# ▸ Linear Regression with Multiple Variables :

1. Suppose m=4 students have taken some classes, and the class had a midterm exam and a final exam. You have collected a dataset of their scores on the two exams, which is as follows: You’d like to use polynomial regression to predict a student’s final exam score from their midterm exam score. Concretely, suppose you want to fit a model of the form , where is the midterm score and x_2 is (midterm score)^2. Further, you plan to use both feature scaling (dividing by the “max-min”, or range, of a feature) and mean normalization.
What is the normalized feature ? (Hint: midterm = 69, final = 78 is training example 4.) Please round off your answer to two decimal places and enter in the text box below.
2. -0.47

1. You run gradient descent for 15 iterations with and compute after each iteration. You find that the value of decreases slowly and is still decreasing after 15 iterations. Based on this, which of the following conclusions seems most plausible?
• Rather than use the current value of α, it’d be more promising to try a larger value of α (say = 1.0).
• Rather than use the current value of α, it’d be more promising to try a smaller value of α (say = 0.1).
• = 0.3 is an effective choice of learning rate.

1. You run gradient descent for 15 iterations with and compute after each iteration. You find that the value of decreases quickly then levels off. Based on this, which of the following conclusions seems most plausible?
• Rather than use the current value of α, it’d be more promising to try a larger value of α (say = 1.0).
• Rather than use the current value of α, it’d be more promising to try a smaller value of α (say = 0.1).
• = 0.3 is an effective choice of learning rate.

1. Suppose you have m = 23 training examples with n = 5 features (excluding the additional all-ones feature for the intercept term, which you should add). The normal equation is . For the given values of m and n, what are the dimensions of , X, and y in this equation?
• X is 23 × 5, y is 23 × 1, θ is 5 × 5
• X is 23 × 6, y is 23 × 6, θ is 6 × 6
• X is 23 × 6, y is 23 × 1, θ is 6 × 1
• X has m rows and n+1 columns (+1 because of the term). y is m-vector. is an (n+1)-vector
• X is 23 × 5, y is 23 × 1, θ is 5 × 1

1. Suppose you have a dataset with m = 1000000 examples and n = 200000 features for each example. You want to use multivariate linear regression to fit the parameters to our data. Should you prefer gradient descent or the normal equation?
• Gradient descent, since it will always converge to the optimal θ.
• Gradient descent, since will be very slow to compute in the normal equation.
• With n = 200000 features, you will have to invert a 200001 x 200001 matrix to compute the normal equation. Inverting such a large matrix is computationally expensive, so gradient descent is a good choice.
• The normal equation, since it provides an efficient way to directly find the solution.
• The normal equation, since gradient descent might be unable to find the optimal θ.

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1. Which of the following are reasons for using feature scaling?
• It is necessary to prevent gradient descent from getting stuck in local optima.
• The cost function for linear regression has no local optima.
• It speeds up solving for θ using the normal equation.
• The magnitute of the feature values are nsignificant in terms of computational cost.
• It prevents the matrix (used in the normal equation) from being non-invertable (singular/degenerate).
• Feature scaling has nothing to do with matrix inversion.
• It speeds up gradient descent by making it require fewer iterations to get to a good solution.
• Feature scaling speeds up gradient descent by avoiding many extra iterations that are required when one or more features takes on much larger values than he rest.

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