Part 1 - Pagerank
Step 1
Form the A matrix. Then, introduce decay and form the B Matrix as we did in the course notes.
A <- matrix(c(0,1/2,1/2,0,0,0,
1/6,1/6,1/6,1/6,1/6,1/6, # replaced row of 0s with uniform probability
1/3,1/3,0,0,1/3,0,
0,0,0,0,1/2,1/2,
0,0,0,1/2,0,1/2,
0,0,0,1,0,0), nrow=6, byrow=TRUE)Creting the A Matrix as it as done in the lecture will leave a “dangling” row in row 2. This will prevent our power iteration method from converging. To remedy this, we can simply use 1/6 for each element in the row, suggesting that there is an equally likely probability for each powition to be selected in the first iteration.
n <- 6
decay <- 0.85
B <- decay * A + ((1-decay)/n)
print(B)## [,1] [,2] [,3] [,4] [,5] [,6]
## [1,] 0.0250000 0.4500000 0.4500000 0.0250000 0.0250000 0.0250000
## [2,] 0.1666667 0.1666667 0.1666667 0.1666667 0.1666667 0.1666667
## [3,] 0.3083333 0.3083333 0.0250000 0.0250000 0.3083333 0.0250000
## [4,] 0.0250000 0.0250000 0.0250000 0.0250000 0.4500000 0.4500000
## [5,] 0.0250000 0.0250000 0.0250000 0.4500000 0.0250000 0.4500000
## [6,] 0.0250000 0.0250000 0.0250000 0.8750000 0.0250000 0.0250000Step 2
Start with a uniform rank vector r and perform power iterations on B till convergence. That is, compute the solution r = Bn × r. Attempt this for a sufficiently large n so that r actually converges. (5 Points)
paste("A",1:6,sep=",")## [1] "A,1" "A,2" "A,3" "A,4" "A,5" "A,6"# Start with uniform rank vector
r <- matrix(c(1/6,1/6,1/6,1/6,1/6,1/6), nrow=1, ncol=6)power_iteration <- function(decayMat, r, iter) {
index <- c(0)
r1 <- c(r[1]) #one element vector with first element of r
r2 <- c(r[2]) #one element vector with second element of r
r3 <- c(r[3]) #one element vector with third element of r
r4 <- c(r[4]) #one element vector with fourth element of r
r5 <- c(r[5]) #one element vector with fifth element of r
r6 <- c(r[6])
power_iteration <- r #set new r equal to starting r
newDecayMat <- decayMat #set new decayMat equal to original decayMat
for (i in 1:iter) { # for each integer between 1 and n (30)
newDecayMat <- newDecayMat%*%decayMat
power_iteration <- r %*% newDecayMat
r1 <- append(r1, power_iteration[1])
r2 <- append(r2, power_iteration[2])
r3 <- append(r3, power_iteration[3])
r4 <- append(r4, power_iteration[4])
r5 <- append(r5, power_iteration[5])
r6 <- append(r6, power_iteration[6])
index <- append(index, i)
}
all_r <- list(r1, r2, r3, r4, r5, r6)
for (i in 1:6){
r_n <- all_r[i]
for (t in 2:length(r_n[[1]])-1) {
current_t <- r_n[[1]][t]
next_t <- r_n[[1]][t+1]
difference <- abs((next_t - current_t)/current_t)
if (difference < 0.0001) {
print(str_interp("element ${i} of r converges after ${t} iterations"))
break
} else {
if (t == length(r_n[[1]])-1) {
print(str_interp("r${i} does not meet difference threshold"))
}
}
}
}
ret = list(power_iteration, newDecayMat)
return(ret)
}response = power_iteration(B, r, 30)## [1] "element 1 of r converges after 17 iterations"
## [1] "element 2 of r converges after 17 iterations"
## [1] "element 3 of r converges after 17 iterations"
## [1] "element 4 of r converges after 14 iterations"
## [1] "element 5 of r converges after 12 iterations"
## [1] "element 6 of r converges after 14 iterations"Based on the above visualization, we can see that convergence occurs roughly around 15 iterations.
power_iteration_r <- response[[1]] #The new value of r
power_iteration_r## [,1] [,2] [,3] [,4] [,5] [,6]
## [1,] 0.05170475 0.07367927 0.05741242 0.3487037 0.1999038 0.2685961newDecayMat <- response[2] #Final iteration of decay matrix
newDecayMat## [[1]]
## [,1] [,2] [,3] [,4] [,5] [,6]
## [1,] 0.05170476 0.07367928 0.05741243 0.3487037 0.1999038 0.2685961
## [2,] 0.05170475 0.07367927 0.05741242 0.3487037 0.1999038 0.2685961
## [3,] 0.05170475 0.07367928 0.05741242 0.3487037 0.1999038 0.2685961
## [4,] 0.05170474 0.07367926 0.05741241 0.3487037 0.1999038 0.2685961
## [5,] 0.05170474 0.07367926 0.05741241 0.3487037 0.1999038 0.2685961
## [6,] 0.05170474 0.07367926 0.05741241 0.3487037 0.1999038 0.2685961Step 3
Compute the eigen-decomposition of B and verify that you indeed get an eigenvalue of 1 as the largest eigenvalue and that its corresponding eigenvector is the same vector that you obtained in the previous power iteration method. Further, this eigenvector has all positive entries and it sums to 1.(10 points)
ev <- eigen(t(B))
as.numeric(ev$values)## [1] 1.00000000 0.57619235 -0.42500000 -0.42500000 -0.34991524 -0.08461044We can see from the above that the largest eigenvalue is 1
sum(ev$vectors[,1])## [1] 2.019902+0iHowever, the sum of the associated eigenvector is greater than 1. This is corrected when transforming the vector to the unit vector (divide each element by the sum of the vector)
eigen_r <- as.numeric(1/(sum(ev$vectors[,1]))*ev$vectors[,1])
sum(eigen_r)## [1] 1eigen_r## [1] 0.05170475 0.07367926 0.05741241 0.34870369 0.19990381 0.26859608power_iteration_r - eigen_r## [,1] [,2] [,3] [,4] [,5]
## [1,] 3.29466e-09 5.724804e-09 3.837685e-09 -6.620217e-09 -1.589199e-09
## [,6]
## [1,] -4.647732e-09The “power_iter_vector” we received after running our power_iteration function is approximately the same as the “eigen_r” vector we received after performing eigenvalue decomposition on our transition matrix.
Step 4
Use the graph package in R and its page.rank method to compute the Page Rank of the graph as given in A. Note that you don’t need to apply decay. The package starts with a connected graph and applies decay internally. Verify that you do get the same PageRank vector as the two approaches above. (10 points)
#Converting to graph from adjacency matrix
webGraph = igraph::graph_from_adjacency_matrix(A,weighted = T)
#Plot the graph
plot(webGraph)
#Resultant vector
pr <- page_rank(webGraph)$vector
pr## [1] 0.05170475 0.07367926 0.05741241 0.34870369 0.19990381 0.26859608round(eigen_r, 5) == round(pr, 5)## [1] TRUE TRUE TRUE TRUE TRUE TRUEPart 2 - Kaggle MNIST
Step 1-2
Go to Kaggle.com and build an account if you do not already have one. It is free
Go to https://www.kaggle.com/c/digit-recognizer/overview, accept the rules of the competition, and download the data. You will not be required to submit work to Kaggle, but you do need the data.
Step 3
Using the training.csv file, plot representations of the first 10 images to understand the data format. Go ahead and divide all pixels by 255 to produce values between 0 and 1. (This is equivalent to min-max scaling.) (5 points)
labels = train_data[,1]
data <- train_data[,-1]/255plot_array <- function(array){
arrayMat <- matrix(array, nrow=28, ncol=28)
mode(arrayMat) = "numeric"
image(arrayMat)
}for (i in 1:10) {
plot_array(data[i,])
}
Step 4
What is the frequency distribution of the numbers in the dataset? (5 points)
table(labels)/42000## labels
## 0 1 2 3 4 5 6
## 0.09838095 0.11152381 0.09945238 0.10359524 0.09695238 0.09035714 0.09850000
## 7 8 9
## 0.10478571 0.09673810 0.09971429While there is some variation, each label roughly corresponds to 10% of the entire training dataset.
Step 5
For each number, provide the mean pixel intensity. What does this tell you? (5 points)
get_number_intensity <- function(target, labels, data){
x = data[labels==target,]
means = rowMeans(x)
return(mean(means))
}for (i in 1:9) {
mean_intensity <- get_number_intensity(i , labels, data)
ret_string = str_interp("Pixel intensity for number ${i} is ${mean_intensity}")
print(ret_string)
}## [1] "Pixel intensity for number 1 is 0.075972720428906"
## [1] "Pixel intensity for number 2 is 0.149415262873165"
## [1] "Pixel intensity for number 3 is 0.141657603055012"
## [1] "Pixel intensity for number 4 is 0.121212097314368"
## [1] "Pixel intensity for number 5 is 0.129231294625887"
## [1] "Pixel intensity for number 6 is 0.138730078688473"
## [1] "Pixel intensity for number 7 is 0.1147021021542"
## [1] "Pixel intensity for number 8 is 0.150981134516322"
## [1] "Pixel intensity for number 9 is 0.12281787715086"My intuition for the mean pixel intensity is that more complex numbers which take up a larger portion of the 28x28 matrix will have a greater mean pixel intensity. This intuition is supported by the fact that 1, a straight line, has the smallest mean pixel intensity. Likewise the number 7 is the second smallest mean pixel intensity. Both of these numbers are comprised of simple straight lines. Alternatively, numbers like 8, 3 and 6, which are typically much curvier, have the highest mean pixel intensities.
Step 6
Reduce the data by using principal components that account for 95% of the variance. How many components did you generate? Use PCA to generate all possible components (100% of the variance). How many components are possible? Why? (5 points)
train_pca <- prcomp(data)train_pca_std <- train_pca$sdev
train_pca_cum_var <- cumsum(train_pca_std^2)/sum(train_pca_std^2)plot(train_pca_cum_var)
which.max(train_pca_cum_var > 0.95)## [1] 154length(train_pca_cum_var)## [1] 784With 154 components we are able to account for 95% variance. There are 784 total components possible, 1 for each column in the original dataset.
Step 7
Plot the first 10 images generated by PCA. They will appear to be noise. Why? (5 points)
train_pca_rot <- train_pca$rotationfor (i in 1:10) {
plot_array(train_pca_rot[,i])
}
Above are the first 10 images generated by PCA. The images appear to be noisy, but this is simply due to the fact that they were generated via PCA. Essentially, each PCA image is the average of the all the entries with that type.
Step 8
Now, select only those images that have labels that are 8’s. Re-run PCA that accounts for all of the variance (100%). Plot the first 10 images. What do you see? (5 points)
x = data[labels==8,]
pca_8 <- prcomp(x)pca_8_std <- pca_8$sdev
pca_8_cum_var <- cumsum(pca_8_std^2)/sum(pca_8_std^2)
plot(pca_8_cum_var)
pca_8_rot <- pca_8$rotation
for (i in 1:10) {
plot_array(pca_8_rot[,i])
}
As expected, multiple representations of the digit 8, blurred due to PCA.
Step 9
An incorrect approach to predicting the images would be to build a linear regression model with y as the digit values and X as the pixel matrix. Instead, we can build a multinomial model that classifies the digits. Build a multinomial model on the entirety of the training set. Then provide its classification accuracy (percent correctly identified) as well as a matrix of observed versus forecast values (confusion matrix). This matrix will be a 10 x 10, and correct classifications will be on the diagonal. (10 points)
test_size <- floor(.2 * nrow(data))
set.seed(42)
test_index <- sample(seq_len(nrow(data)), size = test_size)
train_df <- data[-test_index,]
train_labels <- c(labels[-test_index,])[[1]]
test_df <- data[test_index,]
test_labels <- c(labels[test_index,])[[1]]model <- multinom(train_labels~., train_df, MaxNWts = 100000)## # weights: 7860 (7065 variable)
## initial value 77366.859125
## iter 10 value 22233.410474
## iter 20 value 18330.173557
## iter 30 value 17277.223639
## iter 40 value 16671.149301
## iter 50 value 16142.222215
## iter 60 value 15530.180131
## iter 70 value 14349.926185
## iter 80 value 12674.327346
## iter 90 value 11166.744868
## iter 100 value 10266.648048
## final value 10266.648048
## stopped after 100 iterationspredictions <- predict(model, test_df)test_labels <- as.factor(test_labels)confusionMatrix(predictions,test_labels)## Confusion Matrix and Statistics
##
## Reference
## Prediction 0 1 2 3 4 5 6 7 8 9
## 0 791 0 10 5 1 12 4 1 9 7
## 1 1 906 12 7 3 16 7 20 24 7
## 2 0 4 699 23 4 4 7 7 6 2
## 3 0 9 15 777 1 27 0 3 32 21
## 4 2 1 18 4 763 17 11 13 7 36
## 5 5 2 4 26 0 644 14 4 23 7
## 6 12 2 14 9 4 16 796 2 6 0
## 7 2 1 13 13 3 4 3 816 3 21
## 8 5 10 28 16 5 13 5 2 658 11
## 9 2 0 4 14 23 11 0 47 11 710
##
## Overall Statistics
##
## Accuracy : 0.9
## 95% CI : (0.8934, 0.9063)
## No Information Rate : 0.1113
## P-Value [Acc > NIR] : < 2.2e-16
##
## Kappa : 0.8888
##
## Mcnemar's Test P-Value : NA
##
## Statistics by Class:
##
## Class: 0 Class: 1 Class: 2 Class: 3 Class: 4 Class: 5
## Sensitivity 0.96463 0.9690 0.85557 0.8691 0.94548 0.84293
## Specificity 0.99354 0.9870 0.99248 0.9856 0.98564 0.98887
## Pos Pred Value 0.94167 0.9033 0.92460 0.8780 0.87500 0.88340
## Neg Pred Value 0.99616 0.9961 0.98456 0.9844 0.99416 0.98436
## Prevalence 0.09762 0.1113 0.09726 0.1064 0.09607 0.09095
## Detection Rate 0.09417 0.1079 0.08321 0.0925 0.09083 0.07667
## Detection Prevalence 0.10000 0.1194 0.09000 0.1054 0.10381 0.08679
## Balanced Accuracy 0.97908 0.9780 0.92403 0.9274 0.96556 0.91590
## Class: 6 Class: 7 Class: 8 Class: 9
## Sensitivity 0.93979 0.89180 0.84467 0.86375
## Specificity 0.99139 0.99158 0.98753 0.98522
## Pos Pred Value 0.92451 0.92833 0.87384 0.86375
## Neg Pred Value 0.99324 0.98684 0.98418 0.98522
## Prevalence 0.10083 0.10893 0.09274 0.09786
## Detection Rate 0.09476 0.09714 0.07833 0.08452
## Detection Prevalence 0.10250 0.10464 0.08964 0.09786
## Balanced Accuracy 0.96559 0.94169 0.91610 0.92448Overall accuracy of the model was very strong, at 90%. Reviewing the confusion matrix shows us that numbers 1 and 7 (the simple ones) have the highest rates of precision. The more complex numbers, 8 in particular, score the lowest.
Part 3 - House Prices
Step 1
Descriptive and Inferential Statistics. Provide univariate descriptive statistics and appropriate plots for the training data set. Provide a scatterplot matrix for at least two of the independent variables and the dependent variable. Derive a correlation matrix for any three quantitative variables in the dataset. Test the hypotheses that the correlations between each pairwise set of variables is 0 and provide an 80% confidence interval. Discuss the meaning of your analysis. Would you be worried about familywise error? Why or why not?
# Univariate descriptive statistics
summary(train_data)## Id MSSubClass MSZoning LotFrontage
## Min. : 1.0 Min. : 20.0 Length:1460 Min. : 21.00
## 1st Qu.: 365.8 1st Qu.: 20.0 Class :character 1st Qu.: 59.00
## Median : 730.5 Median : 50.0 Mode :character Median : 69.00
## Mean : 730.5 Mean : 56.9 Mean : 70.05
## 3rd Qu.:1095.2 3rd Qu.: 70.0 3rd Qu.: 80.00
## Max. :1460.0 Max. :190.0 Max. :313.00
## NA's :259
## LotArea Street Alley LotShape
## Min. : 1300 Length:1460 Length:1460 Length:1460
## 1st Qu.: 7554 Class :character Class :character Class :character
## Median : 9478 Mode :character Mode :character Mode :character
## Mean : 10517
## 3rd Qu.: 11602
## Max. :215245
##
## LandContour Utilities LotConfig LandSlope
## Length:1460 Length:1460 Length:1460 Length:1460
## Class :character Class :character Class :character Class :character
## Mode :character Mode :character Mode :character Mode :character
##
##
##
##
## Neighborhood Condition1 Condition2 BldgType
## Length:1460 Length:1460 Length:1460 Length:1460
## Class :character Class :character Class :character Class :character
## Mode :character Mode :character Mode :character Mode :character
##
##
##
##
## HouseStyle OverallQual OverallCond YearBuilt
## Length:1460 Min. : 1.000 Min. :1.000 Min. :1872
## Class :character 1st Qu.: 5.000 1st Qu.:5.000 1st Qu.:1954
## Mode :character Median : 6.000 Median :5.000 Median :1973
## Mean : 6.099 Mean :5.575 Mean :1971
## 3rd Qu.: 7.000 3rd Qu.:6.000 3rd Qu.:2000
## Max. :10.000 Max. :9.000 Max. :2010
##
## YearRemodAdd RoofStyle RoofMatl Exterior1st
## Min. :1950 Length:1460 Length:1460 Length:1460
## 1st Qu.:1967 Class :character Class :character Class :character
## Median :1994 Mode :character Mode :character Mode :character
## Mean :1985
## 3rd Qu.:2004
## Max. :2010
##
## Exterior2nd MasVnrType MasVnrArea ExterQual
## Length:1460 Length:1460 Min. : 0.0 Length:1460
## Class :character Class :character 1st Qu.: 0.0 Class :character
## Mode :character Mode :character Median : 0.0 Mode :character
## Mean : 103.7
## 3rd Qu.: 166.0
## Max. :1600.0
## NA's :8
## ExterCond Foundation BsmtQual BsmtCond
## Length:1460 Length:1460 Length:1460 Length:1460
## Class :character Class :character Class :character Class :character
## Mode :character Mode :character Mode :character Mode :character
##
##
##
##
## BsmtExposure BsmtFinType1 BsmtFinSF1 BsmtFinType2
## Length:1460 Length:1460 Min. : 0.0 Length:1460
## Class :character Class :character 1st Qu.: 0.0 Class :character
## Mode :character Mode :character Median : 383.5 Mode :character
## Mean : 443.6
## 3rd Qu.: 712.2
## Max. :5644.0
##
## BsmtFinSF2 BsmtUnfSF TotalBsmtSF Heating
## Min. : 0.00 Min. : 0.0 Min. : 0.0 Length:1460
## 1st Qu.: 0.00 1st Qu.: 223.0 1st Qu.: 795.8 Class :character
## Median : 0.00 Median : 477.5 Median : 991.5 Mode :character
## Mean : 46.55 Mean : 567.2 Mean :1057.4
## 3rd Qu.: 0.00 3rd Qu.: 808.0 3rd Qu.:1298.2
## Max. :1474.00 Max. :2336.0 Max. :6110.0
##
## HeatingQC CentralAir Electrical 1stFlrSF
## Length:1460 Length:1460 Length:1460 Min. : 334
## Class :character Class :character Class :character 1st Qu.: 882
## Mode :character Mode :character Mode :character Median :1087
## Mean :1163
## 3rd Qu.:1391
## Max. :4692
##
## 2ndFlrSF LowQualFinSF GrLivArea BsmtFullBath
## Min. : 0 Min. : 0.000 Min. : 334 Min. :0.0000
## 1st Qu.: 0 1st Qu.: 0.000 1st Qu.:1130 1st Qu.:0.0000
## Median : 0 Median : 0.000 Median :1464 Median :0.0000
## Mean : 347 Mean : 5.845 Mean :1515 Mean :0.4253
## 3rd Qu.: 728 3rd Qu.: 0.000 3rd Qu.:1777 3rd Qu.:1.0000
## Max. :2065 Max. :572.000 Max. :5642 Max. :3.0000
##
## BsmtHalfBath FullBath HalfBath BedroomAbvGr
## Min. :0.00000 Min. :0.000 Min. :0.0000 Min. :0.000
## 1st Qu.:0.00000 1st Qu.:1.000 1st Qu.:0.0000 1st Qu.:2.000
## Median :0.00000 Median :2.000 Median :0.0000 Median :3.000
## Mean :0.05753 Mean :1.565 Mean :0.3829 Mean :2.866
## 3rd Qu.:0.00000 3rd Qu.:2.000 3rd Qu.:1.0000 3rd Qu.:3.000
## Max. :2.00000 Max. :3.000 Max. :2.0000 Max. :8.000
##
## KitchenAbvGr KitchenQual TotRmsAbvGrd Functional
## Min. :0.000 Length:1460 Min. : 2.000 Length:1460
## 1st Qu.:1.000 Class :character 1st Qu.: 5.000 Class :character
## Median :1.000 Mode :character Median : 6.000 Mode :character
## Mean :1.047 Mean : 6.518
## 3rd Qu.:1.000 3rd Qu.: 7.000
## Max. :3.000 Max. :14.000
##
## Fireplaces FireplaceQu GarageType GarageYrBlt
## Min. :0.000 Length:1460 Length:1460 Min. :1900
## 1st Qu.:0.000 Class :character Class :character 1st Qu.:1961
## Median :1.000 Mode :character Mode :character Median :1980
## Mean :0.613 Mean :1979
## 3rd Qu.:1.000 3rd Qu.:2002
## Max. :3.000 Max. :2010
## NA's :81
## GarageFinish GarageCars GarageArea GarageQual
## Length:1460 Min. :0.000 Min. : 0.0 Length:1460
## Class :character 1st Qu.:1.000 1st Qu.: 334.5 Class :character
## Mode :character Median :2.000 Median : 480.0 Mode :character
## Mean :1.767 Mean : 473.0
## 3rd Qu.:2.000 3rd Qu.: 576.0
## Max. :4.000 Max. :1418.0
##
## GarageCond PavedDrive WoodDeckSF OpenPorchSF
## Length:1460 Length:1460 Min. : 0.00 Min. : 0.00
## Class :character Class :character 1st Qu.: 0.00 1st Qu.: 0.00
## Mode :character Mode :character Median : 0.00 Median : 25.00
## Mean : 94.24 Mean : 46.66
## 3rd Qu.:168.00 3rd Qu.: 68.00
## Max. :857.00 Max. :547.00
##
## EnclosedPorch 3SsnPorch ScreenPorch PoolArea
## Min. : 0.00 Min. : 0.00 Min. : 0.00 Min. : 0.000
## 1st Qu.: 0.00 1st Qu.: 0.00 1st Qu.: 0.00 1st Qu.: 0.000
## Median : 0.00 Median : 0.00 Median : 0.00 Median : 0.000
## Mean : 21.95 Mean : 3.41 Mean : 15.06 Mean : 2.759
## 3rd Qu.: 0.00 3rd Qu.: 0.00 3rd Qu.: 0.00 3rd Qu.: 0.000
## Max. :552.00 Max. :508.00 Max. :480.00 Max. :738.000
##
## PoolQC Fence MiscFeature MiscVal
## Length:1460 Length:1460 Length:1460 Min. : 0.00
## Class :character Class :character Class :character 1st Qu.: 0.00
## Mode :character Mode :character Mode :character Median : 0.00
## Mean : 43.49
## 3rd Qu.: 0.00
## Max. :15500.00
##
## MoSold YrSold SaleType SaleCondition
## Min. : 1.000 Min. :2006 Length:1460 Length:1460
## 1st Qu.: 5.000 1st Qu.:2007 Class :character Class :character
## Median : 6.000 Median :2008 Mode :character Mode :character
## Mean : 6.322 Mean :2008
## 3rd Qu.: 8.000 3rd Qu.:2009
## Max. :12.000 Max. :2010
##
## SalePrice
## Min. : 34900
## 1st Qu.:129975
## Median :163000
## Mean :180921
## 3rd Qu.:214000
## Max. :755000
## There are a lot of features here to consider, with a roughly even split between numeric and categorical data. We will need to think about ways to attack missing values later.
# Scatter plot between 1stFlrSF (first floot square feet) and SalePrice
# We see a positive correlation between independent variable '1stFlrSF' and dependent variable 'SalePrice'. As the first floor square footage increases, so does the price.
train_data %>%
ggplot(aes(x=`1stFlrSF`, y=`SalePrice`)) +
geom_point(position="jitter") +
geom_smooth(method="lm")## `geom_smooth()` using formula 'y ~ x'
# Scatter plot between TotalBsmtSF (basement square feet) and SalePrice
# We see a positive correlation between independent variable 'TotalBsmtSF' and dependent variable 'SalePrice'. As the basement square footage increases, so does the price.
train_data %>%
ggplot(aes(x=`TotalBsmtSF`, y=`SalePrice`)) +
geom_point(position="jitter") +
geom_smooth(method="lm")## `geom_smooth()` using formula 'y ~ x'
correlation <- cor(train_data[,c("LotArea","TotalBsmtSF","1stFlrSF","SalePrice")])
correlation## LotArea TotalBsmtSF 1stFlrSF SalePrice
## LotArea 1.0000000 0.2608331 0.2994746 0.2638434
## TotalBsmtSF 0.2608331 1.0000000 0.8195300 0.6135806
## 1stFlrSF 0.2994746 0.8195300 1.0000000 0.6058522
## SalePrice 0.2638434 0.6135806 0.6058522 1.0000000cor.test(train_data$LotArea, train_data$TotalBsmtSF, conf.level=0.80)##
## Pearson's product-moment correlation
##
## data: train_data$LotArea and train_data$TotalBsmtSF
## t = 10.317, df = 1458, p-value < 2.2e-16
## alternative hypothesis: true correlation is not equal to 0
## 80 percent confidence interval:
## 0.2292786 0.2918400
## sample estimates:
## cor
## 0.2608331cor.test(train_data$TotalBsmtSF, train_data$`1stFlrSF`, conf.level=0.80)##
## Pearson's product-moment correlation
##
## data: train_data$TotalBsmtSF and train_data$`1stFlrSF`
## t = 54.609, df = 1458, p-value < 2.2e-16
## alternative hypothesis: true correlation is not equal to 0
## 80 percent confidence interval:
## 0.8081976 0.8302556
## sample estimates:
## cor
## 0.81953cor.test(train_data$LotArea, train_data$`1stFlrSF`, conf.level=0.80)##
## Pearson's product-moment correlation
##
## data: train_data$LotArea and train_data$`1stFlrSF`
## t = 11.985, df = 1458, p-value < 2.2e-16
## alternative hypothesis: true correlation is not equal to 0
## 80 percent confidence interval:
## 0.2686127 0.3297222
## sample estimates:
## cor
## 0.2994746All of the above tests indicate that we can reject the null hypothesis that the correlation is zero between the features. As we are dealing with many features in this dataset, we should be worried about familywise error.
Part 2
Invert your correlation matrix from above. (This is known as the precision matrix and contains variance inflation factors on the diagonal.) Multiply the correlation matrix by the precision matrix, and then multiply the precision matrix by the correlation matrix. Conduct LU decomposition on the matrix. 5 points
inverseMat <- solve(correlation)
inverseMat## LotArea TotalBsmtSF 1stFlrSF SalePrice
## LotArea 1.1116224106 -0.0005917129 -0.2448005 -0.1446182
## TotalBsmtSF -0.0005917129 3.2603189969 -2.3064606 -0.6029380
## 1stFlrSF -0.2448004553 -2.3064606461 3.2656838 -0.4987333
## SalePrice -0.1446182309 -0.6029379875 -0.4987333 1.7102662mult <- correlation %*% inverseMat
mult## LotArea TotalBsmtSF 1stFlrSF SalePrice
## LotArea 1.000000e+00 2.775558e-17 5.551115e-17 5.551115e-17
## TotalBsmtSF 1.387779e-17 1.000000e+00 1.665335e-16 2.220446e-16
## 1stFlrSF 6.938894e-17 -5.551115e-17 1.000000e+00 2.220446e-16
## SalePrice 2.775558e-17 -3.330669e-16 -5.551115e-17 1.000000e+00lu_decomp <- lu(correlation)expand(lu_decomp)## $L
## 4 x 4 Matrix of class "dtrMatrix" (unitriangular)
## [,1] [,2] [,3] [,4]
## [1,] 1.0000000 . . .
## [2,] 0.2608331 1.0000000 . .
## [3,] 0.2994746 0.7955408 1.0000000 .
## [4,] 0.2638434 0.5845293 0.2916115 1.0000000
##
## $U
## 4 x 4 Matrix of class "dtrMatrix"
## [,1] [,2] [,3] [,4]
## [1,] 1.00000000 0.26083313 0.29947458 0.26384335
## [2,] . 0.93196608 0.74141708 0.54476146
## [3,] . . 0.32048740 0.09345781
## [4,] . . . 0.58470428
##
## $P
## 4 x 4 sparse Matrix of class "pMatrix"
##
## [1,] | . . .
## [2,] . | . .
## [3,] . . | .
## [4,] . . . |Part 3
Many times, it makes sense to fit a closed form distribution to data. Select a variable in the Kaggle.com training dataset that is skewed to the right, shift it so that the minimum value is absolutely above zero if necessary. Then load the MASS package and run fitdistr to fit an exponential probability density function. (See https://stat.ethz.ch/R-manual/Rdevel/library/MASS/html/fitdistr.html ). Find the optimal value of λ for this distribution, and then take 1000 samples from this exponential distribution using this value (e.g., rexp(1000, λ)). Plot a histogram and compare it with a histogram of your original variable. Using the exponential pdf, find the 5th and 95th percentiles using the cumulative distribution function (CDF). Also generate a 95% confidence interval from the empirical data, assuming normality. Finally, provide the empirical 5th percentile and 95th percentile of the data. Discuss. 10 points
MasVnrArea is heavily right skewed. Additionally, there are some missing values that will cause us problems when attempting the fitdistr() method.
sum(is.na(train_data$MasVnrArea))## [1] 8# Replace missing values with the mean of the column
train_data$MasVnrArea[is.na(train_data$MasVnrArea)] <- mean(train_data$MasVnrArea, na.rm = T)lambda <- fitdistr(train_data$MasVnrArea, densfun = "exponential")
lambda[[1]]## rate
## 0.009644572samples <- rexp(1000, lambda[[1]])hist(samples, breaks=50)
lower_conf <- qexp(.05, rate = lambda[[1]])
upper_conf <- qexp(.95, rate = lambda[[1]])
print(c(lower_conf, upper_conf))## [1] 5.318359 310.613285lower_conf <- quantile(data$MasVnrArea, 0.05)
upper_conf <- quantile(data$MasVnrArea, 0.95)
print(c(lower_conf, upper_conf))## 5% 95%
## NA NAPart 4
Build some type of multiple regression model and submit your model to the competition board. Provide your complete model summary and results with analysis. Report your Kaggle.com user name and score. 10 points
First let’s check to see which columns have the most NAs
sapply(train_data, function(x) sum(is.na(x)))## Id MSSubClass MSZoning LotFrontage LotArea
## 0 0 0 259 0
## Street Alley LotShape LandContour Utilities
## 0 1369 0 0 0
## LotConfig LandSlope Neighborhood Condition1 Condition2
## 0 0 0 0 0
## BldgType HouseStyle OverallQual OverallCond YearBuilt
## 0 0 0 0 0
## YearRemodAdd RoofStyle RoofMatl Exterior1st Exterior2nd
## 0 0 0 0 0
## MasVnrType MasVnrArea ExterQual ExterCond Foundation
## 8 0 0 0 0
## BsmtQual BsmtCond BsmtExposure BsmtFinType1 BsmtFinSF1
## 37 37 38 37 0
## BsmtFinType2 BsmtFinSF2 BsmtUnfSF TotalBsmtSF Heating
## 38 0 0 0 0
## HeatingQC CentralAir Electrical 1stFlrSF 2ndFlrSF
## 0 0 1 0 0
## LowQualFinSF GrLivArea BsmtFullBath BsmtHalfBath FullBath
## 0 0 0 0 0
## HalfBath BedroomAbvGr KitchenAbvGr KitchenQual TotRmsAbvGrd
## 0 0 0 0 0
## Functional Fireplaces FireplaceQu GarageType GarageYrBlt
## 0 0 690 81 81
## GarageFinish GarageCars GarageArea GarageQual GarageCond
## 81 0 0 81 81
## PavedDrive WoodDeckSF OpenPorchSF EnclosedPorch 3SsnPorch
## 0 0 0 0 0
## ScreenPorch PoolArea PoolQC Fence MiscFeature
## 0 0 1453 1179 1406
## MiscVal MoSold YrSold SaleType SaleCondition
## 0 0 0 0 0
## SalePrice
## 0We can immediately remove the columns that have over 1000 NA values (or are the ID)
And for the rest we can impute the missing values:
- Using mean imputation for numeric columns.
- For categorical columns, we will replace missing values with “missing”
replace_numeric <- function(x) {
replace(x, is.na(x), mean(x, na.rm = TRUE))
}
replace_category <- function(x) {
replace(x, is.na(x), "missing")
}clean_df <- function(dataframe) {
dataframe <- dataframe %>%
dplyr::select(-Id, -Alley, -PoolQC, -Fence, -MiscFeature)
train_numeric <- dataframe %>%
dplyr::select(is.numeric)
train_category <- dataframe %>%
dplyr::select(!is.numeric)
train_numeric <- replace(train_numeric, TRUE, lapply(train_numeric, replace_numeric))
train_category <- replace(train_category, TRUE, lapply(train_category, replace_category))
train_data_clean <- cbind(train_numeric, train_category)
return(train_data_clean)
}train_data_clean <- clean_df(train_data)## Warning: Predicate functions must be wrapped in `where()`.
##
## # Bad
## data %>% select(is.numeric)
##
## # Good
## data %>% select(where(is.numeric))
##
## ℹ Please update your code.
## This message is displayed once per session.model <- lm(SalePrice~., data=train_data_clean)
summary(model)##
## Call:
## lm(formula = SalePrice ~ ., data = train_data_clean)
##
## Residuals:
## Min 1Q Median 3Q Max
## -180209 -9385 386 9615 180209
##
## Coefficients: (8 not defined because of singularities)
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -6.017e+05 1.055e+06 -0.570 0.568617
## MSSubClass -3.211e+01 8.282e+01 -0.388 0.698306
## LotFrontage 5.448e+01 4.353e+01 1.252 0.210930
## LotArea 7.382e-01 1.094e-01 6.745 2.35e-11 ***
## OverallQual 6.639e+03 1.013e+03 6.551 8.43e-11 ***
## OverallCond 5.752e+03 8.730e+02 6.589 6.58e-11 ***
## YearBuilt 3.267e+02 7.692e+01 4.247 2.33e-05 ***
## YearRemodAdd 9.663e+01 5.539e+01 1.745 0.081308 .
## MasVnrArea 1.941e+01 5.783e+00 3.357 0.000811 ***
## BsmtFinSF1 3.916e+01 5.301e+00 7.388 2.76e-13 ***
## BsmtFinSF2 3.036e+01 9.076e+00 3.345 0.000848 ***
## BsmtUnfSF 2.059e+01 4.856e+00 4.241 2.39e-05 ***
## TotalBsmtSF NA NA NA NA
## `1stFlrSF` 4.671e+01 5.624e+00 8.307 2.59e-16 ***
## `2ndFlrSF` 6.612e+01 5.606e+00 11.795 < 2e-16 ***
## LowQualFinSF 1.032e+01 1.846e+01 0.559 0.576211
## GrLivArea NA NA NA NA
## BsmtFullBath 8.231e+02 1.982e+03 0.415 0.677991
## BsmtHalfBath -2.303e+02 3.029e+03 -0.076 0.939403
## FullBath 3.651e+03 2.209e+03 1.652 0.098697 .
## HalfBath 1.223e+03 2.099e+03 0.583 0.560232
## BedroomAbvGr -3.427e+03 1.363e+03 -2.514 0.012070 *
## KitchenAbvGr -1.303e+04 5.678e+03 -2.295 0.021929 *
## TotRmsAbvGrd 1.072e+03 9.499e+02 1.129 0.259204
## Fireplaces 6.386e+03 2.560e+03 2.494 0.012763 *
## GarageYrBlt -2.068e+01 6.124e+01 -0.338 0.735678
## GarageCars 3.976e+03 2.280e+03 1.744 0.081374 .
## GarageArea 1.820e+01 7.920e+00 2.298 0.021746 *
## WoodDeckSF 1.448e+01 5.853e+00 2.474 0.013497 *
## OpenPorchSF 2.805e+00 1.154e+01 0.243 0.807929
## EnclosedPorch 4.774e+00 1.242e+01 0.384 0.700803
## `3SsnPorch` 3.213e+01 2.251e+01 1.428 0.153686
## ScreenPorch 2.861e+01 1.231e+01 2.324 0.020313 *
## PoolArea 8.283e+01 1.848e+01 4.482 8.08e-06 ***
## MiscVal -4.912e-02 1.419e+00 -0.035 0.972398
## MoSold -4.403e+02 2.453e+02 -1.795 0.072834 .
## YrSold -4.798e+02 5.167e+02 -0.929 0.353275
## MSZoningFV 3.350e+04 1.196e+04 2.800 0.005186 **
## MSZoningRH 2.443e+04 1.191e+04 2.050 0.040533 *
## MSZoningRL 2.645e+04 1.019e+04 2.596 0.009551 **
## MSZoningRM 2.322e+04 9.527e+03 2.438 0.014922 *
## StreetPave 3.223e+04 1.209e+04 2.665 0.007790 **
## LotShapeIR2 3.710e+03 4.238e+03 0.875 0.381526
## LotShapeIR3 4.766e+03 8.909e+03 0.535 0.592758
## LotShapeReg 1.684e+03 1.611e+03 1.045 0.296155
## LandContourHLS 7.642e+03 5.148e+03 1.484 0.137972
## LandContourLow -1.128e+04 6.419e+03 -1.758 0.079082 .
## LandContourLvl 5.419e+03 3.705e+03 1.463 0.143813
## UtilitiesNoSeWa -2.924e+04 2.650e+04 -1.103 0.270212
## LotConfigCulDSac 9.094e+03 3.317e+03 2.742 0.006201 **
## LotConfigFR2 -7.513e+03 4.034e+03 -1.862 0.062797 .
## LotConfigFR3 -1.595e+04 1.262e+04 -1.263 0.206677
## LotConfigInside -1.211e+03 1.794e+03 -0.675 0.499839
## LandSlopeMod 7.361e+03 3.987e+03 1.846 0.065116 .
## LandSlopeSev -4.331e+04 1.146e+04 -3.780 0.000164 ***
## NeighborhoodBlueste -6.013e+02 1.904e+04 -0.032 0.974810
## NeighborhoodBrDale -9.287e+02 1.101e+04 -0.084 0.932813
## NeighborhoodBrkSide -4.637e+03 9.461e+03 -0.490 0.624128
## NeighborhoodClearCr -1.516e+04 9.243e+03 -1.640 0.101285
## NeighborhoodCollgCr -1.041e+04 7.277e+03 -1.431 0.152770
## NeighborhoodCrawfor 1.136e+04 8.587e+03 1.323 0.185949
## NeighborhoodEdwards -2.068e+04 8.021e+03 -2.578 0.010043 *
## NeighborhoodGilbert -1.096e+04 7.699e+03 -1.424 0.154776
## NeighborhoodIDOTRR -1.070e+04 1.075e+04 -0.996 0.319684
## NeighborhoodMeadowV -7.015e+03 1.123e+04 -0.624 0.532436
## NeighborhoodMitchel -2.277e+04 8.187e+03 -2.781 0.005498 **
## NeighborhoodNAmes -1.645e+04 7.846e+03 -2.096 0.036260 *
## NeighborhoodNoRidge 2.522e+04 8.446e+03 2.986 0.002882 **
## NeighborhoodNPkVill 1.362e+04 1.412e+04 0.965 0.334672
## NeighborhoodNridgHt 1.754e+04 7.544e+03 2.324 0.020264 *
## NeighborhoodNWAmes -1.806e+04 8.033e+03 -2.248 0.024784 *
## NeighborhoodOldTown -1.434e+04 9.610e+03 -1.492 0.136004
## NeighborhoodSawyer -1.072e+04 8.152e+03 -1.315 0.188904
## NeighborhoodSawyerW -4.150e+03 7.810e+03 -0.531 0.595224
## NeighborhoodSomerst -2.158e+03 9.038e+03 -0.239 0.811330
## NeighborhoodStoneBr 3.775e+04 8.334e+03 4.530 6.48e-06 ***
## NeighborhoodSWISU -8.225e+03 9.726e+03 -0.846 0.397867
## NeighborhoodTimber -1.065e+04 8.170e+03 -1.304 0.192578
## NeighborhoodVeenker 9.633e+02 1.053e+04 0.091 0.927156
## Condition1Feedr 5.573e+03 5.001e+03 1.114 0.265336
## Condition1Norm 1.476e+04 4.166e+03 3.543 0.000410 ***
## Condition1PosA 7.112e+03 1.002e+04 0.710 0.477796
## Condition1PosN 1.323e+04 7.442e+03 1.778 0.075600 .
## Condition1RRAe -1.654e+04 9.090e+03 -1.820 0.069070 .
## Condition1RRAn 8.259e+03 6.865e+03 1.203 0.229200
## Condition1RRNe -3.861e+03 1.758e+04 -0.220 0.826183
## Condition1RRNn 6.212e+03 1.284e+04 0.484 0.628569
## Condition2Feedr -2.408e+03 2.345e+04 -0.103 0.918233
## Condition2Norm -4.021e+03 2.035e+04 -0.198 0.843411
## Condition2PosA 4.107e+04 3.721e+04 1.104 0.269886
## Condition2PosN -2.375e+05 2.779e+04 -8.545 < 2e-16 ***
## Condition2RRAe -1.221e+05 4.634e+04 -2.635 0.008530 **
## Condition2RRAn -1.585e+04 3.165e+04 -0.501 0.616626
## Condition2RRNn 3.872e+03 2.718e+04 0.142 0.886734
## BldgType2fmCon -4.976e+03 1.250e+04 -0.398 0.690572
## BldgTypeDuplex -9.392e+03 7.374e+03 -1.274 0.203030
## BldgTypeTwnhs -1.983e+04 9.999e+03 -1.983 0.047595 *
## BldgTypeTwnhsE -1.626e+04 8.993e+03 -1.808 0.070860 .
## HouseStyle1.5Unf 1.206e+04 7.924e+03 1.522 0.128228
## HouseStyle1Story 7.201e+03 4.353e+03 1.654 0.098347 .
## HouseStyle2.5Fin -2.313e+04 1.229e+04 -1.881 0.060173 .
## HouseStyle2.5Unf -8.943e+03 9.271e+03 -0.965 0.334890
## HouseStyle2Story -6.048e+03 3.511e+03 -1.723 0.085228 .
## HouseStyleSFoyer 1.370e+03 6.279e+03 0.218 0.827273
## HouseStyleSLvl 2.925e+03 5.563e+03 0.526 0.599164
## RoofStyleGable 6.147e+03 1.847e+04 0.333 0.739366
## RoofStyleGambrel 9.688e+03 2.018e+04 0.480 0.631341
## RoofStyleHip 6.020e+03 1.853e+04 0.325 0.745388
## RoofStyleMansard 1.843e+04 2.147e+04 0.859 0.390753
## RoofStyleShed 9.839e+04 3.473e+04 2.833 0.004687 **
## RoofMatlCompShg 6.837e+05 3.376e+04 20.252 < 2e-16 ***
## RoofMatlMembran 7.756e+05 4.811e+04 16.119 < 2e-16 ***
## RoofMatlMetal 7.469e+05 4.711e+04 15.856 < 2e-16 ***
## RoofMatlRoll 6.718e+05 4.217e+04 15.932 < 2e-16 ***
## RoofMatlTar&Grv 6.879e+05 3.848e+04 17.875 < 2e-16 ***
## RoofMatlWdShake 6.758e+05 3.731e+04 18.115 < 2e-16 ***
## RoofMatlWdShngl 7.361e+05 3.500e+04 21.034 < 2e-16 ***
## Exterior1stAsphShn -2.528e+04 3.317e+04 -0.762 0.446087
## Exterior1stBrkComm -1.215e+04 2.778e+04 -0.438 0.661763
## Exterior1stBrkFace 4.886e+03 1.277e+04 0.383 0.702128
## Exterior1stCBlock -1.889e+04 2.727e+04 -0.693 0.488632
## Exterior1stCemntBd -1.367e+04 1.910e+04 -0.716 0.474428
## Exterior1stHdBoard -1.574e+04 1.296e+04 -1.214 0.224840
## Exterior1stImStucc -4.463e+04 2.788e+04 -1.600 0.109765
## Exterior1stMetalSd -7.602e+03 1.464e+04 -0.519 0.603574
## Exterior1stPlywood -1.729e+04 1.279e+04 -1.352 0.176603
## Exterior1stStone -5.368e+03 2.438e+04 -0.220 0.825810
## Exterior1stStucco -8.274e+03 1.413e+04 -0.585 0.558327
## Exterior1stVinylSd -1.647e+04 1.328e+04 -1.240 0.215212
## Exterior1stWd Sdng -1.571e+04 1.237e+04 -1.270 0.204312
## Exterior1stWdShing -1.183e+04 1.335e+04 -0.886 0.375896
## Exterior2ndAsphShn 1.074e+04 2.229e+04 0.482 0.630021
## Exterior2ndBrk Cmn 8.616e+03 2.018e+04 0.427 0.669423
## Exterior2ndBrkFace 5.453e+03 1.325e+04 0.411 0.680870
## Exterior2ndCBlock NA NA NA NA
## Exterior2ndCmentBd 1.338e+04 1.879e+04 0.712 0.476486
## Exterior2ndHdBoard 1.031e+04 1.246e+04 0.827 0.408290
## Exterior2ndImStucc 2.798e+04 1.427e+04 1.960 0.050169 .
## Exterior2ndMetalSd 6.564e+03 1.426e+04 0.460 0.645457
## Exterior2ndOther -1.419e+04 2.725e+04 -0.521 0.602739
## Exterior2ndPlywood 8.287e+03 1.210e+04 0.685 0.493666
## Exterior2ndStone -9.985e+03 1.729e+04 -0.578 0.563687
## Exterior2ndStucco 8.188e+03 1.361e+04 0.602 0.547503
## Exterior2ndVinylSd 1.474e+04 1.284e+04 1.148 0.251178
## Exterior2ndWd Sdng 1.250e+04 1.198e+04 1.044 0.296821
## Exterior2ndWd Shng 7.176e+03 1.246e+04 0.576 0.564781
## MasVnrTypeBrkFace 6.170e+03 6.811e+03 0.906 0.365212
## MasVnrTypemissing 1.812e+03 1.085e+04 0.167 0.867316
## MasVnrTypeNone 9.548e+03 6.875e+03 1.389 0.165130
## MasVnrTypeStone 1.140e+04 7.211e+03 1.581 0.114192
## ExterQualFa -7.029e+03 1.098e+04 -0.640 0.522140
## ExterQualGd -2.021e+04 4.800e+03 -4.210 2.74e-05 ***
## ExterQualTA -2.087e+04 5.306e+03 -3.933 8.85e-05 ***
## ExterCondFa -3.610e+03 1.810e+04 -0.199 0.841933
## ExterCondGd -7.600e+03 1.729e+04 -0.440 0.660322
## ExterCondPo 4.216e+03 3.140e+04 0.134 0.893203
## ExterCondTA -4.976e+03 1.725e+04 -0.289 0.772974
## FoundationCBlock 2.634e+03 3.184e+03 0.827 0.408248
## FoundationPConc 3.814e+03 3.431e+03 1.112 0.266489
## FoundationSlab -8.872e+03 1.007e+04 -0.881 0.378294
## FoundationStone 7.636e+03 1.106e+04 0.691 0.489980
## FoundationWood -2.995e+04 1.479e+04 -2.025 0.043067 *
## BsmtQualFa -1.274e+04 6.348e+03 -2.006 0.045058 *
## BsmtQualGd -1.864e+04 3.354e+03 -5.556 3.39e-08 ***
## BsmtQualmissing 3.512e+04 3.683e+04 0.954 0.340457
## BsmtQualTA -1.551e+04 4.150e+03 -3.738 0.000194 ***
## BsmtCondGd -4.895e+02 5.287e+03 -0.093 0.926244
## BsmtCondmissing NA NA NA NA
## BsmtCondPo 7.159e+04 2.996e+04 2.389 0.017039 *
## BsmtCondTA 2.839e+03 4.253e+03 0.667 0.504604
## BsmtExposureGd 1.305e+04 3.005e+03 4.343 1.52e-05 ***
## BsmtExposuremissing -1.079e+04 2.316e+04 -0.466 0.641440
## BsmtExposureMn -4.383e+03 3.035e+03 -1.444 0.148959
## BsmtExposureNo -5.703e+03 2.191e+03 -2.602 0.009373 **
## BsmtFinType1BLQ 2.015e+03 2.797e+03 0.720 0.471382
## BsmtFinType1GLQ 5.635e+03 2.522e+03 2.235 0.025617 *
## BsmtFinType1LwQ -3.455e+03 3.755e+03 -0.920 0.357780
## BsmtFinType1missing NA NA NA NA
## BsmtFinType1Rec -3.913e+02 2.993e+03 -0.131 0.896005
## BsmtFinType1Unf 2.877e+03 2.918e+03 0.986 0.324330
## BsmtFinType2BLQ -1.307e+04 7.577e+03 -1.725 0.084801 .
## BsmtFinType2GLQ -2.653e+03 9.367e+03 -0.283 0.777079
## BsmtFinType2LwQ -1.502e+04 7.393e+03 -2.032 0.042411 *
## BsmtFinType2missing -2.811e+04 2.513e+04 -1.119 0.263392
## BsmtFinType2Rec -1.029e+04 7.118e+03 -1.446 0.148414
## BsmtFinType2Unf -8.904e+03 7.576e+03 -1.175 0.240105
## HeatingGasA 1.163e+04 2.572e+04 0.452 0.651218
## HeatingGasW 8.187e+03 2.652e+04 0.309 0.757552
## HeatingGrav 4.763e+03 2.816e+04 0.169 0.865725
## HeatingOthW -1.102e+04 3.154e+04 -0.349 0.726963
## HeatingWall 2.601e+04 2.985e+04 0.871 0.383724
## HeatingQCFa 6.932e+02 4.680e+03 0.148 0.882275
## HeatingQCGd -3.600e+03 2.072e+03 -1.737 0.082553 .
## HeatingQCPo 3.434e+03 2.672e+04 0.129 0.897770
## HeatingQCTA -3.535e+03 2.067e+03 -1.710 0.087513 .
## CentralAirY 2.484e+02 3.886e+03 0.064 0.949043
## ElectricalFuseF -2.531e+02 5.774e+03 -0.044 0.965036
## ElectricalFuseP -8.006e+03 1.859e+04 -0.431 0.666873
## Electricalmissing 1.373e+04 2.423e+04 0.566 0.571265
## ElectricalMix -4.751e+04 4.467e+04 -1.064 0.287710
## ElectricalSBrkr -1.333e+03 2.959e+03 -0.451 0.652417
## KitchenQualFa -1.923e+04 6.217e+03 -3.093 0.002026 **
## KitchenQualGd -2.421e+04 3.494e+03 -6.929 6.84e-12 ***
## KitchenQualTA -2.198e+04 3.936e+03 -5.584 2.89e-08 ***
## FunctionalMaj2 -1.315e+03 1.447e+04 -0.091 0.927601
## FunctionalMin1 9.318e+03 8.646e+03 1.078 0.281342
## FunctionalMin2 1.100e+04 8.656e+03 1.271 0.203869
## FunctionalMod -7.700e+02 1.045e+04 -0.074 0.941269
## FunctionalSev -4.439e+04 2.949e+04 -1.505 0.132548
## FunctionalTyp 2.047e+04 7.501e+03 2.729 0.006443 **
## FireplaceQuFa -1.988e+03 6.871e+03 -0.289 0.772400
## FireplaceQuGd 2.167e+03 5.326e+03 0.407 0.684125
## FireplaceQumissing 8.347e+03 6.248e+03 1.336 0.181830
## FireplaceQuPo 8.983e+03 7.850e+03 1.144 0.252702
## FireplaceQuTA 3.260e+03 5.541e+03 0.588 0.556380
## GarageTypeAttchd 1.960e+04 1.098e+04 1.785 0.074576 .
## GarageTypeBasment 2.271e+04 1.273e+04 1.784 0.074608 .
## GarageTypeBuiltIn 1.886e+04 1.143e+04 1.650 0.099237 .
## GarageTypeCarPort 2.569e+04 1.464e+04 1.755 0.079466 .
## GarageTypeDetchd 2.271e+04 1.099e+04 2.066 0.039038 *
## GarageTypemissing 2.384e+04 2.079e+04 1.147 0.251736
## GarageFinishmissing NA NA NA NA
## GarageFinishRFn -2.732e+03 1.968e+03 -1.388 0.165301
## GarageFinishUnf -1.223e+01 2.424e+03 -0.005 0.995974
## GarageQualFa -1.164e+05 3.013e+04 -3.862 0.000118 ***
## GarageQualGd -1.085e+05 3.089e+04 -3.511 0.000463 ***
## GarageQualmissing NA NA NA NA
## GarageQualPo -1.320e+05 3.847e+04 -3.431 0.000621 ***
## GarageQualTA -1.106e+05 2.983e+04 -3.707 0.000219 ***
## GarageCondFa 1.042e+05 3.475e+04 3.000 0.002756 **
## GarageCondGd 1.037e+05 3.586e+04 2.891 0.003912 **
## GarageCondmissing NA NA NA NA
## GarageCondPo 1.092e+05 3.729e+04 2.929 0.003462 **
## GarageCondTA 1.055e+05 3.442e+04 3.067 0.002213 **
## PavedDriveP -2.750e+03 5.552e+03 -0.495 0.620487
## PavedDriveY 2.161e+01 3.469e+03 0.006 0.995031
## SaleTypeCon 2.626e+04 1.764e+04 1.489 0.136876
## SaleTypeConLD 1.649e+04 9.706e+03 1.699 0.089532 .
## SaleTypeConLI 4.348e+03 1.157e+04 0.376 0.707083
## SaleTypeConLw 1.261e+03 1.220e+04 0.103 0.917688
## SaleTypeCWD 1.566e+04 1.291e+04 1.213 0.225406
## SaleTypeNew 2.281e+04 1.551e+04 1.471 0.141587
## SaleTypeOth 8.426e+03 1.444e+04 0.584 0.559586
## SaleTypeWD 8.561e+01 4.195e+03 0.020 0.983722
## SaleConditionAdjLand 7.512e+03 1.454e+04 0.517 0.605571
## SaleConditionAlloca 6.021e+03 8.633e+03 0.697 0.485646
## SaleConditionFamily -7.042e+02 6.101e+03 -0.115 0.908127
## SaleConditionNormal 5.535e+03 2.895e+03 1.912 0.056130 .
## SaleConditionPartial -2.294e+03 1.492e+04 -0.154 0.877826
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 22770 on 1219 degrees of freedom
## Multiple R-squared: 0.9314, Adjusted R-squared: 0.9178
## F-statistic: 68.92 on 240 and 1219 DF, p-value: < 2.2e-16# Create a second model that only uses the most significant of the features from the first model summary
model2 <- lm(SalePrice ~ LotArea + OverallQual + OverallCond + YearBuilt + YearRemodAdd + MasVnrArea + BsmtFinSF1 + BsmtFinSF2 + BsmtUnfSF + `1stFlrSF` + `2ndFlrSF` + FullBath + BedroomAbvGr + KitchenAbvGr + Fireplaces + GarageCars + GarageArea + WoodDeckSF + ScreenPorch + PoolArea + MoSold + MSZoning + LandContour + Street + LotConfig + LandSlope + Neighborhood + Condition1 + Condition2 + BldgType + RoofStyle + RoofMatl + ExterQual + Foundation + BsmtQual + BsmtCond + BsmtExposure, data = train_data_clean)summary(model2)##
## Call:
## lm(formula = SalePrice ~ LotArea + OverallQual + OverallCond +
## YearBuilt + YearRemodAdd + MasVnrArea + BsmtFinSF1 + BsmtFinSF2 +
## BsmtUnfSF + `1stFlrSF` + `2ndFlrSF` + FullBath + BedroomAbvGr +
## KitchenAbvGr + Fireplaces + GarageCars + GarageArea + WoodDeckSF +
## ScreenPorch + PoolArea + MoSold + MSZoning + LandContour +
## Street + LotConfig + LandSlope + Neighborhood + Condition1 +
## Condition2 + BldgType + RoofStyle + RoofMatl + ExterQual +
## Foundation + BsmtQual + BsmtCond + BsmtExposure, data = train_data_clean)
##
## Residuals:
## Min 1Q Median 3Q Max
## -179169 -10422 563 10262 179169
##
## Coefficients: (1 not defined because of singularities)
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -1.635e+06 1.523e+05 -10.737 < 2e-16 ***
## LotArea 7.391e-01 1.000e-01 7.390 2.56e-13 ***
## OverallQual 7.394e+03 9.648e+02 7.664 3.43e-14 ***
## OverallCond 6.309e+03 7.913e+02 7.973 3.27e-15 ***
## YearBuilt 3.608e+02 6.471e+01 5.575 2.99e-08 ***
## YearRemodAdd 1.276e+02 5.019e+01 2.543 0.011112 *
## MasVnrArea 9.129e+00 4.657e+00 1.960 0.050157 .
## BsmtFinSF1 4.367e+01 4.278e+00 10.208 < 2e-16 ***
## BsmtFinSF2 2.760e+01 5.747e+00 4.802 1.74e-06 ***
## BsmtUnfSF 2.550e+01 4.199e+00 6.072 1.64e-09 ***
## `1stFlrSF` 5.045e+01 4.528e+00 11.142 < 2e-16 ***
## `2ndFlrSF` 5.949e+01 2.835e+00 20.988 < 2e-16 ***
## FullBath 2.105e+03 1.995e+03 1.055 0.291720
## BedroomAbvGr -4.945e+03 1.201e+03 -4.119 4.04e-05 ***
## KitchenAbvGr -1.635e+04 5.215e+03 -3.135 0.001753 **
## Fireplaces 3.110e+03 1.335e+03 2.330 0.019946 *
## GarageCars 3.368e+03 2.160e+03 1.559 0.119186
## GarageArea 1.159e+01 7.391e+00 1.568 0.117136
## WoodDeckSF 1.663e+00 5.793e+00 0.287 0.774112
## ScreenPorch 3.328e+01 1.214e+01 2.741 0.006205 **
## PoolArea 8.988e+01 1.742e+01 5.161 2.82e-07 ***
## MoSold -3.021e+02 2.436e+02 -1.240 0.215112
## MSZoningFV 2.376e+04 1.179e+04 2.016 0.044038 *
## MSZoningRH 2.070e+04 1.190e+04 1.740 0.082169 .
## MSZoningRL 2.260e+04 1.007e+04 2.245 0.024919 *
## MSZoningRM 1.773e+04 9.431e+03 1.880 0.060260 .
## LandContourHLS 1.046e+04 5.129e+03 2.039 0.041691 *
## LandContourLow -1.098e+04 6.287e+03 -1.747 0.080824 .
## LandContourLvl 5.060e+03 3.606e+03 1.403 0.160792
## StreetPave 3.563e+04 1.191e+04 2.992 0.002823 **
## LotConfigCulDSac 4.710e+03 3.137e+03 1.501 0.133466
## LotConfigFR2 -7.271e+03 4.046e+03 -1.797 0.072533 .
## LotConfigFR3 -1.474e+04 1.309e+04 -1.126 0.260500
## LotConfigInside -1.205e+03 1.754e+03 -0.687 0.492083
## LandSlopeMod 5.842e+03 3.929e+03 1.487 0.137253
## LandSlopeSev -3.587e+04 1.077e+04 -3.329 0.000895 ***
## NeighborhoodBlueste 4.480e+03 1.907e+04 0.235 0.814321
## NeighborhoodBrDale -1.054e+03 1.075e+04 -0.098 0.921925
## NeighborhoodBrkSide -7.603e+03 8.912e+03 -0.853 0.393741
## NeighborhoodClearCr -1.922e+04 8.958e+03 -2.145 0.032125 *
## NeighborhoodCollgCr -1.160e+04 7.109e+03 -1.631 0.103074
## NeighborhoodCrawfor 4.847e+03 8.248e+03 0.588 0.556882
## NeighborhoodEdwards -2.132e+04 7.721e+03 -2.761 0.005840 **
## NeighborhoodGilbert -1.594e+04 7.484e+03 -2.130 0.033325 *
## NeighborhoodIDOTRR -9.493e+03 1.029e+04 -0.922 0.356447
## NeighborhoodMeadowV 3.259e+02 1.007e+04 0.032 0.974186
## NeighborhoodMitchel -2.892e+04 7.918e+03 -3.653 0.000269 ***
## NeighborhoodNAmes -2.021e+04 7.556e+03 -2.675 0.007566 **
## NeighborhoodNoRidge 2.311e+04 8.121e+03 2.845 0.004505 **
## NeighborhoodNPkVill 4.509e+03 1.072e+04 0.421 0.673940
## NeighborhoodNridgHt 2.130e+04 7.302e+03 2.917 0.003598 **
## NeighborhoodNWAmes -2.604e+04 7.791e+03 -3.343 0.000853 ***
## NeighborhoodOldTown -1.692e+04 9.225e+03 -1.834 0.066837 .
## NeighborhoodSawyer -1.778e+04 7.917e+03 -2.246 0.024869 *
## NeighborhoodSawyerW -1.064e+04 7.551e+03 -1.409 0.159171
## NeighborhoodSomerst 5.829e+03 8.771e+03 0.665 0.506455
## NeighborhoodStoneBr 3.717e+04 8.087e+03 4.596 4.71e-06 ***
## NeighborhoodSWISU -1.200e+04 9.321e+03 -1.287 0.198267
## NeighborhoodTimber -1.624e+04 8.068e+03 -2.012 0.044371 *
## NeighborhoodVeenker 1.679e+03 1.011e+04 0.166 0.868153
## Condition1Feedr 2.418e+03 4.870e+03 0.497 0.619592
## Condition1Norm 9.363e+03 3.998e+03 2.342 0.019320 *
## Condition1PosA 6.640e+03 9.913e+03 0.670 0.503100
## Condition1PosN 1.196e+04 7.314e+03 1.635 0.102271
## Condition1RRAe -1.511e+04 8.727e+03 -1.732 0.083520 .
## Condition1RRAn 4.366e+03 6.769e+03 0.645 0.519089
## Condition1RRNe -8.146e+03 1.808e+04 -0.450 0.652455
## Condition1RRNn 1.070e+03 1.274e+04 0.084 0.933104
## Condition2Feedr 2.812e+03 2.218e+04 0.127 0.899135
## Condition2Norm 4.295e+02 1.894e+04 0.023 0.981910
## Condition2PosA 2.679e+04 3.175e+04 0.844 0.398908
## Condition2PosN -2.243e+05 2.698e+04 -8.316 < 2e-16 ***
## Condition2RRAe -9.187e+04 4.366e+04 -2.104 0.035555 *
## Condition2RRAn -4.805e+03 3.118e+04 -0.154 0.877546
## Condition2RRNn 9.026e+03 2.611e+04 0.346 0.729596
## BldgType2fmCon -5.498e+03 5.602e+03 -0.981 0.326561
## BldgTypeDuplex -2.350e+03 5.850e+03 -0.402 0.687952
## BldgTypeTwnhs -3.225e+04 5.267e+03 -6.122 1.21e-09 ***
## BldgTypeTwnhsE -2.491e+04 3.569e+03 -6.979 4.66e-12 ***
## RoofStyleGable -9.380e+03 1.829e+04 -0.513 0.608210
## RoofStyleGambrel -5.773e+03 1.984e+04 -0.291 0.771165
## RoofStyleHip -7.922e+03 1.838e+04 -0.431 0.666607
## RoofStyleMansard 7.644e+03 2.111e+04 0.362 0.717255
## RoofStyleShed 6.715e+04 3.474e+04 1.933 0.053434 .
## RoofMatlCompShg 6.721e+05 3.081e+04 21.813 < 2e-16 ***
## RoofMatlMembran 7.302e+05 4.504e+04 16.214 < 2e-16 ***
## RoofMatlMetal 7.110e+05 4.531e+04 15.692 < 2e-16 ***
## RoofMatlRoll 6.732e+05 3.957e+04 17.011 < 2e-16 ***
## RoofMatlTar&Grv 6.520e+05 3.579e+04 18.217 < 2e-16 ***
## RoofMatlWdShake 6.603e+05 3.397e+04 19.440 < 2e-16 ***
## RoofMatlWdShngl 7.436e+05 3.190e+04 23.314 < 2e-16 ***
## ExterQualFa -1.656e+04 9.527e+03 -1.739 0.082304 .
## ExterQualGd -3.497e+04 4.543e+03 -7.696 2.69e-14 ***
## ExterQualTA -3.674e+04 5.066e+03 -7.252 6.91e-13 ***
## FoundationCBlock -2.037e+03 3.082e+03 -0.661 0.508676
## FoundationPConc 3.670e+03 3.389e+03 1.083 0.279074
## FoundationSlab -5.772e+03 9.031e+03 -0.639 0.522798
## FoundationStone -1.915e+03 1.059e+04 -0.181 0.856553
## FoundationWood -1.948e+04 1.473e+04 -1.323 0.186021
## BsmtQualFa -2.043e+04 6.242e+03 -3.273 0.001092 **
## BsmtQualGd -2.901e+04 3.255e+03 -8.912 < 2e-16 ***
## BsmtQualmissing 2.358e+04 2.635e+04 0.895 0.370851
## BsmtQualTA -2.366e+04 4.056e+03 -5.834 6.75e-09 ***
## BsmtCondGd 4.332e+03 5.215e+03 0.831 0.406278
## BsmtCondmissing NA NA NA NA
## BsmtCondPo 1.885e+04 1.957e+04 0.964 0.335466
## BsmtCondTA 6.360e+03 4.090e+03 1.555 0.120186
## BsmtExposureGd 1.243e+04 3.038e+03 4.091 4.54e-05 ***
## BsmtExposuremissing -2.309e+04 2.441e+04 -0.946 0.344482
## BsmtExposureMn -3.751e+03 2.971e+03 -1.263 0.206910
## BsmtExposureNo -7.238e+03 2.053e+03 -3.526 0.000437 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 24200 on 1350 degrees of freedom
## Multiple R-squared: 0.9141, Adjusted R-squared: 0.9072
## F-statistic: 131.9 on 109 and 1350 DF, p-value: < 2.2e-16We can see that the overall Adjusted R2 has decreased by roughly 1%. That being said, we’ve decreased our independent features by 50%, which makes for a much more simplified model.
# Create a third model using only the most significant features from the second model summary
model3 <- lm(SalePrice ~ LotArea + OverallQual + OverallCond + YearBuilt + YearRemodAdd + MasVnrArea + BsmtFinSF1 + BsmtFinSF2 + BsmtUnfSF + `1stFlrSF` + `2ndFlrSF` + BedroomAbvGr + KitchenAbvGr + Fireplaces + ScreenPorch + PoolArea + MSZoning + LandContour + Street + LotConfig + LandSlope + Neighborhood + Condition1 + Condition2 + BldgType + RoofStyle + RoofMatl + ExterQual + Foundation + BsmtQual + BsmtCond + BsmtExposure, data = train_data_clean)summary(model3)##
## Call:
## lm(formula = SalePrice ~ LotArea + OverallQual + OverallCond +
## YearBuilt + YearRemodAdd + MasVnrArea + BsmtFinSF1 + BsmtFinSF2 +
## BsmtUnfSF + `1stFlrSF` + `2ndFlrSF` + BedroomAbvGr + KitchenAbvGr +
## Fireplaces + ScreenPorch + PoolArea + MSZoning + LandContour +
## Street + LotConfig + LandSlope + Neighborhood + Condition1 +
## Condition2 + BldgType + RoofStyle + RoofMatl + ExterQual +
## Foundation + BsmtQual + BsmtCond + BsmtExposure, data = train_data_clean)
##
## Residuals:
## Min 1Q Median 3Q Max
## -184232 -10808 431 10655 184232
##
## Coefficients: (1 not defined because of singularities)
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -1.740e+06 1.503e+05 -11.577 < 2e-16 ***
## LotArea 7.729e-01 1.004e-01 7.696 2.68e-14 ***
## OverallQual 7.631e+03 9.689e+02 7.876 6.89e-15 ***
## OverallCond 6.348e+03 7.955e+02 7.980 3.08e-15 ***
## YearBuilt 4.203e+02 6.388e+01 6.580 6.69e-11 ***
## YearRemodAdd 1.263e+02 5.049e+01 2.501 0.012485 *
## MasVnrArea 1.124e+01 4.678e+00 2.402 0.016438 *
## BsmtFinSF1 4.379e+01 4.313e+00 10.153 < 2e-16 ***
## BsmtFinSF2 2.695e+01 5.789e+00 4.655 3.55e-06 ***
## BsmtUnfSF 2.528e+01 4.228e+00 5.979 2.86e-09 ***
## `1stFlrSF` 5.574e+01 4.350e+00 12.813 < 2e-16 ***
## `2ndFlrSF` 6.297e+01 2.678e+00 23.513 < 2e-16 ***
## BedroomAbvGr -5.254e+03 1.188e+03 -4.423 1.05e-05 ***
## KitchenAbvGr -1.579e+04 5.218e+03 -3.026 0.002525 **
## Fireplaces 2.889e+03 1.339e+03 2.157 0.031217 *
## ScreenPorch 3.450e+01 1.212e+01 2.847 0.004474 **
## PoolArea 8.898e+01 1.753e+01 5.077 4.37e-07 ***
## MSZoningFV 2.015e+04 1.183e+04 1.703 0.088876 .
## MSZoningRH 1.772e+04 1.196e+04 1.481 0.138782
## MSZoningRL 2.067e+04 1.011e+04 2.044 0.041128 *
## MSZoningRM 1.616e+04 9.486e+03 1.703 0.088760 .
## LandContourHLS 1.121e+04 5.150e+03 2.177 0.029687 *
## LandContourLow -9.488e+03 6.309e+03 -1.504 0.132849
## LandContourLvl 5.281e+03 3.621e+03 1.458 0.144990
## StreetPave 3.054e+04 1.194e+04 2.558 0.010650 *
## LotConfigCulDSac 4.199e+03 3.162e+03 1.328 0.184447
## LotConfigFR2 -7.314e+03 4.067e+03 -1.798 0.072379 .
## LotConfigFR3 -1.449e+04 1.319e+04 -1.098 0.272277
## LotConfigInside -1.557e+03 1.766e+03 -0.882 0.378202
## LandSlopeMod 6.061e+03 3.950e+03 1.535 0.125139
## LandSlopeSev -3.956e+04 1.084e+04 -3.650 0.000272 ***
## NeighborhoodBlueste 7.331e+03 1.921e+04 0.382 0.702766
## NeighborhoodBrDale -3.245e+03 1.077e+04 -0.301 0.763150
## NeighborhoodBrkSide -1.077e+04 8.939e+03 -1.205 0.228521
## NeighborhoodClearCr -2.237e+04 8.997e+03 -2.487 0.013019 *
## NeighborhoodCollgCr -1.300e+04 7.112e+03 -1.828 0.067726 .
## NeighborhoodCrawfor 2.033e+03 8.273e+03 0.246 0.805944
## NeighborhoodEdwards -2.489e+04 7.726e+03 -3.222 0.001304 **
## NeighborhoodGilbert -1.794e+04 7.534e+03 -2.381 0.017421 *
## NeighborhoodIDOTRR -1.272e+04 1.029e+04 -1.237 0.216309
## NeighborhoodMeadowV -2.840e+03 1.000e+04 -0.284 0.776441
## NeighborhoodMitchel -3.035e+04 7.951e+03 -3.817 0.000141 ***
## NeighborhoodNAmes -2.199e+04 7.569e+03 -2.906 0.003722 **
## NeighborhoodNoRidge 2.135e+04 8.147e+03 2.621 0.008863 **
## NeighborhoodNPkVill 7.835e+03 1.075e+04 0.729 0.466189
## NeighborhoodNridgHt 2.153e+04 7.298e+03 2.950 0.003237 **
## NeighborhoodNWAmes -2.611e+04 7.837e+03 -3.331 0.000888 ***
## NeighborhoodOldTown -1.868e+04 9.264e+03 -2.016 0.043979 *
## NeighborhoodSawyer -1.962e+04 7.953e+03 -2.466 0.013768 *
## NeighborhoodSawyerW -1.240e+04 7.589e+03 -1.634 0.102432
## NeighborhoodSomerst 7.214e+03 8.786e+03 0.821 0.411775
## NeighborhoodStoneBr 3.538e+04 8.120e+03 4.357 1.42e-05 ***
## NeighborhoodSWISU -1.514e+04 9.342e+03 -1.620 0.105360
## NeighborhoodTimber -1.680e+04 8.120e+03 -2.068 0.038786 *
## NeighborhoodVeenker 2.749e+02 1.016e+04 0.027 0.978419
## Condition1Feedr 1.762e+03 4.908e+03 0.359 0.719585
## Condition1Norm 8.788e+03 4.028e+03 2.182 0.029297 *
## Condition1PosA 5.707e+03 9.989e+03 0.571 0.567832
## Condition1PosN 1.097e+04 7.355e+03 1.491 0.136163
## Condition1RRAe -1.484e+04 8.791e+03 -1.688 0.091558 .
## Condition1RRAn 4.167e+03 6.827e+03 0.610 0.541664
## Condition1RRNe -8.005e+03 1.821e+04 -0.440 0.660274
## Condition1RRNn -2.002e+02 1.284e+04 -0.016 0.987558
## Condition2Feedr 4.468e+03 2.231e+04 0.200 0.841302
## Condition2Norm 4.217e+02 1.907e+04 0.022 0.982358
## Condition2PosA 3.270e+04 3.175e+04 1.030 0.303266
## Condition2PosN -2.259e+05 2.710e+04 -8.334 < 2e-16 ***
## Condition2RRAe -9.498e+04 4.394e+04 -2.162 0.030813 *
## Condition2RRAn -4.909e+03 3.143e+04 -0.156 0.875905
## Condition2RRNn 1.342e+04 2.627e+04 0.511 0.609510
## BldgType2fmCon -5.642e+03 5.636e+03 -1.001 0.317013
## BldgTypeDuplex -2.857e+03 5.894e+03 -0.485 0.628032
## BldgTypeTwnhs -3.458e+04 5.254e+03 -6.582 6.61e-11 ***
## BldgTypeTwnhsE -2.615e+04 3.564e+03 -7.337 3.76e-13 ***
## RoofStyleGable -1.595e+04 1.839e+04 -0.868 0.385668
## RoofStyleGambrel -1.380e+04 1.993e+04 -0.692 0.488843
## RoofStyleHip -1.465e+04 1.848e+04 -0.793 0.427990
## RoofStyleMansard 9.252e+02 2.123e+04 0.044 0.965252
## RoofStyleShed 6.383e+04 3.495e+04 1.826 0.068046 .
## RoofMatlCompShg 6.827e+05 3.035e+04 22.493 < 2e-16 ***
## RoofMatlMembran 7.398e+05 4.488e+04 16.484 < 2e-16 ***
## RoofMatlMetal 7.205e+05 4.511e+04 15.973 < 2e-16 ***
## RoofMatlRoll 6.847e+05 3.915e+04 17.491 < 2e-16 ***
## RoofMatlTar&Grv 6.562e+05 3.552e+04 18.474 < 2e-16 ***
## RoofMatlWdShake 6.685e+05 3.368e+04 19.852 < 2e-16 ***
## RoofMatlWdShngl 7.533e+05 3.147e+04 23.940 < 2e-16 ***
## ExterQualFa -1.808e+04 9.599e+03 -1.884 0.059800 .
## ExterQualGd -3.439e+04 4.579e+03 -7.509 1.07e-13 ***
## ExterQualTA -3.658e+04 5.108e+03 -7.162 1.30e-12 ***
## FoundationCBlock -3.568e+03 3.088e+03 -1.155 0.248135
## FoundationPConc 3.044e+03 3.414e+03 0.892 0.372775
## FoundationSlab -6.187e+03 9.102e+03 -0.680 0.496793
## FoundationStone -1.415e+03 1.068e+04 -0.133 0.894547
## FoundationWood -2.129e+04 1.483e+04 -1.436 0.151293
## BsmtQualFa -1.869e+04 6.273e+03 -2.980 0.002934 **
## BsmtQualGd -2.935e+04 3.276e+03 -8.960 < 2e-16 ***
## BsmtQualmissing 2.405e+04 2.653e+04 0.907 0.364820
## BsmtQualTA -2.384e+04 4.078e+03 -5.846 6.29e-09 ***
## BsmtCondGd 4.951e+03 5.255e+03 0.942 0.346339
## BsmtCondmissing NA NA NA NA
## BsmtCondPo 1.509e+04 1.970e+04 0.766 0.444041
## BsmtCondTA 6.583e+03 4.124e+03 1.596 0.110652
## BsmtExposureGd 1.228e+04 3.058e+03 4.016 6.26e-05 ***
## BsmtExposuremissing -2.337e+04 2.459e+04 -0.950 0.342064
## BsmtExposureMn -3.308e+03 2.991e+03 -1.106 0.268877
## BsmtExposureNo -7.531e+03 2.061e+03 -3.654 0.000268 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 24410 on 1355 degrees of freedom
## Multiple R-squared: 0.9123, Adjusted R-squared: 0.9056
## F-statistic: 135.6 on 104 and 1355 DF, p-value: < 2.2e-16Once again we see a decrease in the overall Adjusted R2 value after removing 4 more independent features from the model.
model4 <- lm(SalePrice ~ LotArea*OverallQual*OverallCond + YearBuilt + BsmtFinSF1 + BsmtUnfSF + `1stFlrSF` + `2ndFlrSF` + BedroomAbvGr + PoolArea + LandSlope + Neighborhood + LotConfig + Condition1 + BldgType + RoofStyle + RoofMatl + ExterQual, data = train_data_clean)summary(model4)##
## Call:
## lm(formula = SalePrice ~ LotArea * OverallQual * OverallCond +
## YearBuilt + BsmtFinSF1 + BsmtUnfSF + `1stFlrSF` + `2ndFlrSF` +
## BedroomAbvGr + PoolArea + LandSlope + Neighborhood + LotConfig +
## Condition1 + BldgType + RoofStyle + RoofMatl + ExterQual,
## data = train_data_clean)
##
## Residuals:
## Min 1Q Median 3Q Max
## -363957 -11843 -852 11396 179813
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -1.730e+06 1.271e+05 -13.606 < 2e-16 ***
## LotArea 9.164e+00 3.043e+00 3.012 0.002642 **
## OverallQual 2.196e+04 5.949e+03 3.692 0.000231 ***
## OverallCond 2.042e+04 6.256e+03 3.264 0.001126 **
## YearBuilt 5.446e+02 5.864e+01 9.288 < 2e-16 ***
## BsmtFinSF1 3.087e+01 3.087e+00 10.001 < 2e-16 ***
## BsmtUnfSF 8.960e+00 2.995e+00 2.992 0.002824 **
## `1stFlrSF` 6.980e+01 3.662e+00 19.064 < 2e-16 ***
## `2ndFlrSF` 6.024e+01 2.732e+00 22.045 < 2e-16 ***
## BedroomAbvGr -5.768e+03 1.288e+03 -4.477 8.19e-06 ***
## PoolArea 9.557e+01 1.941e+01 4.923 9.56e-07 ***
## LandSlopeMod 6.765e+03 3.763e+03 1.798 0.072432 .
## LandSlopeSev -1.902e+04 1.115e+04 -1.706 0.088299 .
## NeighborhoodBlueste -1.154e+03 2.081e+04 -0.055 0.955796
## NeighborhoodBrDale -9.156e+03 1.087e+04 -0.842 0.399915
## NeighborhoodBrkSide -1.411e+04 9.135e+03 -1.544 0.122702
## NeighborhoodClearCr -1.983e+04 9.780e+03 -2.028 0.042770 *
## NeighborhoodCollgCr -1.864e+04 7.738e+03 -2.409 0.016108 *
## NeighborhoodCrawfor -1.007e+03 8.918e+03 -0.113 0.910113
## NeighborhoodEdwards -3.007e+04 8.381e+03 -3.587 0.000346 ***
## NeighborhoodGilbert -2.412e+04 8.247e+03 -2.925 0.003497 **
## NeighborhoodIDOTRR -2.536e+04 9.629e+03 -2.634 0.008543 **
## NeighborhoodMeadowV -1.077e+04 1.036e+04 -1.040 0.298446
## NeighborhoodMitchel -3.268e+04 8.659e+03 -3.774 0.000167 ***
## NeighborhoodNAmes -2.616e+04 8.120e+03 -3.221 0.001305 **
## NeighborhoodNoRidge 1.776e+04 8.795e+03 2.019 0.043655 *
## NeighborhoodNPkVill 6.360e+02 1.179e+04 0.054 0.956996
## NeighborhoodNridgHt 3.011e+04 7.962e+03 3.782 0.000162 ***
## NeighborhoodNWAmes -3.385e+04 8.416e+03 -4.022 6.09e-05 ***
## NeighborhoodOldTown -2.712e+04 8.870e+03 -3.057 0.002276 **
## NeighborhoodSawyer -2.561e+04 8.594e+03 -2.980 0.002933 **
## NeighborhoodSawyerW -2.237e+04 8.275e+03 -2.703 0.006962 **
## NeighborhoodSomerst 2.299e+03 7.748e+03 0.297 0.766708
## NeighborhoodStoneBr 3.943e+04 8.905e+03 4.427 1.03e-05 ***
## NeighborhoodSWISU -2.038e+04 1.010e+04 -2.019 0.043686 *
## NeighborhoodTimber -1.197e+04 8.838e+03 -1.354 0.176009
## NeighborhoodVeenker 3.247e+03 1.120e+04 0.290 0.771940
## LotConfigCulDSac 5.201e+03 3.502e+03 1.485 0.137786
## LotConfigFR2 -6.519e+03 4.484e+03 -1.454 0.146219
## LotConfigFR3 -1.880e+04 1.434e+04 -1.311 0.189978
## LotConfigInside -2.512e+03 1.945e+03 -1.292 0.196718
## Condition1Feedr 1.882e+03 5.281e+03 0.356 0.721571
## Condition1Norm 9.771e+03 4.313e+03 2.265 0.023655 *
## Condition1PosA 3.390e+03 1.103e+04 0.307 0.758579
## Condition1PosN -1.428e+04 7.806e+03 -1.829 0.067570 .
## Condition1RRAe -1.426e+04 9.657e+03 -1.477 0.139979
## Condition1RRAn 7.885e+03 7.076e+03 1.114 0.265317
## Condition1RRNe -1.025e+04 2.021e+04 -0.507 0.612066
## Condition1RRNn 1.045e+04 1.356e+04 0.771 0.440976
## BldgType2fmCon -7.997e+03 5.396e+03 -1.482 0.138590
## BldgTypeDuplex -1.780e+04 4.313e+03 -4.127 3.89e-05 ***
## BldgTypeTwnhs -4.463e+04 5.653e+03 -7.896 5.83e-15 ***
## BldgTypeTwnhsE -3.142e+04 3.664e+03 -8.576 < 2e-16 ***
## RoofStyleGable -2.723e+04 2.025e+04 -1.345 0.178851
## RoofStyleGambrel -2.224e+04 2.194e+04 -1.013 0.311015
## RoofStyleHip -2.398e+04 2.034e+04 -1.179 0.238570
## RoofStyleMansard -5.820e+03 2.331e+04 -0.250 0.802845
## RoofStyleShed -7.698e+03 2.919e+04 -0.264 0.792055
## RoofMatlCompShg 6.519e+05 3.484e+04 18.712 < 2e-16 ***
## RoofMatlMembran 6.569e+05 5.008e+04 13.118 < 2e-16 ***
## RoofMatlMetal 6.503e+05 4.994e+04 13.022 < 2e-16 ***
## RoofMatlRoll 6.573e+05 4.439e+04 14.809 < 2e-16 ***
## RoofMatlTar&Grv 6.223e+05 4.006e+04 15.534 < 2e-16 ***
## RoofMatlWdShake 6.266e+05 3.763e+04 16.653 < 2e-16 ***
## RoofMatlWdShngl 7.186e+05 3.578e+04 20.086 < 2e-16 ***
## ExterQualFa -4.419e+04 9.850e+03 -4.486 7.86e-06 ***
## ExterQualGd -4.457e+04 4.721e+03 -9.441 < 2e-16 ***
## ExterQualTA -4.793e+04 5.330e+03 -8.993 < 2e-16 ***
## LotArea:OverallQual -1.550e+00 4.810e-01 -3.222 0.001302 **
## LotArea:OverallCond -1.881e+00 5.434e-01 -3.462 0.000552 ***
## OverallQual:OverallCond -2.602e+03 1.076e+03 -2.418 0.015737 *
## LotArea:OverallQual:OverallCond 3.352e-01 8.773e-02 3.821 0.000139 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 27240 on 1388 degrees of freedom
## Multiple R-squared: 0.8882, Adjusted R-squared: 0.8825
## F-statistic: 155.3 on 71 and 1388 DF, p-value: < 2.2e-16By introducing some logical feature interactions, and removing many features, we see roughly the same Adjusted R2 as model3. This is favorable, as we now have a much simpler model.
plot(model4 )## Warning: not plotting observations with leverage one:
## 121, 272, 1276, 1299
Now we can test our final model on the test set. We first need to perform the same types of imputation that we did for the training set.
test_set_clean <- clean_df(test_data)test_set_clean[28,]predictions <- predict(model4, test_set_clean)df <- data.frame(Id = 1461:(length(predictions)+1460), SalePrice = predictions)
write.csv(df, "submission.csv", row.names = FALSE)Finally, I want to add a few more interactions to see if I can lift the score even further.
model6 <- lm(SalePrice ~ LotArea*ExterQual + LotArea**2+ OverallQual*OverallCond + YearBuilt + BsmtFinSF1*BsmtUnfSF + `1stFlrSF`*`2ndFlrSF` + BedroomAbvGr + PoolArea + Neighborhood*BldgType + RoofMatl, data = train_data_clean)summary(model6)##
## Call:
## lm(formula = SalePrice ~ LotArea * ExterQual + LotArea^2 + OverallQual *
## OverallCond + YearBuilt + BsmtFinSF1 * BsmtUnfSF + `1stFlrSF` *
## `2ndFlrSF` + BedroomAbvGr + PoolArea + Neighborhood * BldgType +
## RoofMatl, data = train_data_clean)
##
## Residuals:
## Min 1Q Median 3Q Max
## -325559 -11307 0 11374 165846
##
## Coefficients: (61 not defined because of singularities)
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -1.442e+06 1.302e+05 -11.072 < 2e-16 ***
## LotArea -2.746e+00 5.661e-01 -4.851 1.37e-06 ***
## ExterQualFa -8.482e+04 2.241e+04 -3.785 0.000160 ***
## ExterQualGd -8.526e+04 8.812e+03 -9.675 < 2e-16 ***
## ExterQualTA -8.918e+04 9.140e+03 -9.757 < 2e-16 ***
## OverallQual 4.089e+03 2.867e+03 1.426 0.154001
## OverallCond 4.979e+02 2.814e+03 0.177 0.859617
## YearBuilt 5.134e+02 6.008e+01 8.545 < 2e-16 ***
## BsmtFinSF1 3.367e+01 3.651e+00 9.222 < 2e-16 ***
## BsmtUnfSF 1.019e+01 3.123e+00 3.264 0.001125 **
## `1stFlrSF` 6.601e+01 4.058e+00 16.266 < 2e-16 ***
## `2ndFlrSF` 3.360e+01 6.784e+00 4.953 8.22e-07 ***
## BedroomAbvGr -6.088e+03 1.288e+03 -4.727 2.51e-06 ***
## PoolArea 7.681e+01 1.920e+01 4.001 6.64e-05 ***
## NeighborhoodBlueste -1.631e+04 2.769e+04 -0.589 0.555880
## NeighborhoodBrDale -1.411e+04 1.537e+04 -0.918 0.358539
## NeighborhoodBrkSide 2.882e+04 2.726e+04 1.057 0.290532
## NeighborhoodClearCr 2.531e+04 2.746e+04 0.922 0.356812
## NeighborhoodCollgCr 2.924e+04 2.676e+04 1.093 0.274799
## NeighborhoodCrawfor 4.478e+04 2.729e+04 1.641 0.101082
## NeighborhoodEdwards 1.585e+04 2.707e+04 0.585 0.558390
## NeighborhoodGilbert 2.635e+04 2.688e+04 0.980 0.327232
## NeighborhoodIDOTRR 2.028e+04 2.746e+04 0.739 0.460249
## NeighborhoodMeadowV -1.608e+04 1.143e+04 -1.407 0.159628
## NeighborhoodMitchel 1.434e+04 2.713e+04 0.528 0.597278
## NeighborhoodNAmes 1.808e+04 2.693e+04 0.671 0.502068
## NeighborhoodNoRidge 6.667e+04 2.712e+04 2.458 0.014098 *
## NeighborhoodNPkVill -6.331e+03 1.316e+04 -0.481 0.630430
## NeighborhoodNridgHt 8.902e+04 2.699e+04 3.299 0.000997 ***
## NeighborhoodNWAmes 1.276e+04 2.700e+04 0.473 0.636525
## NeighborhoodOldTown 1.697e+04 2.723e+04 0.623 0.533253
## NeighborhoodSawyer 1.974e+04 2.706e+04 0.730 0.465792
## NeighborhoodSawyerW 1.989e+04 2.699e+04 0.737 0.461246
## NeighborhoodSomerst 4.840e+04 2.687e+04 1.802 0.071833 .
## NeighborhoodStoneBr 1.343e+05 2.778e+04 4.835 1.48e-06 ***
## NeighborhoodSWISU 2.088e+04 2.767e+04 0.755 0.450628
## NeighborhoodTimber 3.815e+04 2.708e+04 1.409 0.159190
## NeighborhoodVeenker 2.552e+04 2.845e+04 0.897 0.369841
## BldgType2fmCon -8.489e+04 3.111e+04 -2.728 0.006450 **
## BldgTypeDuplex -1.851e+04 1.307e+04 -1.416 0.156970
## BldgTypeTwnhs -3.619e+04 9.710e+03 -3.727 0.000201 ***
## BldgTypeTwnhsE 1.892e+04 2.741e+04 0.690 0.490237
## RoofMatlCompShg 5.000e+05 3.951e+04 12.654 < 2e-16 ***
## RoofMatlMembran 5.477e+05 4.833e+04 11.334 < 2e-16 ***
## RoofMatlMetal 5.109e+05 4.794e+04 10.657 < 2e-16 ***
## RoofMatlRoll 4.974e+05 4.812e+04 10.338 < 2e-16 ***
## RoofMatlTar&Grv 4.939e+05 4.013e+04 12.308 < 2e-16 ***
## RoofMatlWdShake 4.771e+05 4.140e+04 11.525 < 2e-16 ***
## RoofMatlWdShngl 5.828e+05 4.046e+04 14.405 < 2e-16 ***
## LotArea:ExterQualFa 2.675e+00 2.303e+00 1.162 0.245597
## LotArea:ExterQualGd 3.462e+00 5.808e-01 5.961 3.19e-09 ***
## LotArea:ExterQualTA 3.347e+00 5.769e-01 5.801 8.17e-09 ***
## OverallQual:OverallCond 1.118e+03 4.956e+02 2.255 0.024263 *
## BsmtFinSF1:BsmtUnfSF -9.941e-03 5.144e-03 -1.933 0.053503 .
## `1stFlrSF`:`2ndFlrSF` 2.302e-02 4.905e-03 4.692 2.97e-06 ***
## NeighborhoodBlueste:BldgType2fmCon NA NA NA NA
## NeighborhoodBrDale:BldgType2fmCon NA NA NA NA
## NeighborhoodBrkSide:BldgType2fmCon 7.314e+04 3.648e+04 2.005 0.045191 *
## NeighborhoodClearCr:BldgType2fmCon NA NA NA NA
## NeighborhoodCollgCr:BldgType2fmCon NA NA NA NA
## NeighborhoodCrawfor:BldgType2fmCon 6.040e+04 4.096e+04 1.475 0.140565
## NeighborhoodEdwards:BldgType2fmCon 7.779e+04 3.395e+04 2.291 0.022089 *
## NeighborhoodGilbert:BldgType2fmCon 7.239e+04 4.025e+04 1.799 0.072312 .
## NeighborhoodIDOTRR:BldgType2fmCon 7.001e+04 3.704e+04 1.890 0.058947 .
## NeighborhoodMeadowV:BldgType2fmCon NA NA NA NA
## NeighborhoodMitchel:BldgType2fmCon 7.957e+04 4.055e+04 1.962 0.049930 *
## NeighborhoodNAmes:BldgType2fmCon 8.825e+04 4.104e+04 2.150 0.031698 *
## NeighborhoodNoRidge:BldgType2fmCon NA NA NA NA
## NeighborhoodNPkVill:BldgType2fmCon NA NA NA NA
## NeighborhoodNridgHt:BldgType2fmCon NA NA NA NA
## NeighborhoodNWAmes:BldgType2fmCon NA NA NA NA
## NeighborhoodOldTown:BldgType2fmCon 7.428e+04 3.206e+04 2.317 0.020657 *
## NeighborhoodSawyer:BldgType2fmCon 8.827e+04 4.103e+04 2.151 0.031622 *
## NeighborhoodSawyerW:BldgType2fmCon NA NA NA NA
## NeighborhoodSomerst:BldgType2fmCon NA NA NA NA
## NeighborhoodStoneBr:BldgType2fmCon NA NA NA NA
## NeighborhoodSWISU:BldgType2fmCon 9.594e+04 3.556e+04 2.698 0.007059 **
## NeighborhoodTimber:BldgType2fmCon NA NA NA NA
## NeighborhoodVeenker:BldgType2fmCon NA NA NA NA
## NeighborhoodBlueste:BldgTypeDuplex NA NA NA NA
## NeighborhoodBrDale:BldgTypeDuplex NA NA NA NA
## NeighborhoodBrkSide:BldgTypeDuplex NA NA NA NA
## NeighborhoodClearCr:BldgTypeDuplex NA NA NA NA
## NeighborhoodCollgCr:BldgTypeDuplex NA NA NA NA
## NeighborhoodCrawfor:BldgTypeDuplex -3.000e+04 2.333e+04 -1.286 0.198639
## NeighborhoodEdwards:BldgTypeDuplex 6.125e+03 1.636e+04 0.374 0.708228
## NeighborhoodGilbert:BldgTypeDuplex NA NA NA NA
## NeighborhoodIDOTRR:BldgTypeDuplex 3.184e+04 3.100e+04 1.027 0.304581
## NeighborhoodMeadowV:BldgTypeDuplex NA NA NA NA
## NeighborhoodMitchel:BldgTypeDuplex 1.952e+03 1.662e+04 0.117 0.906496
## NeighborhoodNAmes:BldgTypeDuplex 8.939e+03 1.465e+04 0.610 0.541743
## NeighborhoodNoRidge:BldgTypeDuplex NA NA NA NA
## NeighborhoodNPkVill:BldgTypeDuplex NA NA NA NA
## NeighborhoodNridgHt:BldgTypeDuplex NA NA NA NA
## NeighborhoodNWAmes:BldgTypeDuplex -3.682e+03 2.325e+04 -0.158 0.874216
## NeighborhoodOldTown:BldgTypeDuplex 1.777e+03 2.070e+04 0.086 0.931596
## NeighborhoodSawyer:BldgTypeDuplex -2.441e+04 1.723e+04 -1.416 0.156866
## NeighborhoodSawyerW:BldgTypeDuplex NA NA NA NA
## NeighborhoodSomerst:BldgTypeDuplex NA NA NA NA
## NeighborhoodStoneBr:BldgTypeDuplex NA NA NA NA
## NeighborhoodSWISU:BldgTypeDuplex NA NA NA NA
## NeighborhoodTimber:BldgTypeDuplex NA NA NA NA
## NeighborhoodVeenker:BldgTypeDuplex NA NA NA NA
## NeighborhoodBlueste:BldgTypeTwnhs 7.526e+04 4.757e+04 1.582 0.113851
## NeighborhoodBrDale:BldgTypeTwnhs 4.996e+04 3.289e+04 1.519 0.128978
## NeighborhoodBrkSide:BldgTypeTwnhs NA NA NA NA
## NeighborhoodClearCr:BldgTypeTwnhs NA NA NA NA
## NeighborhoodCollgCr:BldgTypeTwnhs NA NA NA NA
## NeighborhoodCrawfor:BldgTypeTwnhs NA NA NA NA
## NeighborhoodEdwards:BldgTypeTwnhs -1.449e+04 1.878e+04 -0.772 0.440499
## NeighborhoodGilbert:BldgTypeTwnhs NA NA NA NA
## NeighborhoodIDOTRR:BldgTypeTwnhs NA NA NA NA
## NeighborhoodMeadowV:BldgTypeTwnhs 5.423e+04 3.194e+04 1.698 0.089792 .
## NeighborhoodMitchel:BldgTypeTwnhs 2.935e+04 2.870e+04 1.023 0.306525
## NeighborhoodNAmes:BldgTypeTwnhs NA NA NA NA
## NeighborhoodNoRidge:BldgTypeTwnhs NA NA NA NA
## NeighborhoodNPkVill:BldgTypeTwnhs 5.325e+04 3.463e+04 1.538 0.124335
## NeighborhoodNridgHt:BldgTypeTwnhs -3.898e+04 1.445e+04 -2.698 0.007063 **
## NeighborhoodNWAmes:BldgTypeTwnhs NA NA NA NA
## NeighborhoodOldTown:BldgTypeTwnhs NA NA NA NA
## NeighborhoodSawyer:BldgTypeTwnhs NA NA NA NA
## NeighborhoodSawyerW:BldgTypeTwnhs NA NA NA NA
## NeighborhoodSomerst:BldgTypeTwnhs NA NA NA NA
## NeighborhoodStoneBr:BldgTypeTwnhs NA NA NA NA
## NeighborhoodSWISU:BldgTypeTwnhs NA NA NA NA
## NeighborhoodTimber:BldgTypeTwnhs NA NA NA NA
## NeighborhoodVeenker:BldgTypeTwnhs NA NA NA NA
## NeighborhoodBlueste:BldgTypeTwnhsE NA NA NA NA
## NeighborhoodBrDale:BldgTypeTwnhsE NA NA NA NA
## NeighborhoodBrkSide:BldgTypeTwnhsE NA NA NA NA
## NeighborhoodClearCr:BldgTypeTwnhsE NA NA NA NA
## NeighborhoodCollgCr:BldgTypeTwnhsE -4.299e+04 2.889e+04 -1.488 0.136917
## NeighborhoodCrawfor:BldgTypeTwnhsE -1.609e+04 3.210e+04 -0.501 0.616200
## NeighborhoodEdwards:BldgTypeTwnhsE -3.724e+04 3.079e+04 -1.210 0.226676
## NeighborhoodGilbert:BldgTypeTwnhsE NA NA NA NA
## NeighborhoodIDOTRR:BldgTypeTwnhsE NA NA NA NA
## NeighborhoodMeadowV:BldgTypeTwnhsE NA NA NA NA
## NeighborhoodMitchel:BldgTypeTwnhsE -3.515e+04 3.175e+04 -1.107 0.268479
## NeighborhoodNAmes:BldgTypeTwnhsE -2.349e+04 3.342e+04 -0.703 0.482333
## NeighborhoodNoRidge:BldgTypeTwnhsE NA NA NA NA
## NeighborhoodNPkVill:BldgTypeTwnhsE NA NA NA NA
## NeighborhoodNridgHt:BldgTypeTwnhsE -8.234e+04 2.854e+04 -2.885 0.003980 **
## NeighborhoodNWAmes:BldgTypeTwnhsE NA NA NA NA
## NeighborhoodOldTown:BldgTypeTwnhsE NA NA NA NA
## NeighborhoodSawyer:BldgTypeTwnhsE NA NA NA NA
## NeighborhoodSawyerW:BldgTypeTwnhsE -2.866e+04 2.949e+04 -0.972 0.331198
## NeighborhoodSomerst:BldgTypeTwnhsE -4.120e+04 2.837e+04 -1.452 0.146747
## NeighborhoodStoneBr:BldgTypeTwnhsE -1.282e+05 2.954e+04 -4.338 1.54e-05 ***
## NeighborhoodSWISU:BldgTypeTwnhsE NA NA NA NA
## NeighborhoodTimber:BldgTypeTwnhsE NA NA NA NA
## NeighborhoodVeenker:BldgTypeTwnhsE 2.730e+04 3.293e+04 0.829 0.407343
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 26580 on 1370 degrees of freedom
## Multiple R-squared: 0.8949, Adjusted R-squared: 0.8881
## F-statistic: 131 on 89 and 1370 DF, p-value: < 2.2e-16predictions <- predict(model6, test_set_clean)## Warning in predict.lm(model6, test_set_clean): prediction from a rank-deficient
## fit may be misleadingdf <- data.frame(Id = 1461:(length(predictions)+1460), SalePrice = predictions)
write.csv(df, "submission7.csv", row.names = FALSE)
kaggle results