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/*
Copyright (C) 2000, 2001, 2002 RiskMap srl
This file is part of QuantLib, a free-software/open-source library
for financial quantitative analysts and developers - http://quantlib.org/
QuantLib is free software: you can redistribute it and/or modify it under the
terms of the QuantLib license. You should have received a copy of the
license along with this program; if not, please email ferdinando@ametrano.net
The license is also available online at http://quantlib.org/html/license.html
This program is distributed in the hope that it will be useful, but WITHOUT
ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
FOR A PARTICULAR PURPOSE. See the license for more details.
*/
/*! \file statistics.hpp
\brief statistic tools
\fullpath
ql/Math/%statistics.hpp
*/
// $Id: statistics.hpp,v 1.11 2002/01/16 14:42:29 nando Exp $
#ifndef quantlib_statistic_h
#define quantlib_statistic_h
#include <ql/null.hpp>
#include <ql/dataformatters.hpp>
#include <iostream>
namespace QuantLib {
namespace Math {
//! Statistic tool
/*! It can accumulate a set of data and return statistic quantities
as mean, variance, std. deviation, skewness, and kurtosis.
*/
class Statistics {
public:
Statistics();
//! \name Inspectors
//@{
//! number of samples collected
Size samples() const;
//! sum of data weights
double weightSum() const;
/*! returns the mean, defined as
\f[ \langle x \rangle = \frac{\sum w_i x_i}{\sum w_i}. \f]
*/
double mean() const;
/*! returns the variance, defined as
\f[ \frac{N}{N-1} \left\langle \left(
x-\langle x \rangle \right)^2 \right\rangle. \f]
*/
double variance() const;
/*! returns the standard deviation \f$ \sigma \f$, defined as the
square root of the variance.
*/
double standardDeviation() const;
/*! returns the downside variance, defined as
\f[ \frac{N}{N-1} \times \frac{ \sum_{i=1}^{N}
\theta \times x_i^{2}}{ \sum_{i=1}^{N} w_i} \f],
where \f$ \theta \f$ = 0 if x > 0 and \f$ \theta \f$ =1 if x <0
*/
double downsideVariance() const;
/*! returns the downside deviation, defined as the
square root of the downside variance.
*/
double downsideDeviation() const;
/*! returns the error estimate \f$ \epsilon \f$, defined as the
square root of the ratio of the variance to the number of
samples.
*/
double errorEstimate() const;
/*! returns the skewness, defined as
\f[ \frac{N^2}{(N-1)(N-2)} \frac{\left\langle \left(
x-\langle x \rangle \right)^3 \right\rangle}{\sigma^3}. \f]
The above evaluates to 0 for a Gaussian distribution.
*/
double skewness() const;
/*! returns the excess kurtosis, defined as
\f[ \frac{N(N+1)}{(N-1)(N-2)(N-3)}
\frac{\left\langle \left( x-\langle x \rangle \right)^4
\right\rangle}{\sigma^4} - \frac{3(N-1)^2}{(N-2)(N-3)}. \f]
The above evaluates to 0 for a Gaussian distribution.
*/
double kurtosis() const;
/*! returns the minimum sample value */
double min() const;
/*! returns the maximum sample value */
double max() const;
//@}
//! \name Modifiers
//@{
//! adds a datum to the set, possibly with a weight
void add(double value, double weight = 1.0);
//! adds a sequence of data to the set
template <class DataIterator>
void addSequence(DataIterator begin, DataIterator end) {
for (;begin!=end;++begin)
add(*begin);
}
//! adds a sequence of data to the set, each with its weight
template <class DataIterator, class WeightIterator>
void addSequence(DataIterator begin, DataIterator end,
WeightIterator wbegin) {
for (;begin!=end;++begin,++wbegin)
add(*begin, *wbegin);
}
//! resets the data to a null set
void reset();
//@}
private:
Size sampleNumber_;
double sampleWeight_;
double sum_, quadraticSum_, downsideQuadraticSum_,
cubicSum_, fourthPowerSum_;
double min_, max_;
};
// inline definitions
/*! \pre weights must be positive or null */
inline void Statistics::add(double value, double weight) {
QL_REQUIRE(weight>=0.0,
"Statistics::add : negative weight (" +
DoubleFormatter::toString(weight) + ") not allowed");
Size oldSamples = sampleNumber_;
sampleNumber_++;
QL_ENSURE(sampleNumber_ > oldSamples,
"Statistics::add : maximum number of samples reached");
sampleWeight_ += weight;
double temp = weight*value;
sum_ += temp;
temp *= value;
quadraticSum_ += temp;
downsideQuadraticSum_ += value < 0.0 ? temp : 0.0;
temp *= value;
cubicSum_ += temp;
temp *= value;
fourthPowerSum_ += temp;
min_=QL_MIN(value, min_);
max_=QL_MAX(value, max_);
}
inline Size Statistics::samples() const {
return sampleNumber_;
}
inline double Statistics::weightSum() const {
return sampleWeight_;
}
inline double Statistics::mean() const {
QL_REQUIRE(sampleWeight_>0.0,
"Stat::mean() : sampleWeight_=0, unsufficient");
return sum_/sampleWeight_;
}
inline double Statistics::variance() const {
QL_REQUIRE(sampleWeight_>0.0,
"Stat::variance() : sampleWeight_=0, unsufficient");
QL_REQUIRE(sampleNumber_>1,
"Stat::variance() : sample number <=1, unsufficient");
return (sampleNumber_/(sampleNumber_-1.0))*
(quadraticSum_ - sum_*sum_/sampleWeight_)/sampleWeight_;
}
inline double Statistics::standardDeviation() const {
return QL_SQRT(variance());
}
inline double Statistics::downsideVariance() const {
QL_REQUIRE(sampleWeight_>0.0,
"Stat::variance() : sampleWeight_=0, unsufficient");
QL_REQUIRE(sampleNumber_>1,
"Stat::variance() : sample number <=1, unsufficient");
return sampleNumber_/(sampleNumber_-1.0)*
downsideQuadraticSum_ /sampleWeight_;
}
inline double Statistics::downsideDeviation() const {
return QL_SQRT(downsideVariance());
}
inline double Statistics::errorEstimate() const {
double var = variance();
QL_REQUIRE(samples() > 0,
"Statistics: zero samples are not sufficient");
if(QL_FABS(var) < 1e-12) var =0.0;
QL_REQUIRE(var >= 0.0,
"Statistics: variance, " +
DoubleFormatter::toString(var,20)
+" is NEGATIVE");
return QL_SQRT(var/samples());
}
inline double Statistics::skewness() const {
QL_REQUIRE(sampleNumber_>2,
"Stat::skewness() : sample number <=2, unsufficient");
double s = standardDeviation();
if (s==0.0) return 0.0;
double m = mean();
return sampleNumber_*sampleNumber_/
((sampleNumber_-1.0)*(sampleNumber_-2.0)*s*s*s)*
(cubicSum_-3.0*m*quadraticSum_+2.0*m*m*sum_)/sampleWeight_;
}
inline double Statistics::kurtosis() const {
QL_REQUIRE(sampleNumber_>3,
"Stat::kurtosis() : sample number <=3, unsufficient");
double m = mean();
double v = variance();
if (v==0)
return - 3.0*(sampleNumber_-1.0)*(sampleNumber_-1.0) /
((sampleNumber_-2.0)*(sampleNumber_-3.0));
return sampleNumber_*sampleNumber_*(sampleNumber_+1.0) /
((sampleNumber_-1.0)*(sampleNumber_-2.0) *
(sampleNumber_-3.0)*v*v) *
(fourthPowerSum_ - 4.0*m*cubicSum_ + 6.0*m*m*quadraticSum_ -
3.0*m*m*m*sum_)/sampleWeight_ -
3.0*(sampleNumber_-1.0)*(sampleNumber_-1.0) /
((sampleNumber_-2.0)*(sampleNumber_-3.0));
}
inline double Statistics::min() const {
QL_REQUIRE(sampleNumber_>0, "Stat::min_() : empty sample");
return min_;
}
inline double Statistics::max() const {
QL_REQUIRE(sampleNumber_>0, "Stat::max_() : empty sample");
return max_;
}
}
}
#endif
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