STAT Homework # 3 Solution

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Instructions:  You may  discuss the homework  problems  in small groups,  but  you must write up the final solutions and code yourself. Please turn  in your code for the problems  that  involve coding.  However, for the  problems  that  involve coding, you must also provide written  answers: you will receive no credit if you submit code with- out written  answers.  You might want to use Rmarkdown to prepare your assignment.

  1. A random variable X has an Exponential(λ) distribution if its probability den- sity function is of the form

 

f (x) =

(λe−λx        if x > 0

,

0            if x ≤ 0

where λ  > 0 is a parameter.   Furthermore, the  mean  of an  Exponential(λ)

random  variable is 1/λ.

Now, consider a classification problem with K = 2 classes and a single feature X ∈ R.  If an observation  is in class 1 (i.e.  Y  = 1) then  X ∼ Exponential(λ1). And if an observation  is in class 2 (i.e. Y  = 2) then X ∼ Exponential(λ2). Let π1  denote the probability  that  an observation  is in class 1, and let π2  = 1 − π1.

(a)  Derive an expression for Pr(Y  = 1 |  X  = x).  Your answer should be in terms of x, λ1, λ2, π1 , π2.

(b)  Write  a simple expression for the Bayes classifier decision boundary,  i.e., an expression for the set of x such that  Pr(Y  = 1 | X = x) = Pr(Y  = 2 | X = x).

(c)  For part (c)  only, suppose λ1   = 2, λ2   = 7, π1  = 0.5.  Make a plot of feature space.  Clearly label:

  1. the region of feature space corresponding to the Bayes classifier deci- sion boundary,
  2. the region of feature space for which the  Bayes classifier will assign an observation  to class 1,

iii. the  region of feature  space for which the  Bayes classifier will assign an observation  to class 2.

(d)  Now suppose that  we observe n independent training  observations, (x1, y1), . . . , (xn , yn ).

Provide  simple  estimators   for λ1,  λ2,  π1 ,  π2,  in  terms  of the  training

observations.

(e)  Given a test  observation  X = x0 , provide an estimate  of

P (Y  = 1 | X = x0).

Your answer should be written only in terms of the n training observations (x1 , y1), . . . , (xn , yn ), and the test observation  x0, and not in terms of any unknown parameters.

  1. We collect some data for students in a statistics class, with predictors  X1   = number of lectures attended, X2  = average number of hours studied  per week, and  response Y   = receive an  A. We fit a logistic regression model,  and  get coefficient estimates  βˆ0, βˆ1, βˆ2.

(a)  Write out an expression for the probability  that  a student gets an A, as a function of the number of lectures she attended, and the average number of hours she studied  per week. Your answer should be written  in terms of X1 , X2 , βˆ0, βˆ1 , βˆ2.

(b)  Write out an expression for the minimum number of hours a student should study  per week in order to have at least an 80% chance of getting  an A. Your answer should be written  in terms of X1, X2, βˆ0, βˆ1, βˆ2.

(c)  Based  on a student’s  value  of X1   and  X2 , her  predicted  probability  of getting  an A in this  course is 60%.  If she increases her studying  by one hour per week, then  what  will be her predicted  probability  of getting  an A in this course?

  1. When the number  of features  p is large, there  tends  to be a deterioration in the  performance  of K -nearest  neighbors  (KNN)  and  other  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.  We will now investigate  this curse.

(a)  Suppose that  we have a set of observations,  each with measurements  on p = 1 feature,  X .  We assume that  X  is uniformly 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 dis- tributed 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 observations are uniformly distributed on each feature, and again each feature  ranges in value from 0 to 1.  We wish to predict  a test  observa- tion’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 KNN when p is large is that  there are very few training  observations “near” any given test  observation.

(e)  Now suppose that  we wish to make a prediction  for a test  observation  by creating  a p-dimensional hypercube  centered  around  the test  observation that  contains,  on average, 10% of the training  observations.  For p = 1, 2, and 100, what  is the length of each side of the hypercube?  Comment on your answer.

Note:  A hypercube is a generalization  of a cube to an  arbitrary number of dimensions.  When p = 1, a hypercube is simply a line segment,  when p = 2 it is a square.

  1. Pick a data set of your choice. It can be chosen from the ISLR package (but not  one of the  data  sets  explored  in the  Chapter   4 lab,  please!),  or it  can be another  data  set that  you choose.  Choose a binary  qualitative variable  in your data  set to be the  response, Y .  (By binary  qualitative  variable, I mean a qualitative variable  with K  = 2 classes.)  If your data  set doesn’t have any binary  qualitative variables,  then  you can create  one (e.g.  by dichotomizing a continuous  variable:  create  a new variable  that  equals 1 or 0 depending  on whether the continuous  variable takes on values above or below its median).  I suggest selecting a data  set with n     p.

(a)  Describe the data.  What  are the values of n and p? What  are you trying to predict,  i.e.  what  is the  meaning of Y ?  What  is the meaning of the features?

(b)  Split  the  data  into  a training  set  and  a test  set.   Perform  LDA on the training  set in order to predict Y  using the features.  What  is the training error of the model obtained?  what is the test  error?

(c)  Perform QDA on the training  set in order to predict Y  using the features.

What  is the training  error of the model obtained?  what is the test  error?

(d)  Perform logistic regression on the training  set in order to predict  Y  using the features.  What  is the training  error of the model obtained?   what  is the test  error?

(e)  Perform KNN on the training  set in order to predict Y  using the features.

What  is the training  error of the model obtained?  what is the test  error? (f )  Comment on your results.


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