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