Multi Linear Regression Assignment With Outliers
Monesh Sharma
July 23, 2017
Problem Definition
Use data set House Property. Find out which all columns / features impact changes in Price of house.
Create a linear model for all coulmns with corrrelation.
Predict the Price of house with some dummy values.
Data Location
The data for these sales comes from the official public records of home sales in the King County area, Washington State.
The data sets contains 21613 rows. Each represents a home sold from May 2014 through May 2015.
http://your.kingcounty.gov/assessor/eRealProperty/ResGlossaryOfTerms.html.
Data Description
Size: 21613 obs. of 21 variables:
Attributes:
id - Unique ID for each home sold
date - Date of the home sale
price - Price of each home sold
bedrooms - Number of bedrooms
bathrooms - Number of bathrooms, where .5 accounts for a room with a toilet but no shower
sqft_living - Square footage of the apartments interior living space
sqft_lot - Square footage of the land space
floors - Number of floors
waterfront - A variable for whether the apartment was overlooking the waterfront or not
view - An index from 0 to 4 of how good the view of the property was
condition - An index from 1 to 5 on the condition of the apartment,
grade - An index from 1 to 13, where 1-3 falls short of building construction and design, 7 has an average level of construction and design, and 11-13 have a high quality level of construction and design.
sqft_above - The square footage of the interior housing space that is above ground level
sqft_basement - The square footage of the interior housing space that is below ground level
yr_built - The year the house was initially built
yr_renovated - The year of the house’s last renovation
zipcode - What zipcode area the house is in
lat - Lattitude
long - Longitude
sqft_living15 - The square footage of interior housing living space for the nearest 15 neighbors
sqft_lot15 - The square footage of the land lots of the nearest 15 neighbors
Setup
library(tidyr)
library(dplyr)##
## 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(ggplot2)
library(corrgram)
library(gridExtra) ##
## Attaching package: 'gridExtra'## The following object is masked from 'package:dplyr':
##
## combineFunctions
detect_outliers <- function(inp, na.rm=TRUE) {
i.qnt <- quantile(inp, probs=c(.25, .75), na.rm=na.rm)
i.max <- 1.5 * IQR(inp, na.rm=na.rm)
otp <- inp
otp[inp < (i.qnt[1] - i.max)] <- NA
otp[inp > (i.qnt[2] + i.max)] <- NA
#inp <- count(inp[is.na(otp)])
sum(is.na(otp))
}Except_outliers <- function(x, na.rm = TRUE, ...) {
qnt <- quantile(x, probs=c(.25, .75), na.rm = na.rm, ...)
H <- 1.5 * IQR(x, na.rm = na.rm)
y <- x
y[x < (qnt[1] - H)] <- NA
y[x > (qnt[2] + H)] <- NA
y
}Remove_Outliers <- function ( z, na.rm = TRUE){
Out <- Except_outliers(z)
Out <-as.data.frame (Out)
z <- Out$Out[match(z, Out$Out)]
z
}Graph_Boxplot <- function (input, na.rm = TRUE){
Plot <- ggplot(dfrModel, aes(x="", y=input)) +
geom_boxplot(aes(fill=input), color="green") +
labs(title="Outliers")
Plot
}Dataset
dfrModel <- read.csv("./house-data.csv", header=T, stringsAsFactors=F)
head(dfrModel)## id date price bedrooms bathrooms sqft_living
## 1 7129300520 20141013T000000 221900 3 1.00 1180
## 2 6414100192 20141209T000000 538000 3 2.25 2570
## 3 5631500400 20150225T000000 180000 2 1.00 770
## 4 2487200875 20141209T000000 604000 4 3.00 1960
## 5 1954400510 20150218T000000 510000 3 2.00 1680
## 6 7237550310 20140512T000000 1225000 4 4.50 5420
## sqft_lot floors waterfront view condition grade sqft_above sqft_basement
## 1 5650 1 0 0 3 7 1180 0
## 2 7242 2 0 0 3 7 2170 400
## 3 10000 1 0 0 3 6 770 0
## 4 5000 1 0 0 5 7 1050 910
## 5 8080 1 0 0 3 8 1680 0
## 6 101930 1 0 0 3 11 3890 1530
## yr_built yr_renovated zipcode lat long sqft_living15 sqft_lot15
## 1 1955 0 98178 47.5112 -122.257 1340 5650
## 2 1951 1991 98125 47.7210 -122.319 1690 7639
## 3 1933 0 98028 47.7379 -122.233 2720 8062
## 4 1965 0 98136 47.5208 -122.393 1360 5000
## 5 1987 0 98074 47.6168 -122.045 1800 7503
## 6 2001 0 98053 47.6561 -122.005 4760 101930Observation
Since id, unknown let’s drop it.
dfrModel <- select(dfrModel, -c(id))
head(dfrModel)## date price bedrooms bathrooms sqft_living sqft_lot floors
## 1 20141013T000000 221900 3 1.00 1180 5650 1
## 2 20141209T000000 538000 3 2.25 2570 7242 2
## 3 20150225T000000 180000 2 1.00 770 10000 1
## 4 20141209T000000 604000 4 3.00 1960 5000 1
## 5 20150218T000000 510000 3 2.00 1680 8080 1
## 6 20140512T000000 1225000 4 4.50 5420 101930 1
## waterfront view condition grade sqft_above sqft_basement yr_built
## 1 0 0 3 7 1180 0 1955
## 2 0 0 3 7 2170 400 1951
## 3 0 0 3 6 770 0 1933
## 4 0 0 5 7 1050 910 1965
## 5 0 0 3 8 1680 0 1987
## 6 0 0 3 11 3890 1530 2001
## yr_renovated zipcode lat long sqft_living15 sqft_lot15
## 1 0 98178 47.5112 -122.257 1340 5650
## 2 1991 98125 47.7210 -122.319 1690 7639
## 3 0 98028 47.7379 -122.233 2720 8062
## 4 0 98136 47.5208 -122.393 1360 5000
## 5 0 98074 47.6168 -122.045 1800 7503
## 6 0 98053 47.6561 -122.005 4760 101930Summary
#summary(dfrModel)
lapply(dfrModel, FUN=summary)## $date
## Length Class Mode
## 21613 character character
##
## $price
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 75000 321950 450000 540088 645000 7700000
##
## $bedrooms
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.000 3.000 3.000 3.371 4.000 33.000
##
## $bathrooms
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.000 1.750 2.250 2.115 2.500 8.000
##
## $sqft_living
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 290 1427 1910 2080 2550 13540
##
## $sqft_lot
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 520 5040 7618 15107 10688 1651359
##
## $floors
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 1.000 1.000 1.500 1.494 2.000 3.500
##
## $waterfront
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.000000 0.000000 0.000000 0.007542 0.000000 1.000000
##
## $view
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.0000 0.0000 0.0000 0.2343 0.0000 4.0000
##
## $condition
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 1.000 3.000 3.000 3.409 4.000 5.000
##
## $grade
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 1.000 7.000 7.000 7.657 8.000 13.000
##
## $sqft_above
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 290 1190 1560 1788 2210 9410
##
## $sqft_basement
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.0 0.0 0.0 291.5 560.0 4820.0
##
## $yr_built
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 1900 1951 1975 1971 1997 2015
##
## $yr_renovated
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.0 0.0 0.0 84.4 0.0 2015.0
##
## $zipcode
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 98001 98033 98065 98078 98118 98199
##
## $lat
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 47.16 47.47 47.57 47.56 47.68 47.78
##
## $long
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## -122.5 -122.3 -122.2 -122.2 -122.1 -121.3
##
## $sqft_living15
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 399 1490 1840 1987 2360 6210
##
## $sqft_lot15
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 651 5100 7620 12768 10083 871200Data Cleaning
dfrModel$Sold_Yr <- substr(dfrModel$date, 1, 4)
dfrModel$Sold_Yr <- as.integer(dfrModel$Sold_Yr)
dfrModel$Sold_Month <- substr(dfrModel$date, 5, 6)
dfrModel$Sold_Month <- as.integer(dfrModel$Sold_Month)
dfrModel$yrs_Old <- (dfrModel$Sold_Yr - dfrModel$yr_built)
dfrModel <- select(dfrModel, -c(date))
dfrModel$renovated <- ifelse(dfrModel$yr_renovated == 0, 0, 1)
#dfrModel$sqft_lot_new <- (dfrModel$sqft_lot+dfrModel$sqft_lot15)/2
#dfrModel$sqft_living_new <- (dfrModel$sqft_living+dfrModel$sqft_living15)/2
#dfrModel$sqft_area <- (dfrModel$sqft_above+dfrModel$sqft_basement)/2
#dfrModel <- select(dfrModel, -c(sqft_lot, sqft_lot15, sqft_living, sqft_living15, sqft_above, sqft_basement, date))
#dfrModel$price <- Remove_Outliers(dfrModel$price)
#dfrModel$sqft_lot_new <- Remove_Outliers(dfrModel$sqft_lot_new)
#dfrModel <- dfrModel[complete.cases(dfrModel), ]
head(dfrModel)## price bedrooms bathrooms sqft_living sqft_lot floors waterfront view
## 1 221900 3 1.00 1180 5650 1 0 0
## 2 538000 3 2.25 2570 7242 2 0 0
## 3 180000 2 1.00 770 10000 1 0 0
## 4 604000 4 3.00 1960 5000 1 0 0
## 5 510000 3 2.00 1680 8080 1 0 0
## 6 1225000 4 4.50 5420 101930 1 0 0
## condition grade sqft_above sqft_basement yr_built yr_renovated zipcode
## 1 3 7 1180 0 1955 0 98178
## 2 3 7 2170 400 1951 1991 98125
## 3 3 6 770 0 1933 0 98028
## 4 5 7 1050 910 1965 0 98136
## 5 3 8 1680 0 1987 0 98074
## 6 3 11 3890 1530 2001 0 98053
## lat long sqft_living15 sqft_lot15 Sold_Yr Sold_Month yrs_Old
## 1 47.5112 -122.257 1340 5650 2014 10 59
## 2 47.7210 -122.319 1690 7639 2014 12 63
## 3 47.7379 -122.233 2720 8062 2015 2 82
## 4 47.5208 -122.393 1360 5000 2014 12 49
## 5 47.6168 -122.045 1800 7503 2015 2 28
## 6 47.6561 -122.005 4760 101930 2014 5 13
## renovated
## 1 0
## 2 1
## 3 0
## 4 0
## 5 0
## 6 0Observation
Below New Columns have been introduced which may affect the price of the house.
1. Year and Month of Sold from Date
2. No. of years house old, subracting year of sold and year of Built
3. Renovated or not renovated (Categorical 0 and)
#detect_outliers(dfrModel$bedrooms)
#detect_outliers(dfrModel$sqft_living15)
lapply(dfrModel, FUN=detect_outliers)## $price
## [1] 1146
##
## $bedrooms
## [1] 546
##
## $bathrooms
## [1] 571
##
## $sqft_living
## [1] 572
##
## $sqft_lot
## [1] 2425
##
## $floors
## [1] 0
##
## $waterfront
## [1] 163
##
## $view
## [1] 2124
##
## $condition
## [1] 30
##
## $grade
## [1] 1911
##
## $sqft_above
## [1] 611
##
## $sqft_basement
## [1] 496
##
## $yr_built
## [1] 0
##
## $yr_renovated
## [1] 914
##
## $zipcode
## [1] 0
##
## $lat
## [1] 2
##
## $long
## [1] 256
##
## $sqft_living15
## [1] 544
##
## $sqft_lot15
## [1] 2194
##
## $Sold_Yr
## [1] 0
##
## $Sold_Month
## [1] 0
##
## $yrs_Old
## [1] 0
##
## $renovated
## [1] 914** Obsevations**
We can see that no. of Outliers are more.
For this model we are going with Outliers.
lapply(dfrModel, FUN=Graph_Boxplot)## $price
##
## $bedrooms
##
## $bathrooms
##
## $sqft_living
##
## $sqft_lot
##
## $floors
##
## $waterfront
##
## $view
##
## $condition
##
## $grade
##
## $sqft_above
##
## $sqft_basement
##
## $yr_built
##
## $yr_renovated
##
## $zipcode
##
## $lat
##
## $long
##
## $sqft_living15
##
## $sqft_lot15
##
## $Sold_Yr
##
## $Sold_Month
##
## $yrs_Old
##
## $renovated
Observation
Here we see that many predictor variables numeric categoric variable. We should convert these to factor variables
dfrModel1<-dfrModel
dfrModel1$waterfront <- as.factor(dfrModel$waterfront)
levels(dfrModel1$waterfront) <- c("No-Waterfront", "Yes-Waterfront")
dfrModel1$view <- as.factor(dfrModel$view)
levels(dfrModel1$view) <- c("Bad View", "Average View", "Good View", "Very Good View", "Excellent View")
dfrModel1$condition <- as.factor(dfrModel$condition)
levels(dfrModel1$condition) <- c("Bad Condition", "Average Condition", "Good Condition", "Very Good Condition", "Excellent Condition")
dfrModel1$grade <- as.factor(dfrModel$grade)
head(dfrModel1)## price bedrooms bathrooms sqft_living sqft_lot floors waterfront
## 1 221900 3 1.00 1180 5650 1 No-Waterfront
## 2 538000 3 2.25 2570 7242 2 No-Waterfront
## 3 180000 2 1.00 770 10000 1 No-Waterfront
## 4 604000 4 3.00 1960 5000 1 No-Waterfront
## 5 510000 3 2.00 1680 8080 1 No-Waterfront
## 6 1225000 4 4.50 5420 101930 1 No-Waterfront
## view condition grade sqft_above sqft_basement yr_built
## 1 Bad View Good Condition 7 1180 0 1955
## 2 Bad View Good Condition 7 2170 400 1951
## 3 Bad View Good Condition 6 770 0 1933
## 4 Bad View Excellent Condition 7 1050 910 1965
## 5 Bad View Good Condition 8 1680 0 1987
## 6 Bad View Good Condition 11 3890 1530 2001
## yr_renovated zipcode lat long sqft_living15 sqft_lot15 Sold_Yr
## 1 0 98178 47.5112 -122.257 1340 5650 2014
## 2 1991 98125 47.7210 -122.319 1690 7639 2014
## 3 0 98028 47.7379 -122.233 2720 8062 2015
## 4 0 98136 47.5208 -122.393 1360 5000 2014
## 5 0 98074 47.6168 -122.045 1800 7503 2015
## 6 0 98053 47.6561 -122.005 4760 101930 2014
## Sold_Month yrs_Old renovated
## 1 10 59 0
## 2 12 63 1
## 3 2 82 0
## 4 12 49 0
## 5 2 28 0
## 6 5 13 0Correlation
vctCorr = numeric(0)
for (i in names(dfrModel)){
cor.result <- cor(dfrModel$price, as.numeric(dfrModel[,i]))
vctCorr <- c(vctCorr, cor.result)
}
dfrCorr <- vctCorr
names(dfrCorr) <- names(dfrModel)
dfrCorr## price bedrooms bathrooms sqft_living sqft_lot
## 1.000000000 0.308349598 0.525137505 0.702035052 0.089660860
## floors waterfront view condition grade
## 0.256793884 0.266369434 0.397293487 0.036361791 0.667434253
## sqft_above sqft_basement yr_built yr_renovated zipcode
## 0.605567295 0.323816022 0.054011527 0.126433794 -0.053202851
## lat long sqft_living15 sqft_lot15 Sold_Yr
## 0.307003482 0.021626239 0.585378902 0.082447152 0.003576038
## Sold_Month yrs_Old renovated
## -0.010080809 -0.053950773 0.126092263cor(dfrModel$sqft_living, dfrModel$sqft_living15)## [1] 0.7564203cor(dfrModel$sqft_lot, dfrModel$sqft_lot15)## [1] 0.7185568cor(dfrModel$sqft_above, dfrModel$sqft_basement)## [1] -0.05194331cor(dfrModel$sqft_above, dfrModel$sqft_living)## [1] 0.8765966cor(dfrModel$sqft_basement, dfrModel$sqft_living)## [1] 0.435043cor(dfrModel$bedrooms, dfrModel$bathrooms)## [1] 0.5158836cor (dfrModel$view, dfrModel$grade)## [1] 0.2513206** Observations**
As given above after checking the correlation between the Dependent (Price) variable and each independent variable, there are may variables for which correlation is very low.
As per Regression model Assumptions correlation should not be low to continue with the model, So we are removing all the columns for which correlation is less than 0.30.
Below are the columns which are going to remove,
sqft_lot, Floors, Waterfront, Condition, Yr_Built, Yr_Renovated, Zipcode, Long,sqft_lot15, Sold_Yr, Sold_Month, Yrs_Old, Renovate etc.
As well as we can see that correlation between sqft_living and sqft_living15 is high so both are dependent, so as per assumption of regression we can not take both variables, so we are only taking with whom correlation of price is more which is sqft_living.
Same for
correlation for sqft_living and sqft_above 0.8765 so need to drop sqft_above
correlation for bedrooms and bathrooms is 0.5158 so need to drop bedrooms
Data Cleaning 2
dfrModel <- select(dfrModel, -c(sqft_lot, floors, waterfront, condition, yr_built, yr_renovated, zipcode, long, sqft_lot15, Sold_Yr, Sold_Month, yrs_Old, renovated, sqft_living15, sqft_above,sqft_basement, bedrooms))
head(dfrModel)## price bathrooms sqft_living view grade lat
## 1 221900 1.00 1180 0 7 47.5112
## 2 538000 2.25 2570 0 7 47.7210
## 3 180000 1.00 770 0 6 47.7379
## 4 604000 3.00 1960 0 7 47.5208
## 5 510000 2.00 1680 0 8 47.6168
## 6 1225000 4.50 5420 0 11 47.6561Summary
#summary(dfrModel)
lapply(dfrModel, FUN=summary)## $price
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 75000 321950 450000 540088 645000 7700000
##
## $bathrooms
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.000 1.750 2.250 2.115 2.500 8.000
##
## $sqft_living
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 290 1427 1910 2080 2550 13540
##
## $view
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.0000 0.0000 0.0000 0.2343 0.0000 4.0000
##
## $grade
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 1.000 7.000 7.000 7.657 8.000 13.000
##
## $lat
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 47.16 47.47 47.57 47.56 47.68 47.78#detect_outliers(dfrModel$bedrooms)
#detect_outliers(dfrModel$sqft_living15)
lapply(dfrModel, FUN=detect_outliers)## $price
## [1] 1146
##
## $bathrooms
## [1] 571
##
## $sqft_living
## [1] 572
##
## $view
## [1] 2124
##
## $grade
## [1] 1911
##
## $lat
## [1] 2lapply(dfrModel, FUN=Graph_Boxplot)## $price
##
## $bathrooms
##
## $sqft_living
##
## $view
##
## $grade
##
## $lat
Observation
Outliers present in many features.
For this Model we are not removing Outliers.
head(dfrModel)## price bathrooms sqft_living view grade lat
## 1 221900 1.00 1180 0 7 47.5112
## 2 538000 2.25 2570 0 7 47.7210
## 3 180000 1.00 770 0 6 47.7379
## 4 604000 3.00 1960 0 7 47.5208
## 5 510000 2.00 1680 0 8 47.6168
## 6 1225000 4.50 5420 0 11 47.6561Visualize
dfrGraph <- gather(dfrModel, variable, value, -price)
head(dfrGraph)## price variable value
## 1 221900 bathrooms 1.00
## 2 538000 bathrooms 2.25
## 3 180000 bathrooms 1.00
## 4 604000 bathrooms 3.00
## 5 510000 bathrooms 2.00
## 6 1225000 bathrooms 4.50ggplot(dfrGraph) +
geom_jitter(aes(value,price, colour=variable)) +
geom_smooth(aes(value,price, colour=variable), method=lm, se=FALSE) +
facet_wrap(~variable, scales="free_x") +
labs(title="Relation Of Price With Other Features")
Observation
There is some impact of all the features with Price.
Find Best Multi Linear Model
Choose the best linear model by using step(). Choose a model by AIC in a Stepwise Algorithm
In statistics, stepwise regression is a method of fitting regression models in which the choice of predictive variables is carried out by an automatic procedure. In each step, a variable is considered for addition to or subtraction from the set of explanatory variables based on some prespecified criterion.
The Akaike information criterion (AIC) is a measure of the relative quality of statistical models for a given set of data. Given a collection of models for the data, AIC estimates the quality of each model, relative to each of the other models. Hence, AIC provides a means for model selection.
#?step()
stpModel=step(lm(data=dfrModel, price~.), trace=0, steps=10000)
stpSummary <- summary(stpModel)
stpSummary ##
## Call:
## lm(formula = price ~ bathrooms + sqft_living + view + grade +
## lat, data = dfrModel)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1209579 -111673 -15912 77148 4681081
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -3.227e+07 5.202e+05 -62.03 < 2e-16 ***
## bathrooms -2.084e+04 3.056e+03 -6.82 9.32e-12 ***
## sqft_living 1.848e+02 2.988e+00 61.84 < 2e-16 ***
## view 9.879e+04 2.056e+03 48.05 < 2e-16 ***
## grade 8.225e+04 2.046e+03 40.21 < 2e-16 ***
## lat 6.690e+05 1.097e+04 61.01 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 221300 on 21607 degrees of freedom
## Multiple R-squared: 0.6367, Adjusted R-squared: 0.6367
## F-statistic: 7575 on 5 and 21607 DF, p-value: < 2.2e-16Observation
Best results given by price ~ bathrooms + view + grade + lat + sqft_living.
Multiple R square value is also 0.6367 which is showing it as a average to good model.
Make Final Multi Linear Model
#x1 <- dfrModel$bedrooms
x2 <- dfrModel$bathrooms
x3 <- dfrModel$view
x4 <- dfrModel$grade
x5 <- dfrModel$lat
#x6 <- dfrModel$sqft_above
x7 <- dfrModel$sqft_living
y <- dfrModel$price
slmModel <- lm(y~x2+x3+x4+x5+x7, data=dfrModel)Observation
No errors. Model successfully created.
Show Model
# print summary
summary(slmModel)##
## Call:
## lm(formula = y ~ x2 + x3 + x4 + x5 + x7, data = dfrModel)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1209579 -111673 -15912 77148 4681081
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -3.227e+07 5.202e+05 -62.03 < 2e-16 ***
## x2 -2.084e+04 3.056e+03 -6.82 9.32e-12 ***
## x3 9.879e+04 2.056e+03 48.05 < 2e-16 ***
## x4 8.225e+04 2.046e+03 40.21 < 2e-16 ***
## x5 6.690e+05 1.097e+04 61.01 < 2e-16 ***
## x7 1.848e+02 2.988e+00 61.84 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 221300 on 21607 degrees of freedom
## Multiple R-squared: 0.6367, Adjusted R-squared: 0.6367
## F-statistic: 7575 on 5 and 21607 DF, p-value: < 2.2e-16Test Data
# find mpg of a person with weight 3.0
dfrTest <- data.frame(x2=c(1, 1.5, 2.25),x3=c(0, 1, 0), x4=c(7,10,7), x5=c(47.5112, 47.721, 47.721), x7=c(1180, 1300, 2570))
dfrTest ## x2 x3 x4 x5 x7
## 1 1.00 0 7 47.5112 1180
## 2 1.50 1 10 47.7210 1300
## 3 2.25 0 7 47.7210 2570#names(dfrTest) <- c("x1","x2","x3")
#dfrTest Observation
Test Data successfully created.
Predict
result <- predict(slmModel, dfrTest)
print(result)## 1 2 3
## 287196.8 784824.0 658320.8** Observation**
1. Prediction is working fine but have some deviation from true values as r squared values was less.
Summary
- This Model is with the Outliers
- Prediction Model created successfully with five independent variables.
- All the required assumptions for Regression model is tested successfully.
- There must be correlation between dependent and independent variable
- There should not be any correlation between two independent variables.
- Test data has been successfully tested and there are some variance with true values as Multiple R square value is around 0.64 so there might be 36% variance in the predicted output with true output.