source: tags/Mars-V0.9.1/mbase/MMath.cc

/* ======================================================================== *\
!
! *
! * This file is part of MARS, the MAGIC Analysis and Reconstruction
! * Software. It is distributed to you in the hope that it can be a useful
! * and timesaving tool in analysing Data of imaging Cerenkov telescopes.
! * It is distributed WITHOUT ANY WARRANTY.
! *
! * Permission to use, copy, modify and distribute this software and its
! * documentation for any purpose is hereby granted without fee,
! * provided that the above copyright notice appear in all copies and
! * that both that copyright notice and this permission notice appear
! * in supporting documentation. It is provided "as is" without express
! * or implied warranty.
! *
!
!
!   Author(s): Thomas Bretz  3/2004 <mailto:tbretz@astro.uni-wuerzburg.de>
!
!   Copyright: MAGIC Software Development, 2000-2004
!
!
\* ======================================================================== */

/////////////////////////////////////////////////////////////////////////////
//
// MMath
//
/////////////////////////////////////////////////////////////////////////////
#include "MMath.h"

// --------------------------------------------------------------------------
//
// Calculate Significance as
// significance = (s-b)/sqrt(s+k*k*b) mit k=s/b
//
// s: total number of events in signal region
// b: number of background events in signal region
// 
Double_t MMath::Significance(Double_t s, Double_t b)
{
    const Double_t k = b==0 ? 0 : s/b;
    const Double_t f = s+k*k*b;

    return f==0 ? 0 : (s-b)/TMath::Sqrt(f);
}

// --------------------------------------------------------------------------
//
// Symmetrized significance - this is somehow analog to
// SignificanceLiMaSigned
//
// Returns Significance(s,b) if s>b otherwise -Significance(b, s);
// 
Double_t MMath::SignificanceSym(Double_t s, Double_t b)
{
    return s>b ? Significance(s, b) : -Significance(b, s);
}

// --------------------------------------------------------------------------
//
//  calculates the significance according to Li & Ma
//  ApJ 272 (1983) 317, Formula 17
//
//  s                    // s: number of on events
//  b                    // b: number of off events
//  alpha = t_on/t_off;  // t: observation time
//
//  The significance has the same (positive!) value for s>b and b>s.
//
//  Returns -1 if sum<0 or alpha<0 or the argument of sqrt<0
//  Returns  0 if s+b==0, s==0 or b==0
//
// Here is some eMail written by Daniel Mazin about the meaning of the arguments:
//
//  > Ok. Here is my understanding:
//  > According to Li&Ma paper (correctly cited in MMath.cc) alpha is the
//  > scaling factor. The mathematics behind the formula 17 (and/or 9) implies
//  > exactly this. If you scale OFF to ON first (using time or using any other
//  > method), then you cannot use formula 17 (9) anymore. You can just try
//  > the formula before scaling (alpha!=1) and after scaling (alpha=1), you
//  > will see the result will be different.
//
//  > Here are less mathematical arguments:
//
//  >  1) the better background determination you have (smaller alpha) the more
//  > significant is your excess, thus your analysis is more sensitive. If you
//  > normalize OFF to ON first, you loose this sensitivity.
//
//  >  2) the normalization OFF to ON has an error, which naturally depends on
//  > the OFF and ON. This error is propagating to the significance of your
//  > excess if you use the Li&Ma formula 17 correctly. But if you normalize
//  > first and use then alpha=1, the error gets lost completely, you loose
//  > somehow the criteria of goodness of the normalization.
//
Double_t MMath::SignificanceLiMa(Double_t s, Double_t b, Double_t alpha)
{
    const Double_t sum = s+b;

    if (s==0 || b==0 || sum==0)
        return 0;

    if (sum<0 || alpha<=0)
        return -1;

    const Double_t l = s*TMath::Log(s/sum*(alpha+1)/alpha);
    const Double_t m = b*TMath::Log(b/sum*(alpha+1)      );

    return l+m<0 ? -1 : TMath::Sqrt((l+m)*2);
}

// --------------------------------------------------------------------------
//
// Calculates MMath::SignificanceLiMa(s, b, alpha). Returns 0 if the
// calculation has failed. Otherwise the Li/Ma significance which was
// calculated. If s<b a negative value is returned.
//
Double_t MMath::SignificanceLiMaSigned(Double_t s, Double_t b, Double_t alpha)
{
    const Double_t sig = SignificanceLiMa(s, b, alpha);
    if (sig<=0)
        return 0;

    return TMath::Sign(sig, s-alpha*b);
}

// --------------------------------------------------------------------------
//
// Returns: 2/(sigma*sqrt(2))*integral[0,x](exp(-(x-mu)^2/(2*sigma^2)))
//
Double_t MMath::GaussProb(Double_t x, Double_t sigma, Double_t mean)
{
    static const Double_t sqrt2 = TMath::Sqrt(2.);
    return TMath::Erf((x-mean)/(sigma*sqrt2));
}

// --------------------------------------------------------------------------
//
// This function reduces the precision to roughly 0.5% of a Float_t by
// changing its bit-pattern (Be carefull, in rare cases this function must
// be adapted to different machines!). This is usefull to enforce better
// compression by eg. gzip.
//
void MMath::ReducePrecision(Float_t &val)
{
    UInt_t &f = (UInt_t&)val;

    f += 0x00004000;
    f &= 0xffff8000;
}

// -------------------------------------------------------------------------
//
// Quadratic interpolation
//
// calculate the parameters of a parabula such that
//    y(i) = a + b*x(i) + c*x(i)^2
//
// If the determinant==0 an empty TVector3 is returned.
//
TVector3 MMath::GetParab(const TVector3 &x, const TVector3 &y)
{
    Double_t x1 = x(0);
    Double_t x2 = x(1);
    Double_t x3 = x(2);

    Double_t y1 = y(0);
    Double_t y2 = y(1);
    Double_t y3 = y(2);

    const double det =
        + x2*x3*x3 + x1*x2*x2 + x3*x1*x1
        - x2*x1*x1 - x3*x2*x2 - x1*x3*x3;


    if (det==0)
        return TVector3();

    const double det1 = 1.0/det;

    const double ai11 = x2*x3*x3 - x3*x2*x2;
    const double ai12 = x3*x1*x1 - x1*x3*x3;
    const double ai13 = x1*x2*x2 - x2*x1*x1;

    const double ai21 = x2*x2 - x3*x3;
    const double ai22 = x3*x3 - x1*x1;
    const double ai23 = x1*x1 - x2*x2;

    const double ai31 = x3 - x2;
    const double ai32 = x1 - x3;
    const double ai33 = x2 - x1;

    return TVector3((ai11*y1 + ai12*y2 + ai13*y3) * det1,
                    (ai21*y1 + ai22*y2 + ai23*y3) * det1,
                    (ai31*y1 + ai32*y2 + ai33*y3) * det1);
}

Double_t MMath::InterpolParabLin(const TVector3 &vx, const TVector3 &vy, Double_t x)
{
    const TVector3 c = GetParab(vx, vy);
    return c(0) + c(1)*x + c(2)*x*x;
}

Double_t MMath::InterpolParabLog(const TVector3 &vx, const TVector3 &vy, Double_t x)
{
    const Double_t l0 = TMath::Log10(vx(0));
    const Double_t l1 = TMath::Log10(vx(1));
    const Double_t l2 = TMath::Log10(vx(2));

    const TVector3 vx0(l0, l1, l2);
    return InterpolParabLin(vx0, vy, TMath::Log10(x));
}

Double_t MMath::InterpolParabCos(const TVector3 &vx, const TVector3 &vy, Double_t x)
{
    const Double_t l0 = TMath::Cos(vx(0));
    const Double_t l1 = TMath::Cos(vx(1));
    const Double_t l2 = TMath::Cos(vx(2));

    const TVector3 vx0(l0, l1, l2);
    return InterpolParabLin(vx0, vy, TMath::Cos(x));
}