Mixing
Linear fit for correlated data
$begingroup$
Assume I have a set of data points $(x_i,y_i)$ that I want to fit to some model function $f$ which depends on a couple of fit parameters, i.e. $y=f(x;a_1,a_2, ...)$. Usually I'd define a "chi-square" as the sum over the residuals,
$chi^2 = sum_i dfrac{(y_i-f_i)^2}{(Delta y_i)^2}$
where the $Delta y_i$ denote the uncertainties of the data points (due to measurements, Monte-Carlo, etc.) and minimize it using some routine like Levenberg-Marquardt or similar to find the optimal parameters $a_i$.
However, if there is correlations between the data points and I know the full covariance matrix $C$ (calculated, for instance, using the Jackknife method), I'd instead minimize the more general
$chi^2 = sum_{ij} (y_i-f_i), C_{ij}, (y_j-f_j)$
which takes the correlations into account. As an example, $C_{24}<0$ would imply that the points $y_2$ and $y_4$ are anti-correlated (perhaps due to being calculated from the same data set). In the special case of no cross-correlations $C = mathrm{diag}(Delta y_1^{-2},Delta y_2^{-2}, ldots) $ and it reduces to the first equation.
My question now is the following: In the case of a linear regression model, I wouldn't need a minimizer routine, since the best fit parameters can be calculated exactly (see, e.g., Numerical Recipes, chapter 15.2). Is this still possible for the "covariance optimized" $chi^2$ defined above?
optimization covariance correlation regression chi-squared
$endgroup$
migrated from math.stackexchange.com Jan 30 at 6:12
This question came from our site for people studying math at any level and professionals in related fields.
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$begingroup$
Assume I have a set of data points $(x_i,y_i)$ that I want to fit to some model function $f$ which depends on a couple of fit parameters, i.e. $y=f(x;a_1,a_2, ...)$. Usually I'd define a "chi-square" as the sum over the residuals,
$chi^2 = sum_i dfrac{(y_i-f_i)^2}{(Delta y_i)^2}$
where the $Delta y_i$ denote the uncertainties of the data points (due to measurements, Monte-Carlo, etc.) and minimize it using some routine like Levenberg-Marquardt or similar to find the optimal parameters $a_i$.
However, if there is correlations between the data points and I know the full covariance matrix $C$ (calculated, for instance, using the Jackknife method), I'd instead minimize the more general
$chi^2 = sum_{ij} (y_i-f_i), C_{ij}, (y_j-f_j)$
which takes the correlations into account. As an example, $C_{24}<0$ would imply that the points $y_2$ and $y_4$ are anti-correlated (perhaps due to being calculated from the same data set). In the special case of no cross-correlations $C = mathrm{diag}(Delta y_1^{-2},Delta y_2^{-2}, ldots) $ and it reduces to the first equation.
My question now is the following: In the case of a linear regression model, I wouldn't need a minimizer routine, since the best fit parameters can be calculated exactly (see, e.g., Numerical Recipes, chapter 15.2). Is this still possible for the "covariance optimized" $chi^2$ defined above?
optimization covariance correlation regression chi-squared
$endgroup$
migrated from math.stackexchange.com Jan 30 at 6:12
This question came from our site for people studying math at any level and professionals in related fields.
add a comment |
$begingroup$
Assume I have a set of data points $(x_i,y_i)$ that I want to fit to some model function $f$ which depends on a couple of fit parameters, i.e. $y=f(x;a_1,a_2, ...)$. Usually I'd define a "chi-square" as the sum over the residuals,
$chi^2 = sum_i dfrac{(y_i-f_i)^2}{(Delta y_i)^2}$
where the $Delta y_i$ denote the uncertainties of the data points (due to measurements, Monte-Carlo, etc.) and minimize it using some routine like Levenberg-Marquardt or similar to find the optimal parameters $a_i$.
However, if there is correlations between the data points and I know the full covariance matrix $C$ (calculated, for instance, using the Jackknife method), I'd instead minimize the more general
$chi^2 = sum_{ij} (y_i-f_i), C_{ij}, (y_j-f_j)$
which takes the correlations into account. As an example, $C_{24}<0$ would imply that the points $y_2$ and $y_4$ are anti-correlated (perhaps due to being calculated from the same data set). In the special case of no cross-correlations $C = mathrm{diag}(Delta y_1^{-2},Delta y_2^{-2}, ldots) $ and it reduces to the first equation.
My question now is the following: In the case of a linear regression model, I wouldn't need a minimizer routine, since the best fit parameters can be calculated exactly (see, e.g., Numerical Recipes, chapter 15.2). Is this still possible for the "covariance optimized" $chi^2$ defined above?
optimization covariance correlation regression chi-squared
$endgroup$
Assume I have a set of data points $(x_i,y_i)$ that I want to fit to some model function $f$ which depends on a couple of fit parameters, i.e. $y=f(x;a_1,a_2, ...)$. Usually I'd define a "chi-square" as the sum over the residuals,
$chi^2 = sum_i dfrac{(y_i-f_i)^2}{(Delta y_i)^2}$
where the $Delta y_i$ denote the uncertainties of the data points (due to measurements, Monte-Carlo, etc.) and minimize it using some routine like Levenberg-Marquardt or similar to find the optimal parameters $a_i$.
However, if there is correlations between the data points and I know the full covariance matrix $C$ (calculated, for instance, using the Jackknife method), I'd instead minimize the more general
$chi^2 = sum_{ij} (y_i-f_i), C_{ij}, (y_j-f_j)$
which takes the correlations into account. As an example, $C_{24}<0$ would imply that the points $y_2$ and $y_4$ are anti-correlated (perhaps due to being calculated from the same data set). In the special case of no cross-correlations $C = mathrm{diag}(Delta y_1^{-2},Delta y_2^{-2}, ldots) $ and it reduces to the first equation.
My question now is the following: In the case of a linear regression model, I wouldn't need a minimizer routine, since the best fit parameters can be calculated exactly (see, e.g., Numerical Recipes, chapter 15.2). Is this still possible for the "covariance optimized" $chi^2$ defined above?
optimization covariance correlation regression chi-squared
optimization covariance correlation regression chi-squared
asked Jan 28 at 21:29
Beaker
migrated from math.stackexchange.com Jan 30 at 6:12
This question came from our site for people studying math at any level and professionals in related fields.
migrated from math.stackexchange.com Jan 30 at 6:12
This question came from our site for people studying math at any level and professionals in related fields.
add a comment |
add a comment |
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