Lab 1: Introduction To Statistical Learning

Questions

Conceptual Questions

  1. \star (ISLR2, Q2.1) For each of parts (a) through (d), indicate whether we would generally expect the performance of a flexible statistical learning method to be better or worse than an inflexible method. Justify your answer.

    1. The sample size n is extremely large, and the number of predictors p is small.

    2. The number of predictors p is extremely large, and the number of observations n is small.

    3. The relationship between the predictors and response is highly non-linear.

    4. The variance of the error terms, i.e., \sigma^2 = \mathbb{V}(\epsilon), is extremely high.

    Solution

  2. (ISLR2, Q2.2) Explain whether each scenario is a classification or regression problem, and indicate whether we are most interested in inference or prediction. Finally, provide n and p.

    1. We collect a set of data on the top 500 firms in the US. For each firm we record profit, number of employees, industry and the CEO salary. We are interested in understanding which factors affect CEO salary.

    2. We are considering launching a new product and wish to know whether it will be a success or a failure. We collect data on 20 similar products that were previously launched. For each product we have recorded whether it was a success or failure, price charged for the product, marketing budget, competition price, and ten other variables.

    3. We are interested in predicting the % change in the USD/Euro exchange rate in relation to the weekly changes in the world stock markets. Hence we collect weekly data for all of 2012. For each week we record the % change in the USD/Euro, the % change in the US market, the % change in the British market, and the % change in the German market.

    Solution

  3. \star (ISLR2, Q2.3) We now revisit the bias-variance decomposition.

    1. Provide a sketch of typical (squared) bias, variance, training error, test error, and Bayes (or irreducible) error curves, on a single plot, as we go from less flexible statistical learning methods towards more flexible approaches. The x-axis should represent the amount of flexibility in the method, and the y-axis should represent the values for each curve. There should be five curves. Make sure to label each one.

    2. Explain why each of the five curves has the shape displayed in part (a).

    Solution

  4. \star (ISLR2, Q2.5) What are the advantages and disadvantages of a very flexible (versus a less flexible) approach for regression or classification? Under what circumstances might a more flexible approach be preferred to a less flexible approach? When might a less flexible approach be preferred?

    Solution

  5. \star (ISLR2, Q2.7) The table below provides a training data set containing six observations, three predictors, and one qualitative response variable.

    Obs. X_1 X_2 X_3 Y
    1 0 3 0 Red
    2 2 0 0 Red
    3 0 1 3 Red
    4 0 1 2 Green
    5 -1 0 1 Green
    6 1 1 1 Red

    Suppose we wish to use this data set to make a prediction for Y when X_1 = X_2 = X_3 = 0 using K-nearest neighbors.

    1. Compute the Euclidean distance between each observation and the test point, X_1 = X_2 = X_3 = 0.

    2. What is our prediction with K = 1? Why?

    3. What is our prediction with K = 3? Why?

    4. If the Bayes decision boundary in this problem is highly non-linear, then would we expect the best value for K to be large or small? Why?

    Solution

Applied Questions

  1. \star (ISLR2, Q2.8) This exercise relates to the College data set, which can be found in the file College.csv on Website homepage. It contains a number of variables for 777 different universities and colleges in the US. The variables are

    • Private: Public/private indicator

    • Apps: Number of applications received

    • Accept: Number of applicants accepted

    • Enroll: Number of new students enrolled

    • Top10perc: New students from top 10% of high school class

    • Top25perc: New students from top 25% of high school class

    • F.Undergrad: Number of full-time undergraduates

    • P.Undergrad: Number of part-time undergraduates

    • Outstate: Out-of-state tuition

    • Room.Board: Room and board costs

    • Books: Estimated book costs

    • Personal: Estimated personal spending

    • PhD: Percent of faculty with Ph.D.’s

    • Terminal: Percent of faculty with terminal degree

    • S.F.Ratio: Student/faculty ratio

    • perc.alumni: Percent of alumni who donate

    • Expend: Instructional expenditure per student

    • Grad.Rate: Graduation rate

    Before reading the data into R, it can be viewed in Excel or a text editor.

    1. Use the read.csv() function to read the data into R. Call the loaded data college. Make sure that you have the directory set to the correct location for the data.

    2. Look at the data using the View() function. You should notice that the first column is just the name of each university. We don’t really want R to treat this as data. However, it may be handy to have these names for later. Try the following commands:

      rownames(college) <- college[, 1]
View(college)

      You should see that there is now a row.names column with the name of each university recorded. This means that R has given each row a name corresponding to the appropriate university. R will not try to perform calculations on the row names. However, we still need to eliminate the first column in the data where the names are stored. Try

      college <- college[, -1]
View(college)

      Now you should see that the first data column is Private. Note that another column labeled row.names now appears before the Private column. However, this is not a data column but rather the name that R is giving to each row.

      1. Use the summary() function to produce a numerical summary of the variables in the data set.

      2. Use the pairs() function to produce a scatterplot matrix of the first ten columns or variables of the data. Recall that you can reference the first ten columns of a matrix A using A[,1:10].

      3. Use the plot() function to produce side-by-side boxplots of Outstate versus Private.

    4. Create a new qualitative variable, called Elite, by binning the Top10perc variable. We are going to divide universities into two groups based on whether or not the proportion of students coming from the top 10% of their high school classes exceeds 50%.

      Elite <- rep("No", nrow(college))
Elite[college$Top10perc > 50] <- "Yes"
Elite <- as.factor(Elite)
college <- data.frame(college, Elite)

      Use the summary() function to see how many elite universities there are. Now use the plot() function to produce side-by-side boxplots of Outstate versus Elite.

    5. Use the hist() function to produce some histograms with differing numbers of bins for a few of the quantitative variables. You may find the command par(mfrow = c(2, 2)) useful: it will divide the print window into four regions so that four plots can be made simultaneously. Modifying the arguments to this function will divide the screen in other ways.

    6. Continue exploring the data, and provide a brief summary of what you discover.

    Solution

  2. \star (ISLR2, Q2.9) This exercise involves the Auto data set studied in the lab. Make sure that the missing values have been removed from the data.

    1. Which of the predictors are quantitative, and which are qualitative?

    2. What is the range of each quantitative predictor? You can answer this using the range() function.

    3. What is the mean and standard deviation of each quantitative predictor?

    4. Now remove the 10th through 85th observations. What is the range, mean, and standard deviation of each predictor in the subset of the data that remains?

    5. Using the full data set, investigate the predictors graphically, using scatterplots or other tools of your choice. Create some plots highlighting the relationships among the predictors. Comment on your findings.

    6. Suppose that we wish to predict gas mileage (mpg) on the basis of the other variables. Do your plots suggest that any of the other variables might be useful in predicting mpg? Justify your answer.

    Solution

  3. (ISLR2, Q2.10) This exercise involves the Boston housing data set.

    1. To begin, load in the Boston data set. The Boston data set is part of the ISLR2 library.

      library(ISLR2)

      Now the data set is contained in the object Boston.

      Boston

      Read about the data set:

      ?Boston

      How many rows are in this data set? How many columns? What do the rows and columns represent?

    2. Make some pairwise scatterplots of the predictors (columns) in this data set. Describe your findings.

    3. Are any of the predictors associated with per capita crime rate? If so, explain the relationship.

    4. Do any of the census tracts of Boston appear to have particularly high crime rates? Tax rates? Pupil-teacher ratios? Comment on the range of each predictor.

    5. How many of the census tracts in this data set bound the Charles river?

    6. What is the median pupil-teacher ratio among the towns in this data set?

    7. Which census tract of Boston has lowest median value of owneroccupied homes? What are the values of the other predictors for that census tract, and how do those values compare to the overall ranges for those predictors? Comment on your findings.

    8. In this data set, how many of the census tracts average more than seven rooms per dwelling? More than eight rooms per dwelling? Comment on the census tracts that average more than eight rooms per dwelling.

    Solution

Solutions

Conceptual Questions

    1. Better: flexible models are better able to capture all the trends in the large amount of data we have.

    2. Worse: flexible models will tend to overfit the small amount of data we have using the large number of predictors.

    3. Better: inflexible models tend to have a hard time fitting non-linear relationships.

    4. Worse: flexible models will tend to fit the noise, which is not desired.

    1. Regression: the response (CEO salary) is continuous. Inference: We are interested in the factors influencing CEO salary – we don’t want to estimate it using a company’s information! n = 500 (500 companies in the data set) p = 3 (predictors: profit, number of employees, industry; response: CEO salary)

    2. Classification: the response (success or failure) is discrete. Prediction: based on various input factors, we want to estimate how well the product will do n = 20 (20 similar products in the data set) p = 13 (predictors: marketing budget, price charged, competition price, +10 others; response: whether it was a success or failure)

    3. Regression: the response (% change in US dollar) is continuous. Prediction: it’s written in the question! We are interested in predicting changes in the US dollar. n \approx 50 (number of trading weeks in a year) p = 3 (predictors: % change in US market, % change in UK market, % change in DE market; response % change in US dollar)

    1. See the figure

      image

    2. Bias: increasing flexibility reduces model bias
      Training error: increasing flexibility makes the model fit the training data better
      Variance: increasing flexibility makes the model incorporate more noise (in other words, it makes the fit bumpier)
      Test error: concave up, since increasing flexibility makes the model fit more of the trend in the data until it starts fitting the noise in the data
      Bayes’ (irreducible) error: horizontal line, since it’s a constant for all models. When the training error dips below the Bayes’ error, the model is overfitting, so the test error starts to increase

  4. Advantages: can fit a larger variety of (non-linear) models, decreasing bias.
    Disadvantages: can lead to overfitting (hence worse results), requires estimating more parameters, and increasing model variance as it incorporates more noise.
    More flexible models would be preferred if the model is non-linear in nature, or interpretability is not a major issue. Less flexible models are preferred when inference is the goal of the model fitting exercise

    1. See the table below

      Obs. X_1 X_2 X_3 Distance Y
      1 0 3 0 3 Red
      2 2 0 0 2 Red
      3 0 1 3 \sqrt{10} \approx 3.2 Red
      4 0 1 2 \sqrt{5} \approx 2.2 Green
      5 -1 0 1 \sqrt{2} \approx 1.4 Green
      6 1 1 1 \sqrt{3} \approx 1.7 Red

    2. Green. K=1 so we only take the closest observation (5).

    3. K=3 so we consider the closest 3: 5, 6 and 2. The majority are Red, so this classifies as Red.

    4. A smaller K would lead to a more flexible decision boundary, which would account for the non-linearity better.

Applied Questions

Refer to Section 2.3 of ISLR2 for a primer of applied questions.

    1. Note that the dataset is available from the course Website homepage.

      college <- read.csv("College.csv")

    2. The row names need to be changed to college names as follow

      rownames(college) <- college[, 1]
college <- college[, -1]
college$Private <- as.factor(college$Private)
      head(college) # You should instead try `View(college)`
      1. summary(college)
         Private        Apps           Accept          Enroll       Top10perc    
 No :212   Min.   :   81   Min.   :   72   Min.   :  35   Min.   : 1.00  
 Yes:565   1st Qu.:  776   1st Qu.:  604   1st Qu.: 242   1st Qu.:15.00  
           Median : 1558   Median : 1110   Median : 434   Median :23.00  
           Mean   : 3002   Mean   : 2019   Mean   : 780   Mean   :27.56  
           3rd Qu.: 3624   3rd Qu.: 2424   3rd Qu.: 902   3rd Qu.:35.00  
           Max.   :48094   Max.   :26330   Max.   :6392   Max.   :96.00  
   Top25perc      F.Undergrad     P.Undergrad         Outstate    
 Min.   :  9.0   Min.   :  139   Min.   :    1.0   Min.   : 2340  
 1st Qu.: 41.0   1st Qu.:  992   1st Qu.:   95.0   1st Qu.: 7320  
 Median : 54.0   Median : 1707   Median :  353.0   Median : 9990  
 Mean   : 55.8   Mean   : 3700   Mean   :  855.3   Mean   :10441  
 3rd Qu.: 69.0   3rd Qu.: 4005   3rd Qu.:  967.0   3rd Qu.:12925  
 Max.   :100.0   Max.   :31643   Max.   :21836.0   Max.   :21700  
   Room.Board       Books           Personal         PhD        
 Min.   :1780   Min.   :  96.0   Min.   : 250   Min.   :  8.00  
 1st Qu.:3597   1st Qu.: 470.0   1st Qu.: 850   1st Qu.: 62.00  
 Median :4200   Median : 500.0   Median :1200   Median : 75.00  
 Mean   :4358   Mean   : 549.4   Mean   :1341   Mean   : 72.66  
 3rd Qu.:5050   3rd Qu.: 600.0   3rd Qu.:1700   3rd Qu.: 85.00  
 Max.   :8124   Max.   :2340.0   Max.   :6800   Max.   :103.00  
    Terminal       S.F.Ratio      perc.alumni        Expend     
 Min.   : 24.0   Min.   : 2.50   Min.   : 0.00   Min.   : 3186  
 1st Qu.: 71.0   1st Qu.:11.50   1st Qu.:13.00   1st Qu.: 6751  
 Median : 82.0   Median :13.60   Median :21.00   Median : 8377  
 Mean   : 79.7   Mean   :14.09   Mean   :22.74   Mean   : 9660  
 3rd Qu.: 92.0   3rd Qu.:16.50   3rd Qu.:31.00   3rd Qu.:10830  
 Max.   :100.0   Max.   :39.80   Max.   :64.00   Max.   :56233  
   Grad.Rate     
 Min.   : 10.00  
 1st Qu.: 53.00  
 Median : 65.00  
 Mean   : 65.46  
 3rd Qu.: 78.00  
 Max.   :118.00

      2. Private is not numerical, so cannot be used in pairs so we plot from column 2 onward

        pairs(college[, 1:10])

      3. plot(college$Private, college$Outstate)

      4. Elite <- with(college, ifelse(Top10perc > 50, "Yes", "No"))
Elite <- as.factor(Elite)
college <- data.frame(college, Elite)
summary(Elite) # there are 78 elite universities
         No Yes 
699  78 
        plot(college$Elite, college$Outstate)

      5. For instance the following code gives histograms for variables Apps, Accept, Top10perc and Top25perc.

        par(mfrow = c(2, 2))
hist(college$Apps, breaks = 20)
hist(college$Accept, breaks = 20)
hist(college$Top10perc, breaks = 20)
hist(college$Top25perc, breaks = 20)

  2. Note that the dataset is available from the course Website homepage.

    auto <- read.csv("Auto.csv", na.strings = "?")
auto <- na.omit(auto) # remove missing values

    1. Qualitative: name, origin. Quantitative: mpg, cylinders, displacement, horsepower, weight, acceleration, year.

    2. We can use function apply combined with function range:

      quant.var <- c(
  "mpg", "cylinders", "displacement", "horsepower",
  "weight", "acceleration", "year"
)
ranges.df <- apply(auto[, quant.var], 2, range)
rownames(ranges.df) <- c("min", "max")
ranges.df
           mpg cylinders displacement horsepower weight acceleration year
min  9.0         3           68         46   1613          8.0   70
max 46.6         8          455        230   5140         24.8   82

    3. Use a similar application of function apply:

      means.df <- apply(auto[, quant.var], 2, mean)
std.df <- apply(auto[, quant.var], 2, sd)
distns.df <- rbind(means.df, std.df)
rownames(distns.df) <- c("mean", "sd.")
t(distns.df)
                          mean        sd.
mpg            23.445918   7.805007
cylinders       5.471939   1.705783
displacement  194.411990 104.644004
horsepower    104.469388  38.491160
weight       2977.584184 849.402560
acceleration   15.541327   2.758864
year           75.979592   3.683737

    4. Semantic note: the following will remove the 10th to the 85th row, which may not be what we want, since we have already removed some rows to begin with:

      subauto <- auto[-(10:85), ]

      You will find that observation #86 has errantly been removed. That is because the na.omit from earlier removed an observation in this range. It is possible to refer to the rows by observation number, which is a character string. In other words, auto["5",] will give me observation #5, even if 1-4 are missing. This does complicate the procedure, though.

      rid <- rownames(auto)
rid <- rid[as.numeric(rid) < 10 | as.numeric(rid) > 85]
subauto <- auto[rid, ]

      Use apply function:

      subranges.df <- apply(subauto[, quant.var], 2, range)
submeans.df <- apply(subauto[, quant.var], 2, mean)
substd.df <- apply(subauto[, quant.var], 2, sd)
subdistns.df <- rbind(subranges.df, submeans.df, substd.df)
rownames(subdistns.df) <- c("min", "max", "mean", "sd.")
t(subdistns.df)
                      min    max        mean        sd.
mpg            11.0   46.6   24.368454   7.880898
cylinders       3.0    8.0    5.381703   1.658135
displacement   68.0  455.0  187.753943  99.939488
horsepower     46.0  230.0  100.955836  35.895567
weight       1649.0 4997.0 2939.643533 812.649629
acceleration    8.5   24.8   15.718297   2.693813
year           70.0   82.0   77.132492   3.110026

    5. For instance a pairwise plot can be produced using:

      pairs(auto[, -9])

    6. Briefly looking at the pairwise plots, the factors cylinders, displacement, horsepower, weight, and possibly year are worth investigating.

    1. library(ISLR2)
dim(Boston)
      [1] 506  13

      506 rows each representing a town, 13 columns each with some data on the towns.

    2. pairs(Boston)

      Various answers can exist:

      • crim relates to zn, indus, age, dis, rad, tax, ptratio

      • nox relates to age, dis, rad

      • age relates to lstat, medv

      • lstat relates to medv

      • zn: Very low, unless zn is very close to 0. Then crim can be much higher.

      • indus: Very low, unless indus is close to 18%. Then crim can be much higher.

      • age: crim increases as this increases

      • dis: crim decreases as this increases

      • tax: Very low, unless tax is at 666

      • ptratio: Very low, unless ptratio is at 20.2

    4. par(mfrow = c(2, 2))
hist(Boston$crim, breaks = 25)
hist(Boston$tax)
hist(Boston$ptratio)
length(Boston$crim[Boston$crim > 20])
      [1] 18

      • crim: Vast majority of cities have low crime rates, but 18 of them have a crime rate of greater than 20, reaching up to.

      • tax: Divided into two sections: low < 500, high \ge 660.

      • ptratio: Mode at about 20, max at 22, minimum at 12.6.

    5. length(Boston$chas[Boston$chas == 1])
      [1] 35
    6. median(Boston$ptratio)
      [1] 19.05
    7. Boston[Boston$medv == min(Boston$medv), ]

      Crime rates are quite high, indus is on the upper end, all owner-occupied units are built before 1940, both don’t bound the Charles river, both are relatively close to employment centres, they’re both very close to radial highways, pupil/teacher ratio is at the mode, lstat is also on the higher end.

    8. length(Boston$rm[Boston$rm > 7])
      [1] 64
      length(Boston$rm[Boston$rm > 8])
      [1] 13

      crim, lstat relatively low.