Data 605 Final Exam
Jeff Littlejohn
3/20/2022
- Final, Problem 1. 30 Points
- Form the A matrix. Then, introduce decay and form the B matrix as we did in the course notes. (5 Points)
- Start with a uniform rank vector r and perform power iterations on B till convergence. That is, compute the solution \(r = B^n × r\). Attempt this for a sufficiently large n so that r actually converges. (5 Points)
- 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)
- 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)
- Final Problem 2. 40 points.
- 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)
- 4. What is the frequency distribution of the numbers in the dataset? (5 points)
- 5. For each number, provide the mean pixel intensity. What does this tell you? (5 points)
- 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)
- 7. Plot the first 10 images generated by PCA. They will appear to be noise. Why? (5 points)
- 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)
- 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)
- Final Problem 3. 30 points
Final, Problem 1. 30 Points
- Playing with PageRank You’ll verify for yourself that PageRank works by performing calculations on a small universe of web pages. Let’s use the 6 page universe that we had in the previous discussion For this directed graph, perform the following calculations in R.
Form the A matrix. Then, introduce decay and form the B matrix as we did in the course notes. (5 Points)
Build the A matrix in R, changing to columns to reflect reference examples:
A <- matrix (c( 0, 1/2, 1/2, 0, 0, 0,
1/6, 1/6, 1/6, 1/6, 1/6, 1/6, #substitute for dandling node
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), byrow=FALSE, nrow=6)
print(A)## [,1] [,2] [,3] [,4] [,5] [,6]
## [1,] 0.0 0.1666667 0.3333333 0.0 0.0 0
## [2,] 0.5 0.1666667 0.3333333 0.0 0.0 0
## [3,] 0.5 0.1666667 0.0000000 0.0 0.0 0
## [4,] 0.0 0.1666667 0.0000000 0.0 0.5 1
## [5,] 0.0 0.1666667 0.3333333 0.5 0.0 0
## [6,] 0.0 0.1666667 0.0000000 0.5 0.5 0Our second row (now column) has no outlinks, so our web pages might be disconnected in disjoint sets. Replaced with 1/n for each value.
colSums(A)## [1] 1 1 1 1 1 1Introduce decay to form the B matrix:
decay <- 0.85 #from the final
ncol_a <- ncol(A)
B <- decay * A + (1 - decay)/ncol_a
print(B)## [,1] [,2] [,3] [,4] [,5] [,6]
## [1,] 0.025 0.1666667 0.3083333 0.025 0.025 0.025
## [2,] 0.450 0.1666667 0.3083333 0.025 0.025 0.025
## [3,] 0.450 0.1666667 0.0250000 0.025 0.025 0.025
## [4,] 0.025 0.1666667 0.0250000 0.025 0.450 0.875
## [5,] 0.025 0.1666667 0.3083333 0.450 0.025 0.025
## [6,] 0.025 0.1666667 0.0250000 0.450 0.450 0.025Start with a uniform rank vector r and perform power iterations on B till convergence. That is, compute the solution \(r = B^n × r\). Attempt this for a sufficiently large n so that r actually converges. (5 Points)
Our uniform rank vector.
r <- rep(1/ncol_a,ncol_a)
r## [1] 0.1666667 0.1666667 0.1666667 0.1666667 0.1666667 0.1666667Convergence attempt using the transpose.
C_10 <- matrix.power(B, 10) %*% r
C_10## [,1]
## [1,] 0.05205661
## [2,] 0.07428990
## [3,] 0.05782138
## [4,] 0.34797267
## [5,] 0.19975859
## [6,] 0.26810085colSums(C_10)## [1] 1Seems to work. Let’s create a matrix showing page-rank convergence.
convergence_m <-cbind(matrix.power(B, 10) %*% r,
matrix.power(B, 20) %*% r,
matrix.power(B, 30) %*% r,
matrix.power(B, 40) %*% r,
matrix.power(B, 50) %*% r,
matrix.power(B, 60) %*% r)
convergence_m## [,1] [,2] [,3] [,4] [,5] [,6]
## [1,] 0.05205661 0.05170616 0.05170475 0.05170475 0.05170475 0.05170475
## [2,] 0.07428990 0.07368173 0.07367927 0.07367926 0.07367926 0.07367926
## [3,] 0.05782138 0.05741406 0.05741242 0.05741241 0.05741241 0.05741241
## [4,] 0.34797267 0.34870083 0.34870367 0.34870369 0.34870369 0.34870369
## [5,] 0.19975859 0.19990313 0.19990381 0.19990381 0.19990381 0.19990381
## [6,] 0.26810085 0.26859408 0.26859607 0.26859608 0.26859608 0.26859608We didn’t define a convergence value, but, if we set one at the thousandths or tens of thousandths, it looks OK by 30 power iterations.
We’ll save that.
pagerank_1 <- matrix.power(B, 30) %*% rCompute 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)
eig_b <- eigen(B)
eig_b## eigen() decomposition
## $values
## [1] 1.00000000+0i 0.57619235+0i -0.42500000+0i -0.42500000-0i -0.34991524+0i
## [6] -0.08461044+0i
##
## $vectors
## [,1] [,2] [,3]
## [1,] 0.1044385+0i 0.2931457+0i 2.486934e-15+0.0000e+00i
## [2,] 0.1488249+0i 0.5093703+0i -8.528385e-16-6.9832e-23i
## [3,] 0.1159674+0i 0.3414619+0i -1.930646e-15-0.0000e+00i
## [4,] 0.7043472+0i -0.5890805+0i -7.071068e-01+0.0000e+00i
## [5,] 0.4037861+0i -0.1413606+0i 7.071068e-01+0.0000e+00i
## [6,] 0.5425377+0i -0.4135367+0i 0.000000e+00-1.7058e-08i
## [,4] [,5] [,6]
## [1,] 2.486934e-15-0.0000e+00i -0.06471710+0i -0.212296003+0i
## [2,] -8.528385e-16+6.9832e-23i 0.01388698+0i 0.854071294+0i
## [3,] -1.930646e-15+0.0000e+00i 0.07298180+0i -0.363638739+0i
## [4,] -7.071068e-01+0.0000e+00i -0.66058664+0i 0.018399984+0i
## [5,] 7.071068e-01-0.0000e+00i 0.73761812+0i -0.304719509+0i
## [6,] 0.000000e+00+1.7058e-08i -0.09918316+0i 0.008182973+0iVerify corresponding eigenvector - with some help from the internet.
pagerank_2 <- eig_b$vectors[,1]/sum(eig_b$vectors[,1])
sum(pagerank_2)## [1] 1+0iWe’ll save the page rank vector for later. We’ve introduced a complex value - utoh.
Use the graph package in R and its page.rank method to compute the Page Rank of the graph as given in A. Note that you don’t need to apply decay. The package starts with a connected graph and applies decay internally. Verify that you do get the same PageRank vector as the two approaches above. (10 points)
For reasons I don’t entirely understand, the below works when I create A where the page ranks are conveyed as rows rather than columns.
A_row <- matrix (c( 0, 1/2, 1/2, 0, 0, 0,
1/6, 1/6, 1/6, 1/6, 1/6, 1/6, #substitute for dandling node
1/3, 1/3, 0, 0, 1/3, 0,
0, 0, 0, 0, 1/2, 1/2,
0, 0, 0, 1/2, 0, 1/2,
0, 0, 0, 1, 0, 0), byrow=TRUE, nrow=6)igraph_A <- graph_from_adjacency_matrix(A_row, weighted = TRUE)
plot(igraph_A)
The self-link might just have been introduced by (faulty?) handling of the dangling node.
pagerank_3 <- page.rank(igraph_A)$vector
pagerank_3## [1] 0.05170475 0.07367926 0.05741241 0.34870369 0.19990381 0.26859608compare_m <-cbind(pagerank_1,pagerank_2,pagerank_3)
compare_m## pagerank_2 pagerank_3
## [1,] 0.05170475+0i 0.05170475+0i 0.05170475+0i
## [2,] 0.07367927+0i 0.07367926+0i 0.07367926+0i
## [3,] 0.05741242+0i 0.05741241+0i 0.05741241+0i
## [4,] 0.34870367+0i 0.34870369+0i 0.34870369+0i
## [5,] 0.19990381+0i 0.19990381+0i 0.19990381+0i
## [6,] 0.26859607+0i 0.26859608+0i 0.26859608+0iThese tie out - ignoring the complex number issue.
Sources Consulted: https://gowrishankar.info/blog/eigenvalue-eigenvector-eigenspace-and-implementation-of-googles-pagerank-algorithm/ https://medium.com/@arpanspeaks/handling-dangling-nodes-pagerank-14c31d5b6b62 https://igraph.org/r/doc/page_rank.html
Final Problem 2. 40 points.
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.
’MNIST (“Modified National Institute of Standards and Technology”) is the de facto “hello world” dataset of computer vision. Since its release in 1999, this classic dataset of handwritten images has served as the basis for benchmarking classification algorithms. As new machine learning techniques emerge, MNIST remains a reliable resource for researchers and learners alike.”
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## 6 0 0 0 03. 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)
digits_training_df_pix <- digits_training_df[,2:ncol(digits_training_df)]/255
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## 4 0.0000000 0.00000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
## 5 0.8039216 0.99215686 0.9921569 0.9921569 0.9803922 0.8156863 0.4156863
## 6 0.0000000 0.01960784 0.7568627 0.9960784 0.3058824 0.0000000 0.0000000
## pixel573 pixel574 pixel575 pixel576 pixel577 pixel578 pixel579
## 1 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.00000000
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## 3 1.0000000 1.0000000 0.7215686 0.0000000 0.0000000 0.0000000 0.00000000
## 4 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.07058824
## 5 0.4156863 0.4156863 0.7843137 0.9294118 0.9921569 0.9921569 0.99215686
## 6 0.0745098 0.5019608 0.9960784 0.7647059 0.1411765 0.0000000 0.00000000
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## 3 0.0000000 0.0000000 0.00000000 0 0 0 0 0
## 4 0.9960784 0.2862745 0.00000000 0 0 0 0 0
## 5 0.9921569 0.8196078 0.08627451 0 0 0 0 0
## 6 0.0000000 0.0000000 0.00000000 0 0 0 0 0
## pixel588 pixel589 pixel590 pixel591 pixel592 pixel593 pixel594 pixel595
## 1 0 0 0 0 0 0 0.0000000 0.3686275
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## pixel596 pixel597 pixel598 pixel599 pixel600 pixel601 pixel602
## 1 0.9921569 0.9921569 0.9921569 0.3686275 0.0000000 0.0000000 0.0000000
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## 3 0.0000000 0.0000000 0.0000000 0.0000000 0.7254902 0.9960784 0.9960784
## 4 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
## 5 0.9921569 0.9921569 0.9921569 0.9921569 0.9921569 0.9921569 0.9921569
## 6 0.4039216 0.9960784 0.8705882 0.2901961 0.5607843 0.9215686 0.9960784
## pixel603 pixel604 pixel605 pixel606 pixel607 pixel608 pixel609
## 1 0.0000000 0.0000000 0.0000000 0.0000000 0.00000000 0.0000000 0.00000000
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## 3 0.7215686 0.0000000 0.0000000 0.0000000 0.00000000 0.0000000 0.00000000
## 4 0.0000000 0.0000000 0.0000000 0.0000000 0.07058824 0.9960784 0.28627451
## 5 0.9921569 0.9921569 0.9921569 0.9921569 0.99215686 0.8196078 0.08627451
## 6 0.8941176 0.3254902 0.0000000 0.0000000 0.00000000 0.0000000 0.00000000
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## 1 0 0 0 0 0.000000000 0.3490196 0.98431373
## 2 0 0 0 0 0.000000000 0.0000000 0.05098039
## 3 0 0 0 0 0.000000000 0.0000000 0.00000000
## 4 0 0 0 0 0.000000000 0.0000000 0.00000000
## 5 0 0 0 0 0.003921569 0.3568627 0.99215686
## 6 0 0 0 0 0.000000000 0.0000000 0.11764706
## pixel625 pixel626 pixel627 pixel628 pixel629 pixel630 pixel631
## 1 0.9921569 0.9803922 0.5137255 0.0000000 0.0000000 0.0000000 0.0000000
## 2 0.7137255 0.9960784 0.9960784 0.9960784 0.9960784 0.9960784 0.9960784
## 3 0.0000000 0.0000000 0.0000000 0.7254902 0.9960784 0.9960784 0.7215686
## 4 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
## 5 0.9921569 0.9921569 0.9921569 0.9921569 0.9921569 0.9921569 0.9921569
## 6 0.9490196 0.9960784 0.9960784 0.9960784 0.9960784 0.9882353 0.3294118
## pixel632 pixel633 pixel634 pixel635 pixel636 pixel637 pixel638
## 1 0.0000000 0.0000000 0.0000000 0.00000000 0.00000000 0.0000000 0
## 2 0.9960784 0.9960784 0.9529412 0.27450980 0.00000000 0.0000000 0
## 3 0.0000000 0.0000000 0.0000000 0.00000000 0.00000000 0.0000000 0
## 4 0.0000000 0.0000000 0.0000000 0.01960784 0.80784314 0.4156863 0
## 5 0.9921569 0.9921569 0.8352941 0.35294118 0.02745098 0.0000000 0
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## 1 0 0 0 0 0.000000000 0.83921569 0.85490196
## 2 0 0 0 0 0.000000000 0.00000000 0.03137255
## 3 0 0 0 0 0.000000000 0.00000000 0.00000000
## 4 0 0 0 0 0.000000000 0.00000000 0.00000000
## 5 0 0 0 0 0.003921569 0.07058824 0.50588235
## 6 0 0 0 0 0.000000000 0.00000000 0.09019608
## pixel654 pixel655 pixel656 pixel657 pixel658 pixel659 pixel660
## 1 0.3725490 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
## 2 0.2980392 0.5725490 0.9960784 1.0000000 0.9960784 1.0000000 0.5725490
## 3 0.0000000 0.0000000 0.2470588 0.9960784 0.9960784 0.2431373 0.0000000
## 4 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
## 5 0.8156863 0.9921569 0.9921569 0.9921569 0.9921569 0.6235294 0.5058824
## 6 0.2509804 0.6196078 0.7843137 0.6823529 0.2392157 0.0000000 0.0000000
## pixel661 pixel662 pixel663 pixel664 pixel665 pixel666 pixel667 pixel668
## 1 0.0000000 0.00000000 0 0.0000000 0.0000000 0 0 0
## 2 0.0745098 0.05882353 0 0.0000000 0.0000000 0 0 0
## 3 0.0000000 0.00000000 0 0.0000000 0.0000000 0 0 0
## 4 0.0000000 0.00000000 0 0.7294118 0.6235294 0 0 0
## 5 0.3529412 0.01568627 0 0.0000000 0.0000000 0 0 0
## 6 0.0000000 0.00000000 0 0.0000000 0.0000000 0 0 0
## pixel669 pixel670 pixel671 pixel672 pixel673 pixel674 pixel675 pixel676
## 1 0 0 0 0 0 0 0 0
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## 1 0 0 0 0 0 0 0.00000000 0.0000000
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## 4 0 0 0 0 0 0 0.02352941 0.8196078
## 5 0 0 0 0 0 0 0.00000000 0.0000000
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## pixel693 pixel694 pixel695 pixel696 pixel697 pixel698 pixel699 pixel700
## 1 0.0000000 0 0 0 0 0 0 0
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## pixel781 pixel782 pixel783 label
## 1 0 0 0 1
## 2 0 0 0 0
## 3 0 0 0 1
## 4 0 0 0 4
## 5 0 0 0 0
## 6 0 0 0 0Borrowing heavily from Stack Overflow, create a function to rotate our numbers to match presentation.
rotate <- function(matrix){apply(matrix,2, rev)}Visualize with the 255 scale.
par(mfrow=c(2,5))
for (i in 1:10){
digit_image <- rotate(matrix(rev(as.numeric(digits_training_df_pix[i,-c(1, 786)])), nrow = 28))
image(digit_image, col = grey.colors(255), axes=FALSE, xlim=c(0,1), ylim=c(-0.2,1.2))
}
Looking at our first ten labels, this looks good.
digits_training_df_pix$label[1:10]## [1] 1 0 1 4 0 0 7 3 5 34. What is the frequency distribution of the numbers in the dataset? (5 points)
Looking just at the training data set - first column is the label, the digit that was drawn by the user.
train_label <- digits_training_df_pix$label
train_frequency <- as.data.frame(table(train_label)) #table for Freqggplot(train_frequency, aes(x = train_label,y=Freq)) +
geom_histogram(stat='identity') +
labs(title = 'Frequency Distribution of Drawn Digits - Training Set',
x = 'Drawn Digit')
5. For each number, provide the mean pixel intensity. What does this tell you? (5 points)
Create a function to calculate pixel intensity.
pixel_intensity_calc <- function(df) {
digit_rows <- df %>%
rowwise() %>%
dplyr::mutate(pixel_intensity_calc = sum(c_across(starts_with("pixel")))) %>%
ungroup()
return(digit_rows)
}Use the function to calculate its mean.
pixel_intensity_mean <-
pixel_intensity_calc(digits_training_df_pix) %>%
group_by(label) %>%
summarise(pixel_intensity_mean = mean(pixel_intensity_calc)) %>%
ungroup()pixel_intensity_mean## # A tibble: 10 x 2
## label pixel_intensity_mean
## <int> <dbl>
## 1 0 136.
## 2 1 59.6
## 3 2 117.
## 4 3 111.
## 5 4 95.0
## 6 5 101.
## 7 6 109.
## 8 7 89.9
## 9 8 118.
## 10 9 96.3My guess is that circular strokes in drawing digits lead to larger intensities than straight ones.
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)
digit_train_pca <- prcomp(digits_training_df_pix)Establish the standard deviation of the PCA and plot the cumulative variance.
digit_train_pca_sd <- digit_train_pca$sdev
digit_train_pca_cum_var <- cumsum(digit_train_pca_sd^2)/sum(digit_train_pca_sd^2)plot(digit_train_pca_cum_var)
Looks like we hit 95% after 100 but before 200. Let’s try halfway - 140 looks better.
which.max(digit_train_pca_cum_var >= .95)## [1] 139which.max(digit_train_pca_cum_var >= 1)## [1] 705We hit 95% variance at 139 and 100% at 705. 785 components are possible - one for each column in the original data set.
7. Plot the first 10 images generated by PCA. They will appear to be noise. Why? (5 points)
digit_train_pca_rotation <- digit_train_pca$rotationfor (d in 1:10){
plot(4,4, xlim=c(1,28), ylim=c(1,28))
imageShow(array(digit_train_pca_rotation[,d],c(28,28)))
}
Looks a little like an ultrasound…
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)
digits_training_df_pix_8 <- digits_training_df_pix %>% filter(digits_training_df_pix$label == 8)
head(digits_training_df_pix_8$label)## [1] 8 8 8 8 8 8digit_train_pca_8 <- prcomp(digits_training_df_pix_8)Establish the standard deviation of the PCA and plot the cumulative variance.
digit_train_pca_sd_8 <- digit_train_pca_8$sdev
digit_train_pca_cum_var_8 <- cumsum(digit_train_pca_sd_8^2)/sum(digit_train_pca_sd_8^2)plot(digit_train_pca_cum_var_8)
which.max(digit_train_pca_cum_var_8 >= .95)## [1] 135which.max(digit_train_pca_cum_var_8 >= 1)## [1] 537digit_train_pca_rotation_8 <- digit_train_pca_8$rotationfor (d in 1:10){
plot(4,4, xlim=c(1,28), ylim=c(1,28))
imageShow(array(digit_train_pca_rotation_8[,d],c(28,28)))
}
The variance hit 95% slower than I would have thought. But our images are clearly horizontal eights - that look vaguely like anthrax or a worn hippy tattoo.
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)
We split our training set into train and test sets before building the model - which might be breaking the rules.
train_size <- floor(.8 * nrow(digits_training_df_pix))
set.seed(6789)
train_index <- sample(seq_len(nrow(digits_training_df_pix)), size = train_size)
train <- digits_training_df_pix[train_index,]
test <- digits_training_df_pix[-train_index,]train_x <- train[,names(train) != "label"]
train_x <- lapply(train_x, as.numeric)
train_y <- as.factor(train$label)test_x <- test[,names(test) != "label"]
test_x <- lapply(test_x, as.numeric)
test_y <- as.factor(test$label)We will use the multinom function from the nnet package.
digit_mn_model <- multinom(train_y~., train_x, MaxNWts = 100000)## # weights: 7860 (7065 variable)
## initial value 77366.859125
## iter 10 value 22615.011966
## iter 20 value 18121.636376
## iter 30 value 16868.689561
## iter 40 value 16254.576319
## iter 50 value 15907.790404
## iter 60 value 15233.728359
## iter 70 value 13943.579896
## iter 80 value 12150.556849
## iter 90 value 10928.878374
## iter 100 value 10175.200392
## final value 10175.200392
## stopped after 100 iterationsdigit_test_prediction <- predict(digit_mn_model, newdata = test_x)confusionMatrix(digit_test_prediction,test_y)## Confusion Matrix and Statistics
##
## Reference
## Prediction 0 1 2 3 4 5 6 7 8 9
## 0 834 0 13 6 1 9 6 1 8 7
## 1 2 917 20 10 6 9 6 14 41 9
## 2 1 6 705 21 3 4 7 6 11 0
## 3 2 6 14 769 3 30 0 0 27 11
## 4 3 2 22 2 778 8 11 10 8 35
## 5 3 2 7 31 3 595 14 4 27 7
## 6 8 3 8 2 6 16 779 2 6 0
## 7 1 2 8 8 4 3 1 823 3 28
## 8 7 7 27 13 6 11 6 1 656 3
## 9 1 1 10 12 31 16 1 33 25 705
##
## Overall Statistics
##
## Accuracy : 0.9001
## 95% CI : (0.8935, 0.9065)
## No Information Rate : 0.1126
## P-Value [Acc > NIR] : < 2.2e-16
##
## Kappa : 0.8889
##
## Mcnemar's Test P-Value : 1.373e-15
##
## Statistics by Class:
##
## Class: 0 Class: 1 Class: 2 Class: 3 Class: 4 Class: 5
## Sensitivity 0.96752 0.9693 0.84532 0.87986 0.92509 0.84879
## Specificity 0.99323 0.9843 0.99220 0.98764 0.98664 0.98727
## Pos Pred Value 0.94237 0.8868 0.92277 0.89211 0.88510 0.85859
## Neg Pred Value 0.99627 0.9961 0.98311 0.98607 0.99162 0.98625
## Prevalence 0.10262 0.1126 0.09929 0.10405 0.10012 0.08345
## Detection Rate 0.09929 0.1092 0.08393 0.09155 0.09262 0.07083
## Detection Prevalence 0.10536 0.1231 0.09095 0.10262 0.10464 0.08250
## Balanced Accuracy 0.98038 0.9768 0.91876 0.93375 0.95586 0.91803
## Class: 6 Class: 7 Class: 8 Class: 9
## Sensitivity 0.93742 0.92058 0.80788 0.87578
## Specificity 0.99326 0.99227 0.98933 0.98288
## Pos Pred Value 0.93855 0.93417 0.89009 0.84431
## Neg Pred Value 0.99313 0.99056 0.97964 0.98678
## Prevalence 0.09893 0.10643 0.09667 0.09583
## Detection Rate 0.09274 0.09798 0.07810 0.08393
## Detection Prevalence 0.09881 0.10488 0.08774 0.09940
## Balanced Accuracy 0.96534 0.95643 0.89860 0.92933Struggled the mist with 8s followed by 5s. Something about those rounded strokes is difficult to discern.
Sources consulted: https://www.youtube.com/watch?v=OowGKNgdowA https://stackoverflow.com/questions/37953644/r-image-plot-mnist-dataset https://python-course.eu/machine-learning/training-and-testing-with-mnist.php https://stackoverflow.com/questions/21729310/unexpected-apply-function-behaviour-in-r https://stackoverflow.com/questions/24604496/r-backwards-principal-component-calculation https://www.rdocumentation.org/packages/graphics/versions/3.6.2/topics/image https://forum.image.sc/t/trackmate-which-pixels-are-used-to-calculate-mean-pixel-intensity/6078 https://stats.oarc.ucla.edu/r/dae/multinomial-logistic-regression/
Final Problem 3. 30 points
You are to compete in the House Prices: Advanced Regression Techniques competition https://www.kaggle.com/c/house-prices-advanced-regression-techniques . I want you to do the following.
Loading the data.
house_train <- read.csv("adv_house_train.csv")
house_test <- read.csv('adv_house_test.csv')Descriptive and Inferential Statistics.
Provide univariate descriptive statistics and appropriate plots for the training data set. Provide a scatterplot matrix for at least two of the independent variables and the dependent variable. Derive a correlation matrix for any three quantitative variables in the dataset. Test the hypotheses that the correlations between each pairwise set of variables is 0 and provide an 80% confidence interval. Discuss the meaning of your analysis. Would you be worried about familywise error? Why or why not? 5 points
summary(house_train)## 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 X1stFlrSF
## 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
##
## X2ndFlrSF 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 X3SsnPorch 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
## ggpairs(house_train[names(house_train) %in% c("FullBath","SaleCondition","SalePrice")], aes(colour = SaleCondition))## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
That was fun, but let’s get a little more useful.
ggplot(house_train,aes(x=TotRmsAbvGrd,y=SalePrice)) +
geom_point()
house_train_numeric_cols <- unlist(lapply(house_train, is.numeric))
house_train_numeric <- house_train[, house_train_numeric_cols]
corrplot(cor(house_train_numeric), method = "color")
That’s not particulary visible…
Correlation matrix:
cor_m <- cor(house_train[,c("TotalBsmtSF","TotRmsAbvGrd","LotArea","BedroomAbvGr")])
cor_m## TotalBsmtSF TotRmsAbvGrd LotArea BedroomAbvGr
## TotalBsmtSF 1.00000000 0.2855726 0.2608331 0.05044996
## TotRmsAbvGrd 0.28557256 1.0000000 0.1900148 0.67661994
## LotArea 0.26083313 0.1900148 1.0000000 0.11968991
## BedroomAbvGr 0.05044996 0.6766199 0.1196899 1.00000000Overall, we definitely have to worry about collinearities in this data set. Missing values are present. There are weird outliers that may be indicative of poor data quality.
Let’s derive a correlation and test three of the above - starting with the hypothesis that the correlation between each pairwise set of variables is 0.
cor.test(house_train$TotalBsmtSF, house_train$TotRmsAbvGrd, conf.level=0.80)##
## Pearson's product-moment correlation
##
## data: house_train$TotalBsmtSF and house_train$TotRmsAbvGrd
## t = 11.378, df = 1458, p-value < 2.2e-16
## alternative hypothesis: true correlation is not equal to 0
## 80 percent confidence interval:
## 0.2544496 0.3161046
## sample estimates:
## cor
## 0.2855726cor.test(house_train$LotArea, house_train$BedroomAbvGr, conf.level=0.80)##
## Pearson's product-moment correlation
##
## data: house_train$LotArea and house_train$BedroomAbvGr
## t = 4.6033, df = 1458, p-value = 4.52e-06
## alternative hypothesis: true correlation is not equal to 0
## 80 percent confidence interval:
## 0.08647564 0.15263840
## sample estimates:
## cor
## 0.1196899cor.test(house_train$LotArea, house_train$FullBath, conf.level=0.80)##
## Pearson's product-moment correlation
##
## data: house_train$LotArea and house_train$FullBath
## t = 4.851, df = 1458, p-value = 1.36e-06
## alternative hypothesis: true correlation is not equal to 0
## 80 percent confidence interval:
## 0.09286177 0.15892007
## sample estimates:
## cor
## 0.1260306Yes, we are definitely worried about familiy wise. We have many variables - if we’re operating at a significance level of .05, 1 in 20 will appear significant by chance even if there’s no relationship. This level of collinearity only increases the chances of Type I error.
Linear Algebra and Correlation.
Invert your correlation matrix from above. (This is known as the precision matrix and contains variance inflation factors on the diagonal.) Multiply the correlation matrix by the precision matrix, and then multiply the precision matrix by the correlation matrix. Conduct LU decomposition on the matrix. 5 points
cor_m## TotalBsmtSF TotRmsAbvGrd LotArea BedroomAbvGr
## TotalBsmtSF 1.00000000 0.2855726 0.2608331 0.05044996
## TotRmsAbvGrd 0.28557256 1.0000000 0.1900148 0.67661994
## LotArea 0.26083313 0.1900148 1.0000000 0.11968991
## BedroomAbvGr 0.05044996 0.6766199 0.1196899 1.00000000Invert it.
invert_m <- solve(cor_m)
invert_m## TotalBsmtSF TotRmsAbvGrd LotArea BedroomAbvGr
## TotalBsmtSF 1.1939276 -0.502802 -0.25301047 0.31025506
## TotRmsAbvGrd -0.5028020 2.098094 -0.10210698 -1.38202442
## LotArea -0.2530105 -0.102107 1.09123111 -0.04875736
## BedroomAbvGr 0.3102551 -1.382024 -0.04875736 1.92528869Precision.
precision_m <- round(cor_m %*% invert_m)
precision_m## TotalBsmtSF TotRmsAbvGrd LotArea BedroomAbvGr
## TotalBsmtSF 1 0 0 0
## TotRmsAbvGrd 0 1 0 0
## LotArea 0 0 1 0
## BedroomAbvGr 0 0 0 1LU Decomposition.
lu_cor_m <- lu.decomposition(cor_m)
lu_cor_m## $L
## [,1] [,2] [,3] [,4]
## [1,] 1.00000000 0.0000000 0.0000000 0
## [2,] 0.28557256 1.0000000 0.0000000 0
## [3,] 0.26083313 0.1257861 1.0000000 0
## [4,] 0.05044996 0.7210126 0.0253247 1
##
## $U
## [,1] [,2] [,3] [,4]
## [1,] 1 0.2855726 0.2608331 0.05044996
## [2,] 0 0.9184483 0.1155280 0.66221281
## [3,] 0 0.0000000 0.9174343 0.02323375
## [4,] 0 0.0000000 0.0000000 0.51940263Calculus-Based Probability & Statistics.
Many times, it makes sense to fit a closed form distribution to data. Select a variable in the Kaggle.com training dataset that is skewed to the right, shift it so that the minimum value is absolutely above zero if necessary. Then load the MASS package and run fitdistr to fit an exponential probability density function. (See https://stat.ethz.ch/R-manual/R- devel/library/MASS/html/fitdistr.html ). Find the optimal value of λ for this distribution, and then take 1000 samples from this exponential distribution using this value (e.g., rexp(1000, λ)). Plot a histogram and compare it with a histogram of your original variable. Using the exponential pdf, find the 5th and 95th percentiles using the cumulative distribution function (CDF). Also generate a 95% confidence interval from the empirical data, assuming normality. Finally, provide the empirical 5th percentile and 95th percentile of the data. Discuss. 10 points
Because I live in a basement-free state, let’s explore basement size, which likely has a rightward skew.
ggplot(house_train,aes(x=BsmtFinSF1)) +
geom_histogram()## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
No shifting required, as 0 is the natural minimum. Using the MASS package, run fitdistr:
optimal_lambda <- fitdistr(house_train$BsmtFinSF1, densfun = "exponential")
optimal_lambda## rate
## 0.002254081
## (0.000058992)Now we take our samples - using that top number from above, returned by estimate.
lambda_sample <- rexp(1000, optimal_lambda$estimate)
hist(lambda_sample, breaks = 100)
low_lambda <- qexp(.05, rate = optimal_lambda$estimate)
low_lambda## [1] 22.75574high_lambda <- qexp(.95, rate = optimal_lambda$estimate)
high_lambda## [1] 1329.026Back to empirical:
low_emp <- quantile(house_train$BsmtFinSF1,0.05)
low_emp## 5%
## 0high_emp <- quantile(house_train$BsmtFinSF1,0.95)
high_emp## 95%
## 1274Our lambda sample moves its mean slightly to the right, with a slightly less skewed distribution - there’s still a lot of skew!
Modeling.
Build some type of multiple regression model and submit your model to the competition board. Provide your complete model summary and results with analysis. Report your Kaggle.com user name and score. 10 points
One could easily spend 8 hours on this, which is the focus of other MSDS courses. We have two young kids and a busy job - going to set a time cap and then have some fun on Kaggle this summer when we’re not taking any courses.
We’re going to start with all numeric variables and then remove them using backward elimination while selectively adding factor variables based on very rudimentary knowledge of the real estate market.
house_train_quant_col <- house_train %>%
select_all() %>%
map_lgl(is.numeric)
house_train_quant_df <- house_train %>%
select_if(house_train_quant_col)house_lm_quant <- lm(SalePrice~.,data=house_train_quant_df)
summary(house_lm_quant)##
## Call:
## lm(formula = SalePrice ~ ., data = house_train_quant_df)
##
## Residuals:
## Min 1Q Median 3Q Max
## -442182 -16955 -2824 15125 318183
##
## Coefficients: (2 not defined because of singularities)
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -3.351e+05 1.701e+06 -0.197 0.843909
## Id -1.205e+00 2.658e+00 -0.453 0.650332
## MSSubClass -2.001e+02 3.451e+01 -5.797 8.84e-09 ***
## LotFrontage -1.160e+02 6.126e+01 -1.894 0.058503 .
## LotArea 5.422e-01 1.575e-01 3.442 0.000599 ***
## OverallQual 1.866e+04 1.482e+03 12.592 < 2e-16 ***
## OverallCond 5.239e+03 1.368e+03 3.830 0.000135 ***
## YearBuilt 3.164e+02 8.766e+01 3.610 0.000321 ***
## YearRemodAdd 1.194e+02 8.668e+01 1.378 0.168607
## MasVnrArea 3.141e+01 7.022e+00 4.473 8.54e-06 ***
## BsmtFinSF1 1.736e+01 5.838e+00 2.973 0.003014 **
## BsmtFinSF2 8.342e+00 8.766e+00 0.952 0.341532
## BsmtUnfSF 5.005e+00 5.277e+00 0.948 0.343173
## TotalBsmtSF NA NA NA NA
## X1stFlrSF 4.597e+01 7.360e+00 6.246 6.02e-10 ***
## X2ndFlrSF 4.663e+01 6.102e+00 7.641 4.72e-14 ***
## LowQualFinSF 3.341e+01 2.794e+01 1.196 0.232009
## GrLivArea NA NA NA NA
## BsmtFullBath 9.043e+03 3.198e+03 2.828 0.004776 **
## BsmtHalfBath 2.465e+03 5.073e+03 0.486 0.627135
## FullBath 5.433e+03 3.531e+03 1.539 0.124182
## HalfBath -1.098e+03 3.321e+03 -0.331 0.740945
## BedroomAbvGr -1.022e+04 2.155e+03 -4.742 2.40e-06 ***
## KitchenAbvGr -2.202e+04 6.710e+03 -3.282 0.001063 **
## TotRmsAbvGrd 5.464e+03 1.487e+03 3.674 0.000251 ***
## Fireplaces 4.372e+03 2.189e+03 1.998 0.046020 *
## GarageYrBlt -4.728e+01 9.106e+01 -0.519 0.603742
## GarageCars 1.685e+04 3.491e+03 4.827 1.58e-06 ***
## GarageArea 6.274e+00 1.213e+01 0.517 0.605002
## WoodDeckSF 2.144e+01 1.002e+01 2.139 0.032662 *
## OpenPorchSF -2.252e+00 1.949e+01 -0.116 0.907998
## EnclosedPorch 7.295e+00 2.062e+01 0.354 0.723590
## X3SsnPorch 3.349e+01 3.758e+01 0.891 0.373163
## ScreenPorch 5.805e+01 2.041e+01 2.844 0.004532 **
## PoolArea -6.052e+01 2.990e+01 -2.024 0.043204 *
## MiscVal -3.761e+00 6.960e+00 -0.540 0.589016
## MoSold -2.217e+02 4.229e+02 -0.524 0.600188
## YrSold -2.474e+02 8.458e+02 -0.293 0.769917
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 36800 on 1085 degrees of freedom
## (339 observations deleted due to missingness)
## Multiple R-squared: 0.8096, Adjusted R-squared: 0.8034
## F-statistic: 131.8 on 35 and 1085 DF, p-value: < 2.2e-16Removing columns that likely shouldn’t have been there in the first place - like time period sold and Id plus the variables that are clearly less useful based on p-value. Year built is a proxy for age and probably should be transformed as such.
house_lm_quant_2 <- lm(SalePrice ~ MSSubClass + LotArea + OverallQual + OverallCond + YearBuilt + MasVnrArea + BsmtFinSF1 + X1stFlrSF + X2ndFlrSF + BsmtFullBath + BedroomAbvGr + KitchenAbvGr + TotRmsAbvGrd + Fireplaces + GarageCars + WoodDeckSF + ScreenPorch + PoolArea,data=house_train_quant_df)
summary(house_lm_quant_2)##
## Call:
## lm(formula = SalePrice ~ MSSubClass + LotArea + OverallQual +
## OverallCond + YearBuilt + MasVnrArea + BsmtFinSF1 + X1stFlrSF +
## X2ndFlrSF + BsmtFullBath + BedroomAbvGr + KitchenAbvGr +
## TotRmsAbvGrd + Fireplaces + GarageCars + WoodDeckSF + ScreenPorch +
## PoolArea, data = house_train_quant_df)
##
## Residuals:
## Min 1Q Median 3Q Max
## -471826 -16295 -1995 13876 299766
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -8.391e+05 8.879e+04 -9.450 < 2e-16 ***
## MSSubClass -1.632e+02 2.582e+01 -6.322 3.44e-10 ***
## LotArea 4.209e-01 1.002e-01 4.202 2.81e-05 ***
## OverallQual 1.869e+04 1.125e+03 16.619 < 2e-16 ***
## OverallCond 5.078e+03 9.226e+02 5.504 4.40e-08 ***
## YearBuilt 3.918e+02 4.508e+01 8.691 < 2e-16 ***
## MasVnrArea 2.949e+01 5.874e+00 5.021 5.78e-07 ***
## BsmtFinSF1 9.989e+00 2.975e+00 3.358 0.000807 ***
## X1stFlrSF 5.770e+01 4.651e+00 12.407 < 2e-16 ***
## X2ndFlrSF 4.883e+01 4.112e+00 11.874 < 2e-16 ***
## BsmtFullBath 8.955e+03 2.373e+03 3.774 0.000167 ***
## BedroomAbvGr -1.026e+04 1.646e+03 -6.233 6.00e-10 ***
## KitchenAbvGr -1.338e+04 5.098e+03 -2.624 0.008774 **
## TotRmsAbvGrd 5.418e+03 1.214e+03 4.461 8.78e-06 ***
## Fireplaces 2.507e+03 1.734e+03 1.446 0.148471
## GarageCars 1.042e+04 1.695e+03 6.146 1.03e-09 ***
## WoodDeckSF 2.462e+01 7.911e+00 3.112 0.001892 **
## ScreenPorch 5.085e+01 1.689e+01 3.010 0.002657 **
## PoolArea -3.049e+01 2.341e+01 -1.302 0.192979
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 34820 on 1433 degrees of freedom
## (8 observations deleted due to missingness)
## Multiple R-squared: 0.8095, Adjusted R-squared: 0.8071
## F-statistic: 338.4 on 18 and 1433 DF, p-value: < 2.2e-16Now we’ll continue with backward elimination but return to the original dataset, adding factor variables selectively.
house_lm_3 <- lm(SalePrice ~ MSSubClass + LotArea + OverallQual + OverallCond + YearBuilt + X1stFlrSF + X2ndFlrSF + BedroomAbvGr + KitchenAbvGr + TotRmsAbvGrd + WoodDeckSF + ScreenPorch + HeatingQC + ExterQual + RoofMatl + Condition1 + Neighborhood + LandSlope + LotConfig,data=house_train,na.action=na.omit)
summary(house_lm_3)##
## Call:
## lm(formula = SalePrice ~ MSSubClass + LotArea + OverallQual +
## OverallCond + YearBuilt + X1stFlrSF + X2ndFlrSF + BedroomAbvGr +
## KitchenAbvGr + TotRmsAbvGrd + WoodDeckSF + ScreenPorch +
## HeatingQC + ExterQual + RoofMatl + Condition1 + Neighborhood +
## LandSlope + LotConfig, data = house_train, na.action = na.omit)
##
## Residuals:
## Min 1Q Median 3Q Max
## -332580 -13455 -818 12089 257403
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -1.509e+06 1.317e+05 -11.454 < 2e-16 ***
## MSSubClass -1.559e+02 2.547e+01 -6.121 1.21e-09 ***
## LotArea 7.079e-01 1.092e-01 6.482 1.25e-10 ***
## OverallQual 1.112e+04 1.084e+03 10.256 < 2e-16 ***
## OverallCond 6.404e+03 8.330e+02 7.688 2.80e-14 ***
## YearBuilt 5.287e+02 6.214e+01 8.508 < 2e-16 ***
## X1stFlrSF 7.992e+01 3.982e+00 20.072 < 2e-16 ***
## X2ndFlrSF 5.947e+01 3.709e+00 16.035 < 2e-16 ***
## BedroomAbvGr -8.095e+03 1.483e+03 -5.457 5.71e-08 ***
## KitchenAbvGr -1.395e+04 4.498e+03 -3.100 0.001971 **
## TotRmsAbvGrd 2.863e+03 1.074e+03 2.666 0.007756 **
## WoodDeckSF 2.118e+01 6.821e+00 3.105 0.001941 **
## ScreenPorch 5.092e+01 1.433e+01 3.553 0.000394 ***
## HeatingQCFa -1.473e+03 4.759e+03 -0.309 0.757001
## HeatingQCGd -4.178e+03 2.460e+03 -1.698 0.089664 .
## HeatingQCPo 1.440e+04 3.125e+04 0.461 0.645049
## HeatingQCTA -4.321e+03 2.307e+03 -1.873 0.061277 .
## ExterQualFa -4.276e+04 1.039e+04 -4.117 4.07e-05 ***
## ExterQualGd -4.961e+04 4.986e+03 -9.949 < 2e-16 ***
## ExterQualTA -5.066e+04 5.667e+03 -8.941 < 2e-16 ***
## RoofMatlCompShg 4.847e+05 3.234e+04 14.989 < 2e-16 ***
## RoofMatlMembran 5.274e+05 4.570e+04 11.542 < 2e-16 ***
## RoofMatlMetal 5.302e+05 4.607e+04 11.509 < 2e-16 ***
## RoofMatlRoll 4.859e+05 4.392e+04 11.064 < 2e-16 ***
## RoofMatlTar&Grv 4.739e+05 3.343e+04 14.176 < 2e-16 ***
## RoofMatlWdShake 4.660e+05 3.503e+04 13.303 < 2e-16 ***
## RoofMatlWdShngl 5.666e+05 3.397e+04 16.677 < 2e-16 ***
## Condition1Feedr -4.611e+02 5.653e+03 -0.082 0.935009
## Condition1Norm 6.733e+03 4.607e+03 1.462 0.144093
## Condition1PosA -1.734e+03 1.176e+04 -0.147 0.882821
## Condition1PosN -1.637e+04 8.329e+03 -1.965 0.049566 *
## Condition1RRAe -1.630e+04 1.082e+04 -1.507 0.132030
## Condition1RRAn 6.341e+03 7.583e+03 0.836 0.403224
## Condition1RRNe -1.333e+04 2.170e+04 -0.614 0.539247
## Condition1RRNn 4.438e+03 1.450e+04 0.306 0.759566
## NeighborhoodBlueste 3.034e+03 2.222e+04 0.137 0.891405
## NeighborhoodBrDale -1.706e+03 1.082e+04 -0.158 0.874783
## NeighborhoodBrkSide 1.186e+04 9.535e+03 1.244 0.213702
## NeighborhoodClearCr 7.137e+03 1.012e+04 0.705 0.480884
## NeighborhoodCollgCr 7.328e+03 7.998e+03 0.916 0.359719
## NeighborhoodCrawfor 2.475e+04 9.339e+03 2.650 0.008144 **
## NeighborhoodEdwards -3.393e+03 8.666e+03 -0.391 0.695503
## NeighborhoodGilbert -1.760e+02 8.522e+03 -0.021 0.983529
## NeighborhoodIDOTRR 6.281e+02 1.007e+04 0.062 0.950293
## NeighborhoodMeadowV 6.520e+03 1.073e+04 0.607 0.543687
## NeighborhoodMitchel -6.913e+02 8.964e+03 -0.077 0.938541
## NeighborhoodNAmes 2.379e+03 8.415e+03 0.283 0.777455
## NeighborhoodNoRidge 5.318e+04 9.188e+03 5.788 8.79e-09 ***
## NeighborhoodNPkVill 1.130e+04 1.254e+04 0.901 0.367870
## NeighborhoodNridgHt 4.922e+04 8.268e+03 5.953 3.32e-09 ***
## NeighborhoodNWAmes -5.116e+03 8.778e+03 -0.583 0.560094
## NeighborhoodOldTown 4.689e+02 9.261e+03 0.051 0.959624
## NeighborhoodSawyer 3.856e+03 8.898e+03 0.433 0.664839
## NeighborhoodSawyerW 3.948e+03 8.599e+03 0.459 0.646178
## NeighborhoodSomerst 2.155e+04 8.140e+03 2.647 0.008208 **
## NeighborhoodStoneBr 6.008e+04 9.450e+03 6.357 2.78e-10 ***
## NeighborhoodSWISU 4.498e+03 1.060e+04 0.424 0.671290
## NeighborhoodTimber 1.428e+04 9.121e+03 1.565 0.117770
## NeighborhoodVeenker 2.566e+04 1.181e+04 2.174 0.029902 *
## LandSlopeMod 8.412e+03 4.027e+03 2.089 0.036890 *
## LandSlopeSev -2.236e+04 1.163e+04 -1.923 0.054670 .
## LotConfigCulDSac 7.112e+03 3.747e+03 1.898 0.057907 .
## LotConfigFR2 -5.704e+03 4.777e+03 -1.194 0.232621
## LotConfigFR3 -1.384e+04 1.532e+04 -0.903 0.366504
## LotConfigInside -1.842e+03 2.082e+03 -0.885 0.376401
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 29240 on 1395 degrees of freedom
## Multiple R-squared: 0.8704, Adjusted R-squared: 0.8645
## F-statistic: 146.4 on 64 and 1395 DF, p-value: < 2.2e-16Note that the following variables have NAs in the test data but would have been significant if we’d had time to build a more robust model: BsmtExposure, KitchenQual
More variables than we’d like. No attempts at transformation. No exploration of time series. Likely overfitting. Unexplored collinearities present. But let’s see how we do, subsetting our test data to only include the variables used in the model.
plot(house_lm_3)
Our model is far from a perfect one. Residuals are not normally distributed. We see outliers for extreme values. Nonetheless, we will submit.
house_test_subset <- house_test %>% dplyr::select(MSSubClass,LotArea,OverallQual,OverallCond,YearBuilt,X1stFlrSF,X2ndFlrSF,BedroomAbvGr,KitchenAbvGr,TotRmsAbvGrd,WoodDeckSF,ScreenPorch,HeatingQC,ExterQual,RoofMatl,Condition1,Neighborhood,LandSlope,LotConfig)
summary(house_test_subset)## MSSubClass LotArea OverallQual OverallCond
## Min. : 20.00 Min. : 1470 Min. : 1.000 Min. :1.000
## 1st Qu.: 20.00 1st Qu.: 7391 1st Qu.: 5.000 1st Qu.:5.000
## Median : 50.00 Median : 9399 Median : 6.000 Median :5.000
## Mean : 57.38 Mean : 9819 Mean : 6.079 Mean :5.554
## 3rd Qu.: 70.00 3rd Qu.:11518 3rd Qu.: 7.000 3rd Qu.:6.000
## Max. :190.00 Max. :56600 Max. :10.000 Max. :9.000
## YearBuilt X1stFlrSF X2ndFlrSF BedroomAbvGr KitchenAbvGr
## Min. :1879 Min. : 407.0 Min. : 0 Min. :0.000 Min. :0.000
## 1st Qu.:1953 1st Qu.: 873.5 1st Qu.: 0 1st Qu.:2.000 1st Qu.:1.000
## Median :1973 Median :1079.0 Median : 0 Median :3.000 Median :1.000
## Mean :1971 Mean :1156.5 Mean : 326 Mean :2.854 Mean :1.042
## 3rd Qu.:2001 3rd Qu.:1382.5 3rd Qu.: 676 3rd Qu.:3.000 3rd Qu.:1.000
## Max. :2010 Max. :5095.0 Max. :1862 Max. :6.000 Max. :2.000
## TotRmsAbvGrd WoodDeckSF ScreenPorch HeatingQC
## Min. : 3.000 Min. : 0.00 Min. : 0.00 Length:1459
## 1st Qu.: 5.000 1st Qu.: 0.00 1st Qu.: 0.00 Class :character
## Median : 6.000 Median : 0.00 Median : 0.00 Mode :character
## Mean : 6.385 Mean : 93.17 Mean : 17.06
## 3rd Qu.: 7.000 3rd Qu.: 168.00 3rd Qu.: 0.00
## Max. :15.000 Max. :1424.00 Max. :576.00
## ExterQual RoofMatl Condition1 Neighborhood
## Length:1459 Length:1459 Length:1459 Length:1459
## Class :character Class :character Class :character Class :character
## Mode :character Mode :character Mode :character Mode :character
##
##
##
## LandSlope LotConfig
## Length:1459 Length:1459
## Class :character Class :character
## Mode :character Mode :character
##
##
## dim(house_test_subset)## [1] 1459 19house_test_predict <- predict(house_lm_3,house_test_subset)
house_test_predict <- as.data.frame(house_test_predict)
house_test_predict$Id <- house_test$Id
house_test_submission <- house_test_predict[,c(2,1)]
names(house_test_submission)[names(house_test_submission) == "house_test_predict"] <- "SalePrice"
head(house_test_submission,140)## Id SalePrice
## 1 1461 121535.23
## 2 1462 170599.29
## 3 1463 168699.73
## 4 1464 192671.41
## 5 1465 243811.59
## 6 1466 176702.95
## 7 1467 181008.25
## 8 1468 162279.91
## 9 1469 187013.66
## 10 1470 107552.93
## 11 1471 176962.80
## 12 1472 99488.71
## 13 1473 84418.14
## 14 1474 145679.97
## 15 1475 128800.64
## 16 1476 389301.73
## 17 1477 276357.65
## 18 1478 275825.74
## 19 1479 298360.13
## 20 1480 446803.92
## 21 1481 316914.16
## 22 1482 247131.16
## 23 1483 186897.61
## 24 1484 165603.16
## 25 1485 177939.11
## 26 1486 202102.79
## 27 1487 341455.70
## 28 1488 254326.25
## 29 1489 206576.68
## 30 1490 232760.70
## 31 1491 206728.91
## 32 1492 96777.21
## 33 1493 186714.16
## 34 1494 304150.40
## 35 1495 295703.42
## 36 1496 225057.91
## 37 1497 194946.78
## 38 1498 164142.99
## 39 1499 163667.98
## 40 1500 155220.19
## 41 1501 177195.83
## 42 1502 150609.40
## 43 1503 253993.85
## 44 1504 244319.24
## 45 1505 227413.61
## 46 1506 192734.78
## 47 1507 234238.74
## 48 1508 191090.54
## 49 1509 155668.06
## 50 1510 136084.44
## 51 1511 136443.91
## 52 1512 180001.66
## 53 1513 173448.86
## 54 1514 144913.40
## 55 1515 228309.59
## 56 1516 155454.34
## 57 1517 159641.61
## 58 1518 128951.37
## 59 1519 210287.15
## 60 1520 124173.18
## 61 1521 133950.58
## 62 1522 184089.22
## 63 1523 109339.00
## 64 1524 131387.98
## 65 1525 115788.40
## 66 1526 93898.76
## 67 1527 95304.64
## 68 1528 141319.45
## 69 1529 139533.22
## 70 1530 217071.11
## 71 1531 127161.83
## 72 1532 93404.86
## 73 1533 157283.78
## 74 1534 132318.28
## 75 1535 154487.44
## 76 1536 101140.09
## 77 1537 30620.32
## 78 1538 171045.50
## 79 1539 213060.24
## 80 1540 121616.03
## 81 1541 146845.87
## 82 1542 156465.53
## 83 1543 203306.24
## 84 1544 81476.28
## 85 1545 134513.43
## 86 1546 136442.72
## 87 1547 144836.45
## 88 1548 159128.02
## 89 1549 121905.65
## 90 1550 126107.52
## 91 1551 103927.68
## 92 1552 134931.95
## 93 1553 164837.56
## 94 1554 125897.11
## 95 1555 175233.67
## 96 1556 87479.49
## 97 1557 95548.29
## 98 1558 106677.13
## 99 1559 87861.55
## 100 1560 106421.03
## 101 1561 132820.99
## 102 1562 123378.40
## 103 1563 119431.42
## 104 1564 139855.51
## 105 1565 138585.47
## 106 1566 239749.61
## 107 1567 58288.43
## 108 1568 225989.04
## 109 1569 104993.87
## 110 1570 138803.00
## 111 1571 115712.55
## 112 1572 134148.48
## 113 1573 221627.42
## 114 1574 133267.82
## 115 1575 226538.99
## 116 1576 229576.42
## 117 1577 197618.10
## 118 1578 136158.74
## 119 1579 132340.89
## 120 1580 203682.23
## 121 1581 158934.85
## 122 1582 117791.96
## 123 1583 269215.55
## 124 1584 233798.87
## 125 1585 140577.86
## 126 1586 60372.26
## 127 1587 118088.96
## 128 1588 149609.05
## 129 1589 108592.93
## 130 1590 127170.65
## 131 1591 92426.58
## 132 1592 123946.00
## 133 1593 121292.65
## 134 1594 160584.25
## 135 1595 114865.63
## 136 1596 226248.58
## 137 1597 191771.09
## 138 1598 218212.78
## 139 1599 176268.29
## 140 1600 176566.34write_csv(house_test_submission, file = 'ames_house_test_submission.csv')Kaggle username: Jeff Littlejohn Score: 0.28253
Kaggle Screenshot
Leaderboard
Sources consulted: https://rviews.rstudio.com/2019/10/02/multiple-hypothesis-testing/ https://stat.ethz.ch/R-manual/R-devel/library/MASS/html/fitdistr.html https://www.programmingr.com/examples/neat-tricks/sample-r-function/rexp/ https://dplyr.tidyverse.org/reference/select.html