t-, chi-squared, F-, Wishart distributions

Variables

Detailed Description

Most of these distributions are typically used for testing purposes. For such a situation, you don't need the models here. Given a statistic of the right properties, you can find the odds that the statistic is above or below a cutoff on the t-, F, or chi-squared distribution using the apop_test function.

In that world, those three distributions are actually parameter free. The data is assumed to be normalized to be based on a mean zero, variance one process, you get the degrees of freedom from the size of the data, and the distribution is fixed.

For modeling purposes, more could be done. For example, the t-distribution is a favorite proxy for Normal-like situations where there are fat tails relative to the Normal (i.e., high kurtosis). Or, you may just prefer not to take the step of normalizing your data---one could easily rewrite the theorems underlying the t-distribution without the normalizations.

In such a case, the researcher would not want to fix the $df$ , because $df$ indicates the fatness of the tails, which has some optimal value given the data. Thus, there are two modes of use for these distributions:

Parameterized, testing style: the degrees of freedom are determined from the data, and all necessary normalizations are assumed. Thus, this code---

apop_data *t_for_testing = apop_estimate(data, apop_t)

---will return exactly the type of $t$ -distribution one would use for testing. For the Wishart distribution, the $df$ is fixed in this form, but the covariance matrix is estimated from the data.

Descriptive: Just add an apop_mle_settings group, and I'll find the best via maximum likelihood.

Apop_settings_add_group(&apop_t, apop_mle, data);
apop_data *t_for_description = apop_estimate(data, apop_t);

df works for all four distributions here; df2 makes sense only for the $F$ ,

For the Wishart, the degrees of freedom and covariance matrix are always estimated via MLE.

Variable Documentation

apop_model apop_chi_squared

The $\chi^2$ distribution, for descriptive purposes. If you want to test a hypothesis, you probably don't need this, and should instead use apop_test. See notes in t-, chi-squared, F-, Wishart distributions.

apop_model apop_f_distribution

The F distribution, for descriptive purposes. If you want to test a hypothesis, you probably don't need this, and should instead use apop_test. See notes in t-, chi-squared, F-, Wishart distributions.

apop_model apop_t_distribution

The t distribution, for descriptive purposes. If you want to test a hypothesis, you probably don't need this, and should instead use apop_test. See notes in t-, chi-squared, F-, Wishart distributions.

apop_model apop_wishart

The Wishart distribution, which is currently somewhat untested.

Here's the likelihood function. $p$ is the dimension of the data and covariance matrix, $n$ is the degrees of freedom, $\mathbf{V}$ is the $p\times p$ matrix of Wishart parameters, and ${\mathbf{W}}$ is the $p\times p$ matrix whose likelihood is being evaluated. $\Gamma_p(\cdot)$ is the multivariate gamma function.

$P(\mathbf{W}) = \frac{\left|\mathbf{W}\right|^\frac{n-p-1}{2}} {2^\frac{np}{2}\left|{\mathbf V}\right|^\frac{n}{2}\Gamma_p(\frac{n}{2})} \exp\left(-\frac{1}{2}{\rm Tr}({\mathbf V}^{-1}\mathbf{W})\right)$

An example for random draws:

gsl_matrix *rmatrix = gsl_marix_alloc(10, 10);
gsl_rng *r = apop_rng_alloc(8765);
for (int i=0; i< 1e8; i++){
    apop_draw(rmatrix->data, r, apop_wishart);
    do_math_with_matrix(rmatrix);
}

See also notes in t-, chi-squared, F-, Wishart distributions.

Apophenia

t-, chi-squared, F-, Wishart distributions

Variables

Detailed Description

Variable Documentation