1
$\begingroup$

I think the correct answer is $7$ because the general quadratic is $$y_i=ax_i^2 + bx_i + c$$

Using the formula $$\color{red}{\fbox{Number of degrees of freedom = Number of data points - Number of Parameters}}$$

The $3$ parameters are $y_i,x_i^2,x_i$

So $10-3=7$ degrees of freedom.

The correct answer is indeed $7$; But is this because $x_i^2$ 'counts' as $2$ parameters as it's squared (adding the other $x_i$ to make $3$)?

I am somewhat new to this idea of degrees of freedom; so I'm not really sure which method is valid, if any?

$\endgroup$
0

2 Answers 2

4
$\begingroup$

The parameters are $a,b,c$ and not $y_i, x_i, x_i^2$ as you pointed out.

$\endgroup$
3
  • $\begingroup$ Are you sure about this as the $a,b,c$ are just constants say $4x^2 + 3x -2$? It is the $y_i, x_i, x_i^2$ that are varying with each $i$. $\endgroup$
    – BLAZE
    Commented Dec 8, 2015 at 1:22
  • 2
    $\begingroup$ $a$, $b$, and $c$ are the parameters you are trying to estimate. ${y_i, x_i, x_i^2}$ is your data. Your estimators $\hat{a}$, $\hat{b}$, and $\hat{c}$ are functions of your data. Your data has 10 degrees of freedom because it has 10 observations. Your model has 3 degrees of freedom because you are estimating three separate values (some people don't count the constant $c$ and call it 2 degrees of freedom). Your residuals have 10 - 3 = 7 degrees of freedom. There are 10 residuals (i.e. the residual vector has 10 dimensions) but they live in a 7 dimensional subspace! $\endgroup$ Commented Dec 8, 2015 at 3:52
  • $\begingroup$ @right Thanks for your answer, I appreciate that (+1) :) $\endgroup$
    – BLAZE
    Commented Dec 8, 2015 at 7:56
1
$\begingroup$

Several comments:

  • $a$, $b$, and $c$ are the parameters you are trying to estimate.
  • $\{y_i, x_i, x_i^2\}$ is your data.
  • Your estimators $\hat{a}$, $\hat{b}$, and $\hat{c}$ are functions of your data.
  • Your data has 10 degrees of freedom because it has 10 different observations.
  • Your model has 3 degrees of freedom because you are estimating three separate values. (Some people don't count the constant $c$ and call it 2 degrees of freedom.)
  • Your residuals have 10 - 3 = 7 degrees of freedom.
  • The residual vector is a 10 dimensional vector (because there are 10 residuals) but the vector lives in a 7 dimensional subspace (because it only has 7 degrees of freedom)!
$\endgroup$
3
  • $\begingroup$ Very nicely explained, (+1) just one more question; What is a constraint in this context? Are the $\{y_i, x_i, x_i^2\}$ constraints, probability theory is not my strong point. I have seen another equation that states: $\color{blue}{\fbox{Number of degrees of freedom = Number of data points - Number of Constraints}}$? If this is true then it means that a Constraint is simply a synonym for a Parameter, is this correct? $\endgroup$
    – BLAZE
    Commented Dec 8, 2015 at 7:54
  • 1
    $\begingroup$ @BLAZE I'd have to see the context to be able to say anything. It doesn't strike me as completely standard usage/terminology? but maybe in context, I'd get it. $\endgroup$ Commented Dec 8, 2015 at 8:08
  • $\begingroup$ I have found some context for the word 'constraint' please see this question. You have been of great help in answering this question, so thought you might like to try the next one. $\endgroup$
    – BLAZE
    Commented Dec 20, 2015 at 10:36

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .