Suppose you have m data vectors of the form (x,y), where x = R is a...

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Suppose you have m data vectors of the form (x,y), where x = R is a vector of descriptive variables that determine the dependent variable y. We aim to fit a linear model that best adjusts the data. For this, we define the matrix X = Rmx (k+1) to have the vectors of descriptive variables x; as rows, and appending a column of ones to account for the bias term. The linear model is then (1) y(x) = cx, with CR+1 being the vector of parameters of the model. To get the best choice of parameters c for our data, we must minimize the difference between the predicted and expected values. For this, we define the cost function (2) FR+1 R20, C (Xc-y) (Xc- y), = " where y (1 m) is the vector of descriptive variables. 1. (0/5) Show that the cost function can be written as T F(c) cXXc-2yXc - yy. = 2. (0/10) Show that the gradient of F is (3) VF=2(XXc - Xy), so that any equilibrium point should satisfy c= = (XX)xy. -1 T Suppose you have m data vectors of the form (x,y), where x = R is a vector of descriptive variables that determine the dependent variable y. We aim to fit a linear model that best adjusts the data. For this, we define the matrix X = Rmx (k+1) to have the vectors of descriptive variables x; as rows, and appending a column of ones to account for the bias term. The linear model is then (1) y(x) = cx, with CR+1 being the vector of parameters of the model. To get the best choice of parameters c for our data, we must minimize the difference between the predicted and expected values. For this, we define the cost function (2) FR+1 R20, C (Xc-y) (Xc- y), = " where y (1 m) is the vector of descriptive variables. 1. (0/5) Show that the cost function can be written as T F(c) cXXc-2yXc - yy. = 2. (0/10) Show that the gradient of F is (3) VF=2(XXc - Xy), so that any equilibrium point should satisfy c= = (XX)xy. -1 T

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To show that the cost function Fc can be written as Fc cT XT X c 2yT X c yT y we ca... View the full answer