Determine design matrix for regression, estimators for alpha

[nat23]

Ok so i have this problem I am trying to solve it reads as follows:

It is known in advance that for the response variable y, the following functional
equation holds. y = (alpha)x + (Beta)e^x
For x = 1; 2; : : : 9, the following values for y were observed:
x= 4, 5, 6, 7, 8, 9,
y=13, 20, 22, 24, 21, 10

Determine the design matrix for this regression problem, and determine the (least square)-
estimators for alpha and beta.

If someone could direct me in the right direction to get it started I'd totally appreciate
Thanks in advance :wink:

 

[tkhunny]

Deriving Normal Equations requires Partial Derivatives. Are you up to that?

Given the model $\displaystyle y = \alpha x + \beta e^{x}$

We create the function $\displaystyle I(x, y, \alpha, \beta) = \left(y - \alpha x - \beta e^{x}\right)^{2}$

Finding the two partial derivatives, $\displaystyle \frac{\partial I}{\partial \alpha}$ and $\displaystyle \frac{\partial I}{\partial \beta}$, equate these to zero (0) and rearrange to produce:

1) $\displaystyle yx = \alpha x^{2} + \beta xe^{x}$

2) $\displaystyle ye^{x} = \alpha xe^{x} + \beta \left(e^{x}\right)^{2}$

This defines your Normal Equations. Put summations in front of each term, calculate all the data, and solve the two rather simple resulting equations in $\displaystyle \alpha$ and $\displaystyle \beta$.

1) $\displaystyle \sum yx = \alpha \sum x^{2} + \beta \sum xe^{x}$

2) $\displaystyle \sum ye^{x} = \alpha \sum xe^{x} + \beta \sum \left(e^{x}\right)^{2}$

Show us what you get.

 

[nat23]

thank you - i shall be working on it and will def. post what i get..
cheers for the help