/**
* Converts source parameters to natural parameters.
*
* @param L source parameters \f$ \mathbf{\Lambda} = (p_1, \cdots, p_k)\f$
* @return natural parameters \f$ \mathbf{\Theta} = \left( \log \left( \frac{p_i}{p_k} \right)
* \right)_i \f$
*/
public PVector Lambda2Theta(PVector L) {
PVector theta = new PVector(L.getDimension() - 1);
theta.type = Parameter.TYPE.NATURAL_PARAMETER;
for (int i = 0; i < L.getDimension() - 1; i++)
theta.array[i] = Math.log(L.array[i] / L.array[L.getDimension() - 1]);
return theta;
}
/**
* Computes the density value \f$ f(x;\mu) \f$.
*
* @param x point
* @param param parameters (source, natural, or expectation)
* @return \f$ f(x;\mu) = \frac{1}{ (2\pi)^{d/2} } \exp \left( - \frac{(x-\mu)^T (x-\mu)}{2}
* \right) \mbox{ for } x \in \mathds{R}^d \f$
*/
public double density(PVector x, PVector param) {
if (param.type == Parameter.TYPE.SOURCE_PARAMETER) {
double v1 = (x.Minus(param)).InnerProduct(x.Minus(param));
double v2 = Math.exp(-0.5d * v1);
return v2 / Math.pow(2.0d * Math.PI, (double) x.dim / 2.0d);
} else if (param.type == Parameter.TYPE.NATURAL_PARAMETER) return super.density(x, param);
else return super.density(x, Eta2Theta(param));
}
/**
* Converts expectation parameters to source parameters.
*
* @param H natural parameters \f$ \mathbf{H} = (\eta_1, \cdots, \eta_{k-1})\f$
* @return source parameters \f$ \mathbf{\Lambda} = \begin{cases} p_i = \frac{\eta_i}{n} &
* \mbox{if $i<k$}\\ p_k = \frac{n - \sum_{j=1}^{k-1} \eta_j}{n} \end{cases}\f$
*/
public PVector Eta2Lambda(PVector H) {
PVector L = new PVector(H.getDimension() + 1);
L.type = Parameter.TYPE.SOURCE_PARAMETER;
double sum = 0;
for (int i = 0; i < H.getDimension(); i++) {
L.array[i] = H.array[i] / n;
sum += H.array[i];
}
L.array[H.getDimension()] = (n - sum) / n;
return L;
}
/**
* Computes \f$ \nabla G (\mathbf{H})\f$
*
* @param H expectation parameters \f$ \mathbf{H} = (\eta_1, \cdots, \eta_{k-1}) \f$
* @return \f$ \nabla G( \mathbf{H} ) = \left( \log \left( \frac{\eta_i}{n - \sum_{j=1}^{k-1}
* \eta_j} \right) \right)_i \f$
*/
public PVector gradG(PVector H) {
// Sum
double sum = 0;
for (int i = 0; i < H.getDimension(); i++) sum += H.array[i];
// Gradient
PVector gradient = new PVector(H.getDimension());
gradient.type = Parameter.TYPE.NATURAL_PARAMETER;
for (int i = 0; i < H.getDimension(); i++) gradient.array[i] = Math.log(H.array[i] / (n - sum));
// Return
return gradient;
}
/**
* Computes \f$ \nabla F ( \mathbf{\Theta} )\f$.
*
* @param T naturel parameters \f$ \mathbf{\Theta} = (\theta_1, \cdots, \theta_{k-1}) \f$
* @return \f$ \nabla F( \mathbf{\Theta} ) = \left( \frac{n \exp \theta_i}{1 + \sum_{j=1}^{k-1}
* \exp \theta_j} \right)_i \f$
*/
public PVector gradF(PVector T) {
// Sum
double sum = 0;
for (int i = 0; i < T.getDimension(); i++) sum += Math.exp(T.array[i]);
// Gradient
PVector gradient = new PVector(T.getDimension());
gradient.type = Parameter.TYPE.EXPECTATION_PARAMETER;
for (int i = 0; i < T.getDimension(); i++)
gradient.array[i] = (n * Math.exp(T.array[i])) / (1 + sum);
// Return
return gradient;
}
/**
* Computes \f$ G(\mathbf{H})\f$.
*
* @param H expectation parameters \f$ \mathbf{H} = (\eta_1, \cdots, \eta_{k-1}) \f$
* @return \f$ G(\mathbf{H}) = \left( \sum_{i=1}^{k-1} \eta_i \log \eta_i \right) + \left( n -
* \sum_{i=1}^{k-1} \eta_i \right) \log \left( n - \sum_{i=1}^{k-1} \eta_i \right) \f$
*/
public double G(PVector H) {
double sum1 = 0;
double sum2 = 0;
for (int i = 0; i < H.getDimension(); i++) {
sum1 += H.array[i] * Math.log(H.array[i]);
sum2 += H.array[i];
}
return sum1 + (n - sum2) * Math.log(n - sum2);
}
/**
* Computes the density value \f$ f(x) \f$.
*
* @param x point
* @param param parameters (source, natural, or expectation)
* @return \f$ f(x_1,\cdots,x_k;p_1,\cdots,p_k,n) = \frac{n!}{x_1! \cdots x_k!} p_1^{x_1} \cdots
* p_k^{x_k} \f$
*/
public double density(PVector x, PVector param) {
if (param.type == Parameter.TYPE.SOURCE_PARAMETER) {
double prod1 = 1;
double prod2 = 1;
for (int i = 0; i < param.getDimension(); i++) {
prod1 *= fact(x.array[i]);
prod2 *= Math.pow(param.array[i], x.array[i]);
}
return (fact(n) * prod2) / prod1;
} else if (param.type == Parameter.TYPE.NATURAL_PARAMETER) return super.density(x, param);
else return super.density(x, Eta2Theta(param));
}
/**
* Converts natural parameters to source parameters.
*
* @param T natural parameters \f$ \mathbf{\Theta} = ( \theta_1, \cdots, \theta_{k-1} )\f$
* @return source parameters \f$ \mathbf{\Lambda} = \begin{cases} p_i = \frac{\exp \theta_i}{1 +
* \sum_{j=1}^{k-1}(\exp \theta_j)} & \mbox{if $i<k$}\\ p_k = \frac{1}{1 +
* \sum_{j=1}^{k-1}(\exp \theta_j)} \end{cases} \f$
*/
public PVector Theta2Lambda(PVector T) {
// Sums
double sum = 0;
for (int i = 0; i < T.getDimension(); i++) sum += Math.exp(T.array[i]);
// Conversion
PVector lambda = new PVector(T.getDimension() + 1);
lambda.type = Parameter.TYPE.SOURCE_PARAMETER;
for (int i = 0; i < T.getDimension(); i++) lambda.array[i] = Math.exp(T.array[i]) / (1.0 + sum);
lambda.array[T.getDimension()] = 1.0 / (1.0 + sum);
// Return
return lambda;
}
/**
* Computes \f$ G(\mathbf{H})\f$
*
* @param H expectation parameters \f$ \mathbf{H} = \eta \f$
* @return \f$ F(\mathbf{\theta})= \frac{1}{2} \eta^\top\eta + \frac{d}{2}\log 2\pi \f$
*/
public double G(PVector H) {
return 0.5d * (H.InnerProduct(H) + H.dim * Math.log(2 * Math.PI));
}
/**
* Computes \f$ \nabla F ( \mathbf{\Theta} )\f$.
*
* @param T natural \f$ \mathbf{\Theta} = \theta \f$
* @return \f$ \nabla F( \mathbf{\Theta} ) = \theta \f$
*/
public PVector gradF(PVector T) {
PVector gradient = (PVector) T.clone();
gradient.type = Parameter.TYPE.EXPECTATION_PARAMETER;
return gradient;
}
/**
* Computes the log normalizer \f$ F( \mathbf{\Theta} ) \f$.
*
* @param T natural parameters \f$ \mathbf{\Theta} = \theta \f$
* @return \f$ F(\mathbf{\theta}) = \frac{1}{2} \theta^\top\theta + \frac{d}{2}\log 2\pi \f$
*/
public double F(PVector T) {
return 0.5d * (T.InnerProduct(T) + T.dim * Math.log(2 * Math.PI));
}
/**
* Converts expectation parameters to source parameters.
*
* @param H expectation parameters \f$ \mathbf{H} = \eta\f$
* @return source parameters \f$ \mathbf{\Lambda} = \eta \f$
*/
public PVector Eta2Lambda(PVector H) {
PVector L = new PVector(1);
L.array[0] = H.array[0];
L.type = Parameter.TYPE.SOURCE_PARAMETER;
return L;
}
/**
* Converts natural parameters to source parameters.
*
* @param T natural parameters \f$ \mathbf{\Theta} = \theta \f$
* @return source parameters \f$ \mathbf{\Lambda} = \frac{\exp\theta}{1+\exp\theta} \f$
*/
public PVector Theta2Lambda(PVector T) {
PVector L = new PVector(1);
L.array[0] = Math.exp(T.array[0]) / (1 + Math.exp(T.array[0]));
L.type = Parameter.TYPE.SOURCE_PARAMETER;
return L;
}
/**
* Computes the carrier measure \f$ k(x) \f$.
*
* @param x a point
* @return \f$ k(x) = -\frac{1}{2}x^\top x \f$
*/
public double k(PVector x) {
return -0.5d * x.InnerProduct(x);
}
/**
* Converts source parameters to expectation parameters.
*
* @param L source parameters \f$ \mathbf{\Lambda} = \mu \f$
* @return expectation parameters \f$ \mathbf{H} = \mu \f$
*/
public PVector Lambda2Eta(PVector L) {
PVector H = (PVector) L.clone();
H.type = Parameter.TYPE.EXPECTATION_PARAMETER;
return H;
}
/**
* Converts source parameters to natural parameters.
*
* @param L source parameters \f$ \mathbf{\Lambda} = \mu \f$
* @return natural parameters \f$ \mathbf{\Theta} = \mu \f$
*/
public PVector Lambda2Theta(PVector L) {
PVector T = (PVector) L.clone();
T.type = Parameter.TYPE.NATURAL_PARAMETER;
return T;
}
/**
* Computes \f$ F( \mathbf{\Theta} ) \f$.
*
* @param T parameters \f$ \mathbf{\Theta} = (\theta_1, \cdots, \theta_{k-1}) \f$
* @return \f$ F(\mathbf{\Theta}) = n \log \left( 1 + \sum_{i=1}^{k-1} \exp \theta_i \right) -
* \log n! \f$
*/
public double F(PVector T) {
double sum = 0;
for (int i = 0; i < T.getDimension(); i++) sum += Math.exp(T.array[i]);
return n * Math.log(1 + sum) - Math.log(fact(n));
}
/**
* Computes the Kullback-Leibler divergence between two Binomial distributions.
*
* @param LA source parameters \f$ \mathbf{\Lambda}_\alpha \f$
* @param LB source parameters \f$ \mathbf{\Lambda}_\beta \f$
* @return \f$ D_{\mathrm{KL}}(f_1\|f_2) = n p_{\alpha,k} \log \frac{p_{\alpha,k}}{p_{\beta,k}} -
* n \sum_{i=1}^{k-1} p_{\alpha,i} \log \frac{p_{\beta,i}}{p_{\alpha,i}} \f$
*/
public double KLD(PVector LA, PVector LB) {
int k = LA.getDimension() - 1;
double sum = 0;
for (int i = 0; i < k; i++) sum += LA.array[i] * Math.log(LB.array[i] / LA.array[i]);
return n * LA.array[k] * Math.log(LA.array[k] / LB.array[k]) - n * sum;
}
/**
* Computes \f$ \nabla G (\mathbf{H})\f$
*
* @param H expectation parameters \f$ \mathbf{H} = \eta \f$
* @return \f$ \nabla G(\mathbf{H}) = \eta \f$
*/
public PVector gradG(PVector H) {
PVector gradient = (PVector) H.clone();
gradient.type = Parameter.TYPE.NATURAL_PARAMETER;
return gradient;
}
/**
* Converts expectation parameters to source parameters.
*
* @param H expectation parameters \f$ \mathbf{H} = \eta \f$
* @return source parameters \f$ \mathbf{\Lambda} = \eta \f$
*/
public PVector Eta2Lambda(PVector H) {
PVector L = (PVector) H.clone();
L.type = Parameter.TYPE.SOURCE_PARAMETER;
return L;
}
/**
* Computes the sufficient statistic \f$ t(x)\f$.
*
* @param x a point
* @return \f$ t(x) = x \f$
*/
public PVector t(PVector x) {
PVector t = (PVector) x.clone();
t.type = Parameter.TYPE.EXPECTATION_PARAMETER;
return t;
}
/**
* Computes \f$ \nabla G (\mathbf{H})\f$.
*
* @param H expectation parameters \f$ \mathbf{H} = \eta \f$
* @return \f$ \nabla G( \mathbf{H} ) = \log \left( \frac{\eta}{1-\eta} \right) \f$
*/
public PVector gradG(PVector H) {
PVector gradient = new PVector(1);
gradient.array[0] = Math.log(H.array[0] / (1 - H.array[0]));
gradient.type = Parameter.TYPE.NATURAL_PARAMETER;
return gradient;
}
/**
* Converts source parameters to natural parameters.
*
* @param L source parameters \f$ \mathbf{\Lambda} = p \f$
* @return natural parameters \f$ \mathbf{\Theta} = \log \left( \frac{p}{1-p} \right) \f$
*/
public PVector Lambda2Theta(PVector L) {
PVector T = new PVector(1);
T.array[0] = Math.log(L.array[0] / (1 - L.array[0]));
T.type = Parameter.TYPE.NATURAL_PARAMETER;
return T;
}
/**
* Computes the carrier measure \f$ k(x) \f$.
*
* @param x a point
* @return \f$ k(x) = - \sum_{i=1}^{k} \log x_i ! \f$
*/
public double k(PVector x) {
double sum = 0;
for (int i = 0; i < x.getDimension(); i++) sum -= Math.log(fact(x.array[i]));
return sum;
}
/**
* Converts source parameters to expectation parameters.
*
* @param L source parameters \f$ \mathbf{\Lambda} = p \f$
* @return expectation parameters \f$ \mathbf{H} = p \f$
*/
public PVector Lambda2Eta(PVector L) {
PVector H = new PVector(1);
H.array[0] = L.array[0];
H.type = Parameter.TYPE.EXPECTATION_PARAMETER;
return H;
}
/**
* Computes the Kullback-Leibler divergence between two multivariate isotropic Gaussian
* distributions.
*
* @param LP source parameters \f$ \mathbf{\Lambda}_P \f$
* @param LQ source parameters \f$ \mathbf{\Lambda}_Q \f$
* @return \f$ D_{\mathrm{KL}}(f_P \| f_Q) = \frac{1}{2} ( \mu_Q - \mu_P )^\top( \mu_Q - \mu_P )
* \f$
*/
public double KLD(PVector LP, PVector LQ) {
PVector diff = LQ.Minus(LP);
return 0.5d * diff.InnerProduct(diff);
}
/**
* Computes \f$ \nabla F ( \mathbf{\Theta} )\f$.
*
* @param T natural parameters \f$ \mathbf{\Theta} = \theta \f$
* @return \f$ \nabla F( \mathbf{\Theta} ) = \frac{\exp \theta}{1 + \exp \theta} \f$
*/
public PVector gradF(PVector T) {
PVector gradient = new PVector(1);
gradient.array[0] = Math.exp(T.array[0]) / (1 + Math.exp(T.array[0]));
gradient.type = Parameter.TYPE.EXPECTATION_PARAMETER;
return gradient;
}
/**
* Converts source parameters to expectation parameters.
*
* @param L source parameters \f$ \mathbf{\Lambda} = ( p_1, \cdots, p_k )\f$
* @return expectation parameters \f$ \mathbf{H} = \left( n p_i \right)_i\f$
*/
public PVector Lambda2Eta(PVector L) {
PVector H = new PVector(L.getDimension() - 1);
H.type = Parameter.TYPE.EXPECTATION_PARAMETER;
for (int i = 0; i < L.getDimension() - 1; i++) H.array[i] = n * L.array[i];
return H;
}
/**
* Computes the sufficient statistic \f$ t(x)\f$.
*
* @param x a point
* @return \f$ t(x) = x \f$
*/
public PVector t(PVector x) {
PVector t = new PVector(1);
t.array[0] = x.array[0];
t.type = Parameter.TYPE.EXPECTATION_PARAMETER;
return t;
}
/**
* Draws a point from the considered distribution.
*
* @param L source parameters \f$ \mathbf{\Lambda} = \mu \f$
* @return a point.
*/
public PVector drawRandomPoint(PVector L) {
Random rand = new Random();
PVector x = new PVector(L.getDimension());
for (int i = 0; i < L.getDimension(); i++) x.array[i] = L.array[i] + rand.nextGaussian();
return x;
}