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Question 3: In the following marketing set, we have 9 years with the sales in 10 million euro and the advertising expenditure in million euro. Year 1 2 3 Sales Advertisement 23 36 4 5 6 34 55 65 87 108 110 7 121 8 125 9 132 a) Formulate the response vector Y, which has nine entries. 47 56 56 58 64 70 72 b) Formulate the data matrix of X, the first column should be all ones corresponding to the intercept, and the second column should be the predictors. The dimension of X should be 9 x 2. c) Write R code to compute X¹X. d) Write R code to compute 0 = (X¹X)-X¹Y. This is the estimated linear regression coefficient of the linear model with Y as the response and X as the data matrix. e)Run the linear regression using Y as the response and X as the predictor using Im command in R and compare the output with your own calculation. f) Now two additional data points arrived. They are Year 10, Sales 96, and Advertisement 53; Year 11, Sales 120, and Advertisement 63. Please use the online algorithm to update the linear model. Use the two new observations together to perform the sequential learning and update the model using stochastic gradient descent algorithm using the learning rate A = 0.0001. Note here in the updating scheme new = ĝold - AVEn, the term En will be the sum of the squared prediction errors over the two new observations: En = (y10 − 910)² + (y₁ − ŷnı)². In this example, we implement the online algorithm when the new data come in batches, not one at a time.

Question 3: In the following marketing set, we have 9 years with the sales in 10 million euro and the advertising expenditure in million euro. Year 1 2 3 Sales Advertisement 23 36 4 5 6 34 55 65 87 108 110 7 121 8 125 9 132 a) Formulate the response vector Y, which has nine entries. 47 56 56 58 64 70 72 b) Formulate the data matrix of X, the first column should be all ones corresponding to the intercept, and the second column should be the predictors. The dimension of X should be 9 x 2. c) Write R code to compute X¹X. d) Write R code to compute 0 = (X¹X)-X¹Y. This is the estimated linear regression coefficient of the linear model with Y as the response and X as the data matrix. e)Run the linear regression using Y as the response and X as the predictor using Im command in R and compare the output with your own calculation. f) Now two additional data points arrived. They are Year 10, Sales 96, and Advertisement 53; Year 11, Sales 120, and Advertisement 63. Please use the online algorithm to update the linear model. Use the two new observations together to perform the sequential learning and update the model using stochastic gradient descent algorithm using the learning rate A = 0.0001. Note here in the updating scheme new = ĝold - AVEn, the term En will be the sum of the squared prediction errors over the two new observations: En = (y10 − 910)² + (y₁ − ŷnı)². In this example, we implement the online algorithm when the new data come in batches, not one at a time.

College Algebra
College Algebra
7th Edition
ISBN:9781305115545
Author:James Stewart, Lothar Redlin, Saleem Watson
Publisher:James Stewart, Lothar Redlin, Saleem Watson
Chapter6: Matrices And Determinants
Section: Chapter Questions
Problem 6P
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Question 3: In the following marketing set, we have 9 years with the sales in 10 million euro and the advertising expenditure in million euro. Year 1 Sales Advertisement 34 23 36 2 3 4 5 6 7 8 9 a) Formulate the response vector Y, which has nine entries. 55 65 87 108 110 121 125 132 47 56 56 58 64 70 72 b) Formulate the data matrix of X, the first column should be all ones corresponding to the intercept, and the second column should be the predictors. The dimension of X should be 9 x 2. c) Write R code to compute X¹X. d) Write R code to compute = (XX)-X¹Y. This is the estimated linear regression coefficient of the linear model with Y as the response and X as the data matrix. e) Run the linear regression using Y as the response and X as the predictor using Im command in R. and compare the output with your own calculation. f) Now two additional data points arrived. They are Year 10, Sales 96, and Advertisement 53; Year 11, Sales 120, and Advertisement 63. Please use the online algorithm to update the linear model. Use the two new observations together to perform the sequential learning and update the model using stochastic gradient descent algorithm using the learning rate X = 0.0001. Note here in the updating scheme new = gold - AVEn, the term En will be the sum of the squared prediction errors over the two new observations: En (310-910)² + (y₁ - 9₁1)². In this example, we implement the online algorithm when the new data come in batches, not one at a time.
Transcribed Image Text:Question 3: In the following marketing set, we have 9 years with the sales in 10 million euro and the advertising expenditure in million euro. Year 1 Sales Advertisement 34 23 36 2 3 4 5 6 7 8 9 a) Formulate the response vector Y, which has nine entries. 55 65 87 108 110 121 125 132 47 56 56 58 64 70 72 b) Formulate the data matrix of X, the first column should be all ones corresponding to the intercept, and the second column should be the predictors. The dimension of X should be 9 x 2. c) Write R code to compute X¹X. d) Write R code to compute = (XX)-X¹Y. This is the estimated linear regression coefficient of the linear model with Y as the response and X as the data matrix. e) Run the linear regression using Y as the response and X as the predictor using Im command in R. and compare the output with your own calculation. f) Now two additional data points arrived. They are Year 10, Sales 96, and Advertisement 53; Year 11, Sales 120, and Advertisement 63. Please use the online algorithm to update the linear model. Use the two new observations together to perform the sequential learning and update the model using stochastic gradient descent algorithm using the learning rate X = 0.0001. Note here in the updating scheme new = gold - AVEn, the term En will be the sum of the squared prediction errors over the two new observations: En (310-910)² + (y₁ - 9₁1)². In this example, we implement the online algorithm when the new data come in batches, not one at a time.
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