Coursera Machine Learning 第二周 quiz Linear Regression with Multiple Variables 习题答案

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 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 x(4)2?
(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.

 平均值为 (7921+5184+8836+4761)/4=6675.5
Max-Min=8836-4761=4075
(4761-6675.5)/4075=-0.46957
保留两位小数为-0.47 

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?

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

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).

下降太快所以a下降速率过大，a越大下降越快，a小下降慢，而本题中，代价函数快速收敛到最小值，代表此时a最合适

Suppose you have m=14 training
examples with n=3 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 14×4, y is 14×4, θ is 4×4

X is 14×3, y is 14×1, θ is 3×3

X is 14×4, y is 14×1, θ is 4×1

X is 14×3, y is 14×1, θ is 3×1

此题注意，计算X维度时要加上X0=[1，1，1，1..,1];故答案选C

4

Suppose you have a dataset with m=50 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 it will always converge to the optimal θ.

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

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.

训练集有2W，转置的话需要大量的时间，花费太多内存与资源，故选择梯度下降。选择B

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 speeds up solving for θ using
the normal equation.

It is necessary to prevent gradient descent from getting stuck in local optima.

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

迭代次数的减少，加快了正确答案的得出。正规方程对计算只与训练集的大小有关，而与至无关，不能阻止梯度下降局部最优(ps:正规方程没有局部最优)，第四个答案，除非可以减少特征变量，否则不能解决此问题所以选A