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I have two functions that I am trying to re-parameterize in the linear-regression context. The following is the progress I've made on each, but am hesitant about my approach and am looking for some feedback.

(i)

$$t = \begin{cases} b_1 + b_2y, & y < 2 \\ b_3 + b_4y, & y \geq 2\\ \end{cases} $$

In order to express in the form of $t = \textbf{A}\alpha(x)$ I defined the following:

$$ \textbf{A} = (b_1,b_2,b_3,b_4)\\ \alpha(x) = \begin{cases} (1,y,0,0)^T & y < 2 \\ (0,0,1,y)^T, & y \geq 2\\ \end{cases}$$

(ii) $$t = (1+b_1x_1) e^{-x_2+b_2} $$

In order to express in the form of $t = \textbf{A}\alpha(x)$ I defined the following:

$$\textbf{A} = (e^{b_2},b_1e^{b_2})\\ \alpha(x) = (\alpha_1(x),\alpha_2(x))^T\\ \text{where } \alpha_i(x) = (x_1)^{i-1}e^{-x_2} $$

As an extension to (ii), I need to formulate a function that takes $\textbf{A}$ back to $(b_1, b_2)$ and came up with the following:

$(b_1, b_2) = \begin{pmatrix} 0 & e^{-b_2}\\ b_2e^{-b_2} & 0 \end{pmatrix}\textbf{A}^T$

I am skeptical about the logic of this approach because I feel like you would need to already know the coefficients $b_i$ to define the above matrix. Some feedback here would be very helpful.

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