Least Squares Estimation for two parameters

[Dixanadu]

Homework Statement

Hi guys,

so the problem is as follows:

A set of n independent measurements $y_{i}, i=1...n$ are treated as Gaussian, each with standard deviations $\sigma_{i}$. Each measurement corresponds to a value of a control variable $x_{i}$. The expectation value of $y$ is given by
$f(x;\alpha,\beta)=\alpha x +\beta x^{2}$.

1) Find the log-likelihood function for the parameters $\alpha,\beta$.

2) Show that the least-squares estimators for $\alpha,\beta$ can be found from the solution of a system of equations as follows:

$\begin{pmatrix} a & b \\ c & d \end{pmatrix} \left( \begin{array}{c} \alpha \\ \beta \end{array}\right) = \left( \begin{array}{c} e \\ f \end{array} \right)$

and find the quantities a,b,c,d,e and f as functions of $x_{i}, y_{i}, \sigma_{i}$.

Homework Equations

[/B]
least squares estimators are
$\chi^{2}(\alpha,\beta)=\sum_{i=1}^{n}\frac{1}{\sigma_{i}^{2}}(y_{i}-f(x_{i};\alpha,\beta)^{2})$
if the measurements are not independent, then given the covariance matrix $V$, the least squares estimators are given by
$\chi^{2}(\vec{\theta})\sum_{i,j=1}^{N}(y_{i}-f(x_{i};\vec{\theta}))(V^{-1})_{ij}(y_{j}-f(x_{j};\vec{\theta}))$
where the $\vec{\theta}$ is the vector of parameters we wish to estimate.

The Attempt at a Solution

[/B]
Right so I'm pretty sure I've solved the first part:
1)

2)
This is where I get stuck. To find the least squares estimators from the chi-squared thing, I have to put it in matrix form, differentiate, set it equal to 0 and solve the resulting system of equations. So in matrix form, since our measurements are all independent, we have

$\chi^{2}(\alpha,\beta)=(\vec{y}-A\vec{\theta})^{2}(V^{-1})_{ij}$

where $A_{ij}$ is given by $f(x_{i};\vec{\theta})=\sum_{j=1}^{m}a_{j}(x_{i})\theta_{j}=\sum_{j=1}^{m}A_{ij}\theta_{j}$
However, in our case, we already have this quantity because

$\sum_{j=1}^{m}a_{j}(x_{i})\theta_{j}=\alpha x +\beta x^{2}$

aaaand this is my problem - I have no idea how to extract the $A_{ij}$ matrix out of this, and even more confusing is: how is it square? if the i index runs from 1...n (the measurements) and j runs from 1,2 (the number of parameters) then how am I supposed to cast this into the square matrix equation above?

Anyway, I did differentiate the chi-squared thing and set it equal to 0, which gives me

$A\vec{\theta}=\vec{y}$

Which fits the system of equations provided that A is square...I don't see how this works...please help!

 

[STEMucator]

Homework Statement

Hi guys,

so the problem is as follows:

A set of n independent measurements $y_{i}, i=1...n$ are treated as Gaussian, each with standard deviations $\sigma_{i}$. Each measurement corresponds to a value of a control variable $x_{i}$. The expectation value of $y$ is given by
$f(x;\alpha,\beta)=\alpha x +\beta x^{2}$.

1) Find the log-likelihood function for the parameters $\alpha,\beta$.

2) Show that the least-squares estimators for $\alpha,\beta$ can be found from the solution of a system of equations as follows:

$\begin{pmatrix} a & b \\ c & d \end{pmatrix} \left( \begin{array}{c} \alpha \\ \beta \end{array}\right) = \left( \begin{array}{c} e \\ f \end{array} \right)$

and find the quantities a,b,c,d,e and f as functions of $x_{i}, y_{i}, \sigma_{i}$.

Homework Equations

[/B]
least squares estimators are
$\chi^{2}(\alpha,\beta)=\sum_{i=1}^{n}\frac{1}{\sigma_{i}^{2}}(y_{i}-f(x_{i};\alpha,\beta)^{2})$
if the measurements are not independent, then given the covariance matrix $V$, the least squares estimators are given by
$\chi^{2}(\vec{\theta})\sum_{i,j=1}^{N}(y_{i}-f(x_{i};\vec{\theta}))(V^{-1})_{ij}(y_{j}-f(x_{j};\vec{\theta}))$
where the $\vec{\theta}$ is the vector of parameters we wish to estimate.

The Attempt at a Solution

[/B]
Right so I'm pretty sure I've solved the first part:
1)

2)
This is where I get stuck. To find the least squares estimators from the chi-squared thing, I have to put it in matrix form, differentiate, set it equal to 0 and solve the resulting system of equations. So in matrix form, since our measurements are all independent, we have

$\chi^{2}(\alpha,\beta)=(\vec{y}-A\vec{\theta})^{2}(V^{-1})_{ij}$

where $A_{ij}$ is given by $f(x_{i};\vec{\theta})=\sum_{j=1}^{m}a_{j}(x_{i})\theta_{j}=\sum_{j=1}^{m}A_{ij}\theta_{j}$
However, in our case, we already have this quantity because

$\sum_{j=1}^{m}a_{j}(x_{i})\theta_{j}=\alpha x +\beta x^{2}$

aaaand this is my problem - I have no idea how to extract the $A_{ij}$ matrix out of this, and even more confusing is: how is it square? if the i index runs from 1...n (the measurements) and j runs from 1,2 (the number of parameters) then how am I supposed to cast this into the square matrix equation above?

Anyway, I did differentiate the chi-squared thing and set it equal to 0, which gives me

$A\vec{\theta}=\vec{y}$

Which fits the system of equations provided that A is square...I don't see how this works...please help!

I don't know much about this, but I don't think your answer to 1) is correct.

Isn't the likelihood function given by:

$$L(\alpha, \beta; x_1, ..., x_n) = ∏_{j=1}^n f(x_j; \alpha, \beta) = ∏_{j=1}^n \alpha x_j +\beta x^2_j$$

 

[Dixanadu]

The likelihood you wrote down cannot be correct, because maximising the likelihood should be the same as minimising the chi-squared function, so they must be the same up to some constant, which I have found to be -1/2.

 

[STEMucator]

So you want to minimize $\chi^2$ or maximize $- \chi^2$. I agree with your work then except for some subscripts.

The second question indeed amounts to solving the normal equations by taking partials with respect to $\alpha$ and $\beta$, setting them equal to zero and then solving.

 

[Dixanadu]

Hmm okay thank you. So doing that, I get these two equations:

$\frac{\partial\chi^{2}}{\partial \alpha}=2x\sum_{i=1}^{n}\frac{1}{\sigma_{i}^{2}}(\alpha x +\beta x^{2} - y_{i})=0$
and
$\frac{\partial\chi^{2}}{\partial \beta}=4\beta x\sum_{i=1}^{n}\frac{1}{\sigma_{i}^{2}}(\alpha x +\beta x^{2} - y_{i})=0$

How does one solve these? I thought about setting the summand to 0 since the sum is 0 (although I'm not sure you can do that) but then I get two identical equations - what should I do?

 

[STEMucator]

Hmm okay thank you. So doing that, I get these two equations:

$\frac{\partial\chi^{2}}{\partial \alpha}=2x\sum_{i=1}^{n}\frac{1}{\sigma_{i}^{2}}(\alpha x +\beta x^{2} - y_{i})=0$
and
$\frac{\partial\chi^{2}}{\partial \beta}=4\beta x\sum_{i=1}^{n}\frac{1}{\sigma_{i}^{2}}(\alpha x +\beta x^{2} - y_{i})=0$

How does one solve these? I thought about setting the summand to 0 since the sum is 0 (although I'm not sure you can do that) but then I get two identical equations - what should I do?

Your derivatives look wrong for some reason. They should read:

$$\frac{\partial\chi^{2}}{\partial \alpha}= - 2 \displaystyle \sum_i^n \frac{x_i}{\sigma_i^2}(y_i - \alpha x_i - \beta x_i^2) = 0$$

$$\frac{\partial\chi^{2}}{\partial \beta} = - 2 \displaystyle \sum_i^n \left(\frac{x_i}{\sigma_i}\right)^2(y_i - \alpha x_i - \beta x_i^2) = 0$$

Now distribute the sum to the terms... do a little bit of re-arranging and presto a set of normal equations will result. These normal equations can be solved for $\alpha$ and $\beta$ and happen to be functions of the aforementioned variables in the question.

 

[Dixanadu]

Yea I did mess up the derivatives and my subscripts were also wrong - thank you for clarifying :) I rearranged and got something like this for the matrix:

 

[STEMucator]

Looks correct at a first glance.

 

[Dixanadu]

Thank you very much :D i know who to ask for help with stats! thanks :)