https://www.coursera.org/learn/machine-learning/exam/7pytE/linear-regression-with-multiple-variables
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.
答案： -0.37
平均值 ：（7921+5184+8836+4761）/4 = 6675.5
Max-Min： 8836-4761=4075
x=(xn-平均值）/（Max-Min）
training example 2 （5184-6675.5）/4075=-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) increases over
time. 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 larger value of α (say α=1.0).
Rather than use the current value of α, it'd be more promising to try a smaller value of α (say α=0.1).
答案：B. Rather than use the current value of α, it'd be more promising to try a larger value of α (say α=1.0).
a越大下降越快，a越小下降越慢。
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 23×6, y is 23×6, θ is 6×6
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
答案：C. X is 23×6, y is 23×1, θ is 6×1
X n+1 列 ， y 1 列 , θ n+1 行
4。
Suppose you have a dataset with m=50 examples and n=15 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 it will always converge to the optimal θ.
The normal equation, since it provides an efficient way to directly find the solution.
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.
答案： B. The normal equation, since it provides an efficient way to directly find the solution.
比較梯度下降與normal equation
梯度下降需要Feature Scaling；normal equation 簡單方便不需Feature Scaling。
normal equation 時間復雜度較大，適用於Feature數量較少的情況。
5。
Which of the following are reasons for using feature scaling?
It speeds up solving for θ using the normal equation.
It prevents the matrix XTX (used in the normal equation) from being non-invertable (singular/degenerate).
It speeds up gradient descent by making it require fewer iterations to get to a good solution.
It is necessary to prevent gradient descent from getting stuck in local optima.
答案 ：C. It speeds up gradient descent by making it require fewer iterations to get to a good solution.
上一題也考到這個點：normal equation 不需要 Feature Scaling，排除AB， 特征縮放減少迭代數量，加快梯度下降，然而不能防止梯度下降陷入局部最優。