Suppose you are running gradient descent to fit a logistic regression model with e E R+1, Which of the following is a reasonable way to make sure the learning rate c is set properly and that gradient descent is running correctly? Plot J(0) as a function of 0 and make sure it is convex. O b. Plot J(8) as a function of 0 and make sure it is decreasing on every iteration. O* Plot /(0) = -E [y®logho(x") + (1 – y®) log (1 – ħø(x"))] a a function of the number of iterations and make sure J(0) is decreasing on every iteration. O d. Plot (8) = (he(x®) – y®)2 as a function of the number of iterations (i.e. the horizontal axis is the iteration number) and make sure J(0) is decreasing on every iteration.

You are given the true labels and the predicted probabilities from logistic regression model for N test examples. Approximately compute the AUC scores for ROC and PR curves.

This is a binary classification problem, y has two values (0 or 1), and X (feature) has three dimensions. • Use a logistic regression model to project X to y (classify X into two categories: 0 or 1). • The initialization is: w1 = 0, w2 = 0, w3 = 0, b = 0, Learning rate is 2. • You must use Gradient Descent for logistic regression in this question. • The regression should stop after one iteration. Calculation process and formulas must be included in your answer! You must answer this question by manual calculation, but not programming.

Assume the following simple regression model, Y = β0 + β1X + ε ε ∼ N(0, σ^2 ) Now run the following R-code to generate values of σ^2 = sig2, β1 = beta1 and β0 = beta0. Simulate the parameters using the following codes: Code: # Simulation ## set.seed("12345") beta0 <- rnorm(1, mean = 0, sd = 1) ## The true beta0 beta1 <- runif(n = 1, min = 1, max = 3) ## The true beta1 sig2 <- rchisq(n = 1, df = 25) ## The true value of the error variance sigmaˆ2 ## Multiple simulation will require loops ## nsample <- 10 ## Sample size n.sim <- 100 ## The number of simulations sigX <- 0.2 ## The variances of X # # Simulate the predictor variable ## X <- rnorm(nsample, mean = 0, sd = sqrt(sigX)) Q1 Fix the sample size nsample = 10 . Here, the values of X are fixed. You just need to generate ε and Y . Execute 100 simulations (i.e., n.sim = 100). For each simulation, estimate the regression coefficients (β0, β1) and the error variance (σ 2 ). Calculate the mean of...

2. Can you design a binary classification experiment with 100 total population (TP+TN+FP+ FN), with precision (TP/(TP+FP)) of 1/2, with sensitivity (TP/(TP+FN)) of 2/3, and specificity (TN/(FP+TN)) of 3/5? (Please consider the population to consist of 100 individuals.)

In R, you will use simulations to prove that the binomial distribution is correct. Recall that the binomial distribution has two parameters n and p. There are n trials and each has two possible outcomes, with probability p for "success" and 1-p for "failure". The binomial gives the probability distribution for the number of successes in n trials. You will conduct simulations with r replicates, where each simulation replicates does n simulated "coin flips". You will add up the number of successes in each coin flip, and compare the result to the true distribution: Generate n*r values from the uniform(0,1) distribution and arrange these in an rxn matrix. Each value less than p is considered a "success". For each row from part I, count the number of successes. The number of possible successes ranges from 0 to n. Use the table function in R and the value_counts function in Python and to count up the number of replicates with each number of successes. Make a table that compares the...