public Func <double, double, double> GetMetric(bool isReverse)
{
return
((x, y) =>
{
y = Math.Abs(y) > Math.Abs(StretchY) ? Math.Sign(y) * Math.Abs(StretchY) : y;
var offX = x + XOffset;
var offC = XOffset != XShift / b ? XOffset : XShift;
int sign;
if (isReverse)
{
sign = Math.Abs(offX % (-2 * Math.PI / StretchX)) < Math.PI / StretchX ? -1 : 1;
}
else
{
sign = offX % (2 * Math.PI / StretchX) < Math.PI / StretchX ? 1 : -1;
}
var n = Math.Round(offX * StretchX / (2 * Math.PI));
Func <double, double> func = u => StretchY *Math.Cos(StretchX *u + offC) + YShift;
var bound = (sign * Math.Acos((y - YShift) / StretchY) - offC + 2 * Math.PI * n) / StretchX;
var minimaX = x == bound ? x : FindMinimum.OfScalarFunctionConstrained(
u => (u - x) * (u - x) + (func(u) - y) * (func(u) - y),
Math.Min(x, bound), Math.Max(x, bound));
return Math.Abs((minimaX - x) * (minimaX - x) + (func(minimaX) - y) * (func(minimaX) - y));
});
}
public static void TestGEV() // Scratchwork, prototyping, etc.
{
Logger output = new Logger("GEV Test A.csv");
Logger output2 = new Logger("GEV Test B.csv");
//var dist = new ChiSquared(4, Program.rand);
var dist = new Beta(2, 2);
//var dist = new Beta(2, 5);
//var dist = new Beta(2, 1.5);
output.WriteLine($"Distribution: {dist.ToString().Replace(',',' ')}");
//var dist = new Exponential(2, Program.rand);
//var dist = new Gamma(2, 2, Program.rand);
const int sampleSize = 300;
output.WriteLine($"Samplesize: {sampleSize}");
// Report the distribution 1-1/e quantile
double upperQuantile = dist.InverseCumulativeDistribution(1 - 1.0 / sampleSize);
double lowerQuantile = dist.InverseCumulativeDistribution(1.0 / sampleSize);
output.WriteLine($"1-1/samplesize quantile: {upperQuantile}");
output.WriteLine($"1/samplesize quantile: {lowerQuantile}");
// Monte Carlo for the true distribution of the sample maximum
double[] observations = new double[10000];
for (int i = 0; i < observations.Length; i++)
{
double max = double.NegativeInfinity;
for (int j = 0; j < sampleSize; j++)
{
max = Math.Max(max, dist.Sample());
}
observations[i] = max;
}
Sorting.Sort(observations);
ContinuousDistribution MonteCarloDistributionOfTheMaximum = ContinuousDistribution.ECDF(observations, Program.rand);
// --- Find the best fit GEV distribution for this dataset ---
#region Old code
/*
* // Compute location and scale parameter estimates for a given shape parameter Xi using the median and variance
* void EstimateParameters(double shape, double median, double variance, out double location, out double scale)
* {
* if (shape == 0)
* {
* scale = Math.Sqrt(6 * variance) / Math.PI;
* location = median + scale * Math.Log(Math.Log(2));
* return;
* }
* // This scale may or may not work for Xi > 0.5
* scale = Math.Sign(shape) * shape * Math.Sqrt(variance) / Math.Sqrt(SpecialFunctions.Gamma(1 - 2 * shape) - SpecialFunctions.Gamma(1 - shape) * SpecialFunctions.Gamma(1 - shape));
* if (double.IsNaN(scale)) scale = Math.Sqrt(6 * variance) / Math.PI;
* location = median - scale * (Math.Pow(Math.Log(2), -shape) - 1) / shape;
* }*/
#endregion
double FitnessExactModel(GEV model)
{
double val = 0;
for (int i = 0; i < observations.Length; i++)
{
val += Math.Pow(model.CumulativeDistribution(observations[i]) - MonteCarloDistributionOfTheMaximum.CumulativeDensity(observations[i]), 2);
}
return(val);
}
#region Old code
//double medianEst = Statistics.Median(observations);
//double varianceEst = Statistics.VarianceEstimate(observations);
/*
* GEV Optimize(double startingval, out double fitness)
* {
* double locationEst;
* double scaleEst;
* double bestScore = double.PositiveInfinity;
* GEV bestSoFar = null;
* bool increasing = false;
* int sinceImproved = 0;
* double shapeEst = startingval; // Neg or pos will stay that way throughout the optimization
*
* while (true)
* {
* EstimateParameters(shapeEst, medianEst, varianceEst, out locationEst, out scaleEst);
* GEV model = new GEV(locationEst, scaleEst, shapeEst, Program.rand);
* double score = FitnessExactModel(model);
* if (score < bestScore)
* {
* bestScore = score;
* bestSoFar = model;
* sinceImproved = 0;
* }
* else
* {
* increasing ^= true;
* if (++sinceImproved > 10) break;
* }
* if (increasing) shapeEst += 0.3 * startingval;
* else shapeEst *= 0.5;
* }
* fitness = bestScore;
* return bestSoFar;
* }
*
* GEV OptimizeV2(double initialGuess, out double fitness)
* {
* double locationEst, scaleEst;
* double bestScore = double.PositiveInfinity;
* GEV bestSoFar = null;
* double shapeEst = initialGuess;
* double bestShapeSoFar = initialGuess;
* // Grow the estimate by doubling until it is no longer improving
* while (true)
* {
* EstimateParameters(shapeEst, medianEst, varianceEst, out locationEst, out scaleEst);
* GEV model = new GEV(locationEst, scaleEst, shapeEst, Program.rand);
* double score = FitnessExactModel(model);
* if (score < bestScore) // If it improved
* {
* bestScore = score;
* bestSoFar = model;
* bestShapeSoFar = shapeEst;
* }
* else break;
* shapeEst *= 2;
* }
* double magnitude = bestShapeSoFar;
* for (int i = 0; i < 10; i++) // 10 corresponds to 3 correct digits
* {
* double delta = magnitude * Math.Pow(2, -(i + 1)); // Half in size for each iteration
*
* // Three positions: the current one, one lower by delta, and one higher by delta
*
* // Lower Model
* EstimateParameters(bestShapeSoFar - delta, medianEst, varianceEst, out locationEst, out scaleEst);
* GEV lowerModel = new GEV(locationEst, scaleEst, bestShapeSoFar - delta, Program.rand);
* double lowerScore = FitnessExactModel(lowerModel);
*
* // Upper Model
* EstimateParameters(bestShapeSoFar + delta, medianEst, varianceEst, out locationEst, out scaleEst);
* GEV upperModel = new GEV(locationEst, scaleEst, bestShapeSoFar + delta, Program.rand);
* double upperScore = FitnessExactModel(upperModel);
*
* // Move to the best of the three
* double bestfitness = Math.Min(bestScore, Math.Min(upperScore, lowerScore));
* bestScore = bestfitness;
* if (lowerScore == bestfitness)
* {
* bestShapeSoFar = bestShapeSoFar - delta;
* bestSoFar = lowerModel;
* }
* else if (upperScore == bestfitness)
* {
* bestShapeSoFar = bestShapeSoFar + delta;
* bestSoFar = upperModel;
* }
* }
* fitness = bestScore;
* return bestSoFar;
* }
*/
#endregion
GEV OptimizeBFGS(Func <Vector <double>, double> objectiveFunc, double initialShape, double initialScale, double initialLocation)
{
// Formatted by shape, scale, location
var lowerBounds = CreateVector.DenseOfArray(new double[] { -10, Math.Min(-3 * initialScale, 3 * initialScale), Math.Min(-3 * initialLocation, 3 * initialLocation) });
var upperBounds = CreateVector.DenseOfArray(new double[] { 10, Math.Max(-3 * initialScale, 3 * initialScale), Math.Max(-3 * initialLocation, 3 * initialLocation) });
var initialGuess = CreateVector.DenseOfArray(new double[] { initialShape, initialScale, initialLocation });
var min = FindMinimum.OfFunctionConstrained(objectiveFunc, lowerBounds, upperBounds, initialGuess);
return(new GEV(min[2], min[1], min[0], Program.rand));
}
#region Old code
// Optimize for Xi
/*double fitNeg, fitZero, fitPos;
* GEV bestNeg = OptimizeV2(-1, out fitNeg);
* GEV bestPos = OptimizeV2(1, out fitPos);
* double locZero, scaleZero;
* EstimateParameters(0, medianEst, varianceEst, out locZero, out scaleZero);
* GEV zeroModel = new GEV(locZero, scaleZero, 0, Program.rand);
* fitZero = Fitness(zeroModel);
* // Choose the best model of the three
* double minScore = Math.Min(fitNeg, Math.Min(fitPos, fitZero));
* GEV bestModel = null;
* if (fitNeg == minScore) bestModel = bestNeg;
* if (fitPos == minScore) bestModel = bestPos;
* if (fitZero == minScore) bestModel = zeroModel; // Prefer zero, then pos
*
* Console.WriteLine($"Best Negative model: shape: {bestNeg.shape} scale: {bestNeg.scale} location: {bestNeg.location} fitness: {fitNeg}");
* Console.WriteLine($"Best Positive model: shape: {bestPos.shape} scale: {bestPos.scale} location: {bestPos.location} fitness: {fitPos}");
* Console.WriteLine($"Zero model: shape: {zeroModel.shape} scale: {zeroModel.scale} location: {zeroModel.location} fitness: {fitZero}");
*/
#endregion
double scaleGuess = Math.Sqrt(6 * Statistics.VarianceEstimate(observations)) / Math.PI;
double locationGuess = Statistics.Median(observations) + scaleGuess * Math.Log(Math.Log(2));
double shapeGuess = 0.5; // Use Pickands estimator here in the actual model
Func <Vector <double>, double> objectiveFunction = x => FitnessExactModel(new GEV(x[2], x[1], x[0], Program.rand));
GEV bestModelMonteCarlo = OptimizeBFGS(objectiveFunction, shapeGuess, scaleGuess, locationGuess);
output.WriteLine($"MC Exact GEV Model: shape{bestModelMonteCarlo.shape} location{bestModelMonteCarlo.location} scale {bestModelMonteCarlo.scale}");
double[] sample = new double[sampleSize];
dist.Samples(sample); // Take a sample from dist
Sorting.Sort(sample);
// Report the sample min and max
output.WriteLine($"Sample maximum: {sample[sample.Length - 1]}");
//var sorter = new List<double>(sample);
//sorter.Sort();
//sample = sorter.ToArray();
// Smoothed version
//double[] smoothedData = new double[sample.Length - 1];
//for (int i = 0; i < smoothedData.Length; i++) { smoothedData[i] = 0.5 * (sample[i] + sample[i + 1]); }
//var pickandsApprox = new PickandsApproximation(smoothedData, method: PickandsApproximation.FittingMethod.Pickands_SupNorm); // Construct a Pickands tail approx from the sample
var pickandsApprox = new GPDApproximation(sample, method: GPDApproximation.FittingMethod.V4); // Construct a Pickands tail approx from the sample
// Bootstrap observations of the distribution of the sample maximum from the Pickands model
double[] approxObservations = new double[observations.Length];
for (int i = 0; i < approxObservations.Length; i++)
{
double max = double.NegativeInfinity;
for (int j = 0; j < sampleSize; j++)
{
max = Math.Max(max, pickandsApprox.Sample());
}
approxObservations[i] = max;
}
ContinuousDistribution approxECDF = ContinuousDistribution.ECDF(approxObservations); // ECDF of the bootstrapped observations
//scaleGuess = Math.Sqrt(6 * Statistics.Variance(approxObservations)) / Math.PI;
//locationGuess = Statistics.Median(approxObservations) + scaleGuess * Math.Log(Math.Log(2));
// Guess location and scale
shapeGuess = pickandsApprox.c;
if (shapeGuess < 0)
{
double g1 = SpecialFunctions.Gamma(1 - shapeGuess);
double g2 = SpecialFunctions.Gamma(1 - 2 * shapeGuess);
scaleGuess = Math.Sqrt(Statistics.Variance(approxObservations) * shapeGuess * shapeGuess / (g2 - g1 * g1));
locationGuess = Statistics.Mean(approxObservations) - scaleGuess * (g1 - 1) / shapeGuess;
}
else
{
scaleGuess = Math.Sqrt(6 * Statistics.Variance(approxObservations)) / Math.PI;
locationGuess = Statistics.Median(approxObservations) + scaleGuess * Math.Log(Math.Log(2));
}
GEV estimatedGEVUnfitted = new GEV(location: locationGuess, scale: scaleGuess, shape: pickandsApprox.c); // Using the Pickands estimator for shape
output.WriteLine($"UnfittedGEVModel: shape{estimatedGEVUnfitted.shape} location{estimatedGEVUnfitted.location} scale {estimatedGEVUnfitted.scale}");
// Fit the model to the data drawn from the Pickands model
double FitnessApproxModel(GEV model)
{
double val = 0;
for (int i = 0; i < approxObservations.Length; i++)
{
val += Math.Pow(model.CumulativeDistribution(approxObservations[i]) - approxECDF.CumulativeDensity(approxObservations[i]), 2);
}
return(val);
}
objectiveFunction = x => FitnessApproxModel(new GEV(x[2], x[1], x[0], Program.rand));
GEV fittedApproxModel = OptimizeBFGS(objectiveFunction, pickandsApprox.c, scaleGuess, locationGuess);
output.WriteLine($"FittedGEVModel: shape{fittedApproxModel.shape} location{fittedApproxModel.location} scale {fittedApproxModel.scale}");
double[] proportions = Interpolation.Linspace(0.000001, 0.999999, 2000);
double[] observationQuantiles = Interpolation.Linspace(0.000001, 0.999999, 2000);
for (int i = 0; i < observationQuantiles.Length; i++)
{
observationQuantiles[i] = Statistics.Quantile(observations, observationQuantiles[i]);
}
output.WriteLine("Abscissas,Monte Carlo ECDF,GEV Fit of MC ECDF,Estimated ECDF,Estimated GEV Unfitted,Estimated GEV Fitted,,ErrDistExactAbscissas,ErrDistExactValues,ErrDistModelAbscissas,ErrDistModelValues,ErrDistUnfittedAbscissas,ErrDistUnfittedValues");
for (int i = 0; i < observationQuantiles.Length; i++)
{
output.WriteLine($"{observationQuantiles[i]}," +
$"{MonteCarloDistributionOfTheMaximum.CumulativeDensity(observationQuantiles[i])}," +
$"{bestModelMonteCarlo.CumulativeDistribution(observationQuantiles[i])}," +
$"{approxECDF.CumulativeDensity(observationQuantiles[i])}," +
$"{estimatedGEVUnfitted.CumulativeDistribution(observationQuantiles[i])}," +
$"{fittedApproxModel.CumulativeDistribution(observationQuantiles[i])}," +
$"," + // Space
$"{observationQuantiles[i] - upperQuantile}," +
$"{MonteCarloDistributionOfTheMaximum.CumulativeDensity(observationQuantiles[i])}," +
//$"{quantiles[i] - sample[sample.Length - 1]}," +
$"{estimatedGEVUnfitted.InverseCumulativeDistribution(proportions[i]) - estimatedGEVUnfitted.location}," +
$"{proportions[i]}," +
$"{fittedApproxModel.InverseCumulativeDistribution(proportions[i]) - fittedApproxModel.location}," +
$"{proportions[i]}");
}
double[] distributionQuantiles = Interpolation.Linspace(0.000001, 0.999999, 2000);
for (int i = 0; i < distributionQuantiles.Length; i++)
{
distributionQuantiles[i] = dist.InverseCumulativeDistribution(distributionQuantiles[i]);
}
output2.WriteLine("Abscissas,True CDF,Pickands Estimate");
for (int i = 0; i < distributionQuantiles.Length; i++)
{
output2.WriteLine($"{distributionQuantiles[i]}," +
$"{dist.CumulativeDistribution(distributionQuantiles[i])}," +
$"{pickandsApprox.CDF(distributionQuantiles[i])}");
}
#region Temp for figure
output2.WriteLine("");
output2.WriteLine("TrueDist");
output2.WriteLine("\\draw[line width=1.5pt]");
for (int i = 0; i < distributionQuantiles.Length - 1; i++)
{
output2.WriteLine($"({distributionQuantiles[i]},{dist.CumulativeDistribution(distributionQuantiles[i])}) --");
}
output2.WriteLine($"({distributionQuantiles[distributionQuantiles.Length - 1]},{dist.CumulativeDistribution(distributionQuantiles[distributionQuantiles.Length - 1])});");
output2.WriteLine("");
output2.WriteLine("Approx");
output2.WriteLine("\\draw[line width=1.5pt]");
for (int i = 0; i < distributionQuantiles.Length; i++)
{
output2.WriteLine($"({distributionQuantiles[i]},{pickandsApprox.CDF(distributionQuantiles[i])}) --");
}
output2.WriteLine($"({distributionQuantiles[distributionQuantiles.Length - 1]},{pickandsApprox.CDF(distributionQuantiles[distributionQuantiles.Length - 1])});");
#endregion
// Clean up
output.Dispose();
output2.Dispose();
//table.Dispose();
}
/// <summary>
/// Non-linear least-squares fitting the points (x,y) to an arbitrary function y : x -> f(p, x),
/// returning its best fitting parameter p.
/// https://github.com/mathnet/mathnet-numerics/issues/597
/// http://kb.en-mat.com/Curve%20Fitting_%20Linear%20Regression.pdf
/// </summary>
public static double Curve(double[] x, double[] y, Func <double, double, double> f, double initialGuess, double tolerance = 1e-8, int maxIterations = 1000)
{
return(FindMinimum.OfScalarFunction(p => Distance.Euclidean(Generate.Map(x, t => f(p, t)), y), initialGuess, tolerance, maxIterations));
}
public static Vector <double> Solve(
string interpolationName,
InitialTerm.InitialParameter param,
double forward,
double vtarget,
List <double> strikeList)
{
InterpolationName = interpolationName;
Param = param;
StrikeList = strikeList;
K25Call = strikeList[3];
KATM = strikeList[2];
K25Put = strikeList[1];
VolSS25 = param.volMS25;
var tol = 1.0E-5;
var error = 1.0;
var LowerVector = Vector <double> .Build.DenseOfArray(new double[] { 0.00001, 0.00001, -0.99999 });
var UpperVector = Vector <double> .Build.DenseOfArray(new double[] { 100, 100, 0.99999 });
VolParameter = Vector <double> .Build.DenseOfArray(new double[] { 0.1, 0.1, -0.1 });
while (Math.Abs(error) > tol)
{
if (InterpolationName == "SABR")
{
VolParameter = FindMinimum.OfFunctionConstrained(
FuncOfVolCondtion,
LowerVector,
UpperVector,
VolParameter);
//VolParameter = FindMinimum.OfFunction(FuncOfVolCondtion, VolParameter);
}
else
{
VolParameter = FindMinimum.OfFunction(FuncOfVolCondtion, VolParameter);
}
CalcStrikeList(VolParameter);
var vTrial = CalcValueTrial(VolParameter);
error = vTrial - vtarget;
if (error > 0)
{
VolSS25 -= tol / 100;
}
else
{
VolSS25 += tol / 100;
}
}
var volRR25 = VolCalculator(K25Call, VolParameter) - VolCalculator(K25Put, VolParameter);
var Ans = Vector <double> .Build.DenseOfArray(new double[]
{
VolParameter[0],
VolParameter[1],
VolParameter[2],
K25Call,
K25Put,
VolSS25,
volRR25
});
return(Ans);
}
public static double Minimize(Func <double, double> fn, double x0, double tolerance = 1E-8)
{
return(FindMinimum.OfScalarFunction(fn, x0, tolerance));
}
// Data must be sorted before using this method
internal static void ApproximateExcessDistributionParametersBFGS(List <double> data, out double a, out double c, out double u)
{
double Fitness(Vector <double> input) // Input is assumed to be (a,c,u)
{
double sum = 0;
//double weightsum = 0;
// Compute the index of the next largest element that is at or after u in the data
int nextLargestIndex = data.BinarySearch(input[2]);
if (nextLargestIndex < 0)
{
nextLargestIndex = ~nextLargestIndex + 1;
}
int knotCount = data.Count - nextLargestIndex;
/* ECDF version MSE
* for (int i = 0; i < knotCount; i++)
* {
* // The largest deviation should occur at one of the step points
* double GHat = TailCDF(data[nextLargestIndex + i] - input[2], input[0], input[1]); // Args: x - u, a, c
* double residual = i * 1.0 / knotCount - GHat; // Deviation from the top of the step at x_i
* //double weight = knotCount - i + 1;
* sum += residual * residual;
* //sum += Math.Abs(residual) * weight;
* //weightsum += weight;
* }
* return sum / knotCount; // Consider dividing by n or n^2 here
*/
//return sum / (weightsum * knotCount);
// Smoothed version MSE
for (int i = 0; i < knotCount - 1; i++)
{
double GHat = TailCDF(0.5 * (data[nextLargestIndex + i] + data[nextLargestIndex + i + 1]) - input[2], input[0], input[1]);
double residual = (2.0 * i + 3) / (2.0 * knotCount) - GHat;
sum += residual * residual;
}
//return sum / knotCount;
return((1 + Math.Abs(input[1])) * sum / knotCount); // Weighted so that smaller magnitudes of c are preferred
}
// Get Pickands' estimates of a and c for m = n/16 + 1 as starting guesses, consistent with Z4M at the 75th percentile
//double pickandsEstA, pickandsEstC;
EstimateParams(data, data.Count / 16 + 1, out double pickandsEstC, out double pickandsEstA);
double lowerBoundA = 0;
double upperBoundA = 3 * pickandsEstA + 1;
double lowerBoundC = Math.Min(3 * pickandsEstC, -3 * pickandsEstC) - 1;
double upperBoundC = -lowerBoundC;
// Initial guess for u is at data[(3/4)n]
var optimum = FindMinimum.OfFunctionConstrained(Fitness,
lowerBound: CreateVector.DenseOfArray(new double[] { lowerBoundA, lowerBoundC, data[0] }),
upperBound: CreateVector.DenseOfArray(new double[] { upperBoundA, upperBoundC, data[data.Count - 3] }),
initialGuess: CreateVector.DenseOfArray(new double[] { pickandsEstA, pickandsEstC /*Math.Min(pickandsEstC, 0)*/, data[data.Count * 3 / 4] }));
// Return parameters
a = optimum[0];
c = optimum[1];
u = optimum[2];
}
