The decision boundary of logistic regression is always a hyperplane. True O False 2. If f(x) is convex, then g(x) = f(ax) is also convex for any a E R %3D O True
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- For logistic regression, the gradient of the cost function is given by J(0) = (i) E (he (x) – y')x;). Write down mathematical expression(s) for the correct m gradient descent update for logistic regression with a learning rate of a. (In the expression, he(x^) should be replaced by the sigmoid function.)Linear regression aims to fit the parameters based on the training set T.x = 1, 2,...,m} so that the hypothesis function he (x) ...... + Onxn can better predict the output y of a new input vector x. Please derive the stochastic gradient descent update rule which can update repeatedly to minimize the least squares cost function J(0). D = {(x(i),y(¹)), i 00+ 01x₁ + 0₂x₂+... = =Consider linear regression where y is our label vector, X is our data matrix, w is our model weights and o² is a measure of variance Using the squared error cost function has a probabilistic interpretation as: O O O Maximising the probability of the model predicting the input data, assuming our input data follows a Normal distribution N(X; Xw, o²) Maximising the probability of the model predicting the input data given the weights N(X; wy, o²) Minimising the probability of the model predicting the labels, assuming our prediction errors follow a Normal distribution N(y; Xw, o²) Maximising the values of the weights to minimise the input data N (y; w, o²) Maximising the probability of the model predicting the labels, assuming our prediction errors follow a Normal distribution N(y; Xw, o²)
- Why don't we use the ordinary least square to learn a linear regression model for a classification problem (i.e., learning to fit the label of 0 or 1)? Select one: a. Linear regression cannot output value in a probability range. b. OSL will learn a bad linear regression model for a classification. c. Yes, we can. Linear regression is the same as logistic regression. d. Linear regression outputs continuous variable.Logistic regression aims to train the parameters from the training set D = {(x(i),y(i)), i 1,2,...,m, y ¤ {0,1}} so that the hypothesis function h(x) = g(0¹ x) 1 (here g(z) is the logistic or sigmod function g(z) can predict the probability of a 1+ e-z new instance x being labeled as 1. Please derive the following stochastic gradient ascent update rule for a logistic regression problem. 0j = 0j + a(y(¹) — hz(x)))x; ave. =We create a simple regression model and call the fit function as follows: Im=LinearRegression() Im.fit(X,Y) in a multilinear model we proceed in the same way: mlm=LinearRegression() mlm.fit(Z,Y) How does the linear regression model knows if we are doing a simple or multiple linear regression? Answer:
- In class, we have defined the hypothesis for univariate linear regression as he(x) = 00 + 0₁x and the mean-square cost function as 2 J(00,01) Σ (ho(x²) - y(i)) ² 2m i=1 We define the following terms: m `x), y= 10. ;Σy"), xy = - ² x(1) y(i) x²: (x(1)) ² m a. Show that the optimal values of the learning parameters 00 and 0₁ are given by: 00 = y-x xy−xy x²-x² 01 = ху-ху x²-x² b. What is the advantage of expressing the learning parameters in this format? =Linear regression aims to learn the parameters 7 from the training set D = {(f(),y(i)), i {(x(i),y(i)),i = 1,2,...,m} so that the hypothesis ho(x) = ēr i can predict the output y given an input vector š. Please derive the least mean squares and stochastic gradient descent update rule, that is to use gradient descent algorithm to update Ô so as to minimize the least squares cost function JO).Machine Learning You are given the scatter of points (x,y) = (1, 1.5), (4, 3.5), (7, 9), (10, 8). Set up an optimization problem to perform linear regression by hand along the lines of the previous problem by defining theta, etc. Then perform hand calculations to implement linear regression using gradient descent, and report the optimal value of theta that you obtain and what that means. Again, list all the steps - 1, 2,...etc. without omitting any details.
- In R, write a function that produces plots of statistical power versus sample size for simple linear regression. The function should be of the form LinRegPower(N,B,A,sd,nrep), where N is a vector/list of sample sizes, B is the true slope, A is the true intercept, sd is the true standard deviation of the residuals, and nrep is the number of simulation replicates. The function should conduct simulations and then produce a plot of statistical power versus the sample sizes in N for the hypothesis test of whether the slope is different than zero. B and A can be vectors/lists of equal length. In this case, the plot should have separate lines for each pair of A and B values (A[1] with B[1], A[2] with B[2], etc). The function should produce an informative error message if A and B are not the same length. It should also give an informative error message if N only has a single value. Demonstrate your function with some sample plots. Find some cases where power varies from close to zero to near…"When conducting a binary regression with a skewed predictor, it is often easiest to assess the need for x and log(x) by including them both in the model so that their relative contributions can be assessed directly." Show that indeed the log odds are a function of x and log(x) for the gamma distribution.Consider a linear regression setting. Given a model's weights W E RD, we incorporate regularisation into the loss function by adding an la regularisation function of the form-W;|*. Select all true statements from below. a. When q = 1, a solution to this problem tends to be sparse. I.e., most weights are driven to zero with only a few weights that are not close to zero. b. When q = 2, a solution to this problem tends to be sparse. I.e., most weights are driven to zero with only a few weights that are not close to zero. c. When q = 1, the problem can be solved analytically as in closed form. d. When q = 2, the problem can be solved analytically as in closed form.