public static void VarianceGammaTimesGaussianMoments2(double a, double m, double v, out double mu, out double vu)
{
// compute weights
Matrix laplacianMoments = new Matrix(nWeights, nWeights);
DenseVector exactMoments = DenseVector.Constant(laplacianMoments.Rows, 1.0);
// a=10: 7-1
// a=15: 8-1
// a=20: 10-1
// a=21: 10-1
// a=30: 12-1
// get best results if the lead term has flat moment ratio
int jMax = Math.Max(laplacianMoments.Cols, (int)Math.Round(a - 10)) - 1;
jMax = laplacianMoments.Cols - 1;
for (int i = 0; i < exactMoments.Count; i++)
{
//int ii = jMax-i;
int ii = i;
double logMoment = MMath.GammaLn(ii + a) - MMath.GammaLn(a) - MMath.GammaLn(ii + 1);
for (int j = 0; j < laplacianMoments.Cols; j++)
{
int jj = jMax - j;
laplacianMoments[i, j] = Math.Exp(MMath.GammaLn(2 * ii + jj + 1) - MMath.GammaLn(2 * ii + 1) - MMath.GammaLn(jj + 1) - logMoment);
}
}
//Console.WriteLine("exactMoments = {0}, laplacianMoments = ", exactMoments);
//Console.WriteLine(laplacianMoments);
(new LuDecomposition(laplacianMoments)).Solve(exactMoments);
DenseVector weights = exactMoments;
Console.WriteLine("weights = {0}", weights);
double Z0Plus = 0, Z1Plus = 0, Z2Plus = 0;
double Z0Minus = 0, Z1Minus = 0, Z2Minus = 0;
double sqrtV = Math.Sqrt(v);
double InvSqrtV = 1 / sqrtV;
double mPlus = (m - v) * InvSqrtV;
double mMinus = (-m - v) * InvSqrtV;
for (int j = 0; j < weights.Count; j++)
{
int jj = jMax - j;
Z0Plus += weights[j] * MMath.NormalCdfMomentRatio(0 + jj, mPlus) * Math.Pow(sqrtV, 0 + jj);
Z1Plus += weights[j] * MMath.NormalCdfMomentRatio(1 + jj, mPlus) * (1 + jj) * Math.Pow(sqrtV, 1 + jj);
Z2Plus += weights[j] * MMath.NormalCdfMomentRatio(2 + jj, mPlus) * (1 + jj) * (2 + jj) * Math.Pow(sqrtV, 2 + jj);
Z0Minus += weights[j] * MMath.NormalCdfMomentRatio(0 + jj, mMinus) * Math.Pow(sqrtV, 0 + jj);
Z1Minus += weights[j] * MMath.NormalCdfMomentRatio(1 + jj, mMinus) * (1 + jj) * Math.Pow(sqrtV, 1 + jj);
Z2Minus += weights[j] * MMath.NormalCdfMomentRatio(2 + jj, mMinus) * (1 + jj) * (2 + jj) * Math.Pow(sqrtV, 2 + jj);
}
double Z0 = Z0Plus + Z0Minus;
double Z1 = Z1Plus - Z1Minus;
double Z2 = Z2Plus + Z2Minus;
mu = Z1 / Z0;
vu = Z2 / Z0 - mu * mu;
}
// returns (E[x],var(x)) where p(x) =propto VG(x;a) N(x;m,v).
public static void VarianceGammaTimesGaussianMoments3(double a, double m, double v, out double mu, out double vu)
{
// compute weights
// termMoments[i,j] is the ith moment of the jth term
Matrix termMoments = new Matrix(nWeights, nWeights);
DenseVector exactMoments = DenseVector.Constant(termMoments.Rows, 1.0);
for (int i = 0; i < exactMoments.Count; i++)
{
// ii is half of the exponent
int ii = i;
double logMoment = MMath.GammaLn(ii + a) - MMath.GammaLn(a) - MMath.GammaLn(ii + 1);
for (int j = 0; j < termMoments.Cols; j++)
{
// jj is the term shape
int jj = j + 1;
termMoments[i, j] = Math.Exp(MMath.GammaLn(ii + jj) - MMath.GammaLn(ii + 1) - MMath.GammaLn(jj) - logMoment);
}
}
//Console.WriteLine("exactMoments = {0}, termMoments = ", exactMoments);
//Console.WriteLine(termMoments);
(new LuDecomposition(termMoments)).Solve(exactMoments);
DenseVector weights = exactMoments;
Console.WriteLine("weights = {0}", weights);
double Z0Plus = 0, Z1Plus = 0, Z2Plus = 0;
double Z0Minus = 0, Z1Minus = 0, Z2Minus = 0;
for (int j = 0; j < weights.Count; j++)
{
int jj = j + 1;
Z0Plus += weights[j] * NormalVGMomentRatio(0, jj, m - v, v);
Z1Plus += weights[j] * NormalVGMomentRatio(1, jj, m - v, v);
Z2Plus += weights[j] * NormalVGMomentRatio(2, jj, m - v, v);
Z0Minus += weights[j] * NormalVGMomentRatio(0, jj, -m - v, v);
Z1Minus += weights[j] * NormalVGMomentRatio(1, jj, -m - v, v);
Z2Minus += weights[j] * NormalVGMomentRatio(2, jj, -m - v, v);
}
double Z0 = Z0Plus + Z0Minus;
double Z1 = Z1Plus - Z1Minus;
double Z2 = Z2Plus + Z2Minus;
mu = Z1 / Z0;
vu = Z2 / Z0 - mu * mu;
//Console.WriteLine("mu = {0}, vu = {1}", mu, vu);
}
#pragma warning disable 162
#endif
// returns (E[x],var(x)) where p(x) =propto VG(x;a) N(x;m,v).
public static void VarianceGammaTimesGaussianMoments4(double a, double m, double v, out double mu, out double vu)
{
// compute weights
// termMoments[i,j] is the ith moment of the jth term
Matrix termMoments = new Matrix(nWeights, nWeights);
DenseVector weights = DenseVector.Constant(termMoments.Rows, 1.0);
// choose shapes of terms to bracket the true shape 'a'
int shapeMin = Math.Max(1, (int)Math.Round(a - nWeights / 2));
for (int j = 0; j < termMoments.Cols; j++)
{
// jj is the term shape
int jj = shapeMin + j;
double r = jj / a;
double moment = 1;
for (int i = 0; i < termMoments.Rows; i++)
{
termMoments[i, j] = moment;
moment *= r;
}
}
//Console.WriteLine("exactMoments = {0}, termMoments = ", exactMoments);
//Console.WriteLine(termMoments);
(new LuDecomposition(termMoments)).Solve(weights);
Console.WriteLine("weights = {0}", weights);
if (false)
{
// exact formula for the weights
for (int i = 0; i < weights.Count; i++)
{
double sum = 0;
double term = 1;
double numr = a - shapeMin + 1;
double denomr = 0;
for (int j = 0; j < i; j++)
{
numr--;
denomr++;
term *= numr / denomr;
}
denomr = 0;
for (int j = i; j < weights.Count; j++)
{
//Console.WriteLine("term = {0}", term);
sum += term;
//Console.WriteLine("sum({0},{1}) = {2}", i,j, sum);
numr--;
denomr++;
term *= -numr / denomr;
}
weights[i] = sum;
}
Console.WriteLine("weights = {0}", weights);
}
double Z0Plus = 0, Z1Plus = 0, Z2Plus = 0;
double Z0Minus = 0, Z1Minus = 0, Z2Minus = 0;
double[][] momentsPlus = NormalVGMomentRatios(2, shapeMin + weights.Count - 1, m - v, v);
double[][] momentsMinus = NormalVGMomentRatios(2, shapeMin + weights.Count - 1, -m - v, v);
for (int j = 0; j < weights.Count; j++)
{
int jj = shapeMin + j;
Z0Plus += weights[j] * momentsPlus[jj - 1][0];
Z1Plus += weights[j] * momentsPlus[jj - 1][1];
Z2Plus += weights[j] * momentsPlus[jj - 1][2];
Z0Minus += weights[j] * momentsMinus[jj - 1][0];
Z1Minus += weights[j] * momentsMinus[jj - 1][1];
Z2Minus += weights[j] * momentsMinus[jj - 1][2];
}
if (false)
{
double[] binomt = new double[nWeights];
for (int i = 0; i < binomt.Length; i++)
{
binomt[i] = momentsPlus[shapeMin - 1 + i][0];
}
double Z0Plus2 = InterpolateBesselKMoment(a - shapeMin + 1, binomt);
Console.WriteLine("Z0Plus = {0}", Z0Plus);
Console.WriteLine("Z0Plus2 = {0}", Z0Plus2);
}
//Console.WriteLine("Z1Plus = {0}, Z1Minus = {1}", Z1Plus, Z1Minus);
double Z0 = Z0Plus + Z0Minus;
double Z1 = Z1Plus - Z1Minus;
double Z2 = Z2Plus + Z2Minus;
mu = Z1 / Z0;
vu = Z2 / Z0 - mu * mu;
//Console.WriteLine("mu = {0}, vu = {1}", mu, vu);
}