▸ Linear Regression with One Variable :

1. Consider the problem of predicting how well a student does in her second year of college/university, given how well she did in her first year. Specifically, let x be equal to the number of “A” grades (including A-. A and A+ grades) that a student receives in their first year of college (freshmen year). We would like to predict the value of y, which we define as the number of “A” grades they get in their second year (sophomore year).
Here each row is one training example. Recall that in linear regression, our hypothesis is $\inline&space;h\theta(x)&space;=&space;\theta_0&space;+&space;\theta_1x$ to denote the number of training examples.

For the training set given above (note that this training set may also be referenced in other questions in this quiz), what is the value of $\inline&space;m$? In the box below, please enter your answer (which should be a number between 0 and 10).
4


1. Many substances that can burn (such as gasoline and alcohol) have a chemical structure based on carbon atoms; for this reason they are called hydrocarbons. A chemist wants to understand how the number of carbon atoms in a molecule affects how much energy is released when that molecule combusts (meaning that it is burned). The chemist obtains the dataset below. In the column on the right, “kJ/mol” is the unit measuring the amount of energy released.

You would like to use linear regression ($\inline&space;h_\theta(x)&space;=&space;\theta_0&space;+&space;\theta_1x$) to estimate the amount of energy released (y) as a function of the number of carbon atoms (x). Which of the following do you think will be the values you obtain for $\inline&space;\theta_0$ and $\inline&space;\theta_1$ ? You should be able to select the right answer without actually implementing linear regression.

1. For this question, assume that we are using the training set from Q1.
Recall our definition of the cost function was $\inline&space;J(\theta_0,&space;\theta_1&space;)&space;=&space;\frac{1}{2m}&space;\sum_{i=1}^{m}&space;(h&space;(x^{(i)}&space;)&space;-&space;y^{(i)})^2$

What is $\inline&space;J(0,1)$? In the box below,
please enter your answer (Simplify fractions to decimals when entering answer, and ‘.’ as the decimal delimiter e.g., 1.5).

0.5


1. Suppose we set $\inline&space;\theta_0&space;=&space;0,&space;\theta_1&space;=&space;1.5$ in the linear regression hypothesis from Q1. What is $\inline&space;h_\theta(2)$ ?
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1. Suppose we set $\inline&space;\theta_0$ = −2, $\inline&space;\theta_1$ = 0.5 in the linear regression hypothesis from Q1. What is $\inline&space;h_\theta(6)$?
1


1. Let $\inline&space;f$ be some function so that $\inline&space;f(\theta_0&space;,&space;\theta_1&space;)$ outputs a number. For this problem, $\inline&space;f$ is some arbitrary/unknown smooth function (not necessarily the cost function of linear regression, so $\inline&space;f$ may have local optima).
Suppose we use gradient descent to try to minimize $\inline&space;f(\theta_0&space;,&space;\theta_1&space;)$ as a function of $\inline&space;\theta_0$ and $\inline&space;\theta_1$.
Which of the following statements are true? (Check all that apply.)

1. In the given figure, the cost function $\inline&space;J(\theta_0,&space;\theta_1)$ has been plotted against $\inline&space;\theta_0$ and $\inline&space;\theta_1$, as shown in ‘Plot 2’. The contour plot for the same cost function is given in ‘Plot 1’. Based on the figure, choose the correct options (check all that apply).

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1. Suppose that for some linear regression problem (say, predicting housing prices as in the lecture), we have some training set, and for our training set we managed to find some $\inline&space;\theta_0,&space;\theta_1$, such that $\inline&space;J(\theta_0&space;,&space;\theta_1)&space;=&space;0$.
Which of the statements below must then be true? (Check all that apply.)

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2 comments:

1. hi Im asking you this coz this was not on your answer list. can you please give the correct answer for the following?
Let fff be some function so that

f(θ0,θ1)f(\theta_0, \theta_1)f(θ0​,θ1​) outputs a number. For this problem,

fff is some arbitrary/unknown smooth function (not necessarily the

cost function of linear regression, so fff may have local optima).

Suppose we use gradient descent to try to minimize f(θ0,θ1)f(\theta_0, \theta_1)f(θ0​,θ1​)

as a function of θ0\theta_0θ0​ and θ1\theta_1θ1​. Which of the

following statements are true? (Check all that apply.)

If theta_zero and theta_one are initialized so that theta_zero = theta_one, then by symmetry (because we do simultaneous updates to two parameters), after one iteration of the gradient descent, we will still have theta_zero = theta_one

(please give the answer whether above statement is correct (i think its wrong) but with an explanation if possible! thanks!

1. This statement is wrong.
If we initialize \theta_0 & \theta_1 to same value. After a simultaneous update it is not necessary that \theta_0 & \theta_1 will be updated to same value. (That updated value of \theta_0 & \theta_1 will depend on the slope of the curve along x_1 & x_2 respectively)

On other hand, for Neural Networks, Above statement hold true. There we have to break the symmetry. That's why we initialize all the weights randomly. But no need to do it in Gradient descent.

Thanks