Q.5. [2+2+1=5 Marks] Recall in logistic regression, probability of a response variable Y is calculated by...

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Q.5. [2+2+1=5 Marks] Recall in logistic regression, probability of a response variable Y is calculated by P(Y=1) 1/(1+exp(f(x)) where, x is the set of explanatory (independent) variables. Assume, f(x) = w0+w1*x. Note, a commonly used quantity, log odds is defined as loge (P(Y=1)/P(Y=0)). An experiment was designed to examine how well a chemical kills an insect, which feeds on a plant species. The explanatory variable is the concentration (in log of milligrams per liter) of the chemical. At each concentration, approximately 50 insects were exposed. Each insect was either killed or not killed. We summarize the data using the number killed. The response variable for logistic regression is the log odds of the proportion killed. Here are the data: Q.6. Concentration (log) Number of insects Number killed 0.96 50 6 1.33 48 18 1.63 46 30 2.04 49 42 2.32 50 48 A. Estimate the logistic regression parameters w0 and w1. Concentration % Killed log Odds 0.96 1.33 0.12 0.375 -0.51083 -1.99243 1.63 0.652174 0.628609 2.04 0.857143 1.791759 2.32 0.96 3.178054 w0 and w1 can be obtained by minimizing the following w.r.t. w0 and w1 L= (w0+0.96w1+1.99243)² + (w0+1.33w1+0.51083)2 + (w0+1.63w1-0.6281)² + (w0+2.04w1-1.7918)² + (w0+2.32w1-3.1780)2 Solving we obtain, w0 = -5.4664 and w1 = 3.6748 B. If concentration (in log of milligrams per liter) = 1.5, estimate the % of insects killed. Using w0 -5.4664 and w1 = 3.6748, we get % killed = 48.86% [2+2=4 Marks] Consider the following data and regression model given by output, y = w0+w1*x + w2*x². Input x 0 Output y 0 1 1 2 4 3 3 A. Assume learning rate a=0.5. At iteration t, w0=2, w1=1 and w2=0. What will be the values of w0, w1 and w2 at next iteration t+1? Input x Output y X1=X X2=x² he(x)=w0+w1x1+w2x2 0 0 0 0 2 1 1 1 1 3 2 3 2 4 4 4 3 4 16 6 w0 = 2 – 0.5/4 [2+2+1+3] = 1 w1-1-0.5/4[2*0+2*1+1*2+3*4] = -1 w2=0-0.5/4[2*0+2*1+1*4+3*16] = -6.75 B. Assume L1 regularization with regularization constant λ=1 and learning rate a=0.5. At iteration t, w0=2, w1-1 and w2=0. What will be the values of w0, w1 and w2 at next iteration t+1? Treat zero as a positive integer, if needed! Q.5. [2+2+1=5 Marks] Recall in logistic regression, probability of a response variable Y is calculated by P(Y=1) 1/(1+exp(f(x)) where, x is the set of explanatory (independent) variables. Assume, f(x) = w0+w1*x. Note, a commonly used quantity, log odds is defined as loge (P(Y=1)/P(Y=0)). An experiment was designed to examine how well a chemical kills an insect, which feeds on a plant species. The explanatory variable is the concentration (in log of milligrams per liter) of the chemical. At each concentration, approximately 50 insects were exposed. Each insect was either killed or not killed. We summarize the data using the number killed. The response variable for logistic regression is the log odds of the proportion killed. Here are the data: Q.6. Concentration (log) Number of insects Number killed 0.96 50 6 1.33 48 18 1.63 46 30 2.04 49 42 2.32 50 48 A. Estimate the logistic regression parameters w0 and w1. Concentration % Killed log Odds 0.96 1.33 0.12 0.375 -0.51083 -1.99243 1.63 0.652174 0.628609 2.04 0.857143 1.791759 2.32 0.96 3.178054 w0 and w1 can be obtained by minimizing the following w.r.t. w0 and w1 L= (w0+0.96w1+1.99243)² + (w0+1.33w1+0.51083)2 + (w0+1.63w1-0.6281)² + (w0+2.04w1-1.7918)² + (w0+2.32w1-3.1780)2 Solving we obtain, w0 = -5.4664 and w1 = 3.6748 B. If concentration (in log of milligrams per liter) = 1.5, estimate the % of insects killed. Using w0 -5.4664 and w1 = 3.6748, we get % killed = 48.86% [2+2=4 Marks] Consider the following data and regression model given by output, y = w0+w1*x + w2*x². Input x 0 Output y 0 1 1 2 4 3 3 A. Assume learning rate a=0.5. At iteration t, w0=2, w1=1 and w2=0. What will be the values of w0, w1 and w2 at next iteration t+1? Input x Output y X1=X X2=x² he(x)=w0+w1x1+w2x2 0 0 0 0 2 1 1 1 1 3 2 3 2 4 4 4 3 4 16 6 w0 = 2 – 0.5/4 [2+2+1+3] = 1 w1-1-0.5/4[2*0+2*1+1*2+3*4] = -1 w2=0-0.5/4[2*0+2*1+1*4+3*16] = -6.75 B. Assume L1 regularization with regularization constant λ=1 and learning rate a=0.5. At iteration t, w0=2, w1-1 and w2=0. What will be the values of w0, w1 and w2 at next iteration t+1? Treat zero as a positive integer, if needed!

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The L1 regularization term in the gradient descent update will be used to fix this issue A penalty t... View the full answer