Description
Problem 1 (10 code + 10 report = 20 points) Experiment with
Gaussian discriminators
Implement Linear Discriminant Analysis (LDA) and Quadratic Discriminant Analysis (QDA). Refer
to part B.3 slides and handouts. Implement two functions in Python: ldaLearn and qdaLearn which take
a training data set (a feature matrix and labels) and return the means and covariance matrix (or matrices).
Implement two functions ldaTest and qdaTest which return the true labels for a given test data set and
the accuracy using the true labels for the test data. The format of arguments and the outputs is provided
in the base code.
REPORT 1.
Train both methods using the sample training data (sample train). Report the accuracy of LDA and QDA
on the provided test data set (sample test). Also, plot the discriminating boundary for linear and quadratic
discriminators. The code to plot the boundaries is already provided in the base code. Explain why there is
a difference in the two boundaries.
Problem 2 (5 code + 5 report = 10 Points): Experiment with
Linear Regression
Implement ordinary least squares method to estimate regression parameters by minimizing the squared loss.
J(w) = 1
2
X
N
i=1
(yi − w>xi)
2
(1)
In matrix-vector notation, the loss function can be written as:
J(w) = 1
2
(y − Xw)
>(y − Xw) (2)
where X is the input data matrix, y is the target vector, and w is the weight vector for regression.
Note that this is same as maximizing the log-likelihood in the Bayesian setting. You need to implement
the function learnOLERegression. Also implement the function testOLERegression to apply the learnt
weights for prediction on both training and testing data and to calculate the root mean squared error
(RMSE):
RMSE =
vuut
1
N
X
N
i=1
(yi − w>xi)
2 (3)
REPORT 2.
Calculate and report the RMSE for training and test data for two cases: first, without using an intercept
(or bias) term, and second with using an intercept. Which one is better?
Problem 3 (10 code + 10 report = 20 Points): Experiment with
Ridge Regression
Implement parameter estimation for ridge regression by minimizing the regularized squared loss as follows:
J(w) = 1
2
X
N
i=1
(yi − w>xi)
2 +
1
2
λw>w (4)
In matrix-vector notation, the squared loss can be written as:
J(w) = 1
2
(y − Xw)
>(y − Xw) + 1
2
λw>w (5)
You need to implement it in the function learnRidgeRegression.
2
REPORT 3.
Calculate and report the RMSE for training and test data using ridge regression parameters using the the
testOLERegression function that you implemented in Problem 2. Use data with intercept. Plot the errors
on train and test data for different values of λ. Vary λ from 0 (no regularization) to 1 in steps of 0.01.
Compare the relative magnitudes of weights learnt using OLE (Problem 1) and weights learnt using ridge
regression. Compare the two approaches in terms of errors on train and test data. What is the optimal value
for λ and why?
Problem 4 (20 code + 5 report = 25 Points): Using Gradient Descent for Ridge Regression Learning
As discussed in class, regression parameters can be calculated directly using analytical expressions (as in
Problem 2 and 3). To avoid computation of (X>X)
−1
, another option is to use gradient descent to minimize
the loss function (or to maximize the log-likelihood) function. In this problem, you have to implement the
gradient descent procedure for estimating the weights w.
You need to use the minimize function (from the scipy library) which is same as the minimizer that you
used for first assignment. You need to implement a function regressionObjVal to compute the regularized
squared error (See (4)) and its gradient with respect to w. In the main script, this objective function will
be used within the minimizer.
REPORT 4.
Plot the errors on train and test data obtained by using the gradient descent based learning by varying the
regularization parameter λ. Compare with the results obtained in Problem 3.
Problem 5 (10 code + 5 report = 15 Points): Non-linear Regression
In this problem we will investigate the impact of using higher order polynomials for the input features. For
this problem use the third variable as the only input variable:
x t r a i n = x t r a i n [ : , 3 ]
x t e s t = x t e s t [ : , 3 ]
Implement the function mapNonLinear.m which converts a single attribute x into a vector of p attributes,
1, x, x2
, . . . , xp
.
REPORT 5.
Using the λ = 0 and the optimal value of λ found in Problem 3, train ridge regression weights using the
non-linear mapping of the data. Vary p from 0 to 6. Note that p = 0 means using a horizontal line as the
regression line, p = 1 is the same as linear ridge regression. Compute the errors on train and test data.
Compare the results for both values of λ. What is the optimal value of d in terms of test error in each
setting? Plot the curve for the optimal value of p for both values of λ and compare.
Problem 6 (0 code + 10 report = 10 points) Interpreting Results
Using the results obtained for previous 4 problems, make final recommendations for anyone using regression
for predicting diabetes level using the input features.
REPORT 6.
Compare the various approaches in terms of training and testing error. What metric should be used to
choose the best setting?
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