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.)
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(i)
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- 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…HWK 7: Regression 1. For the questions below answer first by techniques that do NOT use the R linear model function (lm) and then compare to what is gotten from the Im() function. You may use R for the calculations not including Im but show how you do them. X = [-2, 1, 2, -1, 0] Y = [25, 18, 13, 23, 16] %D a) Find the slope and intercept of the line b) Find the residuals, the average of the absolute value of the residuals, and the standard deviation of the residuals using the appropriate number of degrees of freedom. c) Calculate the correlation coefficient d) Calculate the F-value and explain what it meansQuestion 3. Regression need answer of part b Consider real-valued variables X and Y. The Y variable is generated, conditional on X, from the fol- lowing process: E~N(0,0²) YaX+e where every e is an independent variable, called a noise term, which is drawn from a Gaussian distri- bution with mean 0, and standard deviation σ. This is a one-feature linear regression model, where a is the only weight parameter. The conditional probability of Y has distribution p(YX, a) ~ N(aX, 0²), so it can be written as p(YX,a) = exp(- (-202 (Y-ax)²) 1 ν2πσ The following questions are all about this model. MLE estimation (a) Assume we have a training dataset of n pairs (X, Y) for i = 1..n, and σ is known. Which ones of the following equations correctly represent the maximum likelihood problem for estimating a? Say yes or no to each one. More than one of them should have the answer "yes." a 1 [Solution: no] arg max > 2πσ 1 [Solution: yes] arg max II a [Solution: no] arg max a [Solution: yes] arg max a 1…
- Solve In R programmning language: Calculate the probability for each of the following events: (a) A standard normally distributed variable is less than -2.5. (b) A normally distributed variable with mean 35 and standard deviation 6 is larger than 42 but less than 45. (c) A normally distributed variable with mean 35 and standard deviation 6 is larger than 40 but less than 41. (d) X < 0.9 when X has the standard uniform distribution (min=0, max=1). (e) 1 < X < 3 in the exp distribution with rate λ = 2.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.Mary: "Before we run the multivariate linear regression, feature scaling should be performed." Give one reason to support Mary's idea. Moreover, should we perform feature scaling before or after the gradient descent?
- 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.Fit an AR(2) model to the cardiovascular mortality series (cmort) discussed in Example 2.2. using linear regression and using Yule-Walker. (a) Compare the parameter estimates obtained by the two methods. (b) Compare the estimated standard errors of the coefficients obtained by linear regression with their corresponding asymptotic approximations, as given in Property 3.10.