01
## factor(Customer.Industry)Retailer -1.820e-01 2.033e-01
## factor(Customer.Industry)Transportation/Freight/Delivery -1.341e-01 3.158e-01
## factor(Customer.Industry)Unspecified 6.473e-03 2.158e-01
## factor(Customer.Industry)Wholesaler -7.862e-02 1.485e-01
## t value Pr(>|t|)
## (Intercept) 20.295 < 2e-16 ***
## Lease...Total.Term..Month. -1.847 0.075699 .
## Property...Size..Sq..Ft.. 7.208 9.44e-08 ***
## Property...Speed.Bay 1.961 0.060249 .
## factor(Property...Year.Built)1984 -2.878 0.007736 **
## factor(Property...Year.Built)1985 -2.394 0.023882 *
## factor(Property...Year.Built)1986 -2.758 0.010312 *
## factor(Property...Year.Built)1987 -4.366 0.000167 ***
## factor(Property...Year.Built)1988 -3.671 0.001051 **
## factor(Property...Year.Built)1989 -4.258 0.000223 ***
## factor(Property...Year.Built)1990 -3.296 0.002752 **
## factor(Property...Year.Built)1995 -2.610 0.014600 *
## factor(Property...Year.Built)1996 -3.061 0.004940 **
## factor(Property...Year.Built)1997 -2.810 0.009110 **
## factor(Property...Year.Built)1998 -2.885 0.007607 **
## factor(Property...Year.Built)1999 -2.687 0.012201 *
## factor(Property...Year.Built)2000 -3.322 0.002572 **
## factor(Property...Year.Built)2001 -3.458 0.001818 **
## factor(Property...Year.Built)2002 -2.475 0.019886 *
## factor(Property...Year.Built)2003 -3.171 0.003763 **
## factor(Property...Year.Built)2005 -3.319 0.002591 **
## factor(Property...Year.Built)2006 -3.283 0.002838 **
## factor(Property...Year.Built)2008 -3.134 0.004123 **
## factor(Property...Year.Built)2014 -3.251 0.003077 **
## factor(Property...Year.Built)2015 -2.696 0.011925 *
## factor(Property...Year.Built)2016 -3.248 0.003105 **
## factor(Transaction.Type)Renewal 1.066 0.295851
## factor(Customer.Industry)Charity/Prof.Services/Other -3.093 0.004570 **
## factor(Customer.Industry)Manufacturer -0.690 0.495889
## factor(Customer.Industry)Retailer -0.895 0.378592
## factor(Customer.Industry)Transportation/Freight/Delivery -0.425 0.674500
## factor(Customer.Industry)Unspecified 0.030 0.976287
## factor(Customer.Industry)Wholesaler -0.529 0.600787
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.2947 on 27 degrees of freedom
## Multiple R-squared: 0.955, Adjusted R-squared: 0.9016
## F-statistic: 17.9 on 32 and 27 DF, p-value: 1.89e-11step.model.backward$anova## Stepwise Model Path
## Analysis of Deviance Table
##
## Initial Model:
## log_totalrent ~ Lease...Total.Term..Month. + Lease...Free.Rent..Month. +
## Property...Size..Sq..Ft.. + Property...Speed.Bay + factor(Property...Year.Built) +
## Property...Office.Area + factor(Transaction.Type) + factor(Customer.Industry)
##
## Final Model:
## log_totalrent ~ Lease...Total.Term..Month. + Property...Size..Sq..Ft.. +
## Property...Speed.Bay + factor(Property...Year.Built) + factor(Transaction.Type) +
## factor(Customer.Industry)
##
##
## Step Df Deviance Resid. Df Resid. Dev AIC
## 1 25 2.248309 -127.0500
## 2 - Lease...Free.Rent..Month. 1 0.05080236 26 2.299111 -127.7093
## 3 - Property...Office.Area 1 0.04599654 27 2.345108 -128.5208library(car)## Loading required package: carData##
## Attaching package: 'car'## The following object is masked from 'package:dplyr':
##
## recode## Check for multicollinearity
vif(fullmodel)## GVIF Df GVIF^(1/(2*Df))
## Lease...Total.Term..Month. 4.135181 1 2.033514
## Lease...Free.Rent..Month. 8.657061 1 2.942288
## Property...Size..Sq..Ft.. 8.248839 1 2.872079
## Property...Speed.Bay 4.315187 1 2.077303
## factor(Property...Year.Built) 12160.405547 22 1.238340
## Property...Office.Area 3.214662 1 1.792948
## factor(Transaction.Type) 3.335258 1 1.826269
## factor(Customer.Industry) 429.208312 6 1.657254##Fit the full model
fullmodel <- lm(totalrent ~ Lease...Total.Term..Month. + Lease...Free.Rent..Month. + Property...Size..Sq..Ft.. + Property...Speed.Bay +
Property...Year.Built + Property...Office.Area + Transaction.Type +
Customer.Industry , data = training_set)
summary(fullmodel)##
## Call:
## lm(formula = totalrent ~ Lease...Total.Term..Month. + Lease...Free.Rent..Month. +
## Property...Size..Sq..Ft.. + Property...Speed.Bay + Property...Year.Built +
## Property...Office.Area + Transaction.Type + Customer.Industry,
## data = training_set)
##
## Residuals:
## Min 1Q Median 3Q Max
## -121285 -40976 0 35722 158048
##
## Coefficients:
## Estimate Std. Error t value
## (Intercept) 3.106e+05 2.382e+05 1.304
## Lease...Total.Term..Month. 1.726e+03 1.146e+03 1.506
## Lease...Free.Rent..Month. -1.102e+04 2.103e+04 -0.524
## Property...Size..Sq..Ft.. 6.029e+00 1.561e-01 38.632
## Property...Speed.Bay -2.349e+03 2.746e+03 -0.855
## Property...Year.Built1984 -1.484e+05 1.824e+05 -0.813
## Property...Year.Built1985 -2.061e+05 1.973e+05 -1.045
## Property...Year.Built1986 -8.819e+04 1.947e+05 -0.453
## Property...Year.Built1987 -1.407e+05 1.778e+05 -0.792
## Property...Year.Built1988 -1.670e+05 1.686e+05 -0.991
## Property...Year.Built1989 -1.027e+05 1.670e+05 -0.615
## Property...Year.Built1990 -9.431e+04 1.574e+05 -0.599
## Property...Year.Built1995 -1.033e+05 1.808e+05 -0.572
## Property...Year.Built1996 1.527e+05 1.767e+05 0.864
## Property...Year.Built1997 -1.695e+05 1.875e+05 -0.904
## Property...Year.Built1998 -2.403e+05 1.825e+05 -1.317
## Property...Year.Built1999 -4.840e+04 1.858e+05 -0.260
## Property...Year.Built2000 -7.296e+04 1.606e+05 -0.454
## Property...Year.Built2001 -6.711e+04 1.581e+05 -0.425
## Property...Year.Built2002 1.081e+04 1.610e+05 0.067
## Property...Year.Built2003 -5.783e+03 1.602e+05 -0.036
## Property...Year.Built2005 -1.681e+05 1.583e+05 -1.062
## Property...Year.Built2006 -1.136e+05 1.646e+05 -0.690
## Property...Year.Built2008 -7.774e+04 1.761e+05 -0.442
## Property...Year.Built2014 -3.610e+04 1.708e+05 -0.211
## Property...Year.Built2015 -1.194e+05 1.818e+05 -0.657
## Property...Year.Built2016 5.688e+05 2.351e+05 2.419
## Property...Office.Area -1.380e+00 6.499e+00 -0.212
## Transaction.TypeRenewal -9.804e+04 4.445e+04 -2.206
## Customer.IndustryCharity/Prof.Services/Other -2.024e+05 1.380e+05 -1.467
## Customer.IndustryManufacturer -4.812e+04 8.218e+04 -0.586
## Customer.IndustryRetailer -1.897e+04 6.610e+04 -0.287
## Customer.IndustryTransportation/Freight/Delivery -1.964e+05 1.012e+05 -1.942
## Customer.IndustryUnspecified -5.143e+04 6.985e+04 -0.736
## Customer.IndustryWholesaler 5.362e+04 5.058e+04 1.060
## Pr(>|t|)
## (Intercept) 0.2041
## Lease...Total.Term..Month. 0.1446
## Lease...Free.Rent..Month. 0.6051
## Property...Size..Sq..Ft.. <2e-16 ***
## Property...Speed.Bay 0.4005
## Property...Year.Built1984 0.4238
## Property...Year.Built1985 0.3061
## Property...Year.Built1986 0.6545
## Property...Year.Built1987 0.4360
## Property...Year.Built1988 0.3314
## Property...Year.Built1989 0.5441
## Property...Year.Built1990 0.5545
## Property...Year.Built1995 0.5727
## Property...Year.Built1996 0.3957
## Property...Year.Built1997 0.3744
## Property...Year.Built1998 0.1997
## Property...Year.Built1999 0.7967
## Property...Year.Built2000 0.6535
## Property...Year.Built2001 0.6748
## Property...Year.Built2002 0.9470
## Property...Year.Built2003 0.9715
## Property...Year.Built2005 0.2984
## Property...Year.Built2006 0.4966
## Property...Year.Built2008 0.6626
## Property...Year.Built2014 0.8343
## Property...Year.Built2015 0.5174
## Property...Year.Built2016 0.0232 *
## Property...Office.Area 0.8336
## Transaction.TypeRenewal 0.0368 *
## Customer.IndustryCharity/Prof.Services/Other 0.1550
## Customer.IndustryManufacturer 0.5634
## Customer.IndustryRetailer 0.7765
## Customer.IndustryTransportation/Freight/Delivery 0.0635 .
## Customer.IndustryUnspecified 0.4684
## Customer.IndustryWholesaler 0.2993
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 94210 on 25 degrees of freedom
## Multiple R-squared: 0.9981, Adjusted R-squared: 0.9955
## F-statistic: 385.9 on 34 and 25 DF, p-value: < 2.2e-16library(leaps)
## regression via subsets
model.regsubsets <- regsubsets(log_totalrent ~ Lease...Total.Term..Month. + Lease...Free.Rent..Month. + Property...Size..Sq..Ft.. + Property...Speed.Bay +
Property...Year.Built + Property...Office.Area + Transaction.Type +
Customer.Industry , data = training_set, method = "exhaustive", nvmax = 15)## Warning in leaps.setup(x, y, wt = wt, nbest = nbest, nvmax = nvmax, force.in =
## force.in, : 3 linear dependencies found## Reordering variables and trying again:model.regsubsets.s <- summary(model.regsubsets)
model.regsubsets.s$which## (Intercept) Lease...Total.Term..Month. Lease...Free.Rent..Month.
## 1 TRUE FALSE FALSE
## 2 TRUE FALSE FALSE
## 3 TRUE TRUE FALSE
## 4 TRUE TRUE FALSE
## 5 TRUE TRUE FALSE
## 6 TRUE TRUE FALSE
## 7 TRUE TRUE FALSE
## 8 TRUE TRUE FALSE
## 9 TRUE TRUE FALSE
## 10 TRUE TRUE FALSE
## 11 TRUE TRUE FALSE
## 12 TRUE TRUE FALSE
## 13 TRUE TRUE FALSE
## 14 TRUE TRUE FALSE
## 15 TRUE TRUE TRUE
## 16 TRUE TRUE TRUE
## Property...Size..Sq..Ft.. Property...Speed.Bay Property...Year.Built1983
## 1 TRUE FALSE FALSE
## 2 TRUE TRUE FALSE
## 3 TRUE TRUE FALSE
## 4 TRUE TRUE FALSE
## 5 TRUE TRUE FALSE
## 6 TRUE TRUE FALSE
## 7 TRUE TRUE FALSE
## 8 TRUE TRUE FALSE
## 9 TRUE TRUE FALSE
## 10 TRUE TRUE FALSE
## 11 TRUE TRUE FALSE
## 12 TRUE TRUE FALSE
## 13 TRUE TRUE FALSE
## 14 TRUE TRUE FALSE
## 15 TRUE TRUE FALSE
## 16 TRUE TRUE FALSE
## Property...Year.Built1984 Property...Year.Built1985
## 1 FALSE FALSE
## 2 FALSE FALSE
## 3 FALSE FALSE
## 4 FALSE FALSE
## 5 FALSE FALSE
## 6 FALSE FALSE
## 7 FALSE FALSE
## 8 FALSE FALSE
## 9 FALSE FALSE
## 10 FALSE FALSE
## 11 FALSE FALSE
## 12 FALSE FALSE
## 13 FALSE FALSE
## 14 FALSE FALSE
## 15 FALSE FALSE
## 16 TRUE FALSE
## Property...Year.Built1986 Property...Year.Built1987
## 1 FALSE FALSE
## 2 FALSE FALSE
## 3 FALSE FALSE
## 4 FALSE FALSE
## 5 FALSE TRUE
## 6 FALSE TRUE
## 7 FALSE TRUE
## 8 FALSE TRUE
## 9 FALSE TRUE
## 10 FALSE TRUE
## 11 FALSE TRUE
## 12 FALSE TRUE
## 13 FALSE TRUE
## 14 FALSE TRUE
## 15 FALSE TRUE
## 16 FALSE TRUE
## Property...Year.Built1988 Property...Year.Built1989
## 1 FALSE FALSE
## 2 FALSE FALSE
## 3 FALSE FALSE
## 4 FALSE TRUE
## 5 FALSE TRUE
## 6 FALSE TRUE
## 7 FALSE TRUE
## 8 TRUE TRUE
## 9 TRUE TRUE
## 10 TRUE TRUE
## 11 TRUE TRUE
## 12 TRUE TRUE
## 13 TRUE TRUE
## 14 TRUE TRUE
## 15 TRUE TRUE
## 16 TRUE TRUE
## Property...Year.Built1990 Property...Year.Built1995
## 1 FALSE FALSE
## 2 FALSE FALSE
## 3 FALSE FALSE
## 4 FALSE FALSE
## 5 FALSE FALSE
## 6 FALSE FALSE
## 7 FALSE FALSE
## 8 FALSE FALSE
## 9 FALSE FALSE
## 10 FALSE FALSE
## 11 FALSE FALSE
## 12 FALSE FALSE
## 13 TRUE FALSE
## 14 TRUE FALSE
## 15 TRUE FALSE
## 16 TRUE FALSE
## Property...Year.Built1996 Property...Year.Built1997
## 1 FALSE FALSE
## 2 FALSE FALSE
## 3 FALSE FALSE
## 4 FALSE FALSE
## 5 FALSE FALSE
## 6 FALSE FALSE
## 7 FALSE FALSE
## 8 FALSE FALSE
## 9 FALSE FALSE
## 10 FALSE FALSE
## 11 FALSE FALSE
## 12 FALSE FALSE
## 13 FALSE FALSE
## 14 FALSE FALSE
## 15 FALSE FALSE
## 16 FALSE FALSE
## Property...Year.Built1998 Property...Year.Built1999
## 1 FALSE FALSE
## 2 FALSE FALSE
## 3 FALSE FALSE
## 4 FALSE FALSE
## 5 FALSE FALSE
## 6 FALSE FALSE
## 7 FALSE FALSE
## 8 FALSE FALSE
## 9 FALSE FALSE
## 10 FALSE FALSE
## 11 FALSE FALSE
## 12 FALSE FALSE
## 13 FALSE FALSE
## 14 FALSE FALSE
## 15 FALSE FALSE
## 16 FALSE FALSE
## Property...Year.Built2000 Property...Year.Built2001
## 1 FALSE FALSE
## 2 FALSE FALSE
## 3 FALSE FALSE
## 4 FALSE FALSE
## 5 FALSE FALSE
## 6 FALSE FALSE
## 7 FALSE FALSE
## 8 FALSE FALSE
## 9 FALSE FALSE
## 10 FALSE FALSE
## 11 FALSE FALSE
## 12 FALSE FALSE
## 13 FALSE FALSE
## 14 FALSE FALSE
## 15 FALSE FALSE
## 16 FALSE FALSE
## Property...Year.Built2002 Property...Year.Built2003
## 1 FALSE FALSE
## 2 FALSE FALSE
## 3 FALSE FALSE
## 4 FALSE FALSE
## 5 FALSE FALSE
## 6 TRUE FALSE
## 7 TRUE FALSE
## 8 TRUE FALSE
## 9 TRUE FALSE
## 10 TRUE FALSE
## 11 TRUE FALSE
## 12 TRUE FALSE
## 13 TRUE FALSE
## 14 TRUE FALSE
## 15 TRUE FALSE
## 16 TRUE FALSE
## Property...Year.Built2004 Property...Year.Built2005
## 1 FALSE FALSE
## 2 FALSE FALSE
## 3 FALSE FALSE
## 4 FALSE FALSE
## 5 FALSE FALSE
## 6 FALSE FALSE
## 7 FALSE FALSE
## 8 FALSE FALSE
## 9 FALSE FALSE
## 10 FALSE FALSE
## 11 FALSE FALSE
## 12 FALSE FALSE
## 13 FALSE FALSE
## 14 FALSE FALSE
## 15 FALSE FALSE
## 16 FALSE FALSE
## Property...Year.Built2006 Property...Year.Built2007
## 1 FALSE FALSE
## 2 FALSE FALSE
## 3 FALSE FALSE
## 4 FALSE FALSE
## 5 FALSE FALSE
## 6 FALSE FALSE
## 7 FALSE FALSE
## 8 FALSE FALSE
## 9 FALSE FALSE
## 10 FALSE FALSE
## 11 FALSE FALSE
## 12 FALSE FALSE
## 13 FALSE FALSE
## 14 FALSE FALSE
## 15 FALSE FALSE
## 16 FALSE FALSE
## Property...Year.Built2008 Property...Year.Built2014
## 1 FALSE FALSE
## 2 FALSE FALSE
## 3 FALSE FALSE
## 4 FALSE FALSE
## 5 FALSE FALSE
## 6 FALSE FALSE
## 7 FALSE FALSE
## 8 FALSE FALSE
## 9 FALSE FALSE
## 10 FALSE FALSE
## 11 FALSE FALSE
## 12 FALSE FALSE
## 13 FALSE FALSE
## 14 FALSE FALSE
## 15 FALSE FALSE
## 16 FALSE FALSE
## Property...Year.Built2015 Property...Year.Built2016 Property...Office.Area
## 1 FALSE FALSE FALSE
## 2 FALSE FALSE FALSE
## 3 FALSE FALSE FALSE
## 4 FALSE FALSE FALSE
## 5 FALSE FALSE FALSE
## 6 FALSE FALSE FALSE
## 7 FALSE FALSE FALSE
## 8 FALSE FALSE FALSE
## 9 FALSE FALSE TRUE
## 10 FALSE FALSE TRUE
## 11 FALSE FALSE TRUE
## 12 TRUE FALSE TRUE
## 13 TRUE FALSE TRUE
## 14 TRUE FALSE TRUE
## 15 FALSE TRUE TRUE
## 16 FALSE TRUE TRUE
## Transaction.TypeRenewal Customer.IndustryCharity/Prof.Services/Other
## 1 FALSE FALSE
## 2 FALSE FALSE
## 3 FALSE FALSE
## 4 FALSE FALSE
## 5 FALSE FALSE
## 6 FALSE FALSE
## 7 TRUE FALSE
## 8 FALSE TRUE
## 9 FALSE TRUE
## 10 FALSE TRUE
## 11 TRUE TRUE
## 12 TRUE TRUE
## 13 TRUE TRUE
## 14 TRUE TRUE
## 15 TRUE TRUE
## 16 TRUE TRUE
## Customer.IndustryManufacturer Customer.IndustryRetailer
## 1 FALSE FALSE
## 2 FALSE FALSE
## 3 FALSE FALSE
## 4 FALSE FALSE
## 5 FALSE FALSE
## 6 FALSE FALSE
## 7 FALSE FALSE
## 8 FALSE FALSE
## 9 FALSE FALSE
## 10 TRUE FALSE
## 11 TRUE FALSE
## 12 TRUE FALSE
## 13 TRUE FALSE
## 14 TRUE FALSE
## 15 TRUE FALSE
## 16 TRUE FALSE
## Customer.IndustryTransportation/Freight/Delivery
## 1 FALSE
## 2 FALSE
## 3 FALSE
## 4 FALSE
## 5 FALSE
## 6 FALSE
## 7 FALSE
## 8 FALSE
## 9 FALSE
## 10 FALSE
## 11 FALSE
## 12 FALSE
## 13 FALSE
## 14 FALSE
## 15 FALSE
## 16 FALSE
## Customer.IndustryUnspecified Customer.IndustryWholesaler
## 1 FALSE FALSE
## 2 FALSE FALSE
## 3 FALSE FALSE
## 4 FALSE FALSE
## 5 FALSE FALSE
## 6 FALSE FALSE
## 7 FALSE FALSE
## 8 FALSE FALSE
## 9 FALSE FALSE
## 10 FALSE FALSE
## 11 FALSE FALSE
## 12 FALSE FALSE
## 13 FALSE FALSE
## 14 FALSE TRUE
## 15 FALSE TRUE
## 16 FALSE TRUEmodel.regsubsets.s$cp## [1] 51.5645497 29.0177868 16.8419196 7.9383514 0.6733363 -3.1476235
## [7] -3.2841546 -2.8310920 -2.0449068 -1.0706246 0.0955142 1.5789278
## [13] 3.2278588 4.8721803 6.3992886 8.0273838model.regsubsets.s$adjr2## [1] 0.7853103 0.8313966 0.8576901 0.8780523 0.8956560 0.9064014 0.9093578
## [8] 0.9110919 0.9121240 0.9127539 0.9129473 0.9123675 0.9113460 0.9102909
## [15] 0.9094962 0.9083927attempt1 <- lm(log_totalrent ~ Lease...Total.Term..Month. + Lease...Free.Rent..Month. + Property...Size..Sq..Ft.. + Property...Speed.Bay +
Property...Year.Built + Property...Office.Area + Transaction.Type +
Customer.Industry , data = training_set)
plot( attempt1$fitted.values,attempt1$residuals, main= "Residual plot for attempt1")
Now after the initial analysis, we can observe that there would be some interaction terms required.
Final Multiple Regression Model
finalmodel <- lm(log_totalrent ~ Lease...Total.Term..Month. + Property...Size..Sq..Ft.. + Property...Speed.Bay +
+ Transaction.Type:Customer.Industry+ Property...Year.Built, data = training_set)
summary(finalmodel)##
## Call:
## lm(formula = log_totalrent ~ Lease...Total.Term..Month. + Property...Size..Sq..Ft.. +
## Property...Speed.Bay + +Transaction.Type:Customer.Industry +
## Property...Year.Built, data = training_set)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.61515 -0.07566 0.00000 0.10563 0.39990
##
## Coefficients: (3 not defined because of singularities)
## Estimate
## (Intercept) 1.423e+01
## Lease...Total.Term..Month. -5.422e-03
## Property...Size..Sq..Ft.. 3.332e-06
## Property...Speed.Bay 1.658e-02
## Property...Year.Built1984 -1.622e+00
## Property...Year.Built1985 -1.468e+00
## Property...Year.Built1986 -1.545e+00
## Property...Year.Built1987 -2.343e+00
## Property...Year.Built1988 -1.814e+00
## Property...Year.Built1989 -2.179e+00
## Property...Year.Built1990 -1.537e+00
## Property...Year.Built1995 -1.431e+00
## Property...Year.Built1996 -1.668e+00
## Property...Year.Built1997 -1.566e+00
## Property...Year.Built1998 -1.591e+00
## Property...Year.Built1999 -1.481e+00
## Property...Year.Built2000 -1.613e+00
## Property...Year.Built2001 -1.671e+00
## Property...Year.Built2002 -1.243e+00
## Property...Year.Built2003 -1.546e+00
## Property...Year.Built2005 -1.618e+00
## Property...Year.Built2006 -1.636e+00
## Property...Year.Built2008 -1.864e+00
## Property...Year.Built2014 -1.691e+00
## Property...Year.Built2015 -1.434e+00
## Property...Year.Built2016 -1.996e+00
## Transaction.TypeOriginal Lease:Customer.Industry3PL 8.305e-04
## Transaction.TypeRenewal:Customer.Industry3PL 3.234e-02
## Transaction.TypeOriginal Lease:Customer.IndustryCharity/Prof.Services/Other -1.282e+00
## Transaction.TypeRenewal:Customer.IndustryCharity/Prof.Services/Other NA
## Transaction.TypeOriginal Lease:Customer.IndustryManufacturer -3.436e-01
## Transaction.TypeRenewal:Customer.IndustryManufacturer 6.453e-02
## Transaction.TypeOriginal Lease:Customer.IndustryRetailer -2.363e-01
## Transaction.TypeRenewal:Customer.IndustryRetailer -6.869e-02
## Transaction.TypeOriginal Lease:Customer.IndustryTransportation/Freight/Delivery -1.749e-01
## Transaction.TypeRenewal:Customer.IndustryTransportation/Freight/Delivery -1.154e-01
## Transaction.TypeOriginal Lease:Customer.IndustryUnspecified -5.297e-02
## Transaction.TypeRenewal:Customer.IndustryUnspecified NA
## Transaction.TypeOriginal Lease:Customer.IndustryWholesaler -8.686e-02
## Transaction.TypeRenewal:Customer.IndustryWholesaler NA
## Std. Error
## (Intercept) 7.402e-01
## Lease...Total.Term..Month. 3.256e-03
## Property...Size..Sq..Ft.. 4.987e-07
## Property...Speed.Bay 8.423e-03
## Property...Year.Built1984 5.952e-01
## Property...Year.Built1985 6.339e-01
## Property...Year.Built1986 5.982e-01
## Property...Year.Built1987 5.694e-01
## Property...Year.Built1988 5.247e-01
## Property...Year.Built1989 5.431e-01
## Property...Year.Built1990 5.033e-01
## Property...Year.Built1995 5.832e-01
## Property...Year.Built1996 5.829e-01
## Property...Year.Built1997 5.916e-01
## Property...Year.Built1998 5.921e-01
## Property...Year.Built1999 6.236e-01
## Property...Year.Built2000 5.188e-01
## Property...Year.Built2001 5.092e-01
## Property...Year.Built2002 5.332e-01
## Property...Year.Built2003 5.233e-01
## Property...Year.Built2005 5.063e-01
## Property...Year.Built2006 5.196e-01
## Property...Year.Built2008 6.065e-01
## Property...Year.Built2014 5.694e-01
## Property...Year.Built2015 5.500e-01
## Property...Year.Built2016 6.599e-01
## Transaction.TypeOriginal Lease:Customer.Industry3PL 2.056e-01
## Transaction.TypeRenewal:Customer.Industry3PL 2.250e-01
## Transaction.TypeOriginal Lease:Customer.IndustryCharity/Prof.Services/Other 4.180e-01
## Transaction.TypeRenewal:Customer.IndustryCharity/Prof.Services/Other NA
## Transaction.TypeOriginal Lease:Customer.IndustryManufacturer 2.866e-01
## Transaction.TypeRenewal:Customer.IndustryManufacturer 2.928e-01
## Transaction.TypeOriginal Lease:Customer.IndustryRetailer 5.108e-01
## Transaction.TypeRenewal:Customer.IndustryRetailer 2.570e-01
## Transaction.TypeOriginal Lease:Customer.IndustryTransportation/Freight/Delivery 4.400e-01
## Transaction.TypeRenewal:Customer.IndustryTransportation/Freight/Delivery 4.222e-01
## Transaction.TypeOriginal Lease:Customer.IndustryUnspecified 1.903e-01
## Transaction.TypeRenewal:Customer.IndustryUnspecified NA
## Transaction.TypeOriginal Lease:Customer.IndustryWholesaler 2.103e-01
## Transaction.TypeRenewal:Customer.IndustryWholesaler NA
## t value
## (Intercept) 19.228
## Lease...Total.Term..Month. -1.665
## Property...Size..Sq..Ft.. 6.682
## Property...Speed.Bay 1.969
## Property...Year.Built1984 -2.725
## Property...Year.Built1985 -2.316
## Property...Year.Built1986 -2.582
## Property...Year.Built1987 -4.115
## Property...Year.Built1988 -3.458
## Property...Year.Built1989 -4.013
## Property...Year.Built1990 -3.055
## Property...Year.Built1995 -2.453
## Property...Year.Built1996 -2.862
## Property...Year.Built1997 -2.647
## Property...Year.Built1998 -2.686
## Property...Year.Built1999 -2.374
## Property...Year.Built2000 -3.109
## Property...Year.Built2001 -3.282
## Property...Year.Built2002 -2.331
## Property...Year.Built2003 -2.953
## Property...Year.Built2005 -3.195
## Property...Year.Built2006 -3.149
## Property...Year.Built2008 -3.074
## Property...Year.Built2014 -2.971
## Property...Year.Built2015 -2.608
## Property...Year.Built2016 -3.026
## Transaction.TypeOriginal Lease:Customer.Industry3PL 0.004
## Transaction.TypeRenewal:Customer.Industry3PL 0.144
## Transaction.TypeOriginal Lease:Customer.IndustryCharity/Prof.Services/Other -3.067
## Transaction.TypeRenewal:Customer.IndustryCharity/Prof.Services/Other NA
## Transaction.TypeOriginal Lease:Customer.IndustryManufacturer -1.199
## Transaction.TypeRenewal:Customer.IndustryManufacturer 0.220
## Transaction.TypeOriginal Lease:Customer.IndustryRetailer -0.463
## Transaction.TypeRenewal:Customer.IndustryRetailer -0.267
## Transaction.TypeOriginal Lease:Customer.IndustryTransportation/Freight/Delivery -0.398
## Transaction.TypeRenewal:Customer.IndustryTransportation/Freight/Delivery -0.273
## Transaction.TypeOriginal Lease:Customer.IndustryUnspecified -0.278
## Transaction.TypeRenewal:Customer.IndustryUnspecified NA
## Transaction.TypeOriginal Lease:Customer.IndustryWholesaler -0.413
## Transaction.TypeRenewal:Customer.IndustryWholesaler NA
## Pr(>|t|)
## (Intercept) 1.14e-15
## Lease...Total.Term..Month. 0.109388
## Property...Size..Sq..Ft.. 8.13e-07
## Property...Speed.Bay 0.061128
## Property...Year.Built1984 0.012061
## Property...Year.Built1985 0.029852
## Property...Year.Built1986 0.016653
## Property...Year.Built1987 0.000423
## Property...Year.Built1988 0.002138
## Property...Year.Built1989 0.000545
## Property...Year.Built1990 0.005620
## Property...Year.Built1995 0.022155
## Property...Year.Built1996 0.008815
## Property...Year.Built1997 0.014418
## Property...Year.Built1998 0.013181
## Property...Year.Built1999 0.026318
## Property...Year.Built2000 0.004936
## Property...Year.Built2001 0.003270
## Property...Year.Built2002 0.028872
## Property...Year.Built2003 0.007132
## Property...Year.Built2005 0.004025
## Property...Year.Built2006 0.004492
## Property...Year.Built2008 0.005365
## Property...Year.Built2014 0.006846
## Property...Year.Built2015 0.015734
## Property...Year.Built2016 0.006017
## Transaction.TypeOriginal Lease:Customer.Industry3PL 0.996812
## Transaction.TypeRenewal:Customer.Industry3PL 0.886962
## Transaction.TypeOriginal Lease:Customer.IndustryCharity/Prof.Services/Other 0.005460
## Transaction.TypeRenewal:Customer.IndustryCharity/Prof.Services/Other NA
## Transaction.TypeOriginal Lease:Customer.IndustryManufacturer 0.242881
## Transaction.TypeRenewal:Customer.IndustryManufacturer 0.827497
## Transaction.TypeOriginal Lease:Customer.IndustryRetailer 0.648060
## Transaction.TypeRenewal:Customer.IndustryRetailer 0.791594
## Transaction.TypeOriginal Lease:Customer.IndustryTransportation/Freight/Delivery 0.694623
## Transaction.TypeRenewal:Customer.IndustryTransportation/Freight/Delivery 0.787112
## Transaction.TypeOriginal Lease:Customer.IndustryUnspecified 0.783227
## Transaction.TypeRenewal:Customer.IndustryUnspecified NA
## Transaction.TypeOriginal Lease:Customer.IndustryWholesaler 0.683408
## Transaction.TypeRenewal:Customer.IndustryWholesaler NA
##
## (Intercept) ***
## Lease...Total.Term..Month.
## Property...Size..Sq..Ft.. ***
## Property...Speed.Bay .
## Property...Year.Built1984 *
## Property...Year.Built1985 *
## Property...Year.Built1986 *
## Property...Year.Built1987 ***
## Property...Year.Built1988 **
## Property...Year.Built1989 ***
## Property...Year.Built1990 **
## Property...Year.Built1995 *
## Property...Year.Built1996 **
## Property...Year.Built1997 *
## Property...Year.Built1998 *
## Property...Year.Built1999 *
## Property...Year.Built2000 **
## Property...Year.Built2001 **
## Property...Year.Built2002 *
## Property...Year.Built2003 **
## Property...Year.Built2005 **
## Property...Year.Built2006 **
## Property...Year.Built2008 **
## Property...Year.Built2014 **
## Property...Year.Built2015 *
## Property...Year.Built2016 **
## Transaction.TypeOriginal Lease:Customer.Industry3PL
## Transaction.TypeRenewal:Customer.Industry3PL
## Transaction.TypeOriginal Lease:Customer.IndustryCharity/Prof.Services/Other **
## Transaction.TypeRenewal:Customer.IndustryCharity/Prof.Services/Other
## Transaction.TypeOriginal Lease:Customer.IndustryManufacturer
## Transaction.TypeRenewal:Customer.IndustryManufacturer
## Transaction.TypeOriginal Lease:Customer.IndustryRetailer
## Transaction.TypeRenewal:Customer.IndustryRetailer
## Transaction.TypeOriginal Lease:Customer.IndustryTransportation/Freight/Delivery
## Transaction.TypeRenewal:Customer.IndustryTransportation/Freight/Delivery
## Transaction.TypeOriginal Lease:Customer.IndustryUnspecified
## Transaction.TypeRenewal:Customer.IndustryUnspecified
## Transaction.TypeOriginal Lease:Customer.IndustryWholesaler
## Transaction.TypeRenewal:Customer.IndustryWholesaler
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.3127 on 23 degrees of freedom
## Multiple R-squared: 0.9568, Adjusted R-squared: 0.8893
## F-statistic: 14.16 on 36 and 23 DF, p-value: 3.405e-09plot(finalmodel$fitted.values , finalmodel$residuals)
### Check for heteroskedasticity :
library(lmtest)## Loading required package: zoo##
## Attaching package: 'zoo'## The following objects are masked from 'package:base':
##
## as.Date, as.Date.numericbptest(finalmodel)##
## studentized Breusch-Pagan test
##
## data: finalmodel
## BP = 31.785, df = 36, p-value = 0.6693#While it doesn’t give us the critical value to compare the test statistic, all you need to look at is the p-value to determine whether or not you should reject the null. If the p-value is less than the level of significance (in this case if the p-value is less than α=0.05), then you reject the null hypothesis. Since 0.01 < 0.05, we can reject the null hypothesis. There is heterskedasticity present.
library(sandwich)
coeftest(finalmodel, vcov = vcovHC(finalmodel, "HC1"))##
## t test of coefficients:
##
## Estimate
## (Intercept) 1.4233e+01
## Lease...Total.Term..Month. -5.4220e-03
## Property...Size..Sq..Ft.. 3.3323e-06
## Property...Speed.Bay 1.6584e-02
## Property...Year.Built1984 -1.6222e+00
## Property...Year.Built1985 -1.4677e+00
## Property...Year.Built1986 -1.5448e+00
## Property...Year.Built1987 -2.3432e+00
## Property...Year.Built1988 -1.8141e+00
## Property...Year.Built1989 -2.1792e+00
## Property...Year.Built1990 -1.5374e+00
## Property...Year.Built1995 -1.4307e+00
## Property...Year.Built1996 -1.6684e+00
## Property...Year.Built1997 -1.5658e+00
## Property...Year.Built1998 -1.5907e+00
## Property...Year.Built1999 -1.4805e+00
## Property...Year.Built2000 -1.6132e+00
## Property...Year.Built2001 -1.6712e+00
## Property...Year.Built2002 -1.2430e+00
## Property...Year.Built2003 -1.5455e+00
## Property...Year.Built2005 -1.6176e+00
## Property...Year.Built2006 -1.6363e+00
## Property...Year.Built2008 -1.8645e+00
## Property...Year.Built2014 -1.6914e+00
## Property...Year.Built2015 -1.4344e+00
## Property...Year.Built2016 -1.9965e+00
## Transaction.TypeOriginal Lease:Customer.Industry3PL 8.3047e-04
## Transaction.TypeRenewal:Customer.Industry3PL 3.2345e-02
## Transaction.TypeOriginal Lease:Customer.IndustryCharity/Prof.Services/Other -1.2818e+00
## Transaction.TypeOriginal Lease:Customer.IndustryManufacturer -3.4357e-01
## Transaction.TypeRenewal:Customer.IndustryManufacturer 6.4531e-02
## Transaction.TypeOriginal Lease:Customer.IndustryRetailer -2.3625e-01
## Transaction.TypeRenewal:Customer.IndustryRetailer -6.8693e-02
## Transaction.TypeOriginal Lease:Customer.IndustryTransportation/Freight/Delivery -1.7492e-01
## Transaction.TypeRenewal:Customer.IndustryTransportation/Freight/Delivery -1.1536e-01
## Transaction.TypeOriginal Lease:Customer.IndustryUnspecified -5.2974e-02
## Transaction.TypeOriginal Lease:Customer.IndustryWholesaler -8.6856e-02
## Std. Error
## (Intercept) 6.4579e-01
## Lease...Total.Term..Month. 3.3435e-03
## Property...Size..Sq..Ft.. 4.7572e-07
## Property...Speed.Bay 1.1560e-02
## Property...Year.Built1984 3.1783e-01
## Property...Year.Built1985 2.8098e-01
## Property...Year.Built1986 2.8985e-01
## Property...Year.Built1987 5.4776e-01
## Property...Year.Built1988 3.1860e-01
## Property...Year.Built1989 3.5795e-01
## Property...Year.Built1990 2.8439e-01
## Property...Year.Built1995 3.2016e-01
## Property...Year.Built1996 3.2288e-01
## Property...Year.Built1997 3.3891e-01
## Property...Year.Built1998 2.9107e-01
## Property...Year.Built1999 3.4457e-01
## Property...Year.Built2000 2.8661e-01
## Property...Year.Built2001 2.1947e-01
## Property...Year.Built2002 3.1861e-01
## Property...Year.Built2003 3.2124e-01
## Property...Year.Built2005 2.5182e-01
## Property...Year.Built2006 2.9903e-01
## Property...Year.Built2008 5.6501e-01
## Property...Year.Built2014 1.5129e-01
## Property...Year.Built2015 3.0777e-01
## Property...Year.Built2016 3.2222e-01
## Transaction.TypeOriginal Lease:Customer.Industry3PL 1.4420e-01
## Transaction.TypeRenewal:Customer.Industry3PL 3.2440e-01
## Transaction.TypeOriginal Lease:Customer.IndustryCharity/Prof.Services/Other 2.8870e-01
## Transaction.TypeOriginal Lease:Customer.IndustryManufacturer 3.2127e-01
## Transaction.TypeRenewal:Customer.IndustryManufacturer 4.8049e-01
## Transaction.TypeOriginal Lease:Customer.IndustryRetailer 2.2015e-01
## Transaction.TypeRenewal:Customer.IndustryRetailer 2.5531e-01
## Transaction.TypeOriginal Lease:Customer.IndustryTransportation/Freight/Delivery 4.0847e-01
## Transaction.TypeRenewal:Customer.IndustryTransportation/Freight/Delivery 3.7247e-01
## Transaction.TypeOriginal Lease:Customer.IndustryUnspecified 2.3168e-01
## Transaction.TypeOriginal Lease:Customer.IndustryWholesaler 1.3722e-01
## t value
## (Intercept) 22.0402
## Lease...Total.Term..Month. -1.6217
## Property...Size..Sq..Ft.. 7.0048
## Property...Speed.Bay 1.4346
## Property...Year.Built1984 -5.1041
## Property...Year.Built1985 -5.2237
## Property...Year.Built1986 -5.3296
## Property...Year.Built1987 -4.2777
## Property...Year.Built1988 -5.6939
## Property...Year.Built1989 -6.0882
## Property...Year.Built1990 -5.4059
## Property...Year.Built1995 -4.4688
## Property...Year.Built1996 -5.1672
## Property...Year.Built1997 -4.6200
## Property...Year.Built1998 -5.4649
## Property...Year.Built1999 -4.2968
## Property...Year.Built2000 -5.6284
## Property...Year.Built2001 -7.6146
## Property...Year.Built2002 -3.9014
## Property...Year.Built2003 -4.8111
## Property...Year.Built2005 -6.4236
## Property...Year.Built2006 -5.4722
## Property...Year.Built2008 -3.2999
## Property...Year.Built2014 -11.1799
## Property...Year.Built2015 -4.6606
## Property...Year.Built2016 -6.1961
## Transaction.TypeOriginal Lease:Customer.Industry3PL 0.0058
## Transaction.TypeRenewal:Customer.Industry3PL 0.0997
## Transaction.TypeOriginal Lease:Customer.IndustryCharity/Prof.Services/Other -4.4400
## Transaction.TypeOriginal Lease:Customer.IndustryManufacturer -1.0694
## Transaction.TypeRenewal:Customer.IndustryManufacturer 0.1343
## Transaction.TypeOriginal Lease:Customer.IndustryRetailer -1.0731
## Transaction.TypeRenewal:Customer.IndustryRetailer -0.2691
## Transaction.TypeOriginal Lease:Customer.IndustryTransportation/Freight/Delivery -0.4282
## Transaction.TypeRenewal:Customer.IndustryTransportation/Freight/Delivery -0.3097
## Transaction.TypeOriginal Lease:Customer.IndustryUnspecified -0.2287
## Transaction.TypeOriginal Lease:Customer.IndustryWholesaler -0.6330
## Pr(>|t|)
## (Intercept) < 2.2e-16
## Lease...Total.Term..Month. 0.1185041
## Property...Size..Sq..Ft.. 3.871e-07
## Property...Speed.Bay 0.1648580
## Property...Year.Built1984 3.605e-05
## Property...Year.Built1985 2.683e-05
## Property...Year.Built1986 2.067e-05
## Property...Year.Built1987 0.0002818
## Property...Year.Built1988 8.489e-06
## Property...Year.Built1989 3.286e-06
## Property...Year.Built1990 1.714e-05
## Property...Year.Built1995 0.0001750
## Property...Year.Built1996 3.084e-05
## Property...Year.Built1997 0.0001200
## Property...Year.Built1998 1.483e-05
## Property...Year.Built1999 0.0002687
## Property...Year.Built2000 9.952e-06
## Property...Year.Built2001 9.900e-08
## Property...Year.Built2002 0.0007184
## Property...Year.Built2003 7.456e-05
## Property...Year.Built2005 1.485e-06
## Property...Year.Built2006 1.457e-05
## Property...Year.Built2008 0.0031308
## Property...Year.Built2014 8.956e-11
## Property...Year.Built2015 0.0001084
## Property...Year.Built2016 2.541e-06
## Transaction.TypeOriginal Lease:Customer.Industry3PL 0.9954544
## Transaction.TypeRenewal:Customer.Industry3PL 0.9214413
## Transaction.TypeOriginal Lease:Customer.IndustryCharity/Prof.Services/Other 0.0001880
## Transaction.TypeOriginal Lease:Customer.IndustryManufacturer 0.2959753
## Transaction.TypeRenewal:Customer.IndustryManufacturer 0.8943318
## Transaction.TypeOriginal Lease:Customer.IndustryRetailer 0.2943448
## Transaction.TypeRenewal:Customer.IndustryRetailer 0.7902853
## Transaction.TypeOriginal Lease:Customer.IndustryTransportation/Freight/Delivery 0.6724629
## Transaction.TypeRenewal:Customer.IndustryTransportation/Freight/Delivery 0.7595722
## Transaction.TypeOriginal Lease:Customer.IndustryUnspecified 0.8211608
## Transaction.TypeOriginal Lease:Customer.IndustryWholesaler 0.5330073
##
## (Intercept) ***
## Lease...Total.Term..Month.
## Property...Size..Sq..Ft.. ***
## Property...Speed.Bay
## Property...Year.Built1984 ***
## Property...Year.Built1985 ***
## Property...Year.Built1986 ***
## Property...Year.Built1987 ***
## Property...Year.Built1988 ***
## Property...Year.Built1989 ***
## Property...Year.Built1990 ***
## Property...Year.Built1995 ***
## Property...Year.Built1996 ***
## Property...Year.Built1997 ***
## Property...Year.Built1998 ***
## Property...Year.Built1999 ***
## Property...Year.Built2000 ***
## Property...Year.Built2001 ***
## Property...Year.Built2002 ***
## Property...Year.Built2003 ***
## Property...Year.Built2005 ***
## Property...Year.Built2006 ***
## Property...Year.Built2008 **
## Property...Year.Built2014 ***
## Property...Year.Built2015 ***
## Property...Year.Built2016 ***
## Transaction.TypeOriginal Lease:Customer.Industry3PL
## Transaction.TypeRenewal:Customer.Industry3PL
## Transaction.TypeOriginal Lease:Customer.IndustryCharity/Prof.Services/Other ***
## Transaction.TypeOriginal Lease:Customer.IndustryManufacturer
## Transaction.TypeRenewal:Customer.IndustryManufacturer
## Transaction.TypeOriginal Lease:Customer.IndustryRetailer
## Transaction.TypeRenewal:Customer.IndustryRetailer
## Transaction.TypeOriginal Lease:Customer.IndustryTransportation/Freight/Delivery
## Transaction.TypeRenewal:Customer.IndustryTransportation/Freight/Delivery
## Transaction.TypeOriginal Lease:Customer.IndustryUnspecified
## Transaction.TypeOriginal Lease:Customer.IndustryWholesaler
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Cross Validation:
library(caret)
plot(finalmodel)
anova(fullmodel, finalmodel)
predictions = finalmodel %>% predict(testing_set, interval = "prediction")
RMSE(predictions, testing_set$log_totalrent)
22.39863/mean(testing_set$log_totalrent)
R2(predictions, testing_set$log_totalrent)
mydata <- cbind(testing_set, predictions)
p <- ggplot(mydata, aes( (Property...Size..Sq..Ft../10000), (log_totalrent))) +
geom_point() +
stat_smooth(method = lm)
p + geom_line(aes(y = (lwr)), color = "red", linetype = "dashed")+
geom_line(aes(y = (upr)), color = "red", linetype = "dashed")From the output above, the R2 is 0.84, meaning that the observed and the predicted outcome values are highly correlated, which is good. The prediction error RMSE is 22.29, representing an error rate of 1.58% which is good.
Clearly we can observe that it violating the assumptions of normality in the predicted dataset and the MLR is not an ideal regression model for this.
ggplot(prologis_data, aes(x = prologis_data$Property...Size..Sq..Ft.., y = prologis_data$log_totalrent)) +
geom_point() +
stat_smooth()## Warning: Use of `prologis_data$Property...Size..Sq..Ft..` is discouraged. Use
## `Property...Size..Sq..Ft..` instead.## Warning: Use of `prologis_data$log_totalrent` is discouraged. Use
## `log_totalrent` instead.## Warning: Use of `prologis_data$Property...Size..Sq..Ft..` is discouraged. Use
## `Property...Size..Sq..Ft..` instead.## Warning: Use of `prologis_data$log_totalrent` is discouraged. Use
## `log_totalrent` instead.## `geom_smooth()` using method = 'loess' and formula 'y ~ x'
Polynomial Model
We can see that there needs to be polyomial terms added to the dataset from the graph above.
Once you have detected a non-linear relationship in your data, the polynomial terms may not be flexible enough to capture the relationship, and spline terms require specifying the knots.
Generalized additive models, or GAM, are a technique to automatically fit a spline regression. This can be done using the mgcv R package:
library(mgcv)## Loading required package: nlme##
## Attaching package: 'nlme'## The following object is masked from 'package:dplyr':
##
## collapse## This is mgcv 1.8-35. For overview type 'help("mgcv-package")'.library(itsadug)## Warning: package 'itsadug' was built under R version 4.1.2## Loading required package: plotfunctions## Warning: package 'plotfunctions' was built under R version 4.1.2##
## Attaching package: 'plotfunctions'## The following object is masked from 'package:ggplot2':
##
## alpha## Loaded package itsadug 2.4 (see 'help("itsadug")' ).model <- bam(log_totalrent ~ s(Lease...Total.Term..Month.) + s(Property...Size..Sq..Ft..) + s(Property...Speed.Bay) + (Transaction.Type:Customer.Industry)+ Property...Year.Built , data = training_set, discrete=TRUE)
summary(model)##
## Family: gaussian
## Link function: identity
##
## Formula:
## log_totalrent ~ s(Lease...Total.Term..Month.) + s(Property...Size..Sq..Ft..) +
## s(Property...Speed.Bay) + (Transaction.Type:Customer.Industry) +
## Property...Year.Built
##
## Parametric coefficients:
## Estimate
## (Intercept) 0.0000
## Property...Year.Built1984 -0.6370
## Property...Year.Built1985 -0.5624
## Property...Year.Built1986 -0.4729
## Property...Year.Built1987 -0.5210
## Property...Year.Built1988 -0.5124
## Property...Year.Built1989 -0.3716
## Property...Year.Built1990 -0.4843
## Property...Year.Built1995 -0.4709
## Property...Year.Built1996 -0.2281
## Property...Year.Built1997 -0.4314
## Property...Year.Built1998 -0.4080
## Property...Year.Built1999 -0.3677
## Property...Year.Built2000 -0.3638
## Property...Year.Built2001 -0.4140
## Property...Year.Built2002 -0.4028
## Property...Year.Built2003 -0.3011
## Property...Year.Built2005 -0.3828
## Property...Year.Built2006 -0.5052
## Property...Year.Built2008 -0.2312
## Property...Year.Built2014 -0.3017
## Property...Year.Built2015 -0.4091
## Property...Year.Built2016 -0.3843
## Transaction.TypeOriginal Lease:Customer.Industry3PL 14.3470
## Transaction.TypeRenewal:Customer.Industry3PL 14.2914
## Transaction.TypeOriginal Lease:Customer.IndustryCharity/Prof.Services/Other 13.7774
## Transaction.TypeRenewal:Customer.IndustryCharity/Prof.Services/Other 0.0000
## Transaction.TypeOriginal Lease:Customer.IndustryManufacturer 14.2818
## Transaction.TypeRenewal:Customer.IndustryManufacturer 14.2124
## Transaction.TypeOriginal Lease:Customer.IndustryRetailer 14.3336
## Transaction.TypeRenewal:Customer.IndustryRetailer 14.2901
## Transaction.TypeOriginal Lease:Customer.IndustryTransportation/Freight/Delivery 14.2889
## Transaction.TypeRenewal:Customer.IndustryTransportation/Freight/Delivery 14.1775
## Transaction.TypeOriginal Lease:Customer.IndustryUnspecified 14.3604
## Transaction.TypeRenewal:Customer.IndustryUnspecified 0.0000
## Transaction.TypeOriginal Lease:Customer.IndustryWholesaler 14.3839
## Transaction.TypeRenewal:Customer.IndustryWholesaler 14.3000
## Std. Error
## (Intercept) 0.0000
## Property...Year.Built1984 0.1980
## Property...Year.Built1985 0.2110
## Property...Year.Built1986 0.2050
## Property...Year.Built1987 0.2124
## Property...Year.Built1988 0.1832
## Property...Year.Built1989 0.2112
## Property...Year.Built1990 0.1819
## Property...Year.Built1995 0.2160
## Property...Year.Built1996 0.2357
## Property...Year.Built1997 0.2124
## Property...Year.Built1998 0.2297
## Property...Year.Built1999 0.2266
## Property...Year.Built2000 0.2068
## Property...Year.Built2001 0.1929
## Property...Year.Built2002 0.1803
## Property...Year.Built2003 0.2009
## Property...Year.Built2005 0.2113
## Property...Year.Built2006 0.1949
## Property...Year.Built2008 0.2592
## Property...Year.Built2014 0.2426
## Property...Year.Built2015 0.2071
## Property...Year.Built2016 0.5276
## Transaction.TypeOriginal Lease:Customer.Industry3PL 0.1954
## Transaction.TypeRenewal:Customer.Industry3PL 0.1992
## Transaction.TypeOriginal Lease:Customer.IndustryCharity/Prof.Services/Other 0.1114
## Transaction.TypeRenewal:Customer.IndustryCharity/Prof.Services/Other 0.0000
## Transaction.TypeOriginal Lease:Customer.IndustryManufacturer 0.1955
## Transaction.TypeRenewal:Customer.IndustryManufacturer 0.2166
## Transaction.TypeOriginal Lease:Customer.IndustryRetailer 0.2452
## Transaction.TypeRenewal:Customer.IndustryRetailer 0.2106
## Transaction.TypeOriginal Lease:Customer.IndustryTransportation/Freight/Delivery 0.2404
## Transaction.TypeRenewal:Customer.IndustryTransportation/Freight/Delivery 0.2356
## Transaction.TypeOriginal Lease:Customer.IndustryUnspecified 0.1771
## Transaction.TypeRenewal:Customer.IndustryUnspecified 0.0000
## Transaction.TypeOriginal Lease:Customer.IndustryWholesaler 0.2047
## Transaction.TypeRenewal:Customer.IndustryWholesaler 0.1908
## t value
## (Intercept) NaN
## Property...Year.Built1984 -3.218
## Property...Year.Built1985 -2.666
## Property...Year.Built1986 -2.307
## Property...Year.Built1987 -2.453
## Property...Year.Built1988 -2.797
## Property...Year.Built1989 -1.759
## Property...Year.Built1990 -2.662
## Property...Year.Built1995 -2.180
## Property...Year.Built1996 -0.968
## Property...Year.Built1997 -2.031
## Property...Year.Built1998 -1.776
## Property...Year.Built1999 -1.623
## Property...Year.Built2000 -1.759
## Property...Year.Built2001 -2.146
## Property...Year.Built2002 -2.234
## Property...Year.Built2003 -1.499
## Property...Year.Built2005 -1.812
## Property...Year.Built2006 -2.592
## Property...Year.Built2008 -0.892
## Property...Year.Built2014 -1.244
## Property...Year.Built2015 -1.976
## Property...Year.Built2016 -0.728
## Transaction.TypeOriginal Lease:Customer.Industry3PL 73.419
## Transaction.TypeRenewal:Customer.Industry3PL 71.760
## Transaction.TypeOriginal Lease:Customer.IndustryCharity/Prof.Services/Other 123.656
## Transaction.TypeRenewal:Customer.IndustryCharity/Prof.Services/Other NaN
## Transaction.TypeOriginal Lease:Customer.IndustryManufacturer 73.071
## Transaction.TypeRenewal:Customer.IndustryManufacturer 65.629
## Transaction.TypeOriginal Lease:Customer.IndustryRetailer 58.452
## Transaction.TypeRenewal:Customer.IndustryRetailer 67.847
## Transaction.TypeOriginal Lease:Customer.IndustryTransportation/Freight/Delivery 59.450
## Transaction.TypeRenewal:Customer.IndustryTransportation/Freight/Delivery 60.172
## Transaction.TypeOriginal Lease:Customer.IndustryUnspecified 81.084
## Transaction.TypeRenewal:Customer.IndustryUnspecified NaN
## Transaction.TypeOriginal Lease:Customer.IndustryWholesaler 70.284
## Transaction.TypeRenewal:Customer.IndustryWholesaler 74.963
## Pr(>|t|)
## (Intercept) NaN
## Property...Year.Built1984 0.00522
## Property...Year.Built1985 0.01663
## Property...Year.Built1986 0.03437
## Property...Year.Built1987 0.02566
## Property...Year.Built1988 0.01267
## Property...Year.Built1989 0.09711
## Property...Year.Built1990 0.01676
## Property...Year.Built1995 0.04413
## Property...Year.Built1996 0.34726
## Property...Year.Built1997 0.05876
## Property...Year.Built1998 0.09419
## Property...Year.Built1999 0.12368
## Property...Year.Built2000 0.09715
## Property...Year.Built2001 0.04713
## Property...Year.Built2002 0.03966
## Property...Year.Built2003 0.15290
## Property...Year.Built2005 0.08828
## Property...Year.Built2006 0.01933
## Property...Year.Built2008 0.38514
## Property...Year.Built2014 0.23098
## Property...Year.Built2015 0.06522
## Property...Year.Built2016 0.47666
## Transaction.TypeOriginal Lease:Customer.Industry3PL < 2e-16
## Transaction.TypeRenewal:Customer.Industry3PL < 2e-16
## Transaction.TypeOriginal Lease:Customer.IndustryCharity/Prof.Services/Other < 2e-16
## Transaction.TypeRenewal:Customer.IndustryCharity/Prof.Services/Other NaN
## Transaction.TypeOriginal Lease:Customer.IndustryManufacturer < 2e-16
## Transaction.TypeRenewal:Customer.IndustryManufacturer < 2e-16
## Transaction.TypeOriginal Lease:Customer.IndustryRetailer < 2e-16
## Transaction.TypeRenewal:Customer.IndustryRetailer < 2e-16
## Transaction.TypeOriginal Lease:Customer.IndustryTransportation/Freight/Delivery < 2e-16
## Transaction.TypeRenewal:Customer.IndustryTransportation/Freight/Delivery < 2e-16
## Transaction.TypeOriginal Lease:Customer.IndustryUnspecified < 2e-16
## Transaction.TypeRenewal:Customer.IndustryUnspecified NaN
## Transaction.TypeOriginal Lease:Customer.IndustryWholesaler < 2e-16
## Transaction.TypeRenewal:Customer.IndustryWholesaler < 2e-16
##
## (Intercept)
## Property...Year.Built1984 **
## Property...Year.Built1985 *
## Property...Year.Built1986 *
## Property...Year.Built1987 *
## Property...Year.Built1988 *
## Property...Year.Built1989 .
## Property...Year.Built1990 *
## Property...Year.Built1995 *
## Property...Year.Built1996
## Property...Year.Built1997 .
## Property...Year.Built1998 .
## Property...Year.Built1999
## Property...Year.Built2000 .
## Property...Year.Built2001 *
## Property...Year.Built2002 *
## Property...Year.Built2003
## Property...Year.Built2005 .
## Property...Year.Built2006 *
## Property...Year.Built2008
## Property...Year.Built2014
## Property...Year.Built2015 .
## Property...Year.Built2016
## Transaction.TypeOriginal Lease:Customer.Industry3PL ***
## Transaction.TypeRenewal:Customer.Industry3PL ***
## Transaction.TypeOriginal Lease:Customer.IndustryCharity/Prof.Services/Other ***
## Transaction.TypeRenewal:Customer.IndustryCharity/Prof.Services/Other
## Transaction.TypeOriginal Lease:Customer.IndustryManufacturer ***
## Transaction.TypeRenewal:Customer.IndustryManufacturer ***
## Transaction.TypeOriginal Lease:Customer.IndustryRetailer ***
## Transaction.TypeRenewal:Customer.IndustryRetailer ***
## Transaction.TypeOriginal Lease:Customer.IndustryTransportation/Freight/Delivery ***
## Transaction.TypeRenewal:Customer.IndustryTransportation/Freight/Delivery ***
## Transaction.TypeOriginal Lease:Customer.IndustryUnspecified ***
## Transaction.TypeRenewal:Customer.IndustryUnspecified
## Transaction.TypeOriginal Lease:Customer.IndustryWholesaler ***
## Transaction.TypeRenewal:Customer.IndustryWholesaler ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Approximate significance of smooth terms:
## edf Ref.df F p-value
## s(Lease...Total.Term..Month.) 1.000 1.000 0.406 0.533
## s(Property...Size..Sq..Ft..) 7.207 7.795 98.676 <2e-16 ***
## s(Property...Speed.Bay) 1.343 1.585 0.385 0.557
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Rank: 61/64
## R-sq.(adj) = 0.993 Deviance explained = 99.8%
## fREML = 5.3422 Scale est. = 0.0079524 n = 60plot(model, all.terms=TRUE, rug=FALSE)
#predictions <- model %>% predict(testing_set)
#data.frame(
#RMSE = RMSE(predictions, testing_set$log_totalrent),
#R2 = R2(predictions, testing_set$log_totalrent)
#)
plot_parametric(model, pred=list(Transaction.Type=c('Renewal', 'Original Lease')))## Summary:
## * Property...Year.Built : factor; set to the value(s): 1988.
## * Transaction.Type : factor; set to the value(s): Original Lease, Renewal.
## * Customer.Industry : factor; set to the value(s): Wholesaler.
## * Lease...Total.Term..Month. : numeric predictor; set to the value(s): 61.
## * Property...Size..Sq..Ft.. : numeric predictor; set to the value(s): 186300.
## * Property...Speed.Bay : numeric predictor; set to the value(s): 58.
## * NOTE : No random effects in the model to cancel.
## 
plot_parametric(model, pred=list(Customer.Industry=c('3PL', 'Charity/Prof.Services/Other', 'Transportation/Freight/Delivery' , 'Retailer', 'Unspecified', 'Wholesaler' )))## Summary:
## * Property...Year.Built : factor; set to the value(s): 1988.
## * Transaction.Type : factor; set to the value(s): Original Lease.
## * Customer.Industry : factor; set to the value(s): 3PL, Charity/Prof.Services/Other, Retailer, Transportation/Freight/Delivery, Unspecified, Wholesaler.
## * Lease...Total.Term..Month. : numeric predictor; set to the value(s): 61.
## * Property...Size..Sq..Ft.. : numeric predictor; set to the value(s): 186300.
## * Property...Speed.Bay : numeric predictor; set to the value(s): 58.
## * NOTE : No random effects in the model to cancel.
## 
#plot_data(model, view="Property...Year.Built", split_by="Transaction.Type", cex=.5)
## Finally we check the relative importance of our model:
#predictions = model %>% predict(testing_set, interval = "prediction")
#RMSE(predictions, testing_set$log_totalrent)
#5.425138/mean(testing_set$log_totalrent)
#R2(predictions, testing_set$log_totalrent)
coeftest(model, vcov = vcovHC(model, "HC1"))## Warning in model.matrix.default(fm, data, contrasts.arg): variable
## 'Transaction.Type' is absent, its contrast will be ignored## Warning in model.matrix.default(fm, data, contrasts.arg): variable
## 'Customer.Industry' is absent, its contrast will be ignored## Warning in model.matrix.default(fm, data, contrasts.arg): variable
## 'Property...Year.Built' is absent, its contrast will be ignored## Warning in model.matrix.default(fm, data, contrasts.arg): variable
## 'Transaction.Type' is absent, its contrast will be ignored## Warning in model.matrix.default(fm, data, contrasts.arg): variable
## 'Property...Year.Built' is absent, its contrast will be ignored## Warning in model.matrix.default(fm, data, contrasts.arg): variable
## 'Customer.Industry' is absent, its contrast will be ignored## Warning in model.matrix.default(fm, data, contrasts.arg): variable
## 'Transaction.Type' is absent, its contrast will be ignored## Warning in model.matrix.default(fm, data, contrasts.arg): variable
## 'Customer.Industry' is absent, its contrast will be ignored## Warning in model.matrix.default(fm, data, contrasts.arg): variable
## 'Property...Year.Built' is absent, its contrast will be ignored## Warning in model.matrix.default(fm, data, contrasts.arg): variable
## 'Transaction.Type' is absent, its contrast will be ignored## Warning in model.matrix.default(fm, data, contrasts.arg): variable
## 'Property...Year.Built' is absent, its contrast will be ignored## Warning in model.matrix.default(fm, data, contrasts.arg): variable
## 'Customer.Industry' is absent, its contrast will be ignored## Warning in sqrt(omega): NaNs produced## Warning in model.matrix.default(fm, data, contrasts.arg): variable
## 'Transaction.Type' is absent, its contrast will be ignored## Warning in model.matrix.default(fm, data, contrasts.arg): variable
## 'Customer.Industry' is absent, its contrast will be ignored## Warning in model.matrix.default(fm, data, contrasts.arg): variable
## 'Property...Year.Built' is absent, its contrast will be ignored## Warning in model.matrix.default(fm, data, contrasts.arg): variable
## 'Transaction.Type' is absent, its contrast will be ignored## Warning in model.matrix.default(fm, data, contrasts.arg): variable
## 'Property...Year.Built' is absent, its contrast will be ignored## Warning in model.matrix.default(fm, data, contrasts.arg): variable
## 'Customer.Industry' is absent, its contrast will be ignored##
## z test of coefficients:
##
## Estimate
## (Intercept) 0.0000e+00
## Property...Year.Built1984 -6.3698e-01
## Property...Year.Built1985 -5.6241e-01
## Property...Year.Built1986 -4.7286e-01
## Property...Year.Built1987 -5.2098e-01
## Property...Year.Built1988 -5.1244e-01
## Property...Year.Built1989 -3.7163e-01
## Property...Year.Built1990 -4.8434e-01
## Property...Year.Built1995 -4.7092e-01
## Property...Year.Built1996 -2.2806e-01
## Property...Year.Built1997 -4.3137e-01
## Property...Year.Built1998 -4.0800e-01
## Property...Year.Built1999 -3.6769e-01
## Property...Year.Built2000 -3.6376e-01
## Property...Year.Built2001 -4.1402e-01
## Property...Year.Built2002 -4.0282e-01
## Property...Year.Built2003 -3.0110e-01
## Property...Year.Built2005 -3.8283e-01
## Property...Year.Built2006 -5.0522e-01
## Property...Year.Built2008 -2.3124e-01
## Property...Year.Built2014 -3.0171e-01
## Property...Year.Built2015 -4.0911e-01
## Property...Year.Built2016 -3.8429e-01
## Transaction.TypeOriginal Lease:Customer.Industry3PL 1.4347e+01
## Transaction.TypeRenewal:Customer.Industry3PL 1.4291e+01
## Transaction.TypeOriginal Lease:Customer.IndustryCharity/Prof.Services/Other 1.3777e+01
## Transaction.TypeRenewal:Customer.IndustryCharity/Prof.Services/Other 0.0000e+00
## Transaction.TypeOriginal Lease:Customer.IndustryManufacturer 1.4282e+01
## Transaction.TypeRenewal:Customer.IndustryManufacturer 1.4212e+01
## Transaction.TypeOriginal Lease:Customer.IndustryRetailer 1.4334e+01
## Transaction.TypeRenewal:Customer.IndustryRetailer 1.4290e+01
## Transaction.TypeOriginal Lease:Customer.IndustryTransportation/Freight/Delivery 1.4289e+01
## Transaction.TypeRenewal:Customer.IndustryTransportation/Freight/Delivery 1.4178e+01
## Transaction.TypeOriginal Lease:Customer.IndustryUnspecified 1.4360e+01
## Transaction.TypeRenewal:Customer.IndustryUnspecified 0.0000e+00
## Transaction.TypeOriginal Lease:Customer.IndustryWholesaler 1.4384e+01
## Transaction.TypeRenewal:Customer.IndustryWholesaler 1.4300e+01
## s(Lease...Total.Term..Month.).1 3.1265e-08
## s(Lease...Total.Term..Month.).2 -2.7646e-08
## s(Lease...Total.Term..Month.).3 -1.7344e-08
## s(Lease...Total.Term..Month.).4 -1.8344e-08
## s(Lease...Total.Term..Month.).5 -1.4584e-08
## s(Lease...Total.Term..Month.).6 1.6960e-08
## s(Lease...Total.Term..Month.).7 -1.0242e-08
## s(Lease...Total.Term..Month.).8 1.1175e-07
## s(Lease...Total.Term..Month.).9 2.7576e-02
## s(Property...Size..Sq..Ft..).1 -1.4301e+00
## s(Property...Size..Sq..Ft..).2 -3.2084e+00
## s(Property...Size..Sq..Ft..).3 -9.7465e-01
## s(Property...Size..Sq..Ft..).4 -1.8902e+00
## s(Property...Size..Sq..Ft..).5 -1.4539e+00
## s(Property...Size..Sq..Ft..).6 1.3591e+00
## s(Property...Size..Sq..Ft..).7 -1.0584e+00
## s(Property...Size..Sq..Ft..).8 3.8868e+00
## s(Property...Size..Sq..Ft..).9 2.8781e+00
## s(Property...Speed.Bay).1 1.0377e-02
## s(Property...Speed.Bay).2 6.3200e-03
## s(Property...Speed.Bay).3 6.5931e-03
## s(Property...Speed.Bay).4 -6.9459e-03
## s(Property...Speed.Bay).5 2.0085e-03
## s(Property...Speed.Bay).6 9.0063e-03
## s(Property...Speed.Bay).7 6.9554e-03
## s(Property...Speed.Bay).8 4.0825e-02
## s(Property...Speed.Bay).9 -3.6304e-02
## Std. Error
## (Intercept) NaN
## Property...Year.Built1984 NaN
## Property...Year.Built1985 NaN
## Property...Year.Built1986 NaN
## Property...Year.Built1987 NaN
## Property...Year.Built1988 NaN
## Property...Year.Built1989 NaN
## Property...Year.Built1990 NaN
## Property...Year.Built1995 NaN
## Property...Year.Built1996 NaN
## Property...Year.Built1997 NaN
## Property...Year.Built1998 NaN
## Property...Year.Built1999 NaN
## Property...Year.Built2000 NaN
## Property...Year.Built2001 NaN
## Property...Year.Built2002 NaN
## Property...Year.Built2003 NaN
## Property...Year.Built2005 NaN
## Property...Year.Built2006 NaN
## Property...Year.Built2008 NaN
## Property...Year.Built2014 NaN
## Property...Year.Built2015 NaN
## Property...Year.Built2016 NaN
## Transaction.TypeOriginal Lease:Customer.Industry3PL NaN
## Transaction.TypeRenewal:Customer.Industry3PL NaN
## Transaction.TypeOriginal Lease:Customer.IndustryCharity/Prof.Services/Other NaN
## Transaction.TypeRenewal:Customer.IndustryCharity/Prof.Services/Other NaN
## Transaction.TypeOriginal Lease:Customer.IndustryManufacturer NaN
## Transaction.TypeRenewal:Customer.IndustryManufacturer NaN
## Transaction.TypeOriginal Lease:Customer.IndustryRetailer NaN
## Transaction.TypeRenewal:Customer.IndustryRetailer NaN
## Transaction.TypeOriginal Lease:Customer.IndustryTransportation/Freight/Delivery NaN
## Transaction.TypeRenewal:Customer.IndustryTransportation/Freight/Delivery NaN
## Transaction.TypeOriginal Lease:Customer.IndustryUnspecified NaN
## Transaction.TypeRenewal:Customer.IndustryUnspecified NaN
## Transaction.TypeOriginal Lease:Customer.IndustryWholesaler NaN
## Transaction.TypeRenewal:Customer.IndustryWholesaler NaN
## s(Lease...Total.Term..Month.).1 NaN
## s(Lease...Total.Term..Month.).2 NaN
## s(Lease...Total.Term..Month.).3 NaN
## s(Lease...Total.Term..Month.).4 NaN
## s(Lease...Total.Term..Month.).5 NaN
## s(Lease...Total.Term..Month.).6 NaN
## s(Lease...Total.Term..Month.).7 NaN
## s(Lease...Total.Term..Month.).8 NaN
## s(Lease...Total.Term..Month.).9 NaN
## s(Property...Size..Sq..Ft..).1 NaN
## s(Property...Size..Sq..Ft..).2 NaN
## s(Property...Size..Sq..Ft..).3 NaN
## s(Property...Size..Sq..Ft..).4 NaN
## s(Property...Size..Sq..Ft..).5 NaN
## s(Property...Size..Sq..Ft..).6 NaN
## s(Property...Size..Sq..Ft..).7 NaN
## s(Property...Size..Sq..Ft..).8 NaN
## s(Property...Size..Sq..Ft..).9 NaN
## s(Property...Speed.Bay).1 NaN
## s(Property...Speed.Bay).2 NaN
## s(Property...Speed.Bay).3 NaN
## s(Property...Speed.Bay).4 NaN
## s(Property...Speed.Bay).5 NaN
## s(Property...Speed.Bay).6 NaN
## s(Property...Speed.Bay).7 NaN
## s(Property...Speed.Bay).8 NaN
## s(Property...Speed.Bay).9 NaN
## z value
## (Intercept) NaN
## Property...Year.Built1984 NaN
## Property...Year.Built1985 NaN
## Property...Year.Built1986 NaN
## Property...Year.Built1987 NaN
## Property...Year.Built1988 NaN
## Property...Year.Built1989 NaN
## Property...Year.Built1990 NaN
## Property...Year.Built1995 NaN
## Property...Year.Built1996 NaN
## Property...Year.Built1997 NaN
## Property...Year.Built1998 NaN
## Property...Year.Built1999 NaN
## Property...Year.Built2000 NaN
## Property...Year.Built2001 NaN
## Property...Year.Built2002 NaN
## Property...Year.Built2003 NaN
## Property...Year.Built2005 NaN
## Property...Year.Built2006 NaN
## Property...Year.Built2008 NaN
## Property...Year.Built2014 NaN
## Property...Year.Built2015 NaN
## Property...Year.Built2016 NaN
## Transaction.TypeOriginal Lease:Customer.Industry3PL NaN
## Transaction.TypeRenewal:Customer.Industry3PL NaN
## Transaction.TypeOriginal Lease:Customer.IndustryCharity/Prof.Services/Other NaN
## Transaction.TypeRenewal:Customer.IndustryCharity/Prof.Services/Other NaN
## Transaction.TypeOriginal Lease:Customer.IndustryManufacturer NaN
## Transaction.TypeRenewal:Customer.IndustryManufacturer NaN
## Transaction.TypeOriginal Lease:Customer.IndustryRetailer NaN
## Transaction.TypeRenewal:Customer.IndustryRetailer NaN
## Transaction.TypeOriginal Lease:Customer.IndustryTransportation/Freight/Delivery NaN
## Transaction.TypeRenewal:Customer.IndustryTransportation/Freight/Delivery NaN
## Transaction.TypeOriginal Lease:Customer.IndustryUnspecified NaN
## Transaction.TypeRenewal:Customer.IndustryUnspecified NaN
## Transaction.TypeOriginal Lease:Customer.IndustryWholesaler NaN
## Transaction.TypeRenewal:Customer.IndustryWholesaler NaN
## s(Lease...Total.Term..Month.).1 NaN
## s(Lease...Total.Term..Month.).2 NaN
## s(Lease...Total.Term..Month.).3 NaN
## s(Lease...Total.Term..Month.).4 NaN
## s(Lease...Total.Term..Month.).5 NaN
## s(Lease...Total.Term..Month.).6 NaN
## s(Lease...Total.Term..Month.).7 NaN
## s(Lease...Total.Term..Month.).8 NaN
## s(Lease...Total.Term..Month.).9 NaN
## s(Property...Size..Sq..Ft..).1 NaN
## s(Property...Size..Sq..Ft..).2 NaN
## s(Property...Size..Sq..Ft..).3 NaN
## s(Property...Size..Sq..Ft..).4 NaN
## s(Property...Size..Sq..Ft..).5 NaN
## s(Property...Size..Sq..Ft..).6 NaN
## s(Property...Size..Sq..Ft..).7 NaN
## s(Property...Size..Sq..Ft..).8 NaN
## s(Property...Size..Sq..Ft..).9 NaN
## s(Property...Speed.Bay).1 NaN
## s(Property...Speed.Bay).2 NaN
## s(Property...Speed.Bay).3 NaN
## s(Property...Speed.Bay).4 NaN
## s(Property...Speed.Bay).5 NaN
## s(Property...Speed.Bay).6 NaN
## s(Property...Speed.Bay).7 NaN
## s(Property...Speed.Bay).8 NaN
## s(Property...Speed.Bay).9 NaN
## Pr(>|z|)
## (Intercept) NaN
## Property...Year.Built1984 NaN
## Property...Year.Built1985 NaN
## Property...Year.Built1986 NaN
## Property...Year.Built1987 NaN
## Property...Year.Built1988 NaN
## Property...Year.Built1989 NaN
## Property...Year.Built1990 NaN
## Property...Year.Built1995 NaN
## Property...Year.Built1996 NaN
## Property...Year.Built1997 NaN
## Property...Year.Built1998 NaN
## Property...Year.Built1999 NaN
## Property...Year.Built2000 NaN
## Property...Year.Built2001 NaN
## Property...Year.Built2002 NaN
## Property...Year.Built2003 NaN
## Property...Year.Built2005 NaN
## Property...Year.Built2006 NaN
## Property...Year.Built2008 NaN
## Property...Year.Built2014 NaN
## Property...Year.Built2015 NaN
## Property...Year.Built2016 NaN
## Transaction.TypeOriginal Lease:Customer.Industry3PL NaN
## Transaction.TypeRenewal:Customer.Industry3PL NaN
## Transaction.TypeOriginal Lease:Customer.IndustryCharity/Prof.Services/Other NaN
## Transaction.TypeRenewal:Customer.IndustryCharity/Prof.Services/Other NaN
## Transaction.TypeOriginal Lease:Customer.IndustryManufacturer NaN
## Transaction.TypeRenewal:Customer.IndustryManufacturer NaN
## Transaction.TypeOriginal Lease:Customer.IndustryRetailer NaN
## Transaction.TypeRenewal:Customer.IndustryRetailer NaN
## Transaction.TypeOriginal Lease:Customer.IndustryTransportation/Freight/Delivery NaN
## Transaction.TypeRenewal:Customer.IndustryTransportation/Freight/Delivery NaN
## Transaction.TypeOriginal Lease:Customer.IndustryUnspecified NaN
## Transaction.TypeRenewal:Customer.IndustryUnspecified NaN
## Transaction.TypeOriginal Lease:Customer.IndustryWholesaler NaN
## Transaction.TypeRenewal:Customer.IndustryWholesaler NaN
## s(Lease...Total.Term..Month.).1 NaN
## s(Lease...Total.Term..Month.).2 NaN
## s(Lease...Total.Term..Month.).3 NaN
## s(Lease...Total.Term..Month.).4 NaN
## s(Lease...Total.Term..Month.).5 NaN
## s(Lease...Total.Term..Month.).6 NaN
## s(Lease...Total.Term..Month.).7 NaN
## s(Lease...Total.Term..Month.).8 NaN
## s(Lease...Total.Term..Month.).9 NaN
## s(Property...Size..Sq..Ft..).1 NaN
## s(Property...Size..Sq..Ft..).2 NaN
## s(Property...Size..Sq..Ft..).3 NaN
## s(Property...Size..Sq..Ft..).4 NaN
## s(Property...Size..Sq..Ft..).5 NaN
## s(Property...Size..Sq..Ft..).6 NaN
## s(Property...Size..Sq..Ft..).7 NaN
## s(Property...Size..Sq..Ft..).8 NaN
## s(Property...Size..Sq..Ft..).9 NaN
## s(Property...Speed.Bay).1 NaN
## s(Property...Speed.Bay).2 NaN
## s(Property...Speed.Bay).3 NaN
## s(Property...Speed.Bay).4 NaN
## s(Property...Speed.Bay).5 NaN
## s(Property...Speed.Bay).6 NaN
## s(Property...Speed.Bay).7 NaN
## s(Property...Speed.Bay).8 NaN
## s(Property...Speed.Bay).9 NaNThe prediction error RMSE is 5.425138, representing an error rate of 0.38% which is good.