Cousera-stanford-机器学习练习-第二周-Linear Regression with Multiple Variables

Linear Regression with Multiple Variables

5 试题

1。

Suppose m=4 students have taken some class, 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:

midterm exam    (midterm exam)2    final exam
89    7921    96
72    5184    74
94    8836    87
69    4761    78

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 hθ(x)=θ0+θ1x1+θ2x2, where x1 is the midterm score and x2 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 x2(2)? (Hint: midterm = 72, final = 74 is training example 2.) Please round off your answer to two decimal places and enter in the text box below.

解：归一化公式为x=(xn-μ）/s 其中μ是平均值,s是值的范围(max-min)。可以算出答案为（5184-（7921+5184+8836+4761）/4）/(8836-4761)=-0.37

2。

You run gradient descent for 15 iterations with α=0.3 and compute J(θ) after each iteration. You find that the value of J(θ) decreases quickly then levels off. Based on this, which of the following conclusions seems most plausible?

α=0.3 is an effective choice of learning rate.

Rather than use the current value of α, it'd be more promising to try a smaller value of α (say α=0.1).

Rather than use the current value of α, it'd be more promising to try a larger value of α (say α=1.0).

解：一个英文翻译注意点：rather than表示而不是，后接被否定的项。本题中梯度下降法可以快速下降到最小值，所以此时的α是一个合适的值。

3。

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 θ=(XTX)−1XTy. For the given values of m and n, what are the dimensions of θ, X, and y in this equation?

X is 23×5, y is 23×1, θ is 5×5

X is 23×6, y is 23×1, θ is 6×1

X is 23×5, y is 23×1, θ is 5×1

X is 23×6, y is 23×6, θ is 6×6

解：首先可确定y是一列，排除D，x是n+1列，θ是n+1行，故选B。

4。

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 θ to our data. Should you prefer gradient descent or the normal equation?

Gradient descent, since (XTX)−1 will be very slow to compute in the normal equation.

The normal equation, since gradient descent might be unable to find the optimal θ.

Gradient descent, since it will always converge to the optimal θ.

The normal equation, since it provides an efficient way to directly find the solution.

解：当数量大于10000时一般选择梯度下降法，因为正规方程时间复杂度约为O（n^3）。

5。

Which of the following are reasons for using feature scaling?

It speeds up gradient descent by making it require fewer iterations to get to a good solution.

It is necessary to prevent the normal equation from getting stuck in local optima.

It speeds up gradient descent by making each iteration of gradient descent less expensive to compute.

It prevents the matrix XTX (used in the normal equation) from being non-invertable (singular/degenerate).

解：这题是多选题，但实际上只有一个选项正确。特征缩放将各种数值全归一为一个数量级大小的值。它对梯度下降法的帮助是：减少迭代次数，从而加速程序。正如视频中所说，正规方程不需要特征缩放，因此有关正规方程的两个选项都不选。