Data Mining Assignment 1

Description

• Finding the most specific/general hypothesis [Warm-UP]

Create a dataset D = (x_i, y_i)^N_i=1 where each x_i is given in two dimension x_i = (x¹_i, x²_i) and each y_i is a binary label i.e. y_i = {0, 1}. First generate the vector Y where each example will

take y_i = 1 with probability 1/2 and y_i = 0 with probability 1/2 with N = 30. Now fixing

the y_i, sample the X matrix as follows: [20 Marks]

• If y_i = 1 then x¹_i ∼ U(2, 7) and x²_i ∼ U(4, 6). where U(a, b) represent the uniform distribution between a, b.

• If y_i = 0 then x¹_i ∼ U(0, 2) ∪ U(7, 9) and x²_i ∼ U(1, 3) ∪ U(6, 8).

Implement the following with respect to the above dataset:

1. Color code the examples with y_i = 1 as red and y_i = 0 as green and plot the dataset.

2. Write a program to find most specific and most general hypothesis when hypothesis class is considered as all possible Rectangles. Plot both the obtained hypothesis along with the dataset.

3. Write a program to find most specific and most general hypothesis when hypothesis class is considered as all possible Circles. Plot both the hypothesis obtained along with the dataset.

4. Mention any observations corresponding to second and third points.

• Polynomial Regression in One Dimension [Easy]

In this question, we will repeat the experiments discussed in the class with respect to poly-

nomial regression but with a different function. [30 Marks]

1. Generate 20 data points from function f(x) = cos(2πx) + ₂^x_π + noise where noise∼ N (0, 0.004) with x ranging from 0 to 2π.

2. Fit a polynomial regression with optimal weight vector w∗ and plot the curves for dif-ferent degree of polynomials M = 1, 2, 3, 5, 7, 10. Explain your observations by plotting the data points generated and the curve obtained for different values of M.

3. Repeat the previous experiments with more number of data points and report your findings.

• Linear Ridge Regression in Multiple Dimension [Medium]

We will be understanding the concept of linear regression along with the regularization parameter on the dataset given below which has multiple attributes. Medical Dataset (https://www.kaggle.com/sudhirnl7/linear-regression-tutorial/data): The medical cost dataset comprises of independent attributes like age, sex, BMI (body mass index), children, smoker, and region. The charge/cost is a dependent feature. Our goal is to predict the individual

medical costs billed by the health insurance. [50 Marks]

1. Feature Normalization: As discussed in the class, we first have to standardize all the features by subtracting with the mean and dividing by the standard deviation. Verify your technique by computing the mean and variance of the transformed data and check if the mean is 0 and variance is 1.

2. K−Fold Cross Validation Randomly partition the data into a training, validation, and test set. Fix 20% of the instances into the test set. For the remaining data perform the below experiments with K−fold cross validation. You can take the value of K to be 10.

3. Ridge-Regression: Here, implement your own function ridgereg(X, Y, λ) that calcu-lates the linear least square solution with the ridge regression penalty parameter λ and return regression weights. Use gradient descent technique to find these weights. Imple-ment predridgereg(X, weights) that returns Y given the input X with learnt weights.

4. Plot the mean square error for each of the dataset obtained from K−fold cross validation with respect to different λ values. Explain your finding and suggest what value of λ will you choose based on the obtained plot.

5. Plot the training error, variance and test error against different values of λ. Explain your finding and suggest what value of λ will you choose based on the obtained plot. Explain your result in the context of bias variance trade off. Does this value coincide with the previous question?

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