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.
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- 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 worQuestion 45. For what ultimate purposes may algorithms like Nelder-Mead, Newton-Raphson or gradient-descentbe used for?a) To find the minimum of a function.b) To find all zeros of a function.c) To evaluate the derivative of a function.d) To solve a generalised regression probleGenerate 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
- Given 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?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 calculationsIn 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…
- Please don't use handwritting for this question How would you modify the dynamic programming algorithm for the coin collecting problem if some cells on the board are inaccessible for the robot? Apply your algorithm to the board below, where the inaccessible cells are shown by X’s. How many optimal paths are there for this board? You need to provide 1) a modified recurrence relation, 2) a pseudo code description of the algorithm, and 3) a table that stores solutions to the subproblems.Suppose 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 ?1. Describe how linear regression can be used on the exponential function in a meaningful way noting that it is not a linear function. 2. Describe two different aspects between closed (bracketed) and open root-finding methods.
- In random forest construction, how many decision tress we need to use to get a good result? How can we use random forest algorithm for regression problem?Explain the role of residual (error) in gradient boosting algorithm? Do not give long answers. Just be to the pointCan you please follow up on the question and answer the second part: 'Propose a transformation of this likelihood function whose maximum is the same and can be computed easily'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).