JMARKOV::JMarkovPathGenerator Class Reference

The JMarkovPathGenerator generates ensembles of photon paths using a Markov Chain Monte Carlo (MCMC). More...

#include <JMarkovPathGenerator.hh>

Public Member Functions
	JMarkovPathGenerator ()
	standard constructor More...
std::vector< JPhotonPath >	generateEnsemble (int n, const JPhotonPath &start_path, JSourceModel *src, JScatteringModel *sm, JTargetModel *trg, double lambda_abs, int nsteps_burn_in, int nsteps_save)
	Generate an ensemble of n paths with a fixed number of scatterings by MCMC-sampling the given scattering model. More...
int	getNsteps ()
	get the number of Markov steps taken in the last call to generateEnsemble (after burn-in) More...
int	getNacceptedSteps ()
	get the number of accepted steps taken during the last call to generateEnsemble (after burn-in) More...
int	getNrejectedSteps ()
	get the number of rejected steps during the last call to generateEnsemble (after burn-in) More...
double	getFractionAccepted ()
	get the fraction of accepted steps during the last call to generateEnsemble (after burn-in) More...
double	getFractionAccepted_radial ()
	get the fraction of steps that were accepted when a radial step was performed during the last call to generateEnsemble (after burn-in) More...
double	getFractionAccepted_tangential ()
	get the fraction of steps that were accepted when a tangential step was performed during the last call to generateEnsemble (after burn-in) More...
bool	doMarkovStep (JSourceModel *src, JScatteringModel *sm, JTargetModel *trg, double lambda_abs, JPhotonPath &p)
	make one Markov chain step for a path More...
virtual int	randomPathChange (JPhotonPath &p, JTargetModel *trg)
void	setTargetStepSize_deg (double val)
	set the average step size in degrees for the impact point on the target More...
void	setRadialStepSize_m (double val)
	set the average step size in [m] in the radial direction for the scattering vertices More...
void	setTangentialStepSize_deg (double val)
	Set the average step size theta in degrees for steps in the tangential direction for the scattering vertices. More...
void	setCoordinateRemapping (bool val=true)
	activate or deactive coordinate remapping More...
bool	getCoordinateRemapping ()
	returns true when coordinate remapping is activated, false otherwise More...

Private Member Functions
double	getR0 ()
	return R0; the length scale used in the coordinate remapping More...
double	getRemappedDistance (double r)
	return the distance between the remapped vertex and the previous vertex given the distance between the not-remapped vertex and the previous vertex More...
double	getUnmappedDistance (double rprime)
	inverse of getRemappedDistance More...
JPosition3D	getRemappedPosition (const JPosition3D &xprev, const JPosition3D &x)
	Return the remapped vertex position of x. More...
JPosition3D	getUnmappedPosition (const JPosition3D &xprev, const JPosition3D &xprime)
	Inverse of getRemappedPosition. More...
JPhotonPath	getRemappedPhotonPath (const JPhotonPath &p)
	returns a remapped version of the photon path More...
JPhotonPath	getUnmappedPhotonPath (const JPhotonPath &pprime)
	inverse of getRemappedPhotonPath More...
double	getRemappingCorrection (const JPosition3D &xprev, const JPosition3D &xprime)
	Returns the conversion factor J needed to compute the path probability density in the remapped coordinates, for a given vertex. More...
double	getRemappingCorrection (const JPhotonPath &p, const JPhotonPath &pprime)
	Return the remapping correction for an entire photon path (product of the remapping correction for the individual scattering vertices). More...

Private Attributes
bool	apply_coordinate_remapping
int	naccepted_r
int	nrejected_r
int	naccepted_t
int	nrejected_t
double	stepsize_angle_target
double	stepsize_r
double	stepsize_angle

Detailed Description

The JMarkovPathGenerator generates ensembles of photon paths using a Markov Chain Monte Carlo (MCMC).

In the Markov chain, the number of scatterings does not change. Therefore the ensembles have to be generated separately for each given number of scatterings.

The small modification of the path that is proposed in each step, is defined by the function 'randomPathChange'. This function is virtual so that it can be overrided in a derived class. The initial and final vertex are fixed and one of the other vertices is displaced in a random direction following an exponential distribution.

The source, target and scattering models are used to calculate the probability density of the path.

Random numbers are drawn from gRandom.

By default, the coordinates are remapped such that the 1/r^2 singularities in the path probability density disappear. This option can be turned off (but do not do it, unless you want to convince yourself that it is necessary) with the setCoordinateRemapping method.

Definition at line 50 of file JMarkovPathGenerator.hh.

Constructor & Destructor Documentation

JMARKOV::JMarkovPathGenerator::JMarkovPathGenerator ( )

inline

standard constructor

Definition at line 54 of file JMarkovPathGenerator.hh.

                           {
      naccepted_r = -1 ;
      nrejected_r = -1 ;
      naccepted_t = -1 ;
      nrejected_t = -1 ;
      setTargetStepSize_deg(1.0) ;
      setRadialStepSize_m(1.0) ;
      setTangentialStepSize_deg(1.0) ;
      setCoordinateRemapping(true) ;
    }

Member Function Documentation

std::vector< JPhotonPath > JMARKOV::JMarkovPathGenerator::generateEnsemble	(	int	n,
		const JPhotonPath &	start_path,
		JSourceModel *	src,
		JScatteringModel *	sm,
		JTargetModel *	trg,
		double	lambda_abs,
		int	nsteps_burn_in,
		int	nsteps_save
	)

Generate an ensemble of n paths with a fixed number of scatterings by MCMC-sampling the given scattering model.

The paths start at the source (a well-defined position) and end on the target (either a fixed position or, in the case of a finite source, the surface of a sphere).

start_path is a photon path that is used a starting point for the MCMC. It has to be supplied by the user (due to the various source and target profiles it is not straightforward to generate one automatically) and must have a path probability density != 0. The number of scatterings is also defined by the start_path.

nsteps_burn_in is the number of steps taken for burn-in.

nsteps_save determines the number of MCMC steps between the paths that are saved to the ensemble.

Definition at line 329 of file JMarkovPathGenerator.hh.

                                                                                                                                                                                                                           {
    vector<JPhotonPath> result ;

    JPhotonPath testpath(start_path) ;

    // check that the starting path is okay
    double dist = testpath[0].getDistanceSquared(src->getPosition()) ;
    if( dist != 0 ) {
      cerr << "ERROR in generateEnsemble: testpath position " << testpath[0] 
           << "does not match source position " << src->getPosition() << "!" << endl ;
      return result ;
    }
    {
      double rho = getPhotonPathProbabilityDensity(testpath,src,sm,trg,lambda_abs) ;
      if( rho  == 0 ) {
        cerr << "ERROR in generateEnsemble: testpath probability density = 0" << endl ;
        return result ;
      }
    }

    // convert the path to remapped coordinates
    if( apply_coordinate_remapping ) testpath = getRemappedPhotonPath(testpath) ;

    // burn-in
    int i = 0 ;
    while( i!=nsteps_burn_in ) {
      doMarkovStep(src,sm,trg,lambda_abs,testpath) ; 
      ++i ;
    }

    // ensemble generation
    i = 0 ;
    naccepted_r = 0 ;
    nrejected_r = 0 ;
    naccepted_t = 0 ;
    nrejected_t = 0 ;
    while( i!=n ) {
      // take nsteps_save steps
      int j = 0 ;
      while( j!=nsteps_save ) {
        if( doMarkovStep(src,sm,trg,lambda_abs,testpath) ) {
          ++j ;
        }
      }
      // save the path to the ensemble
      if( apply_coordinate_remapping ) {
        result.push_back( getUnmappedPhotonPath(testpath) ) ;
      } else {
        result.push_back( testpath ) ;
      }
      ++i ;
    }
    
    return result ;
  }

int JMARKOV::JMarkovPathGenerator::getNsteps ( )

inline

get the number of Markov steps taken in the last call to generateEnsemble (after burn-in)

Definition at line 86 of file JMarkovPathGenerator.hh.

{ return naccepted_r + naccepted_t + nrejected_r + nrejected_t ; }

int JMARKOV::JMarkovPathGenerator::getNacceptedSteps ( )

inline

get the number of accepted steps taken during the last call to generateEnsemble (after burn-in)

Definition at line 89 of file JMarkovPathGenerator.hh.

{ return naccepted_r + naccepted_t ; }

int JMARKOV::JMarkovPathGenerator::getNrejectedSteps ( )

inline

get the number of rejected steps during the last call to generateEnsemble (after burn-in)

Definition at line 92 of file JMarkovPathGenerator.hh.

{ return nrejected_r + nrejected_t ; }

double JMARKOV::JMarkovPathGenerator::getFractionAccepted ( )

inline

get the fraction of accepted steps during the last call to generateEnsemble (after burn-in)

Definition at line 95 of file JMarkovPathGenerator.hh.

                                 { 
      if( getNsteps() <= 0 ) return 0 ;
      return 1.0*getNacceptedSteps()/getNsteps() ; 
    }

double JMARKOV::JMarkovPathGenerator::getFractionAccepted_radial ( )

inline

get the fraction of steps that were accepted when a radial step was performed during the last call to generateEnsemble (after burn-in)

Definition at line 101 of file JMarkovPathGenerator.hh.

                                        { 
      if( naccepted_r + nrejected_r <= 0 ) return 0 ;
      return 1.0*naccepted_r/(naccepted_r+nrejected_r) ; 
    }

double JMARKOV::JMarkovPathGenerator::getFractionAccepted_tangential ( )

inline

get the fraction of steps that were accepted when a tangential step was performed during the last call to generateEnsemble (after burn-in)

Definition at line 107 of file JMarkovPathGenerator.hh.

                                            { 
      if( naccepted_t+nrejected_t <= 0 ) return 0 ;
      return 1.0*naccepted_t/(naccepted_t+nrejected_t) ; 
    }

bool JMARKOV::JMarkovPathGenerator::doMarkovStep	(	JSourceModel *	src,
		JScatteringModel *	sm,
		JTargetModel *	trg,
		double	lambda_abs,
		JPhotonPath &	p
	)

make one Markov chain step for a path

Definition at line 385 of file JMarkovPathGenerator.hh.

                                                                                                                                         {
    int nscat = p.n-2 ;
    if( nscat == 0 && !trg->getRadius()>0 ) {
      return false ;
    }

    // copy path
    JPhotonPath copy = p ;

    double rhoBefore ;
    if( !apply_coordinate_remapping ) {
      rhoBefore = getPhotonPathProbabilityDensity(copy,src,sm,trg,lambda_abs) ;
    } else {
      // we are working in remapped coordinates, so we have to calculate a corrected
      // probability density
      JPhotonPath copy_unmapped = getUnmappedPhotonPath(copy) ;
      rhoBefore = getPhotonPathProbabilityDensity(copy_unmapped,src,sm,trg,lambda_abs) ;
      rhoBefore *= getRemappingCorrection(copy_unmapped,copy) ;
    }

    if( rhoBefore == 0 ) {
      cerr << "FATAL ERROR: starting probability density = 0" << endl ;
      exit(1) ;
    }

    // apply random change to the path
    int type = randomPathChange(copy,trg) ;

    // recalculate log L
    double rhoAfter ;
    if( !apply_coordinate_remapping ) {
      rhoAfter = getPhotonPathProbabilityDensity(copy,src,sm,trg,lambda_abs) ;
    } else {
      // we are working in remapped coordinates, so we have to calculate a corrected
      // probability density
      JPhotonPath copy_unmapped = getUnmappedPhotonPath(copy) ;
      rhoAfter = getPhotonPathProbabilityDensity(copy_unmapped,src,sm,trg,lambda_abs) ;
      rhoAfter *= getRemappingCorrection(copy_unmapped,copy) ;
    }

    // use Metropolis-Hastings algorithm to keep or reject change
    if( rhoAfter > rhoBefore ) {
      // accept the change
      p = copy ;
      if( type <  2 ) ++naccepted_r ;
      if( type == 2 ) ++naccepted_t ;
      return true ;
    } else {
      // accept the change with some probability
      double P = rhoAfter/rhoBefore ;
      if( gRandom->Uniform()<P ) {
        p = copy ;
        if( type <  2 ) ++naccepted_r ;
        if( type == 2 ) ++naccepted_t ;
        return true ;
      }
    }
    if( type <  2 ) ++nrejected_r ;
    if( type == 2 ) ++nrejected_t ;
    return false ;
  }

int JMARKOV::JMarkovPathGenerator::randomPathChange ( JPhotonPath &  p, JTargetModel *  trg  )

virtual

Definition at line 264 of file JMarkovPathGenerator.hh.

                                                                                {
    int nscat = p.n-2 ;
    int type = 0 ;

    if( nscat>0 ) {
      // pick vertex from 1 to nscat+1 (so excluding the start and end point)
      int nv = gRandom->Integer(nscat)+1 ;

      // distance w.r.t. previous vertex
      double r = p[nv-1].getDistance(p[nv]) ;

      if( gRandom->Uniform() > 0.5 ) {
        // with P=1/2, take a step in the radial direction
        type = 1 ;

        // to avoid nasty phase space Jacobian issues, I use this rather primitive method
        // where I simply generate a new position with a step in a random direction, then
        // only apply the radial part of the change
        //
        // NOTE: simply choosing 
        // *) r+dr with P proportional to  (r+dr)^2 
        // *) r-dr with P proportional to  (r-dr)^2
        // does not work. Trust me. I've tried.
        double _r = gRandom->Exp(stepsize_r) ;
        double _x, _y, _z ;
        gRandom->Sphere( _x, _y, _z, _r ) ;
        double _rnew = ( JPosition3D(_x,_y,_z) + JPosition3D(0,0,r) ).getLength() ;
        double dr = _rnew - r ;

        p[nv] = p[nv] + dr * JPosition3D(JVersor3D(p[nv]-p[nv-1])) ;

      } else {
        // with P=1/2, take a step in the tangential direction
        type = 2 ;

        // for large radii, limit the maximum angle over which we can move
        // so that the average change in distance does not exceed stepsize_r too much
        double _stepsize_angle = min(stepsize_angle,stepsize_r/r) ;
        double theta = gRandom->Exp(_stepsize_angle) ;
        double phi   = gRandom->Uniform(2.0*M_PI) ;
        JVersor3D dir( sin(theta)*cos(phi), sin(theta)*sin(phi), cos(theta) ) ;
        JVersor3D xdir( p[nv] - p[nv-1] ) ;
        JRotation3D rot(xdir) ;
        dir = dir.rotate_back(rot) ;
        p[nv] = p[nv-1] + r * JVector3D(dir) ;
      }
    }

    // in case of a finite target, also move the end point
    if( trg->getRadius() > 0 ) {
      const double dct = 1-cos(stepsize_angle_target) ;
      double ct = -2 ;
      while( ct<=-1 ) ct = 1-gRandom->Exp(dct) ;
      double theta = acos(ct) ;
      double phi = gRandom->Uniform(2*M_PI) ;
      JVersor3D original_dir( p.back()-trg->getPosition() ) ;
      const JRotation3D R( original_dir ) ;
      JVersor3D new_dir( JAngle3D(theta,phi) ) ;
      new_dir = new_dir.rotate_back(R) ;
      p.back() = trg->getPosition() + trg->getRadius() * JVector3D(new_dir) ;
    }

    return type ;
  }

void JMARKOV::JMarkovPathGenerator::setTargetStepSize_deg ( double  val )

inline

set the average step size in degrees for the impact point on the target

Definition at line 118 of file JMarkovPathGenerator.hh.

                                             {
      stepsize_angle_target = val * M_PI / 180 ;
    }

void JMARKOV::JMarkovPathGenerator::setRadialStepSize_m ( double  val )

inline

set the average step size in [m] in the radial direction for the scattering vertices

Definition at line 123 of file JMarkovPathGenerator.hh.

                                           {
      stepsize_r = val ;
    }

void JMARKOV::JMarkovPathGenerator::setTangentialStepSize_deg ( double  val )

inline

Set the average step size theta in degrees for steps in the tangential direction for the scattering vertices.

When r * theta is larger than the radial step size, the effective theta will be temporarily set to a smaller value to avoid taking steps that are too large.

Definition at line 133 of file JMarkovPathGenerator.hh.

                                                 {
      stepsize_angle = val * M_PI / 180 ;
    }

void JMARKOV::JMarkovPathGenerator::setCoordinateRemapping ( bool  val = true )

inline

activate or deactive coordinate remapping

Definition at line 138 of file JMarkovPathGenerator.hh.

                                                 {
      apply_coordinate_remapping = val ;
    }

bool JMARKOV::JMarkovPathGenerator::getCoordinateRemapping ( )

inline

returns true when coordinate remapping is activated, false otherwise

Definition at line 143 of file JMarkovPathGenerator.hh.

{ return apply_coordinate_remapping ; }

double JMARKOV::JMarkovPathGenerator::getR0 ( )

inlineprivate

return R0; the length scale used in the coordinate remapping

Definition at line 149 of file JMarkovPathGenerator.hh.

{ return 3 * stepsize_r ; }

double JMARKOV::JMarkovPathGenerator::getRemappedDistance ( double  r )

inlineprivate

return the distance between the remapped vertex and the previous vertex given the distance between the not-remapped vertex and the previous vertex

Definition at line 155 of file JMarkovPathGenerator.hh.

                                         {
      if( 3.0*sqrt(3)*r < getR0() ) return pow(r*getR0()*getR0(),1.0/3.0) ;
      return r + 2.0 / (3.0*sqrt(3.0))*getR0() ;
    }

double JMARKOV::JMarkovPathGenerator::getUnmappedDistance ( double  rprime )

inlineprivate

inverse of getRemappedDistance

Definition at line 161 of file JMarkovPathGenerator.hh.

                                              {
      if( sqrt(3.0)*rprime < getR0() ) return rprime*rprime*rprime/(getR0()*getR0()) ;
      return rprime - 2.0/(3.0*sqrt(3.0))*getR0() ;
    }

JPosition3D JMARKOV::JMarkovPathGenerator::getRemappedPosition ( const JPosition3D &  xprev, const JPosition3D &  x  )

inlineprivate

Return the remapped vertex position of x.

xprev is the (not remapped!) coordinate of the previous vertex in the photon path.

Definition at line 170 of file JMarkovPathGenerator.hh.

                                                                                      {
      double r = x.getDistance(xprev) ;
      double rprime = getRemappedDistance(r) ;
      return xprev + (rprime/r)*(x-xprev) ;
    }

JPosition3D JMARKOV::JMarkovPathGenerator::getUnmappedPosition ( const JPosition3D &  xprev, const JPosition3D &  xprime  )

inlineprivate

Inverse of getRemappedPosition.

Again, xprev is the (not remapped!) coordinate of the previous vertex in the photon path.

Definition at line 180 of file JMarkovPathGenerator.hh.

                                                                                           {
      double rprime = xprime.getDistance(xprev) ;
      double r = getUnmappedDistance(rprime) ;
      return xprev + (r/rprime)*(xprime-xprev) ;
    }

JPhotonPath JMARKOV::JMarkovPathGenerator::getRemappedPhotonPath ( const JPhotonPath &  p )

inlineprivate

returns a remapped version of the photon path

Definition at line 187 of file JMarkovPathGenerator.hh.

                                                             {
      JPhotonPath pprime(p) ;
      for( int i=1; i<(int)pprime.size()-1; ++i ) {
        pprime[i] = getRemappedPosition(p[i-1],p[i]) ;
      }
      return pprime ;
    }

JPhotonPath JMARKOV::JMarkovPathGenerator::getUnmappedPhotonPath ( const JPhotonPath &  pprime )

inlineprivate

inverse of getRemappedPhotonPath

Definition at line 196 of file JMarkovPathGenerator.hh.

                                                                  {
      JPhotonPath p(pprime) ;
      for( int i=1; i<(int)p.size()-1; ++i ) {
        p[i] = getUnmappedPosition(p[i-1],p[i]) ;
      }
      return p ;
    }

double JMARKOV::JMarkovPathGenerator::getRemappingCorrection ( const JPosition3D &  xprev, const JPosition3D &  xprime  )

inlineprivate

Returns the conversion factor J needed to compute the path probability density in the remapped coordinates, for a given vertex.

xprime is the remapped vertex position xprev is the (not remapped!) position of the previous vertex

In casual notation (the ' indicates remapped coordinates):

rho(x) dV = rho(x) r^2 dr dOmega = rho(x) r^2 / r'^2 dr/dr' r'^2 dr' dOmega = rho( x(x') ) r^2 / r'^2 dr/dr' dV' <— since dOmega == dOmega' := J x rho( x(x') ) dV' := rho'(x') dV' Where J := r^2 / r'^2 dr/dr'

The point of this is that we typically pick r proportional to r'^3, so that dr/dr' is proportional to r'^2, which cancels the factor 1/r'^2 in rho'. The new factor r^2 then cancels the typical 1/r^2 singularity present in rho.

Definition at line 224 of file JMarkovPathGenerator.hh.

                                                                                         {
      double rprime = xprev.getDistance(xprime) ;
      double r = getUnmappedDistance(rprime) ;
      if( sqrt(3.0)*rprime >= getR0() ) {
        // J = r^2 / r'^2 * dr/dr'
        // dr/dr' = 1
        return r*r/(rprime*rprime) ;
      }
      // J = r^2 / r'^2 * dr/dr'
      // dr/dr' = 3 * r'^2/R0^2
      // ==> J = 3 * r^2 / R0^2 
      return 3*r*r/(getR0()*getR0()) ;
    }

double JMARKOV::JMarkovPathGenerator::getRemappingCorrection ( const JPhotonPath &  p, const JPhotonPath &  pprime  )

inlineprivate

Return the remapping correction for an entire photon path (product of the remapping correction for the individual scattering vertices).

p is the photon path in normal coordinates pprime is the remapped version of p

Definition at line 245 of file JMarkovPathGenerator.hh.

                                                                                     {
      double J = 1 ;
      for( int i=1; i<(int)p.size()-1; ++i ) {
        J *= getRemappingCorrection(p[i-1],pprime[i]) ;
      }
      return J ;
    }

Member Data Documentation

bool JMARKOV::JMarkovPathGenerator::apply_coordinate_remapping

private

Definition at line 254 of file JMarkovPathGenerator.hh.

int JMARKOV::JMarkovPathGenerator::naccepted_r

private

Definition at line 255 of file JMarkovPathGenerator.hh.

int JMARKOV::JMarkovPathGenerator::nrejected_r

private

Definition at line 256 of file JMarkovPathGenerator.hh.

int JMARKOV::JMarkovPathGenerator::naccepted_t

private

Definition at line 257 of file JMarkovPathGenerator.hh.

int JMARKOV::JMarkovPathGenerator::nrejected_t

private

Definition at line 258 of file JMarkovPathGenerator.hh.

double JMARKOV::JMarkovPathGenerator::stepsize_angle_target

private

Definition at line 259 of file JMarkovPathGenerator.hh.

double JMARKOV::JMarkovPathGenerator::stepsize_r

private

Definition at line 260 of file JMarkovPathGenerator.hh.

double JMARKOV::JMarkovPathGenerator::stepsize_angle

private

Definition at line 261 of file JMarkovPathGenerator.hh.

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The documentation for this class was generated from the following file:

• JMarkovPathGenerator.hh