FEM21039 - Spline Regression Techniques Linear Cubic Splines Assignment

Assignment Task

Question 1. (12 pts) The file Q1.csv contains observations ( y ₁ , x ₁ ) , . . . , ( y _n , x _n ), where Y is the numerical response variable and X is the explanatory variable. Let f be the conditional expectation function of Y given X .

For parts (1), (2), (4), when you are asked to fit a spline regression model, you need to

give the explicit formula for the conditional expectation;

plot the fitted regression line overlapped with the observed data points and indicate where the knots are (similar to the figures in Lecture 6).

For part (3), you only need to plot the fitted regression lines.

(3 pts) Find the best fitting linear spline model to estimate f .

(3 pts) Find the best fitting cubic spline model to estimate f .

(3 pts) Fit three smoothing splines to estimate f with smoothing parameter λ = 10 ^− 7 ,

(3 pts) We see that the scatter plot of the data is quite flat when x > Fit a special cubic spline f ˆ (with one knot at x = 5) to estimate f such that

f ˆis linear for x > 5;

f ˆis cubic for x ≤ 5;

f ˆhas continuous second derivative at the knot x =

2 FEM21039 ASSIGNMENT

Question 2. (12 pts) The file Q2.csv contains observations ( y _i , x _i 1 , x _i 2 )’s from the joint distribution of ( Y, X ₁ , X ₂ ), where X ₁ and X ₂ are numerical variables, and Y takes value in 0 or 1. We would like to estimate

p ( x ) = p ( x ₁ , x ₂ ) = P ( Y = 1 | X ₁ = x ₁ , X ₂ = x ₂ )

through kernel density classification. We estimate the conditional pdf’s

f ₁ ( x ) = f ( x | Y = 1) , f ₀ ( x ) = f ( x | Y = 0) ,

Plot the prediction yˆ _i ’s for all the data points (x _i 1 , x _i 2 )’s in the following fashion.

• Denote a data point with a circle if y _i = 1 and a cross if y _i =
• Colour the data point with red if yˆ _i = 1, with blue if yˆ _i = 0 How many data points have you misclassified?

• (4 pts) Is this estimator equivalent to a Naive Bayes estimator? Why or why not?

(Optional, for your own amusement: repeat the estimations for pˆ and yˆ _i with a linear

logistic regression or linear discriminant analysis. Compare the results.)

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