$[$Prompt $1]$ For Questions $1-7,$ we apply tools for linear regression to fit a quadratic polynomial to a given dataset. Recall that

hat$()=(M_x^T{M}_x{)}^{-1}{M}_x^{T}y,$ where hat$()=\left[\begin{array}{c}\text{h}\text{a}\text{t}({}_0)\\ \text{h}\text{a}\text{t}({}_1)\\ \text{h}\text{a}\text{t}({}_2)\end{array}\right]$

In what follows, given a dataset $D_{Tr}=\{(x_i,y_i){\}}_i=1^4=\{(1,2),(2,5),(3,13),(4,20)\},$ we will take steps to find the coefficients of a quadratic mode hat$(Y)=$hat$({}_0)+$hat$({}_1)x+$hat$({}_2)x^2.$

Step $1$: What is $M_x^T?$ Choose one of the options below.

$\left[\begin{array}{cccccccccc}1& 1& 1& 1& 4& 9& 16& 2& 3& 4\end{array}\right]$

$1$

$1$

$1$

$\left[\begin{array}{cccccccccc}1& 1& 1& 1& 2& 3& 4& 4& 9& 16\end{array}\right]$

$1$

$1$

$\left[\begin{array}{ccccccccc}1& 1& 1& 2& 4& 3& 9& 4& 16\end{array}\right]$

$1$

$1$

$1$

$\left[\begin{array}{ccccccccc}1& 1& 1& 4& 2& 9& 3& 16& 4\end{array}\right]$

$1$

$1$

$1$

$[$Refer to Prompt $1]$ What is $M_x^T{M}_x?$

[12344916111

0001]

$0$

$1$

$0$

$1$

$1$

$\left[\begin{array}{ccccccc}4& 10& 30& 30& 100& 100& 354\end{array}\right]$

$30$

$10$

$\left[\begin{array}{ccccccc}4& 30& 10& 354& 100& 100& 30\end{array}\right]$

$10$

$30$

$\left[\begin{array}{ccccccc}30& 15& 16& 30& 10& 1& 30\end{array}\right]$

$1$

$10$

$[$Refer to Prompt $1]$ What is $(M_x^T{M}_x{)}^{\{-1\}}?$

Hint: Because this is not a $22$ matrix, try using numpy in Python on your own Jupyter notebook $($see starter code below$)$:

$1$ import numpy $asnp$

$2$ help$(np.$array$)$

$3$ help$(np.$transpose$)$

$4$ help$(np.$linalg.inv$)$

import numpy $asnp$

help$(np.$array$)$

help$(np.$transpose$)$

help$(np.$linalg.inv$)$

$\left[\begin{array}{ccccccc}7\mathrm{.}75& -6\mathrm{.}75& 1\mathrm{.}25& 6\mathrm{.}45& -1\mathrm{.}25& -1\mathrm{.}25& 0\mathrm{.}25\end{array}\right]$

$1\mathrm{.}25$

$1\mathrm{.}25$

$-6\mathrm{.}75$

$\left[\begin{array}{ccccccc}7\mathrm{.}75& 1\mathrm{.}25& -6\mathrm{.}75& 0\mathrm{.}25& -1\mathrm{.}25& -1\mathrm{.}25& 6\mathrm{.}45\end{array}\right]$

$-6\mathrm{.}75$

$1\mathrm{.}25$

$\left[\begin{array}{ccccccc}-6\mathrm{.}75& -1\mathrm{.}25& 0\mathrm{.}25& 1\mathrm{.}25& -6\mathrm{.}25& -1\mathrm{.}25& 1\mathrm{.}45\end{array}\right]$

$-6\mathrm{.}75$

$0\mathrm{.}25$

$\left[\begin{array}{ccccccc}7\mathrm{.}75& 1\mathrm{.}25& -6\mathrm{.}75& 1\mathrm{.}25& -6\mathrm{.}25& -1\mathrm{.}25& 1\mathrm{.}45\end{array}\right]$

$-6\mathrm{.}75$

$0\mathrm{.}25$

$[$Refer to Prompt $1]$ What is $M_x^{T}y?$

$\left[\begin{array}{c}459\end{array}\right]$

$40$

$131$

$\left[\begin{array}{c}131\end{array}\right]$

$40$

$459$

$\left[\begin{array}{c}40\end{array}\right]$

$459$

$131$

$\left[\begin{array}{c}40\end{array}\right]$

$131$

$459$

$[$Refer to Prompt $1]$ What is hat$({)}_0?($Answer as a real number with one decimal point.$)$

Type your answer...

$1$ point

$[$Refer to Prompt $1]$ What is hat$({)}_1?($Answer as a real number with one decimal point.$)$

Type your answer...

$1$ point

$[$Refer to Prompt $1]$ What is hat$({)}_2?($Answer as a real number with one decimal point.$)$

$1$ point

What would be your prediction hat$(Y)$ for $x=2?$

$5$

$2$

$5\mathrm{.}9$

$25$

hat$()=(M_x^T{M}_x{)}^{-1}{M}_x^{T}y,$ where hat$()=\left[\begin{array}{c}\text{h}\text{a}\text{t}({)}_0\\ \text{h}\text{a}\text{t}({)}_1\\ \text{h}\text{a}\text{t}({}_2)\end{array}\right]$

In what follows, given a dataset $D_{Tr}=\{(x_i,y_i){\}}_i=1^4=\{(1,2),(2,5),(3,13),(4,20)\},$ we will take steps to find the coefficients of a quadratic mode hat$(Y)=$hat$({}_0)+$hat$({}_1)x+$hat$({}_2)x^2.$

Step $1$: What is $M_x^T?$ Choose one of the options below.

$\left[\begin{array}{cccccccccc}1& 1& 1& 1& 4& 9& 16& 2& 3& 4\end{array}\right]$

$1$

$1$

$1$

$\left[\begin{array}{cccccccccc}1& 1& 1& 1& 2& 3& 4& 4& 9& 16\end{array}\right]$

$1$

$1$

$\left[\begin{array}{ccccccccc}1& 1& 1& 2& 4& 3& 9& 4& 16\end{array}\right]$

$1$

$1$

$1$

$\left[\begin{array}{ccccccccc}1& 1& 1& 4& 2& 9& 3& 16& 4\end{array}\right]$

$1$

$1$

$1$

$[$Refer to Prompt $1]$ What is $M_x^T{M}_x?$

[12344916111

0001]

$0$

$1$

$0$

$1$

$1$

$\left[\begin{array}{ccccccc}4& 10& 30& 30& 100& 100& 354\end{array}\right]$

$30$

$10$

$\left[\begin{array}{ccccccc}4& 30& 10& 354& 100& 100& 30\end{array}\right]$

$10$

$30$

$\left[\begin{array}{ccccccc}30& 15& 16& 30& 10& 1& 30\end{array}\right]$

$1$

$10$

$[$Refer to Prompt $1]$ What is $(M_x^T{M}_x{)}^{\{-1\}}?$

Hint: Because this is not a $22$ matrix, try using numpy in Python on your own Jupyter notebook $($see starter code below$)$:

$])$:$\}$

$1\mathrm{.}25$

$-6\mathrm{.}75$

$\left[\begin{array}{ccccccc}7\mathrm{.}75& 1\mathrm{.}25& -6\mathrm{.}75& 0\mathrm{.}25& -1\mathrm{.}25& -1\mathrm{.}25& 6\mathrm{.}45\end{array}\right]$

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