STATS 769

STATISTICS

Data Science Practice

SECOND SEMESTER, 2017

1.                                                                                                                                    [5 marks]

Figure 1 shows the content of an HTML file, "2017-07-29 .html". This is asimplified extract from the HOT 100 songs web site that was used for the web scraping lab.

Write five R expressions that use functions from the xml2 package (and XPath expres- sions) to perform the following steps:

•  Read the HTML file into R.

• Extract the song title from the HTML (output shown below).

[1]  "Despacito"

• Extract the artist name from the HTML (output shown below; note that white space has been removed).

[1]  "Luis  Fonsi  &  Daddy  Yankee  Featuring  Justin  Bieber"

• Extract the song rank from the HTML (output shown below).

[1]  "1"

• Extract the rank from the previous week from the HTML (output shown below; note that the result is more than one character value).

[1]  "Last  Week"  "1"

Figure 1: The HTML file "2017-07-29 .html".

2.                                                                                                                                   [5 marks]

Write a paragraph explaining the purpose of the flatten() function from the jsonlite package. You should provide at least one example of its use.

3.                                                                                                                                 [10 marks]

Explain what each of the following shell commands is doing and, where there is output, what the output means.  These commands were all run on one of the virtual machines that were used in the course.

pmur002@stats769prd01:~/$ mkdir  exam pmur002@stats769prd01:~/$  cd  exam

pmur002@stats769prd01:~/exam$  ls  -1  /course/AT/BUSDATA/  |   wc  -l 98973

pmur002@stats769prd01:~/exam$  ls  -l  /course/AT/BUSDATA/  |   awk  ' {  print($5)  } '   >  sizes .txt pmur002@stats769prd01:~/exam$  head  sizes .txt

343

345

345

345

436

437

438

531

438

pmur002@stats769prd01:~/exam$  grep  --no-filename  ' ,6215, '   \

>  /course/AT/BUSDATA/trip_updates_20170401* .csv  >  bus-6215-2017-04-01 .csv

pmur002@stats769prd01:~/exam$  head  bus-6215-2017-04-01 .csv

8300033770-20170322104732_v52 .21,30002-20170322104732_v52 .21,6215,NA,252,6,7168,1490975922   8300033770-20170322104732_v52 .21,30002-20170322104732_v52 .21,6215,275,NA,7,8502,1490976035   8300033770-20170322104732_v52 .21,30002-20170322104732_v52 .21,6215,275,NA,7,8502,1490976035   8300033770-20170322104732_v52 .21,30002-20170322104732_v52 .21,6215,275,NA,7,8502,1490976035   8300033770-20170322104732_v52 .21,30002-20170322104732_v52 .21,6215,NA,293,9,8516,1490976233   8300033770-20170322104732_v52 .21,30002-20170322104732_v52 .21,6215,299,NA,9,8516,1490976239   8300033770-20170322104732_v52 .21,30002-20170322104732_v52 .21,6215,NA,319,10,8524,1490976349 8300033770-20170322104732_v52 .21,30002-20170322104732_v52 .21,6215,388,NA,10,8524,1490976418 8300033770-20170322104732_v52 .21,30002-20170322104732_v52 .21,6215,388,NA,10,8524,1490976418 8300033770-20170322104732_v52 .21,30002-20170322104732_v52 .21,6215,NA,403,11,8532,1490976523

4.                                                                                                                                   [10 marks]

The following R code was run on one of the virtual machines that were used in the course to investigate how much memory would be required to load a large CSV file with several million rows into R. The plot that this code produces is also shown.

Explain what the code is doing and discuss whether this will lead to a good estimate of the memory required to read the complete CSV into R. Is there another way to estimate the memory required (without reading the entire CSV file into R) ?

numLines  <-  10^(1:5)

samples  <-  lapply(numLines,

function(i)  {

read .csv("/course/AT/alldata .csv",

nrows=i,  stringsAsFactors=FALSE) })

plot(numLines,  sapply(samples,  object .size),  log="xy", xlab="number  of  lines",  ylab="data  frame  size")

5.                                                                                                                                 [10 marks]

The following R code was run on one of the virtual machines that were used in the course to measure how much time is required to read diferent subsets of a large CSV file into R.

sapply(numLines,

function(i)  {

system.time(read .csv("/course/AT/alldata .csv",

nrows=i,  stringsAsFactors=FALSE))[1]

})

The result of running this code is shown below.

user .self  user .self  user .self  user .self  user .self 0.001         0.001         0.005         0.036         0.417

The following code was run to perform profiling.  The profiling result is shown below the code.

library(profvis) p  <-  profvis({

lapply(numLines,

function(i)  {

read .csv("/course/AT/alldata .csv",

nrows=i,  stringsAsFactors=FALSE)

})  })

htmlwidgets::saveWidget(p,  "profile.html")

Explain what the timing and profiling results mean.  Suggest how you could make the code run faster.

6.                                                                                                                                 [10 marks]

Write R code to perform a parallel version of the lapply() call from Question 4. Discuss the advantages and disadvantages of using the mclapply() (forking) approach compared to the makeCluster()  (socket) approach for this task.  Also discuss whether load balancing would make sense for this task.

7.                                                                                                                                 [10 marks]

Suppose we estimate the regression coefficients in a linear regression by minimizing

subject to

for a particular value of s. As we increases from 0, indicate which of the following is correct. Justify your answer.

a. The training residual sum of squares (SS residual) will increase initially, and then eventually start decreasing in an inverted U shape.

b. The training SSresidual  will decrease initially, and then eventually start increasing in a U shape.

c. The training SS residual  will steadily increase. d. The training SS residual  will steadily decrease. e. The training SS residual  will remain constant.

8.                                                                                                                                 [10 marks]

The table below provides a training data set containing six observations, three predictors, and one qualitative response variable.

Suppose we wish to use this data set to make a prediction for Y when X1 = X2 = X3 = 0 using a K-NN classifier.

a. Compute the Euclidean distance between each observation and the test point, X1 = X2 = X3 = 0. (recall that the Euclidean  distance  between  two points  (x1 ;x2 ;x3 ) and  (y1 ;y2 ;y3)

is √ (x1 — y1)2 +(x2 — y2)2 +(x3 — y3)2 ).

b. What is our prediction with K = 1? Why? c. What is our prediction with K = 3? Why?

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

9.                                                                                                                                 [10 marks]

I collect a set of data (n = 100 observations) containing a single predictor and a quantitative response. I then fita linear regression model to the data, as well as a separate cubic regression, i.e. Y = b0 + b1X + b2X2 + b3X3 + ε .

a.  Suppose that the true relationship between X  and Y  is  linear, i.e.  Y = b0 + b1X + ε . Consider the training residual sum of squares (SS resdiual) for the linear regression, and also the training SS residual  for the cubic regression. Would we expect one to be lower than the other, would we expect them to be the same, or is there not enough information to tell? Justify your answer.

b. Answer part (a) using test SS residual  rather than training SS residual.

10.                                                                                                                                 [10 marks]

Explain how k-fold cross-validation is implemented.  And what are the advantages and dis- advantages of k-fold cross validation relative to the validation set approach?

11.                                                                                                                                 [10 marks]

When the number of features p is large, there tends to be a deterioration in the performance of K-NN and other local approaches that perform. prediction using only observations that are near the test observation for which a prediction must be made. This phenomenon is known as the curse of dimensionality, and it ties into the fact that non-parametric approaches often perform poorly when p is large.

a.  Suppose that we have a set of observations, each with measurements on p = 1 feature, X . We assume that X  is uniformly (evenly) distributed on [0; 1].  Associated with each observation is a response value.  Suppose that we wish to predict a test observation’s response using only observations that are within 10% of the range of X  closest to that test observation.  For instance, in order to predict the response for a test observation with X = 0.6, we will use observations in the range [0.55; 0.65]. On average, what fraction of the available observations will we use to make the prediction?

b.  Now suppose that we have a set of observations, each with measurements on p = 2 features, X1  and X2 .  We assume that (X1 ;X2) are uniformly distributed on  [0; 1] × [0; 1]. We wish to predict a test observation’s response using only observations that are within 10% of the range of X1  and within 10% of the range of X2  closest to that test observation. For instance, in order to predict the response for a test observation with X1 = 0.6 and X2  = 0.35,  we  will use observations in the range  [0.55; 0.65]  for X1   and  in the range [0.3; 0.4] for X2 .  On average, what fraction of the available observations will we use to make the prediction?

c.  Now suppose that we have a set of observations on p = 100 features.  Again the observa- tions are uniformly distributed on each feature, and again each feature ranges in value from 0 to 1. We wish to predict a test observation’s response using observations within the 10% of each feature’s range that is closest to that test observation.  What fraction of the available observations will we use to make the prediction?

d. Using your answers to parts (a)–(c), argue that a drawback of K-NN when p is large is that there are very few training observations “near” any given test observation.