LFD Book Forum - Questions 1-4: Clarification

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- - Questions 1-4: Clarification (http://book.caltech.edu/bookforum/showthread.php?t=4298)

Questions 1-4: Clarification

Sorry to ask about what may be obvious to all:

No regularization, just linear regression with transformed input variables...
So the models k=3..7 correspond to

$\phi_3 = 1 + x_1 + x_2 + x_1^2$

$\phi_4 = 1 + x_1 + x_2 + x_1^2 + x_2^2$

etc.?

Re: Questions 1-4: Clarification

Quote:

Originally Posted by Kekeli (Post 10874)

So the models k=3..7 correspond to

$\phi_3 = 1 + x_1 + x_2 + x_1^2$

$\phi_4 = 1 + x_1 + x_2 + x_1^2 + x_2^2$

etc.?

The transformed space in the case of

would be

${\bf z}=(\phi_0({\bf x}),\phi_1({\bf x}),\phi_2({\bf x}),\phi_3({\bf x}))=(1,x_1,x_2,x_1^2)$

The model (hypothesis set) would be a linear combination of these coordinates,

$h({\bf x})=\tilde{w}_0 \phi_0({\bf x})+\tilde{w}_1 \phi_1({\bf x})+\tilde{w}_2\phi_2({\bf x})+\tilde{w}_3\phi_3({\bf x})$

Re: Questions 1-4: Clarification

thank you!
to beat a dead horse:
for Q1&Q2, for each model, train with the 1st 25 points and, using the weights, eval Ein with the last 10 points, and Eout with the out.dta points...
for Q3&Q4, rinse/repeat with a different split for training and validation data

[p.s., really enjoying the course, and appreciate your time and consideration in the forum!]

Re: Questions 1-4: Clarification

Quote:

Originally Posted by Kekeli (Post 10876)

for Q1&Q2, for each model, train with the 1st 25 points and, using the weights, eval Ein with the last 10 points, and Eout with the out.dta points

Correct, except that it is $E_{\rm val}$ with the last 10 points.

Re: Questions 1-4: Clarification

Hello:

I'm having some doubts with these questions as well. For questions 1 and 2, I trained on the first 25 samples of in.dta, validated on the last 10 samples of the same data file, and finally evaluated Eout for each of the five models with the 250 samples in out.dta.

The five models are linear regression models with 4 to 7 weigths using the first 4 through all seven nonlinear transformations explained in the question. As before (with homework 6), once a linear model predicts a certain Y value, the sign operation is taken on this sum $Y=\sum_{i=1}^k w_i\cdot \phi_i(x_n)$ to see how each point is classified. The classification error is simply the ration of misclassified points in the entire test/validation data set.

I know I must be doing something wrong because when I see question 5, my Eout's are much higher than any of the options given. After all, when you have a 4 to 7 parameter model and you're only training with 10 or 25 samples, you'd expect an outrageous Eout. ... What is the mistake in my procedure?

Re: Questions 1-4: Clarification

Please disregard my previous post.

I made a stupid programming error in my code. As I was generating the predictions for each model on the test/validation data sets, I was comparing that output to the wrong "Y" variable. Once corrected, the out of sample errors of questions 2 and 4 are within the options given in question 5.

To those who may be struggling with these questions,

Remeber that in questions 1 and 2 you train with the first 25 samples of "in.dta" and validate with the last 10 samples. For questions 3 and 4 you do the opposite: train with the last 10 samples and validate with the first 25.
When you calculate the validation and test set errors, remember to compare each model's output with the correct Y's.