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data1<- read.csv(file.choose())
summary(data1)## ï..Price County Size Elevation
## Min. : 1.70 Min. :0.0000 Min. : 6.90 Min. : 0.000
## 1st Qu.: 5.35 1st Qu.:0.0000 1st Qu.: 20.35 1st Qu.: 2.000
## Median :11.70 Median :1.0000 Median : 51.40 Median : 4.000
## Mean :11.95 Mean :0.6129 Mean : 139.97 Mean : 4.645
## 3rd Qu.:16.05 3rd Qu.:1.0000 3rd Qu.: 104.10 3rd Qu.: 7.000
## Max. :37.20 Max. :1.0000 Max. :1695.20 Max. :20.000
## Sewer Date Flood Distance
## Min. : 0 Min. :-103.00 Min. :0.0000 Min. : 0.000
## 1st Qu.: 0 1st Qu.: -63.50 1st Qu.:0.0000 1st Qu.: 0.850
## Median : 900 Median : -59.00 Median :0.0000 Median : 4.900
## Mean : 1981 Mean : -58.65 Mean :0.1613 Mean : 5.132
## 3rd Qu.: 3450 3rd Qu.: -51.00 3rd Qu.:0.0000 3rd Qu.: 5.500
## Max. :10000 Max. : -4.00 Max. :1.0000 Max. :16.500#drawing a histogram to check how the price of plot is distributed
hist(data1$ï..Price)
#Histogram is skewed. Looks the average price of the plot could be around 10k/acre
#but to have a better histogram; lets take a log to base 10 for Price.
#just transforming the price factor
logprice <- log(data1$ï..Price)
hist(logprice, main= "History of Price taking log")
#taking lof of price helps to have a better histogram
#calling libraries
library(car)## Warning: package 'car' was built under R version 3.4.2
library(corrplot) ## Warning: package 'corrplot' was built under R version 3.4.2## corrplot 0.84 loadedlibrary(visreg) ## Warning: package 'visreg' was built under R version 3.4.2library(rgl)## Warning: package 'rgl' was built under R version 3.4.2library(knitr)## Warning: package 'knitr' was built under R version 3.4.2library(scatterplot3d) #for 3Dfigures
library(GGally)## Warning: package 'GGally' was built under R version 3.4.2#drawing a scatter plot to check how the correlation is
#distributed in the group.
#Price is corelated to distance, elevation, sewer, date.
plot(data1, pch=10, col="blue")
ggpairs(data1)
#Lets draw a correlation plot in numbers
cordata1 <- cor(data1)
corrplot(cordata1, method= "number")
#there is some correlation between independant variables
#distance & country; flood & country; elevation & country
#drawing a scatter plot
?scatterplot## starting httpd help server ...## donescatterplot(data1$ï..Price, data1$Date, main="Price vs Date over time")
scatterplot(data1$ï..Price, data1$Size)
scatterplot(data1$ï..Price, data1$Elevation, main="Price Vs Elevation")
scatterplot(data1$ï..Price, data1$Distance)
#just to see how the data is correlated
attach(data1)
set.seed(1)
# Center predictors.
elevation.c <- scale(Elevation, center = T, scale = F)
date.c <- scale(Date, center = T, scale = F)
flood.c <- scale(Flood, center = T, scale = F)
distance.c <- scale(Distance, center = T, scale = F)
#just to try if center predictor will be of help
# bind these new variables into newdata and display a summary.
new.c.vars <- cbind(elevation.c, date.c, flood.c, distance.c)
data2 <- cbind(data1, new.c.vars)
data2## ï..Price County Size Elevation Sewer Date Flood Distance 1
## 1 4.5 1 138.4 10 3000 -103 0 0.3 5.3548387
## 2 10.6 1 52.0 4 0 -103 0 2.5 -0.6451613
## 3 1.7 0 16.1 0 2640 -98 1 10.3 -4.6451613
## 4 5.0 0 1695.2 1 3500 -93 0 14.0 -3.6451613
## 5 5.0 0 845.0 1 1000 -92 1 14.0 -3.6451613
## 6 3.3 1 6.9 2 10000 -86 0 0.0 -2.6451613
## 7 5.7 1 105.9 4 0 -68 0 0.0 -0.6451613
## 8 6.2 1 56.6 4 0 -64 0 0.0 -0.6451613
## 9 19.4 1 51.4 20 1300 -63 0 1.2 15.3548387
## 10 3.2 1 22.1 0 6000 -62 0 0.0 -4.6451613
## 11 4.7 1 22.1 0 6000 -61 0 0.0 -4.6451613
## 12 6.9 1 27.7 3 4500 -60 0 0.0 -1.6451613
## 13 8.1 1 18.6 5 5000 -59 0 0.5 0.3548387
## 14 11.6 1 69.9 8 0 -59 0 4.4 3.3548387
## 15 19.3 1 145.7 10 0 -59 0 4.2 5.3548387
## 16 11.7 1 77.2 9 0 -59 0 4.5 4.3548387
## 17 13.3 1 26.2 8 0 -59 0 4.7 3.3548387
## 18 15.1 1 102.3 6 0 -59 0 4.9 1.3548387
## 19 12.4 1 49.5 11 0 -59 0 4.6 6.3548387
## 20 15.3 1 12.2 8 0 -59 0 5.0 3.3548387
## 21 12.2 0 320.6 0 4000 -54 0 16.5 -4.6451613
## 22 18.1 1 9.9 5 0 -54 0 5.2 0.3548387
## 23 16.8 1 15.3 2 0 -53 0 5.5 -2.6451613
## 24 5.9 0 55.2 0 1320 -49 1 11.9 -4.6451613
## 25 4.0 0 116.2 2 900 -45 1 5.5 -2.6451613
## 26 37.2 0 15.0 5 0 -39 0 7.2 0.3548387
## 27 18.2 0 23.4 5 4420 -39 0 5.5 0.3548387
## 28 15.1 0 132.8 2 2640 -35 0 10.2 -2.6451613
## 29 22.9 0 12.0 5 3400 -16 0 5.5 0.3548387
## 30 15.2 0 67.0 2 900 -5 1 5.5 -2.6451613
## 31 21.9 0 30.8 2 900 -4 0 5.5 -2.6451613
## 2 3 4
## 1 -44.3548387 -0.1612903 -4.83225806
## 2 -44.3548387 -0.1612903 -2.63225806
## 3 -39.3548387 0.8387097 5.16774194
## 4 -34.3548387 -0.1612903 8.86774194
## 5 -33.3548387 0.8387097 8.86774194
## 6 -27.3548387 -0.1612903 -5.13225806
## 7 -9.3548387 -0.1612903 -5.13225806
## 8 -5.3548387 -0.1612903 -5.13225806
## 9 -4.3548387 -0.1612903 -3.93225806
## 10 -3.3548387 -0.1612903 -5.13225806
## 11 -2.3548387 -0.1612903 -5.13225806
## 12 -1.3548387 -0.1612903 -5.13225806
## 13 -0.3548387 -0.1612903 -4.63225806
## 14 -0.3548387 -0.1612903 -0.73225806
## 15 -0.3548387 -0.1612903 -0.93225806
## 16 -0.3548387 -0.1612903 -0.63225806
## 17 -0.3548387 -0.1612903 -0.43225806
## 18 -0.3548387 -0.1612903 -0.23225806
## 19 -0.3548387 -0.1612903 -0.53225806
## 20 -0.3548387 -0.1612903 -0.13225806
## 21 4.6451613 -0.1612903 11.36774194
## 22 4.6451613 -0.1612903 0.06774194
## 23 5.6451613 -0.1612903 0.36774194
## 24 9.6451613 0.8387097 6.76774194
## 25 13.6451613 0.8387097 0.36774194
## 26 19.6451613 -0.1612903 2.06774194
## 27 19.6451613 -0.1612903 0.36774194
## 28 23.6451613 -0.1612903 5.06774194
## 29 42.6451613 -0.1612903 0.36774194
## 30 53.6451613 0.8387097 0.36774194
## 31 54.6451613 -0.1612903 0.36774194names(data2)[9:12] <- c("elevation.c", "date.c", "flood.c", "distance.c" )
#Model1 - plain comparing with independent variables
mod1 <- lm(data2$ï..Price~ elevation.c+date.c+flood.c+distance.c)
summary(mod1)##
## Call:
## lm(formula = data2$ï..Price ~ elevation.c + date.c + flood.c +
## distance.c)
##
## Residuals:
## Min 1Q Median 3Q Max
## -5.311 -2.954 -1.040 1.273 18.915
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 11.9516 0.8958 13.342 3.86e-13 ***
## elevation.c 0.6861 0.2326 2.950 0.00664 **
## date.c 0.1907 0.0372 5.125 2.41e-05 ***
## flood.c -6.9243 2.7845 -2.487 0.01964 *
## distance.c 0.5933 0.2285 2.596 0.01531 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 4.988 on 26 degrees of freedom
## Multiple R-squared: 0.6378, Adjusted R-squared: 0.582
## F-statistic: 11.44 on 4 and 26 DF, p-value: 1.718e-05#Rsquare is 0.582
#taking log of income
mod2 <- lm(log(data2$ï..Price)~ elevation.c+date.c+flood.c+distance.c)
summary(mod2)##
## Call:
## lm(formula = log(data2$ï..Price) ~ elevation.c + date.c + flood.c +
## distance.c)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.5099 -0.2090 -0.1097 0.1721 1.0031
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2.25944 0.06478 34.879 < 2e-16 ***
## elevation.c 0.07459 0.01682 4.435 0.000149 ***
## date.c 0.01857 0.00269 6.902 2.5e-07 ***
## flood.c -0.77886 0.20136 -3.868 0.000659 ***
## distance.c 0.05908 0.01653 3.575 0.001401 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.3607 on 26 degrees of freedom
## Multiple R-squared: 0.7806, Adjusted R-squared: 0.7468
## F-statistic: 23.12 on 4 and 26 DF, p-value: 3.049e-08#increased R square to 0.746. Taking log of price is giving a better model
#testing without center predictors
mod3 <- lm(data2$ï..Price~ Elevation+ Date+ Flood+Distance)
summary(mod3)##
## Call:
## lm(formula = data2$ï..Price ~ Elevation + Date + Flood + Distance)
##
## Residuals:
## Min 1Q Median 3Q Max
## -5.311 -2.954 -1.040 1.273 18.915
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 18.0191 2.9536 6.101 1.90e-06 ***
## Elevation 0.6861 0.2326 2.950 0.00664 **
## Date 0.1907 0.0372 5.125 2.41e-05 ***
## Flood -6.9243 2.7845 -2.487 0.01964 *
## Distance 0.5933 0.2285 2.596 0.01531 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 4.988 on 26 degrees of freedom
## Multiple R-squared: 0.6378, Adjusted R-squared: 0.582
## F-statistic: 11.44 on 4 and 26 DF, p-value: 1.718e-05#R square is 0.58
#removing log reduced the R square
plot(mod2, pch=10, which = 1)
dwt(mod2)## lag Autocorrelation D-W Statistic p-value
## 1 -0.1952503 2.37622 0.398
## Alternative hypothesis: rho != 0??dwtest
qqPlot(mod2)
outlierTest(mod2)## rstudent unadjusted p-value Bonferonni p
## 2 3.704906 0.0010527 0.032634help("outlier.test")
#Lets take all the factors and see which has more P value
mod4 <- lm(data2$ï..Price ~ ., data=data1)
summary(mod4)##
## Call:
## lm(formula = data2$ï..Price ~ ., data = data1)
##
## Residuals:
## Min 1Q Median 3Q Max
## -5.169 -2.957 -0.256 2.070 13.031
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2.364e+01 3.829e+00 6.174 2.68e-06 ***
## County -8.789e+00 3.652e+00 -2.407 0.024532 *
## Size -6.043e-03 3.501e-03 -1.726 0.097702 .
## Elevation 5.193e-01 2.386e-01 2.177 0.040030 *
## Sewer -9.573e-04 4.169e-04 -2.296 0.031126 *
## Date 8.508e-02 4.865e-02 1.749 0.093646 .
## Flood -1.202e+01 2.989e+00 -4.020 0.000536 ***
## Distance 1.858e-01 3.395e-01 0.547 0.589386
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 4.431 on 23 degrees of freedom
## Multiple R-squared: 0.747, Adjusted R-squared: 0.67
## F-statistic: 9.703 on 7 and 23 DF, p-value: 1.351e-05#We will take the variables for which the pvalue is less than 0.05
#looks like distance & size have more than 0.05 p value.
#R square is 0.747. Therefore, our error is more than 1/4
#lets try with a different value eliminating the distance & size
#and see if our R square increases
mod5 <- lm(data1$ï..Price ~ . -Distance - Size, data=data1)
summary(mod5)##
## Call:
## lm(formula = data1$ï..Price ~ . - Distance - Size, data = data1)
##
## Residuals:
## Min 1Q Median 3Q Max
## -5.5688 -2.7883 -0.3453 1.9312 14.4498
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2.497e+01 2.597e+00 9.614 7.05e-10 ***
## County -7.439e+00 2.383e+00 -3.122 0.00450 **
## Elevation 5.291e-01 2.397e-01 2.207 0.03671 *
## Sewer -9.513e-04 3.800e-04 -2.504 0.01919 *
## Date 1.247e-01 3.840e-02 3.249 0.00330 **
## Flood -1.064e+01 2.871e+00 -3.707 0.00105 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 4.518 on 25 degrees of freedom
## Multiple R-squared: 0.7143, Adjusted R-squared: 0.6571
## F-statistic: 12.5 on 5 and 25 DF, p-value: 3.831e-06#R square of 0.657. Lets see if this can be slightly improved
?par
par(mfrow= c(2,2))
plot(mod5)
#26 row seems to be an outlier; lets try removing this row 26 alone
#to increase our model
data3 <-data1[-26,]
data3## ï..Price County Size Elevation Sewer Date Flood Distance
## 1 4.5 1 138.4 10 3000 -103 0 0.3
## 2 10.6 1 52.0 4 0 -103 0 2.5
## 3 1.7 0 16.1 0 2640 -98 1 10.3
## 4 5.0 0 1695.2 1 3500 -93 0 14.0
## 5 5.0 0 845.0 1 1000 -92 1 14.0
## 6 3.3 1 6.9 2 10000 -86 0 0.0
## 7 5.7 1 105.9 4 0 -68 0 0.0
## 8 6.2 1 56.6 4 0 -64 0 0.0
## 9 19.4 1 51.4 20 1300 -63 0 1.2
## 10 3.2 1 22.1 0 6000 -62 0 0.0
## 11 4.7 1 22.1 0 6000 -61 0 0.0
## 12 6.9 1 27.7 3 4500 -60 0 0.0
## 13 8.1 1 18.6 5 5000 -59 0 0.5
## 14 11.6 1 69.9 8 0 -59 0 4.4
## 15 19.3 1 145.7 10 0 -59 0 4.2
## 16 11.7 1 77.2 9 0 -59 0 4.5
## 17 13.3 1 26.2 8 0 -59 0 4.7
## 18 15.1 1 102.3 6 0 -59 0 4.9
## 19 12.4 1 49.5 11 0 -59 0 4.6
## 20 15.3 1 12.2 8 0 -59 0 5.0
## 21 12.2 0 320.6 0 4000 -54 0 16.5
## 22 18.1 1 9.9 5 0 -54 0 5.2
## 23 16.8 1 15.3 2 0 -53 0 5.5
## 24 5.9 0 55.2 0 1320 -49 1 11.9
## 25 4.0 0 116.2 2 900 -45 1 5.5
## 27 18.2 0 23.4 5 4420 -39 0 5.5
## 28 15.1 0 132.8 2 2640 -35 0 10.2
## 29 22.9 0 12.0 5 3400 -16 0 5.5
## 30 15.2 0 67.0 2 900 -5 1 5.5
## 31 21.9 0 30.8 2 900 -4 0 5.5str(data3)## 'data.frame': 30 obs. of 8 variables:
## $ ï..Price : num 4.5 10.6 1.7 5 5 3.3 5.7 6.2 19.4 3.2 ...
## $ County : int 1 1 0 0 0 1 1 1 1 1 ...
## $ Size : num 138.4 52 16.1 1695.2 845 ...
## $ Elevation: int 10 4 0 1 1 2 4 4 20 0 ...
## $ Sewer : int 3000 0 2640 3500 1000 10000 0 0 1300 6000 ...
## $ Date : int -103 -103 -98 -93 -92 -86 -68 -64 -63 -62 ...
## $ Flood : int 0 0 1 0 1 0 0 0 0 0 ...
## $ Distance : num 0.3 2.5 10.3 14 14 0 0 0 1.2 0 ...str(data1)## 'data.frame': 31 obs. of 8 variables:
## $ ï..Price : num 4.5 10.6 1.7 5 5 3.3 5.7 6.2 19.4 3.2 ...
## $ County : int 1 1 0 0 0 1 1 1 1 1 ...
## $ Size : num 138.4 52 16.1 1695.2 845 ...
## $ Elevation: int 10 4 0 1 1 2 4 4 20 0 ...
## $ Sewer : int 3000 0 2640 3500 1000 10000 0 0 1300 6000 ...
## $ Date : int -103 -103 -98 -93 -92 -86 -68 -64 -63 -62 ...
## $ Flood : int 0 0 1 0 1 0 0 0 0 0 ...
## $ Distance : num 0.3 2.5 10.3 14 14 0 0 0 1.2 0 ...mod6 <- lm(log(ï..Price) ~ . -Distance - Size, data= data3)
summary(mod6)##
## Call:
## lm(formula = log(ï..Price) ~ . - Distance - Size, data = data3)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.54690 -0.21040 0.01803 0.23982 0.56446
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3.419e+00 2.005e-01 17.056 6.42e-15 ***
## County -3.592e-01 1.857e-01 -1.934 0.065025 .
## Elevation 4.525e-02 1.763e-02 2.567 0.016920 *
## Sewer -9.915e-05 2.848e-05 -3.482 0.001926 **
## Date 1.403e-02 2.825e-03 4.965 4.54e-05 ***
## Flood -9.153e-01 2.198e-01 -4.164 0.000347 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.3321 on 24 degrees of freedom
## Multiple R-squared: 0.8041, Adjusted R-squared: 0.7633
## F-statistic: 19.7 on 5 and 24 DF, p-value: 8.585e-08#We have an improved R Square of 0.763. The looks like a better model
#trying it with step function.
step(lm(ï..Price ~ ., data=data3))## Start: AIC=75.45
## ï..Price ~ County + Size + Elevation + Sewer + Date + Flood +
## Distance
##
## Df Sum of Sq RSS AIC
## - County 1 8.266 225.90 74.566
## <none> 217.63 75.448
## - Size 1 18.032 235.66 75.836
## - Sewer 1 26.544 244.17 76.901
## - Distance 1 27.596 245.23 77.030
## - Elevation 1 100.809 318.44 84.867
## - Flood 1 116.091 333.72 86.273
## - Date 1 127.501 345.13 87.282
##
## Step: AIC=74.57
## ï..Price ~ Size + Elevation + Sewer + Date + Flood + Distance
##
## Df Sum of Sq RSS AIC
## - Size 1 12.890 238.79 74.231
## <none> 225.90 74.566
## - Sewer 1 18.645 244.54 74.946
## - Distance 1 85.238 311.14 82.171
## - Elevation 1 98.595 324.49 83.432
## - Flood 1 126.247 352.14 85.885
## - Date 1 308.603 534.50 98.404
##
## Step: AIC=74.23
## ï..Price ~ Elevation + Sewer + Date + Flood + Distance
##
## Df Sum of Sq RSS AIC
## <none> 238.79 74.231
## - Sewer 1 20.87 259.66 74.745
## - Distance 1 78.70 317.48 80.777
## - Elevation 1 101.70 340.49 82.875
## - Flood 1 115.21 354.00 84.043
## - Date 1 451.51 690.29 104.078##
## Call:
## lm(formula = ï..Price ~ Elevation + Sewer + Date + Flood + Distance,
## data = data3)
##
## Coefficients:
## (Intercept) Elevation Sewer Date Flood
## 17.9571991 0.5422328 -0.0004005 0.1627357 -6.1840320
## Distance
## 0.4229008mod9 <- lm(ï..Price ~ . -Distance - Size, data= data3)
summary(mod9)##
## Call:
## lm(formula = ï..Price ~ . - Distance - Size, data = data3)
##
## Residuals:
## Min 1Q Median 3Q Max
## -5.0186 -2.2651 -0.3114 2.1549 5.1596
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 22.0187525 1.9634490 11.214 5.01e-11 ***
## County -4.4613706 1.8189990 -2.453 0.02183 *
## Elevation 0.5086667 0.1726287 2.947 0.00704 **
## Sewer -0.0006846 0.0002789 -2.455 0.02173 *
## Date 0.1308357 0.0276699 4.728 8.28e-05 ***
## Flood -7.6795702 2.1524916 -3.568 0.00156 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.252 on 24 degrees of freedom
## Multiple R-squared: 0.7747, Adjusted R-squared: 0.7278
## F-statistic: 16.51 on 5 and 24 DF, p-value: 4.372e-07#We have an R square of 0.7278. since, we have a better results of mod6, we will choose mod6
?predict
lesiesaltprice <- predict(mod6)
lesiesaltprice## 1 2 3 4 5 6 7
## 1.7704282 1.7963983 0.8676524 1.8130681 1.1596736 0.9528662 2.2873697
## 8 9 10 11 12 13 14
## 2.3434807 2.9525663 1.5956419 1.6096697 1.9081652 1.9631118 2.5946080
## 15 16 17 18 19 20 21
## 2.6851022 2.6398551 2.5946080 2.5041137 2.7303493 2.5946080 2.2653279
## 22 23 24 25 27 28 29
## 2.5290053 2.4072917 1.6858916 1.8741403 2.6603364 2.7571948 3.0841087
## 30 31
## 2.4352504 3.3645778Add a new chunk by clicking the Insert Chunk button on the toolbar or by pressing Ctrl+Alt+I.
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