Machine Vision Lab 4-Multivariate Linear Regression Solved

Question 1 – Multiple Linear Regression Models

In the practical lab folder you will find a data file called boston.csv. This is a well-known regression dataset.

The following is a description of each of the features:

1. CRIM 2. ZN
3. INDUS 4. CHAS 5. NOX 6. RM

per capita crime rate by town
proportion of residential land zoned for lots over 25,000 sq.ft.

proportion of non-retail business acres per town
Charles River dummy variable (= 1 if tract bounds river; 0 otherwise)

nitric oxides concentration (parts per 10 million) average number of rooms per dwelling
proportion of owner-occupied units built prior to 1940

7. AGE
8. DIS
9. RAD
10. TAX
11. PTRATIO pupil-teacher ratio by town
12. B 1000(Bk – 0.63)^2 where Bk is the proportion of black population 13. LSTAT % lower status of the population

weighted distances to five Boston employment centres index of accessibility to radial highways

full-value property-tax rate per $10,000

14. MEDV Median value of owner-occupied homes in $1000’s Instructions:

1. 1)  The objective of this question is to build and assess the performance of a MLR model for the Boston house prices dataset. You will find a template for your code in the Canvas folder. In the code you will notice that rather than having a single coefficient value (as was the case in linear regression), we design to code so that it can scale for multiple coefficients. One for each feature we read in from the dataset.
2. 2)  You should complete the hypothesis method. It takes in the following arguments:

a. A 2D NumPy array containing all the feature data

2. A array containing each of the coefficients for each feature in our dataset
3. The bias (a single numerical value)

The objective of the function is to calculate the predicted output for each training example. In this simple implementation you can use a single for loop. Remember your hypothesis for an MLR is:

h(x)=𝜆𝜆𝑥𝑥+𝜆𝜆𝑥𝑥+𝜆𝜆𝑥𝑥 +…..+𝜆𝜆𝑥𝑥+𝑏𝑏 11 22 33 𝑛𝑛𝑛𝑛

3) Complete the gradient calculations in the gradient_descent function. The function calculates the gradient for the bias and the updated bias value. You should also update each of the coefficients using the same gradient descent update rule. Calculate the gradient and use this to update the coefficient. You will need one for loop for iterating with gradient descent and in this basic implementation you can use a second (inner) for loop for updating the value of each of the weights (coefficients). Once complete you should now be able to run your MLR algorithm.

When you train the model on the Boston dataset you should be obtaining an R2 performance (on the training set) of approximately 0.73. In this case we only look at the performance of the model on the training set. In the next question you should adopt your code to assess it’s performance on a test set.

Question 2 – Performance on Regression Data with Test Set

I have provided another dataset in a .zip file called Dataset.zip that contains both a train and test file.

Run your MLR (developed in Q1) on the training set to build your model.

Adjust your existing code to calculate the R2 accuracy of the model on the test set (you will need to complete the function calculateTestR2). You should be able to obtain a very high R2 >0.98.