December 7, 2004 --- Class 28 --- More on Chi-squared Distribution
Activities:
We showed how the chi squared distribution can be found by
integrating random variables in spherical coordinates. If we integrate
all the variables but the radial one, we find that the radial variable
is equivalent to the random variable chi-squared. In fact, we don't
even have to do the angular integral because it contributes to the
normalization and we can just normalize our probability distribution
function so that when we integrate it from 0 to infinity, we get one.
However, we derived the angular integral which is the surface area
of an n-dimensional sphere by completing the integration over the
radial variable and using the fact that we started with the
normalized integral.
You can also find in ~sg/gausdist/dec2.math a Mathematica expression
for the chi-squared distribution:
chisq[x_, n_] := (x^((n - 2)/2)*Exp[-(x/2)])/(Gamma[n/2]*2^(n/2))
When making histograms of chi square or graphs in mathematica, we find
that the peak in the function occurs where x is about n, the number of
degrees of freedom.
We compared a historgram from a numerical experiment with a
Mathematica plot of the chisq function above and found good agreement.
Confidence Levels
We considered the fact that the quantity \chi^2 can be used to
determine the goodness of fit. Since \chi^2 should be the sum of
the squares of N normalized Gaussian variables, it has a known
distribution. If we find that our fit gives a small value of
\chi^2, that is good. If we get a large value, maybe our model is
not the correct description of the data. If the model has p
parameters that we adjust, the fit is said to have N-p degrees of
freedom. A program confidence2, whose source code is in
~sg/src/confidence2_2.c can be used to determine the confidence
level of a fit. The confidence level depends on \chi^2 and the
number of degrees of freedom. The confidence level is the
probability that \chi^2 would exceed the value for which we are
computing the confidence level. As an example:
confidence2
6.3 5
6.300000e+00 5.000000e+00 2.781123e-01
This represents the screen display for considering \chi^2=6.3 for
5 degrees of freedom. The confidence level is 0.278 or nearly 28%.
Thus, for a fit with this confidence level, we would realize that
even for a correct model the computed value of \chi^2 would be
larger than this 28% of the time. Usually, statistician like to
rule things out at the 90% or 95% confidence level. So, when your
confidence level gets below 5-10%, you begin to wonder if you
really have the right model for describing your data.
BTW, confidence2 takes its input from the standard input, so hit
Control-D or Control-C to terminate its running.
Confidence Level From Mathmatica
Since we have already considered how to get Mathmematica to calculate
the normalized chisq distribution, we can just use numerical
integration to compute the confidence level:
chisq[x_, n_] := (x^((n - 2)/2)*Exp[-(x/2)])/(Gamma[n/2]*2^(n/2))
confidence[chisquared_, n_] := NIntegrate[ chisq[y,n], {y,chisquared,Infinity}]