Introduction
Part 2 of the Data 605 final. Digit Recognize from Kaggle Dataset.
library(tidyverse)## Warning: package 'tidyverse' was built under R version 4.1.3## -- Attaching packages --------------------------------------- tidyverse 1.3.1 --## v ggplot2 3.3.5 v purrr 0.3.4
## v tibble 3.1.6 v dplyr 1.0.8
## v tidyr 1.2.0 v stringr 1.4.0
## v readr 2.1.2 v forcats 0.5.1## Warning: package 'ggplot2' was built under R version 4.1.3## Warning: package 'tibble' was built under R version 4.1.3## Warning: package 'tidyr' was built under R version 4.1.3## Warning: package 'readr' was built under R version 4.1.3## Warning: package 'purrr' was built under R version 4.1.3## Warning: package 'dplyr' was built under R version 4.1.3## Warning: package 'forcats' was built under R version 4.1.3## -- Conflicts ------------------------------------------ tidyverse_conflicts() --
## x dplyr::filter() masks stats::filter()
## x dplyr::lag() masks stats::lag()library(OpenImageR)## Warning: package 'OpenImageR' was built under R version 4.1.3library(nnet)## Warning: package 'nnet' was built under R version 4.1.3library(caret)## Warning: package 'caret' was built under R version 4.1.3## Loading required package: lattice##
## Attaching package: 'caret'## The following object is masked from 'package:purrr':
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## liftExcercise 1
Go to Kaggle.com and build an account if you do not already have one. It is free.
Excercise 2
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.”
Excercise 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.)
df <-read.csv("digit-recognizer_train.csv")
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## 5 0 0 0 0
## 6 0 0 0 0#Removing first column since it is not a pixel variable
df1 <- df %>%
select(-label)
df1 <- df1/255
head(df1)## pixel0 pixel1 pixel2 pixel3 pixel4 pixel5 pixel6 pixel7 pixel8 pixel9 pixel10
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## 6 0 0 0 0 0.000000000 0.00000000 0.0000000
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## 6 0 0 0df2 <- cbind(df$label, df1[1:784])
df2 <- df2 %>% rename(label = `df$label`)
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## 4 0 0 0
## 5 0 0 0
## 6 0 0 0Excercise 4
What is the frequency distribution of the numbers in the dataset?
ggplot(df, aes(x=label))+
geom_bar(colour="black", fill="steelblue")
## Excercise 5 For each number, provide the mean pixel intensity. What
does this tell you? Based on the results below we can see that it takes
the most pixel to create an 8. The least required was a 1.
mean_pixel <- c()
#9
df_9i <- df2 %>%
filter(label == 9)%>%
select(-label)
df_9i <- df_9i %>% mutate(Pixel_Sum = rowSums(df_9i))
mean_pixel[9] <- mean(df_9i$Pixel_Sum)
#8
df_8i <- df2 %>%
filter(label == 8)%>%
select(-label)
df_8i <- df_8i %>% mutate(Pixel_Sum = rowSums(df_8i))
mean_pixel[8] <- mean(df_8i$Pixel_Sum)
#7
df_7i <- df2 %>%
filter(label == 7)%>%
select(-label)
df_7i <- df_7i %>% mutate(Pixel_Sum = rowSums(df_7i))
mean_pixel[7] <- mean(df_7i$Pixel_Sum)
#6
df_6i <- df2 %>%
filter(label == 6)%>%
select(-label)
df_6i <- df_6i %>% mutate(Pixel_Sum = rowSums(df_6i))
mean_pixel[6] <-mean(df_6i$Pixel_Sum)
#5
df_5i <- df2 %>%
filter(label == 5)%>%
select(-label)
df_5i <- df_5i %>% mutate(Pixel_Sum = rowSums(df_5i))
mean_pixel[5] <-mean(df_5i$Pixel_Sum)
#4
df_4i <- df2 %>%
filter(label == 4)%>%
select(-label)
df_4i <- df_4i %>% mutate(Pixel_Sum = rowSums(df_4i))
mean_pixel[4] <-mean(df_4i$Pixel_Sum)
#3
df_3i <- df2 %>%
filter(label == 3)%>%
select(-label)
df_3i <- df_3i %>% mutate(Pixel_Sum = rowSums(df_3i))
mean_pixel[3] <-mean(df_3i$Pixel_Sum)
#2
df_2i <- df2 %>%
filter(label == 2)%>%
select(-label)
df_2i <- df_2i %>% mutate(Pixel_Sum = rowSums(df_2i))
mean_pixel[2] <-mean(df_2i$Pixel_Sum)
#1
df_1i <- df2 %>%
filter(label == 1)%>%
select(-label)
df_1i <- df_1i %>% mutate(Pixel_Sum = rowSums(df_1i))
mean_pixel[1] <-mean(df_1i$Pixel_Sum)
num <- c(1:9)
cbind(num,mean_pixel)## num mean_pixel
## [1,] 1 59.56261
## [2,] 2 117.14157
## [3,] 3 111.05956
## [4,] 4 95.03028
## [5,] 5 101.31733
## [6,] 6 108.76438
## [7,] 7 89.92645
## [8,] 8 118.36921
## [9,] 9 96.28922Excercise 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? There is one principal component per variable in the data. Since each variable affects the variability of the data, there is a principal component for each pixel.
df3 <- df2 %>% select(-label)
df4 <- cov(df3)
digitPr <- prcomp(df4)
#summary(digitPr); omitted this command since output exceeds print limit.
str(digitPr)## List of 5
## $ sdev : num [1:784] 0.166 0.1351 0.1158 0.1011 0.0883 ...
## $ rotation: num [1:784, 1:784] 1.47e-18 1.67e-15 0.00 -1.39e-17 1.04e-17 ...
## ..- attr(*, "dimnames")=List of 2
## .. ..$ : chr [1:784] "pixel0" "pixel1" "pixel2" "pixel3" ...
## .. ..$ : chr [1:784] "PC1" "PC2" "PC3" "PC4" ...
## $ center : Named num [1:784] 0 0 0 0 0 0 0 0 0 0 ...
## ..- attr(*, "names")= chr [1:784] "pixel0" "pixel1" "pixel2" "pixel3" ...
## $ scale : logi FALSE
## $ x : num [1:784, 1:784] 0.0764 0.0764 0.0764 0.0764 0.0764 ...
## ..- attr(*, "dimnames")=List of 2
## .. ..$ : chr [1:784] "pixel0" "pixel1" "pixel2" "pixel3" ...
## .. ..$ : chr [1:784] "PC1" "PC2" "PC3" "PC4" ...
## - attr(*, "class")= chr "prcomp"Excercise 7
Plot the first 10 images generated by PCA. They will appear to be noise. Why? There is noise becuase the PCA analysis includes all of the numbers in the data set.
img <- matrix(digitPr$rotation[,1], nrow = 28, ncol = 28)
image(img, useRaster=TRUE, axes=FALSE)
img2 <- matrix(digitPr$rotation[,2], nrow = 28, ncol = 28)
image(img2, useRaster=TRUE, axes=FALSE)
img3 <- matrix(digitPr$rotation[,3], nrow = 28, ncol = 28)
image(img3, useRaster=TRUE, axes=FALSE)
img4 <- matrix(digitPr$rotation[,4], nrow = 28, ncol = 28)
image(img4, useRaster=TRUE, axes=FALSE)
img5 <- matrix(digitPr$rotation[,5], nrow = 28, ncol = 28)
image(img5, useRaster=TRUE, axes=FALSE)
img6 <- matrix(digitPr$rotation[,6], nrow = 28, ncol = 28)
image(img6, useRaster=TRUE, axes=FALSE)
img7 <- matrix(digitPr$rotation[,7], nrow = 28, ncol = 28)
image(img7, useRaster=TRUE, axes=FALSE)
img8 <- matrix(digitPr$rotation[,8], nrow = 28, ncol = 28)
image(img8, useRaster=TRUE, axes=FALSE)
img9 <- matrix(digitPr$rotation[,9], nrow = 28, ncol = 28)
image(img9, useRaster=TRUE, axes=FALSE)
img10 <- matrix(digitPr$rotation[,10], nrow = 28, ncol = 28)
image(img10, useRaster=TRUE, axes=FALSE)
## Excercise 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? After we filter the data see with only
8’s we can see that 8’s are in fact produced in the image.
df5 <- df2 %>% filter(label == 8) %>% select(-label)
df6 <- cov(df5)
digitPr8 <- prcomp(df6)
img <- matrix(digitPr8$rotation[,1], nrow = 28, ncol = 28)
image(img, useRaster=TRUE, axes=FALSE)
img2 <- matrix(digitPr8$rotation[,2], nrow = 28, ncol = 28)
image(img2, useRaster=TRUE, axes=FALSE)
img3 <- matrix(digitPr$rotation[,3], nrow = 28, ncol = 28)
image(img3, useRaster=TRUE, axes=FALSE)
img4 <- matrix(digitPr8$rotation[,4], nrow = 28, ncol = 28)
image(img4, useRaster=TRUE, axes=FALSE)
img5 <- matrix(digitPr8$rotation[,5], nrow = 28, ncol = 28)
image(img5, useRaster=TRUE, axes=FALSE)
img6 <- matrix(digitPr8$rotation[,6], nrow = 28, ncol = 28)
image(img6, useRaster=TRUE, axes=FALSE)
img7 <- matrix(digitPr8$rotation[,7], nrow = 28, ncol = 28)
image(img7, useRaster=TRUE, axes=FALSE)
img8 <- matrix(digitPr8$rotation[,8], nrow = 28, ncol = 28)
image(img8, useRaster=TRUE, axes=FALSE)
img9 <- matrix(digitPr8$rotation[,9], nrow = 28, ncol = 28)
image(img9, useRaster=TRUE, axes=FALSE)
img10 <- matrix(digitPr8$rotation[,10], nrow = 28, ncol = 28)
image(img10, useRaster=TRUE, axes=FALSE)
Excercise 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.
df2_mlogit <- multinom(label~., data=df2, MaxNWts = 10000000)## # weights: 7860 (7065 variable)
## initial value 96708.573906
## iter 10 value 25322.714106
## iter 20 value 20402.086316
## iter 30 value 19312.872829
## iter 40 value 18703.256586
## iter 50 value 18197.815143
## iter 60 value 17732.985798
## iter 70 value 16739.962157
## iter 80 value 14961.658448
## iter 90 value 13446.085942
## iter 100 value 12442.636014
## final value 12442.636014
## stopped after 100 iterationsstr(df2_mlogit)## List of 26
## $ n : num [1:3] 785 0 10
## $ nunits : int 796
## $ nconn : num [1:797] 0 0 0 0 0 0 0 0 0 0 ...
## $ conn : num [1:7860] 0 1 2 3 4 5 6 7 8 9 ...
## $ nsunits : num 786
## $ decay : num 0
## $ entropy : logi FALSE
## $ softmax : logi TRUE
## $ censored : logi FALSE
## $ value : num 12443
## $ wts : num [1:7860] 0 0 0 0 0 0 0 0 0 0 ...
## $ convergence : int 1
## $ fitted.values: num [1:42000, 1:10] 0.000701 0.998131 0.000274 0.054333 0.998741 ...
## ..- attr(*, "dimnames")=List of 2
## .. ..$ : chr [1:42000] "1" "2" "3" "4" ...
## .. ..$ : chr [1:10] "0" "1" "2" "3" ...
## $ residuals : num [1:42000, 1:10] -0.000701 0.001869 -0.000274 -0.054333 0.001259 ...
## ..- attr(*, "dimnames")=List of 2
## .. ..$ : chr [1:42000] "1" "2" "3" "4" ...
## .. ..$ : chr [1:10] "0" "1" "2" "3" ...
## $ lev : chr [1:10] "0" "1" "2" "3" ...
## $ call : language multinom(formula = label ~ ., data = df2, MaxNWts = 1e+07)
## $ terms :Classes 'terms', 'formula' language label ~ pixel0 + pixel1 + pixel2 + pixel3 + pixel4 + pixel5 + pixel6 + pixel7 + pixel8 + pixel9 + pixel10 + | __truncated__ ...
## .. ..- attr(*, "variables")= language list(label, pixel0, pixel1, pixel2, pixel3, pixel4, pixel5, pixel6, pixel7, pixel8, pixel9, pixel10, pixel11| __truncated__ ...
## .. ..- attr(*, "factors")= int [1:785, 1:784] 0 1 0 0 0 0 0 0 0 0 ...
## .. .. ..- attr(*, "dimnames")=List of 2
## .. .. .. ..$ : chr [1:785] "label" "pixel0" "pixel1" "pixel2" ...
## .. .. .. ..$ : chr [1:784] "pixel0" "pixel1" "pixel2" "pixel3" ...
## .. ..- attr(*, "term.labels")= chr [1:784] "pixel0" "pixel1" "pixel2" "pixel3" ...
## .. ..- attr(*, "order")= int [1:784] 1 1 1 1 1 1 1 1 1 1 ...
## .. ..- attr(*, "intercept")= int 1
## .. ..- attr(*, "response")= int 1
## .. ..- attr(*, ".Environment")=<environment: R_GlobalEnv>
## .. ..- attr(*, "predvars")= language list(label, pixel0, pixel1, pixel2, pixel3, pixel4, pixel5, pixel6, pixel7, pixel8, pixel9, pixel10, pixel11| __truncated__ ...
## .. ..- attr(*, "dataClasses")= Named chr [1:785] "numeric" "numeric" "numeric" "numeric" ...
## .. .. ..- attr(*, "names")= chr [1:785] "label" "pixel0" "pixel1" "pixel2" ...
## $ weights : num [1:42000, 1] 1 1 1 1 1 1 1 1 1 1 ...
## ..- attr(*, "dimnames")=List of 2
## .. ..$ : chr [1:42000] "1" "2" "3" "4" ...
## .. ..$ : NULL
## $ deviance : num 24885
## $ rank : int 705
## $ lab : chr [1:10] "0" "1" "2" "3" ...
## $ coefnames : chr [1:785] "(Intercept)" "pixel0" "pixel1" "pixel2" ...
## $ vcoefnames : chr [1:785] "(Intercept)" "pixel0" "pixel1" "pixel2" ...
## $ xlevels : Named list()
## $ edf : num 6345
## $ AIC : num 37575
## - attr(*, "class")= chr [1:2] "multinom" "nnet"#In an earlier iteration dataframe df3 is without the labels
predicted <- predict(df2_mlogit,df3)
predicted_df <- as.data.frame(predicted)
label <- df$label
#Error message produced for the below command. Both vectors 42000 obs. and were numbers 1 through 9. Further investigation required.
##confusionMatrix(label,predicted_df)