Linear regression aims to fit the parameters based on the training set Tx D = {(x(i),y(i)), i = 1, 2,...,m} so that the hypothesis function he (x) 00+ 01x₁ + 0₂x₂+.. + 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). ...... = =
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- GD algorithm Consider Linear Regression with single variable (univariate) problem. What will be the (approximate if can’t say accurately) values of derivatives of cost/loss function ‘J’ w.r.t. all the parameters by considering one at a time, and why? What is the significance and/or usage of these θj* for the cost function ‘J’ and hypothesis ‘h’? Given a dataset where first column is the label ‘y’ while other columns represent factors ‘xi’ as follows: X = [ 1 0 1 0 1 0 ] Using GD algorithm, find the linear model. Show all the calculationsWhy 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.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…
- 1. The Linear Discriminant Analysis method for classification was proposed by Edgar Anderson Ronald Fisher Gareth James Thomas Bayes The logit transformation, used in logistic regression, is a map FromRtoR From R to [0,1] From [0,1] to R From [0,1] to [0,1] True or false: Linear Discriminant analysis is based on the use of Bayes theorem to calculate posterior probabilities. The best classification method for a problem is the one that minimizes the training set mean square error (MSE). In general, the variance of a classification procedure increases with the flexibility of the method. In logistic regression, the regressors or explanatory variables can only be numerical variables.A Ridge Linear Regression adds the sum of the squared values of the coefficients to the loss function to penalize large coefficients. Group of answer choices True FalseSuppose you are using a Linear SVM classifier with 2 class classification problem. Now you have been given the following data in which some points are circled red that are representing support vectors. a) Draw the decision boundary of linear SVM. Give a brief explanation. b) Suppose instead of SVM, we use regularized logistic regression to learn the classifier circle the points such that removing that example from the training set and running regularized logistic regression, we would get a different decision boundary than training with regularized logistic regression on the full sample . why ?
- We are intrested in predicting the percentage of people commuting to work by walking given some input variables. Each observation corresponds to a different city and each input variable summarizes some characteristic of a given city, such as density, urban sprawl and average income per capita. This is 1. not a machine learning problem. Only social scientists would be interested in such a problem. 2. both a classification and a regression problem as it depends on the way one codes the output variable as either 0, 1 or a a particular number in the [0,1] interval. 3. a regression problem. The output variable is continuous. 4. a classification problem. Walking to work is a discrete variable and can only take two values: to walk to work and not to walk to worGiven a two-category classification problem under the univariate case, where there are two training sets (one for each category) as follows: D₁ = (-3,-1,0,4} D₂ = {-2,1,2,3,6,8} Given the test example x = 5, please answer the following questions: have and a) Assume that the likelihood function of each category has certain paramétric form. Specifically, we p(x | w₁) N, 07) p(x₂)~ N(μ₂, 02). Which category should we decide on when maximum-likelihood estimation is employed to make the prediction?(control variates) Reproduce the class example of estimating int 0 ^ 1 2 dz 1+x by the MC approach using 100 uniform random variables and after that by using a control variate with function g(U) = 1 + U as suggested in class. Compare the results.
- What is the primary objective of linear regression in machine learning? A) Maximizing accuracy of predictions.B) Minimizing the number of features.C) Minimizing the sum of squared differences between predicted and actual values. D) Maximizing the correlation coefficient between variables.Assume that your hypothesis function is of the form f(x) = w0 + w1x and that the current values of w0 and w1 are 1 and 2 respectively. Further assume that you are using a learning rate (alpha) of 0.001 What is the gradient update for w0 (only the change) associated with the point (1, 12)?Generate 100 synthetic data points (x,y) as follows: x is uniform over [0,1]10 and y = P10 i=1 i ∗ xi + 0.1 ∗ N(0,1) where N(0,1) is the standard normal distribution. Implement full gradient descent and stochastic gradient descent, and test them on linear regression over the synthetic data points. Subject: Python Programming