Implements the abstract class ContinuousDistributionMulti for the Dirichlet distribution with parameters ( \(\alpha_1\),…, \(\alpha_d\)), \(\alpha_i > 0\). More...
Public Member Functions | |
| DirichletDist (double[] alpha) | |
| double | density (double[] x) |
| double [] | getMean () |
| double [][] | getCovariance () |
| double [][] | getCorrelation () |
| double [] | getAlpha () |
| Returns the parameters ( \(\alpha_1\), …, \(\alpha_d\)) of this object. | |
| double | getAlpha (int i) |
| Returns the \(i\)th component of the alpha vector. | |
| void | setParams (double[] alpha) |
| Sets the parameters ( \(\alpha_1\), …, \(\alpha_d\)) of this object. | |
 Public Member Functions inherited from ContinuousDistributionMulti | |
| abstract double | density (double[] x) |
| Returns \(f(x_1, x_2, …, x_d)\), the probability density of \(X\) evaluated at the point \(x\), where \(x = \{x_1, x_2, …, x_d\}\). More... | |
| int | getDimension () |
| Returns the dimension \(d\) of the distribution. | |
| abstract double [] | getMean () |
| Returns the mean vector of the distribution, defined as \(\mu_i = E[X_i]\). | |
| abstract double [][] | getCovariance () |
| Returns the variance-covariance matrix of the distribution, defined as \(\sigma_{ij} = E[(X_i - \mu_i)(X_j - \mu_j)]\). | |
| abstract double [][] | getCorrelation () |
| Returns the correlation matrix of the distribution, defined as \(\rho_{ij} = \sigma_{ij}/\sqrt{\sigma_{ii}\sigma_{jj}}\). | |
Static Public Member Functions | |
| static double | density (double[] alpha, double[] x) |
| Computes the density ( fDirichlet ) of the Dirichlet distribution with parameters ( \(\alpha_1\), …, \(\alpha_d\)). | |
| static double [][] | getCovariance (double[] alpha) |
| Computes the covariance matrix of the Dirichlet distribution with parameters ( \(\alpha_1\), …, \(\alpha_d\)). | |
| static double [][] | getCorrelation (double[] alpha) |
| Computes the correlation matrix of the Dirichlet distribution with parameters ( \(\alpha_1\), …, \(\alpha_d\)). | |
| static double [] | getMLE (double[][] x, int n, int d) |
| Estimates the parameters [ \(\hat{\alpha_1},…,\hat{\alpha_d}\)] of the Dirichlet distribution using the maximum likelihood method. More... | |
| static double [] | getMean (double[] alpha) |
| Computes the mean \(E[X] = \alpha_i / \alpha_0\) of the Dirichlet distribution with parameters ( \(\alpha_1\), …, \(\alpha_d\)), where \(\alpha_0 = \sum_{i=1}^d \alpha_i\). | |
Protected Attributes | |
| double [] | alpha |
| Protected Attributes inherited from ContinuousDistributionMulti | |
| int | dimension |
Detailed Description
Implements the abstract class ContinuousDistributionMulti for the Dirichlet distribution with parameters ( \(\alpha_1\),…, \(\alpha_d\)), \(\alpha_i > 0\).
\[ f(x_1,…, x_d) = \frac{\Gamma(\alpha_0) \prod_{i=1}^d x_i^{\alpha_i - 1}}{\prod_{i=1}^d \Gamma(\alpha_i)} \tag{fDirichlet} \]
where \(x_i \ge0\), \(\sum_{i=1}^d x_i = 1\), \(\alpha_0 = \sum_{i=1}^d \alpha_i\), and \(\Gamma\) is the Gamma function.
Member Function Documentation
◆ getMLE()
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static |
Estimates the parameters [ \(\hat{\alpha_1},…,\hat{\alpha_d}\)] of the Dirichlet distribution using the maximum likelihood method.
It uses the \(n\) observations of \(d\) components in table \(x[i][j]\), \(i = 0, 1, …, n-1\) and \(j = 0, 1, …, d-1\). The equations of the maximum likelihood are defined in [12] (Technical appendix)
\begin{align*} L(\hat{\alpha}_1,\hat{\alpha}_2,…,\hat{\alpha}_k) & = n \left( G(\alpha_0) - \sum_{i=1}^k G(\hat{\alpha}_i) \right) + \sum_{i=1}^k (\hat{\alpha}_i - 1) Z_i \end{align*}
where \(G\) is the logarithm of the gamma function and
\begin{align*} \alpha_0 & = \sum_{i=1}^k \hat{\alpha}_i \\ Z_i & = \sum_{j=1}^n \ln(X_{i,j}) \qquad\mbox{for }i=1,…,k. \end{align*}
- Parameters
-
x the list of observations to use to evaluate parameters n the number of observations to use to evaluate parameters d the dimension of each vector
- Returns
- returns the parameter [ \(\hat{\alpha_1},…,\hat{\alpha_d}\)]
The documentation for this class was generated from the following file:
- DirichletDist.java
