Manually train a hypothesis function h(x) = g(Ō¹x) based on the following training instances using stochastic gradient ascent rule. The initial values of parameters are 0o = 0.1,0₁ = 0.1,0₂ = 0.1. The learning rate a is 0.1. Please update each parameter at least five times. X1 0 0 1 1 X2 0 1 0 1 y 1 1 0 0
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- please try to simulate the probability of rolling a Die with Sample Space* S={1,2,3,4,5,6} and the probability of each sample point has a 1/6 chance of occurring, i.e., you need to verify that your simulation converges to 1/6 when you select one point of sample space. When X is a random variable for sample point of rolling a Die, Pr(X<=4)=2/3. Please verify this result by simulation. Please let me know how to make an Excel file as stated above.The task is to implement density estimation using the K-NN method. Obtain an iidsample of N ≥ 1 points from a univariate normal (Gaussian) distribution (let us callthe random variable X) centered at 1 and with variance 2. Now, empirically obtain anestimate of the density from the sample points using the K-NN method, for any valueof K, where 1 ≤ K ≤ N. Produce one plot for each of the following cases (each plotshould show the following three items: the N data points (instances or realizations ofX) and the true and estimated densities versus x for a large number – e.g., 1000, 10000– of discrete, linearly-spaced x values): (i) K = N = 1, (ii) K = 2, N = 10, (iii) K = 10,N = 10, (iv) K = 10, N= 1000, (v) K = 100, N= 1000, (vi) K = N = 50,000. Pleaseprovide appropriate axis labels and legends. Thus there should be a total of six figures(plots),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. =
- Considering the problem of performing rapid COVID test among a large group of people. Assuming there are exactly 1 positive case among N people, and you are asked to find the positive case with the least number tests. 1. A straight forward approach is to test every single person, which requires N tests in total. Can you design a testing framework that works better than the greedy one under, even under the worst-case scenario? Write the pseudo code for your approach and briefly explain the time complexity of your approach. 2. Based on your framework, if there are 1,000 people, what are the maximum number of tests that need to be performed in order to identify the positive case (number of tests required in the worst case)?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…Implement a simple linear regression model using Python without using any machine learning libraries like scikit-learn. Your model should take a dataset of input features X and corresponding target values y, and it should output the coefficients w and b for the linear equation y =wX + b
- implement Bresenham line drawing algorithm:For Slope |m|<1:Either value of x is increasedOR both x and y is increased using decision parameter.A machine learning model is learning parameters w₁ and w₂. It turns out that for the latest data inputted, the error function is: E=(w₁-1)²+(W₂-1)² The current values of the parameters are w₁-2 and w₂=0.9. Gradient descent is applied to update the parameters. Which statement is true? 7 0 w1 2 W2 3 4 0 W2 W1 decreases and decreases, change in w₁ is bigger. W₁ decreases and w₂ increases, change in w₂ is bigger. W₁ decreases and w₂ increases, change in w₁ is bigger. W₁ decreases and w₂ increases, both by the same amount. 18 16 14 12 10 8 6 4 HH +++ 2This is a Machine Learning question : Given a dataset, (1,+), (7, - ), (2, +), (6, -), (5, +), (9, -), (11, +) You are supposed to find a threshold function that minimizes the error in the given dataset. Threshold functions look like this: f( x | a,b ) = sign(a-x).b where a is a real number and b is in {1,-1}. How many possible values should you consider to solve this problem? What is the value of a and b in the case that minimizes the error? What is the minimum error? Show how you found it. Note: Please first sort the dataset, and use the mid-point between two adjacent points as the threshold value to identify the misclassified points. Explain in detail how you have identified the misclassified points using the threshold value and the values of b.
- This is a Machine Learning question : Given a dataset, (1,+), (7, - ), (2, +), (6, -), (5, +), (9, -), (11, +) You are supposed to find a threshold function that minimizes the error in the given dataset. Threshold functions look like this: f( x | a,b ) = sign(a-x).b where a is a real number and b is in {1,-1}. How many possible values should you consider to solve this problem? What is the value of a and b in the case that minimizes the error? What is the minimum error? Show how you found it.We all know that when the temperature of a metal increases, it begins to expand. So, we experimented with exposing a metal rod to different temperatures and recorded its length as follows: 20 30 50 55 60 Temp Length 0.5 25 35 40 45 65 1.8 6.2 6.5 7.8 9.4 9.8 10.9 Required: 1. Implement and plot a simple linear regression for the above data, where the temperature is “x", and the length is “y"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