lsst::meas::modelfit::TruncatedGaussian Class Reference

Represents a multidimensional Gaussian function truncated at zero. More...

#include <TruncatedGaussian.h>

Public Types
enum	SampleStrategy { DIRECT_WITH_REJECTION , ALIGN_AND_WEIGHT }
	Enum that describes different ways of sampling from a multidimensional TruncatedGaussian. More...
typedef TruncatedGaussianSampler	Sampler
typedef TruncatedGaussianLogEvaluator	LogEvaluator
typedef TruncatedGaussianEvaluator	Evaluator

Public Member Functions
Sampler	sample (SampleStrategy strategy) const
	Create a Sampler object that uses the given strategy. More...
Sampler	sample (Scalar minRejectionEfficiency=0.1) const
	Create a Sampler object that determines the strategy to use automatically. More...
LogEvaluator	evaluateLog () const
	Create a LogEvaluator object that can be used to efficiently evaluate the -log of the function. More...
Evaluator	evaluate () const
	Create an Evaluator object that can be used to efficiently evaluate the function. More...
int	getDim () const
	Return the dimensionality of the function. More...
Vector	maximize () const
	Return the location of the maximum of the truncated Gaussian. More...
Scalar	getUntruncatedFraction () const
	Return the fraction of the Gaussian integral that was truncated by the bounds. More...
Scalar	getLogPeakAmplitude () const
	Return the -log of the peak amplitude of the untruncated function. More...
Scalar	getLogIntegral () const
	Return the -log of the integral of the truncated function. More...
	~TruncatedGaussian ()

Static Public Member Functions
static TruncatedGaussian	fromSeriesParameters (Scalar q0, Vector const &gradient, Matrix const &hessian)
	Create from the first and second logarithmic derivatives of the Gaussian. More...
static TruncatedGaussian	fromStandardParameters (Vector const &mean, Matrix const &covariance)
	Create from the "standard" mean and covariance parameters of the normal distribution. More...

Friends
class	TruncatedGaussianSampler
class	TruncatedGaussianLogEvaluator

Detailed Description

Represents a multidimensional Gaussian function truncated at zero.

This is typically used to represent the posterior probability of amplitude parameters, given a flat prior; we require that the amplitudes each be positive but otherwise do not modify the Gaussian likelihood.

Currently only 1 and 2 dimensions are supported, and all dimensions must be truncated. Computing integrals is the only operation for which > 2-d is not implemented, but the integrals must be computed upon construction, so we can't support any other operations for > 2-d either.

Many operations on TruncatedGaussians are defined in -log space, as underflow/overflow problems will often occur in the non-log forms.

See Truncated Gaussian Math for implementation notes

Definition at line 59 of file TruncatedGaussian.h.

Member Typedef Documentation

Evaluator

typedef TruncatedGaussianEvaluator lsst::meas::modelfit::TruncatedGaussian::Evaluator

Definition at line 89 of file TruncatedGaussian.h.

LogEvaluator

typedef TruncatedGaussianLogEvaluator lsst::meas::modelfit::TruncatedGaussian::LogEvaluator

Definition at line 88 of file TruncatedGaussian.h.

Sampler

typedef TruncatedGaussianSampler lsst::meas::modelfit::TruncatedGaussian::Sampler

Definition at line 87 of file TruncatedGaussian.h.

Member Enumeration Documentation

SampleStrategy

enum lsst::meas::modelfit::TruncatedGaussian::SampleStrategy

Enum that describes different ways of sampling from a multidimensional TruncatedGaussian.

Enumerator
DIRECT_WITH_REJECTION	Draw from the untruncated Gaussian, and discard negative draws. This approach is fast for minimally-truncated Gaussians, and produces ideally distributed samples with unit weights (because we discard with replacement). This approach can be extremely inefficient when the untruncated fraction is small, and cannot be used when the Hessian matrix is singular (as this sets the untruncated fraction to zero).
ALIGN_AND_WEIGHT	Create a similar Gaussian with no x-y covariance, and importance sample by drawing from the independent 1-d truncated Gaussians. This approach is faster for Gaussians that are heavily truncated, but it requires the use of per-sample weights and may produce "noisier" samples (in that regions with low weight may have many samples, while regions with high weight have few). It may also be slower for minimally truncated Gaussians because each 1-d variate requires an inverse erfc evaluation. In 1-d, this produces unit weights and ideally distributed samples, but still uses an inverse erfc approach rather than rejection to create samples.

Definition at line 63 of file TruncatedGaussian.h.

                        {
 
        DIRECT_WITH_REJECTION, 
        ALIGN_AND_WEIGHT       
    };

Constructor & Destructor Documentation

~TruncatedGaussian()

lsst::meas::modelfit::TruncatedGaussian::~TruncatedGaussian

(

)

Member Function Documentation

evaluate()

TruncatedGaussian::Evaluator lsst::meas::modelfit::TruncatedGaussian::evaluate ( ) const

inline

Create an Evaluator object that can be used to efficiently evaluate the function.

Definition at line 355 of file TruncatedGaussian.h.

{ return Evaluator(*this); }

evaluateLog()

TruncatedGaussian::LogEvaluator lsst::meas::modelfit::TruncatedGaussian::evaluateLog ( ) const

inline

Create a LogEvaluator object that can be used to efficiently evaluate the -log of the function.

Definition at line 353 of file TruncatedGaussian.h.

{ return LogEvaluator(*this); }

fromSeriesParameters()

static TruncatedGaussian lsst::meas::modelfit::TruncatedGaussian::fromSeriesParameters ( Scalar  q0, Vector const &  gradient, Matrix const &  hessian  )

static

Create from the first and second logarithmic derivatives of the Gaussian.

This causes the TruncatedGaussian to model the function:

\[ f(\alpha) = 1_{\alpha \ge 0}(\alpha) \; e^{-q(0) -g^T \alpha -\frac{1}{2}\alpha^T H \alpha} = 1_{\alpha \ge 0}(\alpha) \; e^{-q(\alpha)} \]

Where \(1_{\alpha \ge 0}(\alpha)\) is the indicator function

\[ 1_{\alpha \ge 0}(\alpha) \equiv \begin{cases} 1 & \text{if $\alpha \ge 0$}\\ 0 & \text{if $\alpha \lt 0$} \end{cases} \]

Note that this implies

\[ \left.\frac{\partial q}{\partial \alpha} \right|_{\alpha=0} = g \]

\[ \left.\frac{\partial^2 q}{\partial \alpha^2} \right|_{\alpha=0} = H \]

with (for \(\alpha \ge 0\))

\[ q(\alpha) \equiv -\ln f(\alpha) \]

Note that unlike that constructed by fromStandardParameters(), in this case \(f(\alpha)\) is NOT normalized to integrate to one.

Parameters

[in]	q0	Zeroth-order term in the expansion
[in]	gradient	Vector of first derivatives \(g\)
[in]	hessian	Matrix of second derivatives \(H\) (symmetric)

fromStandardParameters()

static TruncatedGaussian lsst::meas::modelfit::TruncatedGaussian::fromStandardParameters ( Vector const &  mean, Matrix const &  covariance  )

static

Create from the "standard" mean and covariance parameters of the normal distribution.

This causes the TruncatedGaussian to model the function:

\[ f(\alpha) = \frac{1_{\alpha \ge 0}(\alpha)}{k\left|2\pi\Sigma\right|^{1/2}} e^{-\frac{1}{2}(\alpha-\mu)^T \Sigma^{-1} (\alpha-\mu)} = \frac{1_{\alpha \ge 0}(\alpha)}{k}\;\Phi(\alpha-\mu,\Sigma) \]

Where \(1_{\alpha \ge 0}(\alpha)\) is the indicator function

\[ 1_{\alpha \ge 0}(\alpha) \equiv \begin{cases} 1 & \text{if $\alpha \ge 0$}\\ 0 & \text{if $\alpha \lt 0$} \end{cases} \]

and \(k\) is the normalization constant that \(f(\alpha)\) integrates to unity:

\[ k \equiv \int_0^{\infty}\!\!\Phi(\alpha-\mu,\Sigma) d\alpha \]

Note that unlike that constructed by fromSeriesParameters(), in this case \(f(\alpha\) is normalized to integrate to one.

Parameters

[in]	mean	Mean vector \(\mu\)
[in]	covariance	Covariance matrix \(\Sigma\) (symmetric)

getDim()

int lsst::meas::modelfit::TruncatedGaussian::getDim

(

)

const

Return the dimensionality of the function.

getLogIntegral()

Scalar lsst::meas::modelfit::TruncatedGaussian::getLogIntegral

(

)

const

Return the -log of the integral of the truncated function.

More precisely, this is simply

\[ -\ln\int_{-\infty}^{\infty} f(\alpha) d\alpha \]

In series form, this is equivalent to

\[ -\ln\int_0^{\infty} e^{-q(\alpha)} d\alpha \]

In standard parameter form it is 0 by construction.

getLogPeakAmplitude()

Scalar lsst::meas::modelfit::TruncatedGaussian::getLogPeakAmplitude

(

)

const

Return the -log of the peak amplitude of the untruncated function.

In series form, this is equal to \(q(\mu)\) where \(\mu=-H^{-1}g\). In standard parameter form, this is \(\frac{1}{2}\ln\left|2\pi\Sigma\right| - \ln k\).

getUntruncatedFraction()

Scalar lsst::meas::modelfit::TruncatedGaussian::getUntruncatedFraction

(

)

const

Return the fraction of the Gaussian integral that was truncated by the bounds.

In series form (see fromSeriesParameters), this is

\[ \frac{\int_0^{\infty} e^{-q(\alpha)} d\alpha}{\int_{-infty}^{\infty} e^{-q(\alpha)} d\alpha} \]

while in standard parameter form it is simply the normalization \(k\) described in fromStandardParameters().

maximize()

Vector lsst::meas::modelfit::TruncatedGaussian::maximize

(

)

const

Return the location of the maximum of the truncated Gaussian.

This is simply the untruncated location parameter mu if all of its elements are positive; otherwise, one or more elements will be zero (and the rest may not be same as the elements of mu).

sample() [1/2]

TruncatedGaussian::Sampler lsst::meas::modelfit::TruncatedGaussian::sample ( SampleStrategy  strategy ) const

inline

Create a Sampler object that uses the given strategy.

Warning

do not call this method with DIRECT_WITH_REJECTION unless you are certain the Gaussian is far from degenerate, as an endless loop may result in this case. It is generally safer to use the overload that selects the strategy automatically.

Definition at line 342 of file TruncatedGaussian.h.

                                                                                       {
    return Sampler(*this, strategy);
}

sample() [2/2]

TruncatedGaussian::Sampler lsst::meas::modelfit::TruncatedGaussian::sample ( Scalar  minRejectionEfficiency = 0.1 ) const

inline

Create a Sampler object that determines the strategy to use automatically.

If the efficiency (number of accepted samples divided by number of total samples) is would (on average) be greater than the given value, DIRECT_WITH_REJECTION is used. If not, ALIGN_AND_WEIGHT is used. Note that the sampling effeciency for DIRECT_WITH_REJECTION is exactly the same as the "untruncated fraction".

Definition at line 346 of file TruncatedGaussian.h.

                                                                                             {
    return Sampler(
        *this,
        (getUntruncatedFraction() < minRejectionEfficiency) ? ALIGN_AND_WEIGHT : DIRECT_WITH_REJECTION
    );
}

Friends And Related Function Documentation

TruncatedGaussianLogEvaluator

friend class TruncatedGaussianLogEvaluator

friend

Definition at line 233 of file TruncatedGaussian.h.

TruncatedGaussianSampler

friend class TruncatedGaussianSampler

friend

Definition at line 232 of file TruncatedGaussian.h.

---

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

• /j/snowflake/release/lsstsw/stack/lsst-scipipe-0.7.0/Linux64/meas_modelfit/22.0.1-4-g44f2e3d+9e4ab0f4fa/include/lsst/meas/modelfit/TruncatedGaussian.h