MS6711 Data Mining
Homework 2
Instruction
This homework contains both coding and non-coding questions. Please submit two files,
1. One word or pdf document of answers and plots of ALL questions without coding details.
2. One jupyter notebook of your codes.
3. Questions 1 and 2 are about concepts, 3 - 6 are about coding.
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Problem 1 [20 points]
We perform best subset, forward stepwise and backward stepwise selection on the same dataset with p
predictors. For each approach, we obtain p + 1 models containing 0, 1, 2, · · · , p predictors. Explain your
answer.
1. Which of the three models with same number of k predictors has smallest training RSS?
2. Which of the three models with same number of k predictors has smallest testing RSS? (best
subset, forward, backward, or cannot determine?)
3. True or False: The predictors in the k-variable model identified by forward stepwise are a subset of
the predictors in the (k + 1)-variable model identified by forward stepwise selection.
4. True or False: The predictors in the k-variable model identified by best subset are a subset of the
predictors in the (k + 1)-variable model identified by best subset selection.
5. True or False: The lasso, relative to OLS, is less flexible and hence will give improved prediction
accuracy when its increase in bias is less than its decrease in variance.
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Problem 2 [20 points]
Suppose we estimate Lasso by minimizing
||Y − Xβ||2
2 + λ||β||1
for a particular value of λ. For part 1 to 5, indicate which of (a) to (e) is correct and explain your answer.
1. As we increase λ from 0, the training RSS will
(a) Increase initially, and then eventually start decreasing in an inverted U shape.
(b) Decrease initially, and then eventually start increasing in a U shape.
(c) Steadily increase.
(d) Steadily decrease.
(e) Remain constant.
2. Repeat 1. for test RSS.
3. Repeat 1. for variance.
4. Repeat 1. for (squared) bias.
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Problem 3 [20 points]
These data record the level of atmospheric ozone concentration from eight daily meteorological mea surements made in the Los Angeles basin in 1976. We have the 330 complete cases1. We want to find
climate/weather factors that impact ozone readings. Ozone is a hazardous byproduct of burning fossil
fuels and can harm lung function. The data set for this problem is:
Variable name Definition
ozone Long Maximum Ozone
vh Vandenberg 500 mb Height
wind Wind speed (mph)
humidity Humidity (%)
temp Sandburg AFB Temperature
ibh Inversion Base Height
dpg Daggot Pressure Gradint
ibt Inversion Base Temperature
vis Visibility (miles)
doy Day of the Year
[Note: I would recommend you use R for this question, since python does not have package for
forward / backward selection. See the code example on Canvas. Or you may use the sample python code
I provided.]
1. Report result of linear regression using all variables. Note that ozone is the response variable to
predict. What variables are significant?
2. Report the selected variables using the following model selection approaches.
(a) All subset selection.
(b) Forward stepwise
(c) Backward stepwise
3. Compare the outcome of these methods with the significant variables found in the full linear regres sion in question 1.
4. Potentially, other transformation of covariates might be important. What happens if you do all
subset selection using both the original variables and their square? That is, for all variables, include
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both
X, X2
in the linear regression model for all subset selection.
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Problem 4 [20 points]
In this exercise, we will predict the number of applications received using the other variables in the College
data set.
Private Public/private school 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 1 = 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.
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
1. Split the data set into a training set and a test set.
2. Fit a linear regression model using OLS on the training set, and report the test error obtained.
3. Fit a ridge regression model on the training set, with λ chosen by cross-validation. Report the test
error obtained.
4. Fit a lasso model on the training set, with λ chosen by cross-validation. Report the test error
obtained, along with the number of non-zero coefficient estimates.
5. Fit a PCR model on the training set, with number of components chosen by cross-validation. Report
the test error obtained, along with the value of M selected by cross-validation.
6. Fit a PLS model on the training set, with number of components chosen by cross-validation. Report
the test error obtained, along with the value of number of components selected by cross-validation.
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Problem 5 [20 points]
We will now try to predict per capita crime rate in the Boston data set.
crim per capita crime rate by town.
zn proportion of residential land zoned for lots over 25,000 sq.ft.
indus proportion of non-retail business acres per town.
chas Charles River dummy variable (= 1 if tract bounds river; 0 otherwise).
nox nitrogen oxides concentration (parts per 10 million).
rm 1 = average number of rooms per dwelling.
age proportion of owner-occupied units built prior to 1940.
dis weighted mean of distances to five Boston employment centres.
rad index of accessibility to radial highways.
tax full-value property-tax rate per $10,000.
ptratio pupil-teacher ratio by town.
black 1000(Bk − 0.63)2 where Bk is the proportion of blacks by town.
lstat lower status of the population (percent).
medv median value of owner-occupied homes in $1000s.
1. Try out some of the regression methods explored in this chapter, such as best subset selection, the
lasso, ridge regression, PCR and partial least squares. Present and discuss results for the approaches
that you consider.
2. Propose a model (or set of models) that seem to perform well on this data set, and justify your
answer. Make sure that you are evaluating model performance using validation set error, cross validation, or some other reasonable alternative, as opposed to using training error.
3. Does your chosen model involve all of the features in the data set? Why or why not?
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Problem 6 [20 points]
In a bike sharing system the process of obtaining membership, rental, and bike return is automated
via a network of kiosk locations throughout a city. In this problem, you will try to combine historical
usage patterns with weather data to forecast bike rental demand in the Capital Bikeshare program in
Washington, D.C.
You are provided hourly rental data collected from the Capital Bikeshare system spanning two years.
The file Bike train.csv, as the training set, contains data for the first 19 days of each month, while
Bike test.csv, as the test set, contains data from the 20th to the end of the month. The dataset includes
the following information:
daylabel day number ranging from 1 to 731
year, month, day, hour hourly date
season 1=spring,2=summer,3=fall,4=winter
holiday whether the day is considered a holiday
workingday whether the day is neither a weekend nor a holiday
weather 1 = clear, few clouds, partly cloudy
2 = mist + cloudy, mist + broken clouds, mist + few clouds, mist
3 = light snow, light rain + thunderstorm + scattered clouds, light rain
4 = 4 = heavy rain + ice pallets + thunderstorm + mist, snow + fog
temp temperature in Celsius
atemp ’feels like’ temperature in Celsius
humidity relative humidity
wind speed wind speed
count number of total rentals, outcome variable to predict
Predictions will be evaluated using the root mean squared error (RMSE), calculated as
RMSE =
v
u
u t
n
1
nX
i=1
(yi − ybi)
2
where yi
is the true count, ybi
is the prediction, and n is the number of entries to be evaluated.
Build a model on train dataset to predict the bikeshare counts for the hours recorded in the test
dataset. Report your prediction RMSE on testing set.
Some tips
• This is a relatively open question, you may use any model you learnt from this class.
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• It will be helpful to examine the data graphically to spot any seasonal pattern or temporal trend.
• There is one day in the training data with weird atemp record and another day with abnormal
humidity. Find those rows and think about what you want to do with them. Is there anything
unusual in the test data?
• It might be helpful to transform the count to log(count + 1). If you did that, do not forget to
transform your predicted values back to count.
• Think about how you would include each predictor into the model, as continuous or as categorical?
• Is there any transformation of the predictors or interactions between them that you think might be
helpful?
Try to summarize your exploration of the data, and modeling process. You may fit a few models and
chose one from them. You will receive points based on your write-up and test RMSE. This is not a
competition among the class to achieve the minimal RMSE, but your result should be in a reasonable
range.
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