1. Assume we have a training set and a test set as follows: Training set Index...

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1. Assume we have a training set and a test set as follows: Training set Index y Test set I Index I y 1 5.86 0.74 1 5.80 0.93 2 1.34 1.18 2 0.57 1.87 3 3.65 0.51 3 4.30 -0.06 4 4.69 -0.48 4 6.55 1.60 5 4.13 -0.07 5 0.82 1.22 6 5.87 0.37 6 3.72 0.90 7 7.91 1.35 7 5.80 0.93 8 5.57 0.30 8 3.26 1.53 9 7.30 1.64 9 6.75 1.73 10 7.89 1.75 10 4.77 -0.51 Let's try to find some linear models to fit the training data. We use the RSS objective function J(0) = ||f(x) - Y||- (a) Calculate the analytic solution of the linear model f(x) = 0 + 0₁x. Then, plot the line of your obtained linear model together with the data points in training set. (5%) (b) Suppose we want to increase the model complexity, by considering y as a linear function of both x and x²: f(x) = 0 + 0₁x+02x². In this case, calculate the analytic solution of the model and plot the curve of the model, together with the data points in training set. (5%) x(1) (hint: in this case, X = : x(1)2 θο and =₁ 1 x(10) x(10) 2 Page 1 of 5 (c) Let's further increase the model complexity, by assuming y is related to higher-order forms of x, ie., fe(x)=80+0₁×+ 0²x² +03x² + 84x4. Again, calculate the analytic solution of the model, plot the curve of the function, together with the data points in training set. (5%) 1 (hint: in this case, X = : x(1) (2 x(1)² X(1)³ X(1)⭑ 100\ 01 1 x (10) x(10) 2 x(10)4 and = 82) x(10) 83 04 (d) Observe the above three functions, please point out which could be faced with underfitting, which could be faced with overfitting, and which one is relatively a good one? Then, you can calculate the values of prediction error on the test data to verify your thoughts. (10%) 1. Assume we have a training set and a test set as follows: Training set Index y Test set I Index I y 1 5.86 0.74 1 5.80 0.93 2 1.34 1.18 2 0.57 1.87 3 3.65 0.51 3 4.30 -0.06 4 4.69 -0.48 4 6.55 1.60 5 4.13 -0.07 5 0.82 1.22 6 5.87 0.37 6 3.72 0.90 7 7.91 1.35 7 5.80 0.93 8 5.57 0.30 8 3.26 1.53 9 7.30 1.64 9 6.75 1.73 10 7.89 1.75 10 4.77 -0.51 Let's try to find some linear models to fit the training data. We use the RSS objective function J(0) = ||f(x) - Y||- (a) Calculate the analytic solution of the linear model f(x) = 0 + 0₁x. Then, plot the line of your obtained linear model together with the data points in training set. (5%) (b) Suppose we want to increase the model complexity, by considering y as a linear function of both x and x²: f(x) = 0 + 0₁x+02x². In this case, calculate the analytic solution of the model and plot the curve of the model, together with the data points in training set. (5%) x(1) (hint: in this case, X = : x(1)2 θο and =₁ 1 x(10) x(10) 2 Page 1 of 5 (c) Let's further increase the model complexity, by assuming y is related to higher-order forms of x, ie., fe(x)=80+0₁×+ 0²x² +03x² + 84x4. Again, calculate the analytic solution of the model, plot the curve of the function, together with the data points in training set. (5%) 1 (hint: in this case, X = : x(1) (2 x(1)² X(1)³ X(1)⭑ 100\ 01 1 x (10) x(10) 2 x(10)4 and = 82) x(10) 83 04 (d) Observe the above three functions, please point out which could be faced with underfitting, which could be faced with overfitting, and which one is relatively a good one? Then, you can calculate the values of prediction error on the test data to verify your thoughts. (10%)

Answer rating: 100% (QA)

import numpy as np Test set data Xtest nparray580 057 430 655 082 372 580 326 675 477 Ytest ... View the full answer