Class MMath

This class provides mathematical constants and special functions, analogous to System.Math. It cannot be instantiated and consists of static members only.

Inheritance

Object

MMath

Inherited Members

Object.Equals(Object)

Object.Equals(Object, Object)

Object.GetHashCode()

Object.GetType()

Object.MemberwiseClone()

Object.ReferenceEquals(Object, Object)

Object.ToString()

Namespace: Microsoft.ML.Probabilistic.Math

Assembly: Microsoft.ML.Probabilistic.dll

Syntax

public static class MMath

Remarks

In order to provide the highest accuracy, some routines return their results in log form or logit form. These transformations expand the domain to cover the full range of double-precision values, ensuring all bits of the representation are utilized. A good example of this is the NormalCdf function, whose value lies between 0 and 1. Numbers between 0 and 1 use only a small fraction of the capacity of a double-precision number. The function NormalCdfLogit transforms the result p according to log(p/(1-p)), providing full use of the range from -Infinity to Infinity and (potentially) much higher precision.

To get maximal use out of these transformations, you want to stay in the expanded form as long as possible. Every time you transform into a smaller domain, you lose precision. Thus helper functions are provided which allow you to perform common tasks directly in the log form and logit form. For logs, you have addition. For logit, you have averaging.

Fields

CbrtUlp1

Math.Pow(Ulp(1.0), 1.0 / 3)

Declaration

public static readonly double CbrtUlp1

Field Value

Type	Description
Double

Digamma1

Digamma(1)

Declaration

public const double Digamma1 = -0.57721566490153287

Field Value

Type	Description
Double

EulerGamma

The Euler-Mascheroni Constant.

Declaration

public const double EulerGamma = 0.57721566490153287

Field Value

Type	Description
Double

InvSqrt2PI

1.0/Math.Sqrt(2*Math.PI)

Declaration

public const double InvSqrt2PI = 0.3989422804014327

Field Value

Type	Description
Double

Ln2

Math.Log(2)

Declaration

public const double Ln2 = 0.69314718055994529

Field Value

Type	Description
Double

LnPI

Math.Log(Math.PI)

Declaration

public const double LnPI = 1.1447298858494

Field Value

Type	Description
Double

LnSqrt2PI

Math.Log(Math.Sqrt(2*Math.PI)).

Declaration

public const double LnSqrt2PI = 0.91893853320467278

Field Value

Type	Description
Double

Sqrt2

Math.Sqrt(2)

Declaration

public const double Sqrt2 = 1.4142135623730951

Field Value

Type	Description
Double

Sqrt2PI

Math.Sqrt(2*Math.PI)

Declaration

public const double Sqrt2PI = 2.5066282746310007

Field Value

Type	Description
Double

Sqrt3

Math.Sqrt(3)

Declaration

public const double Sqrt3 = 1.7320508075688772

Field Value

Type	Description
Double

SqrtHalf

Math.Sqrt(0.5)

Declaration

public const double SqrtHalf = 0.70710678118654757

Field Value

Type	Description
Double

SqrtUlp1

Math.Sqrt(Ulp(1.0))

Declaration

public static readonly double SqrtUlp1

Field Value

Type	Description
Double

Ulp1

Ulp(1.0) == NextDouble(1.0) - 1.0

Declaration

public static readonly double Ulp1

Field Value

Type	Description
Double

Zeta2

Zeta(2) = Trigamma(1) = pi^2/6.

Declaration

public const double Zeta2 = 1.6449340668482264

Field Value

Type	Description
Double

Methods

AbsDiff(Double, Double)

Returns the distance between two numbers.

Declaration

public static double AbsDiff(double x, double y)

Parameters

Type	Name	Description
Double	x
Double	y

Returns

Type	Description
Double	abs(x - y). Matching infinities give zero.

AbsDiff(Double, Double, Double)

Returns the relative distance between two numbers.

Declaration

public static double AbsDiff(double x, double y, double rel)

Parameters

Type	Name	Description
Double	x
Double	y
Double	rel	An offset to avoid division by zero.

Returns

Type	Description
Double	abs(x - y)/(abs(x) + rel). Matching infinities give zero.

Remarks

This routine is often used to measure the error of y in estimating x.

AreEqual(Double, Double)

Returns true if two numbers are equal when represented in double precision.

Declaration

public static bool AreEqual(double x, double y)

Parameters

Type	Name	Description
Double	x
Double	y

Returns

Type	Description
Boolean

Average(Double, Double)

Returns (a+b)/2, avoiding overflow. The result is guaranteed to be between a and b.

Declaration

public static double Average(double a, double b)

Parameters

Type	Name	Description
Double	a
Double	b

Returns

Type	Description
Double

Average(Int64, Int64)

Returns (a+b)/2, avoiding overflow. The result is guaranteed to be between a and b.

Declaration

public static long Average(long a, long b)

Parameters

Type	Name	Description
Int64	a
Int64	b

Returns

Type	Description
Int64

BesselI(Double, Double)

Modified Bessel function of the first kind

Declaration

public static double BesselI(double a, double x)

Parameters

Type	Name	Description
Double	a	Order parameter. Any real number except a negative integer.
Double	x	Argument of the Bessel function. Non-negative real number.

Returns

Type	Description
Double	BesselI(a,x)

Remarks

Reference: "A short note on parameter approximation for von Mises-Fisher distributions, And a fast implementation of Is(x)" Suvrit Sra Computational Statistics, 2011 http://people.kyb.tuebingen.mpg.de/suvrit/papers/vmfFinal.pdf

Beta(Double, Double, Double, Double)

Computes the regularized incomplete beta function: int_0^x t^(a-1) (1-t)^(b-1) dt / Beta(a,b)

Declaration

public static double Beta(double x, double a, double b, double epsilon = 1E-15)

Parameters

Type	Name	Description
Double	x	The first argument - any real number between 0 and 1.
Double	a	The second argument - any real number greater than 0.
Double	b	The third argument - any real number greater than 0.
Double	epsilon	A tolerance for terminating the series calculation.

Returns

Type	Description
Double	The incomplete beta function at (x, a, b).

Remarks

The beta function is obtained by setting x to 1.

ChooseLn(Double, Double)

Evaluates the natural logarithm of Gamma(n+1)/(Gamma(k+1)*Gamma(n-k+1))

Declaration

public static double ChooseLn(double n, double k)

Parameters

Type	Name	Description
Double	n
Double	k

Returns

Type	Description
Double

ChooseLn(Double, Double[])

Evaluates the natural logarithm of Gamma(n+1)/(prod_i Gamma(k[i]+1))

Declaration

public static double ChooseLn(double n, double[] k)

Parameters

Type	Name	Description
Double	n
Double[]	k

Returns

Type	Description
Double

DifferenceOfExp(Double, Double)

Computes exp(x)-exp(y) to high accuracy.

Declaration

public static double DifferenceOfExp(double x, double y)

Parameters

Type	Name	Description
Double	x	Any real number
Double	y	Any real number

Returns

Type	Description
Double	exp(x)-exp(y)

DiffLogSumExp(Double, Double, Double)

Computes log(exp(x)+exp(a))-log(exp(x)+exp(b)) to high accuracy.

Declaration

public static double DiffLogSumExp(double x, double a, double b)

Parameters

Type	Name	Description
Double	x	Any real number from -Inf to Inf, or NaN.
Double	a	A finite real number.
Double	b	A finite real number.

Returns

Type	Description
Double	log(exp(x)+exp(a))-log(exp(x)+exp(b))

Remarks

This function provides higher accuracy than a direct evaluation of LogSumExp(x,a)-LogSumExp(x,b), particularly when x is large.

Digamma(Double)

Evaluates Digamma(x), the derivative of ln(Gamma(x)).

Declaration

public static double Digamma(double x)

Parameters

Type	Name	Description
Double	x	Any real value.

Returns

Type	Description
Double	Digamma(x).

Digamma(Double, Double)

Derivative of the natural logarithm of the multivariate Gamma function.

Declaration

public static double Digamma(double x, double d)

Parameters

Type	Name	Description
Double	x	A real value >= 0
Double	d	The dimension, an integer > 0

Returns

Type	Description
Double	digamma_d(x)

Erfc(Double)

Computes the complementary error function. This function is defined by 2/sqrt(pi) * integral from x to infinity of exp (-t^2) dt.

Declaration

public static double Erfc(double x)

Parameters

Type	Name	Description
Double	x	Any real value.

Returns

Type	Description
Double	The complementary error function at x.

ErfcInv(Double)

Computes the inverse of the complementary error function, i.e. erfcinv(erfc(x)) == x.

Declaration

public static double ErfcInv(double y)

Parameters

Type	Name	Description
Double	y	A real number between 0 and 2.

Returns

Type	Description
Double	A number x such that erfc(x) == y.

ExpMinus1(Double)

Computes the exponential of x and subtracts 1.

Declaration

public static double ExpMinus1(double x)

Parameters

Type	Name	Description
Double	x	Any real number from -Inf to Inf, or NaN.

Returns

Type	Description
Double	exp(x)-1

Remarks

This function is more accurate than a direct evaluation of exp(x)-1 when x is small. It is the inverse function to Log1Plus: ExpMinus1(Log1Plus(x)) == x.

ExpMinus1RatioMinus1RatioMinusHalf(Double)

Computes ((exp(x)-1)/x - 1)/x - 0.5

Declaration

public static double ExpMinus1RatioMinus1RatioMinusHalf(double x)

Parameters

Type	Name	Description
Double	x	Any real number from -Inf to Inf, or NaN.

Returns

Type	Description
Double	((exp(x)-1)/x - 1)/x - 0.5

Gamma(Double)

Evaluates Gamma(x), defined as the integral from 0 to x of t^(x-1)*exp(-t) dt.

Declaration

public static double Gamma(double x)

Parameters

Type	Name	Description
Double	x	Any real value.

Returns

Type	Description
Double	Gamma(x).

GammaLn(Double)

Computes the natural logarithm of the Gamma function.

Declaration

public static double GammaLn(double x)

Parameters

Type	Name	Description
Double	x	A real value >= 0.

Returns

Type	Description
Double	ln(Gamma(x)).

Remarks

This function provides higher accuracy than Math.Log(Gamma(x)), which may fail for large x.

GammaLn(Double, Double)

Computes the natural logarithm of the multivariate Gamma function.

Declaration

public static double GammaLn(double x, double d)

Parameters

Type	Name	Description
Double	x	A real value >= 0.
Double	d	The dimension, an integer > 0.

Returns

Type	Description
Double	ln(Gamma_d(x))

Remarks

The multivariate Gamma function is defined as Gamma_d(x) = pi^(d*(d-1)/4)*prod_(i=1..d) Gamma(x + (1-i)/2)

GammaLower(Double, Double)

Compute the regularized lower incomplete Gamma function: int_0^x t^(a-1) exp(-t) dt / Gamma(a)

Declaration

public static double GammaLower(double a, double x)

Parameters

Type	Name	Description
Double	a	The shape parameter, > 0
Double	x	The upper bound of the integral, >= 0

Returns

Type	Description
Double

GammaUpper(Double, Double, Boolean)

Compute the regularized upper incomplete Gamma function: int_x^inf t^(a-1) exp(-t) dt / Gamma(a)

Declaration

public static double GammaUpper(double a, double x, bool regularized = true)

Parameters

Type	Name	Description
Double	a	The shape parameter. Must be > 0 if regularized is true or x is 0.
Double	x	The lower bound of the integral, >= 0
Boolean	regularized	If true, result is divided by Gamma(a)

Returns

Type	Description
Double

GammaUpperLogScale(Double, Double)

Computes log(x^a e^(-x)/Gamma(a)) to high accuracy.

Declaration

public static double GammaUpperLogScale(double a, double x)

Parameters

Type	Name	Description
Double	a	A positive real number
Double	x

Returns

Type	Description
Double

GammaUpperScale(Double, Double)

Computes x^a e^(-x)/Gamma(a) to high accuracy.

Declaration

public static double GammaUpperScale(double a, double x)

Parameters

Type	Name	Description
Double	a	A positive real number
Double	x

Returns

Type	Description
Double

IndexOfMaximum<T>(IEnumerable<T>)

Returns the index of the maximum element, or -1 if empty.

Declaration

public static int IndexOfMaximum<T>(IEnumerable<T> list)
    where T : IComparable<T>

Parameters

Type	Name	Description
IEnumerable<T>	list

Returns

Type	Description
Int32

Type Parameters

Name	Description
T

IndexOfMaximumDouble(IList<Double>)

Returns the index of the maximum element, or -1 if empty.

Declaration

public static int IndexOfMaximumDouble(IList<double> list)

Parameters

Type	Name	Description
IList<Double>	list

Returns

Type	Description
Int32

IndexOfMinimum<T>(IEnumerable<T>)

Returns the index of the minimum element, or -1 if empty.

Declaration

public static int IndexOfMinimum<T>(IEnumerable<T> list)
    where T : IComparable<T>

Parameters

Type	Name	Description
IEnumerable<T>	list

Returns

Type	Description
Int32

Type Parameters

Name	Description
T

Log1MinusExp(Double)

Computes log(1 - exp(x)) to high accuracy.

Declaration

public static double Log1MinusExp(double x)

Parameters

Type	Name	Description
Double	x	A non-positive real number: -Inf <= x <= 0, or NaN.

Returns

Type	Description
Double	log(1-exp(x)), which is always <= 0.

Remarks

This function provides higher accuracy than a direct evaluation of log(1-exp(x)), particularly when x < -5 or x > -1e-5.

Log1Plus(Double)

Computes the natural logarithm of 1+x.

Declaration

public static double Log1Plus(double x)

Parameters

Type	Name	Description
Double	x	A real number in the range -1 <= x <= Inf, or NaN.

Returns

Type	Description
Double	log(1+x), which is always >= 0.

Remarks

This function provides higher accuracy than a direct evaluation of log(1+x), particularly when x is small.

Log1PlusExp(Double)

Computes log(1 + exp(x)) to high accuracy.

Declaration

public static double Log1PlusExp(double x)

Parameters

Type	Name	Description
Double	x	Any real number from -Inf to Inf, or NaN.

Returns

Type	Description
Double	log(1+exp(x)), which is always >= 0.

Remarks

This function provides higher accuracy than a direct evaluation of log(1+exp(x)), particularly when x < -36 or x > 50.

Log1PlusExpGaussian(Double, Double)

Evaluates E[log(1+exp(x))] under a Gaussian distribution with specified mean and variance.

Declaration

public static double Log1PlusExpGaussian(double mean, double variance)

Parameters

Type	Name	Description
Double	mean
Double	variance

Returns

Type	Description
Double

LogDifferenceOfExp(Double, Double)

Computes log(exp(x) - exp(y)) to high accuracy.

Declaration

public static double LogDifferenceOfExp(double x, double y)

Parameters

Type	Name	Description
Double	x	Any real number from -Inf to Inf, or NaN. Must be greater or equal to y.
Double	y	Any real number from -Inf to Inf, or NaN. Must be less or equal to x.

Returns

Type	Description
Double

Remarks

This function provides higher accuracy than a direct evaluation of log(exp(x)-exp(y)).

LogExpMinus1(Double)

Computes log(exp(x)-1) for non-negative x.

Declaration

public static double LogExpMinus1(double x)

Parameters

Type	Name	Description
Double	x	A non-negative real number: 0 <= x <= Inf, or NaN.

Returns

Type	Description
Double	log(exp(x)-1)

Remarks

This function is more accurate than a direct evaluation of log(exp(x)-1) when x < 1e-3 or x > 50. It is the inverse function to Log1PlusExp: LogExpMinus1(Log1PlusExp(x)) == x.

Logistic(Double)

Computes the logistic function 1/(1+exp(-x)).

Declaration

public static double Logistic(double x)

Parameters

Type	Name	Description
Double	x	Any real number from -Inf to Inf, or NaN.

Returns

Type	Description
Double	1/(1+exp(-x)).

LogisticGaussian(Double, Double)

Calculate sigma(m,v) = \int N(x;m,v) logistic(x) dx

Declaration

public static double LogisticGaussian(double mean, double variance)

Parameters

Type	Name	Description
Double	mean	Mean
Double	variance	Variance

Returns

Type	Description
Double	The value of this special function.

Remarks

Note 1-LogisticGaussian(m,v) = LogisticGaussian(-m,v) which is more accurate.

For large v we can use the big v approximation \sigma(m,v)=normcdf(m/sqrt(v+pi^2/3)). For small and moderate v we use Gauss-Hermite quadrature. For moderate v we first find the mode of the (log concave) function since this may be quite far from m.

LogisticGaussianDerivative(Double, Double)

Calculate \sigma'(m,v)=\int N(x;m,v)logistic'(x) dx

Declaration

public static double LogisticGaussianDerivative(double mean, double variance)

Parameters

Type	Name	Description
Double	mean	Mean.
Double	variance	Variance.

Returns

Type	Description
Double	The value of this special function.

Remarks

For large v we can use the big v approximation \sigma'(m,v)=N(m,0,v+pi^2/3). For small and moderate v we use Gauss-Hermite quadrature. For moderate v we first find the mode of the (log concave) function since this may be quite far from m.

LogisticGaussianDerivative2(Double, Double)

Calculate \sigma''(m,v)=\int N(x;m,v)logistic''(x) dx

Declaration

public static double LogisticGaussianDerivative2(double mean, double variance)

Parameters

Type	Name	Description
Double	mean	Mean.
Double	variance	Variance.

Returns

Type	Description
Double	The value of this special function.

Remarks

For large v we can use the big v approximation \sigma'(m,v)=-m/(v+pi^2/3)*N(m,0,v+pi^2/3). For small and moderate v we use Gauss-Hermite quadrature. The function is multimodal so mode finding is difficult and probably won't help.

LogisticGaussianRatio(Double, Double, Int32)

Calculate (kth derivative of LogisticGaussian)exp(0.5mean^2/variance)

Declaration

public static double LogisticGaussianRatio(double mean, double variance, int k)

Parameters

Type	Name	Description
Double	mean
Double	variance
Int32	k

Returns

Type	Description
Double

LogisticLn(Double)

Compute the natural logarithm of the logistic function, i.e. -log(1+exp(-x)).

Declaration

public static double LogisticLn(double x)

Parameters

Type	Name	Description
Double	x	Any real number from -Inf to Inf, or NaN.

Returns

Type	Description
Double	-log(1+exp(-x)).

Remarks

This function provides higher accuracy than a direct evaluation of -log(1+exp(-x)), which can fail for x < -50 and x > 36.

Logit(Double)

Computes the log-odds function log(p/(1-p)).

Declaration

public static double Logit(double p)

Parameters

Type	Name	Description
Double	p	Any number between 0 and 1, inclusive.

Returns

Type	Description
Double	log(p/(1-p))

Remarks

This function is the inverse of the logistic function, i.e. Logistic(Logit(p)) == p.

LogitFromLog(Double)

Compute log(p/(1-p)) from log(p).

Declaration

public static double LogitFromLog(double logp)

Parameters

Type	Name	Description
Double	logp	Any number between -infinity and 0, inclusive.

Returns

Type	Description
Double	log(exp(logp)/(1-exp(logp))) = -log(exp(-logp)-1).

LogRatio(Double, Double)

Computes log(numerator/denominator) to high accuracy.

Declaration

public static double LogRatio(double numerator, double denominator)

Parameters

Type	Name	Description
Double	numerator	Any positive real number.
Double	denominator	Any positive real number.

Returns

Type	Description
Double	log(numerator/denominator)

LogSumExp(IEnumerable<Double>)

Returns the log of the sum of exponentials of a list of doubles

Declaration

public static double LogSumExp(IEnumerable<double> list)

Parameters

Type	Name	Description
IEnumerable<Double>	list

Returns

Type	Description
Double

LogSumExp(Double, Double)

Computes log(exp(x) + exp(y)) to high accuracy.

Declaration

public static double LogSumExp(double x, double y)

Parameters

Type	Name	Description
Double	x	Any real number from -Inf to Inf, or NaN.
Double	y	Any real number from -Inf to Inf, or NaN.

Returns

Type	Description
Double	log(exp(x)+exp(y)), which is always >= max(x,y).

Remarks

This function provides higher accuracy than a direct evaluation of log(exp(x)+exp(y)).

LogSumExpSparse(IEnumerable<Double>)

Returns the log of the sum of exponentials of a list of doubles

Declaration

public static double LogSumExpSparse(IEnumerable<double> list)

Parameters

Type	Name	Description
IEnumerable<Double>	list

Returns

Type	Description
Double

Max(IEnumerable<Double>)

Returns the maximum of a list of doubles

Declaration

public static double Max(IEnumerable<double> list)

Parameters

Type	Name	Description
IEnumerable<Double>	list

Returns

Type	Description
Double

Median(Double[])

Returns the median of the array elements.

Declaration

public static double Median(double[] array)

Parameters

Type	Name	Description
Double[]	array

Returns

Type	Description
Double	The median ignoring NaNs.

Median(Double[], Int32, Int32)

Returns the median of elements in a subrange of an array.

Declaration

public static double Median(double[] array, int start, int length)

Parameters

Type	Name	Description
Double[]	array
Int32	start	Starting index of the range.
Int32	length	The number of elements in the range.

Returns

Type	Description
Double	The median of array[start:(start+length-1)], ignoring NaNs.

Min(IEnumerable<Double>)

Returns the minimum of a list of doubles

Declaration

public static double Min(IEnumerable<double> list)

Parameters

Type	Name	Description
IEnumerable<Double>	list

Returns

Type	Description
Double

NextDouble(Double)

Returns the smallest double precision number greater than value, if one exists. Otherwise returns value.

Declaration

public static double NextDouble(double value)

Parameters

Type	Name	Description
Double	value

Returns

Type	Description
Double

NextDoubleWithPositiveDifference(Double)

Returns the smallest double precision number such that when value is subtracted from it, the result is strictly greater than zero, if one exists. Otherwise returns value.

Declaration

public static double NextDoubleWithPositiveDifference(double value)

Parameters

Type	Name	Description
Double	value

Returns

Type	Description
Double

NormalCdf(Double)

Computes the cumulative Gaussian distribution, defined as the integral from -infinity to x of N(t;0,1) dt.
For example, NormalCdf(0) == 0.5.

Declaration

public static double NormalCdf(double x)

Parameters

Type	Name	Description
Double	x	Any real number.

Returns

Type	Description
Double	The cumulative Gaussian distribution at x.

NormalCdf(Double, Double, Double)

Computes the cumulative bivariate normal distribution.

Declaration

public static double NormalCdf(double x, double y, double r)

Parameters

Type	Name	Description
Double	x	First upper limit.
Double	y	Second upper limit.
Double	r	Correlation coefficient.

Returns

Type	Description
Double	phi(x,y,r)

Remarks

The cumulative bivariate normal distribution is defined as int_(-inf)^x int_(-inf)^y N([x;y],[0;0],[1 r; r 1]) dx dy where N([x;y],[0;0],[1 r; r 1]) = exp(-0.5*(x^2+y^2-2xyr)/(1-r^2))/(2pi*sqrt(1-r^2)).

NormalCdf(Double, Double, Double, Double)

Declaration

public static ExtendedDouble NormalCdf(double x, double y, double r, double sqrtomr2)

Parameters

Type	Name	Description
Double	x
Double	y
Double	r
Double	sqrtomr2

Returns

Type	Description
ExtendedDouble

NormalCdfDiff(Double, Double)

Computes NormalCdf(x) - NormalCdf(y) to high accuracy.

Declaration

public static ExtendedDouble NormalCdfDiff(double x, double y)

Parameters

Type	Name	Description
Double	x
Double	y

Returns

Type	Description
ExtendedDouble	The difference.

NormalCdfExtended(Double)

Computes the cumulative Gaussian distribution, defined as the integral from -infinity to x of N(t;0,1) dt.
For example, NormalCdf(0) == 0.5.

Declaration

public static ExtendedDouble NormalCdfExtended(double x)

Parameters

Type	Name	Description
Double	x	Any real number.

Returns

Type	Description
ExtendedDouble	The cumulative Gaussian distribution at x.

NormalCdfExtended(Double, Double, Double)

Computes the cumulative bivariate normal distribution.

Declaration

public static ExtendedDouble NormalCdfExtended(double x, double y, double r)

Parameters

Type	Name	Description
Double	x	First upper limit.
Double	y	Second upper limit.
Double	r	Correlation coefficient.

Returns

Type	Description
ExtendedDouble	phi(x,y,r)

Remarks

The cumulative bivariate normal distribution is defined as int_(-inf)^x int_(-inf)^y N([x;y],[0;0],[1 r; r 1]) dx dy where N([x;y],[0;0],[1 r; r 1]) = exp(-0.5*(x^2+y^2-2xyr)/(1-r^2))/(2pi*sqrt(1-r^2)).

NormalCdfIntegral(Double, Double, Double)

Declaration

public static double NormalCdfIntegral(double x, double y, double r)

Parameters

Type	Name	Description
Double	x
Double	y
Double	r

Returns

Type	Description
Double

NormalCdfIntegral(Double, Double, Double, Double)

Computes the integral of the cumulative bivariate normal distribution wrt x, from -infinity to x.

Declaration

public static ExtendedDouble NormalCdfIntegral(double x, double y, double r, double sqrtomr2)

Parameters

Type	Name	Description
Double	x	First upper limit.
Double	y	Second upper limit.
Double	r	Correlation coefficient.
Double	sqrtomr2	sqrt(1-r*r)

Returns

Type	Description
ExtendedDouble	r such that r*exp(exponent) is the integral

NormalCdfIntegralRatio(Double, Double, Double)

Computes the integral of the cumulative bivariate normal distribution wrt x, divided by the cumulative bivariate normal distribution.

Declaration

public static double NormalCdfIntegralRatio(double x, double y, double r)

Parameters

Type	Name	Description
Double	x	First upper limit.
Double	y	Second upper limit.
Double	r	Correlation coefficient.

Returns

Type	Description
Double

NormalCdfIntegralRatio(Double, Double, Double, Double)

Computes the integral of the cumulative bivariate normal distribution wrt x, divided by the cumulative bivariate normal distribution.

Declaration

public static double NormalCdfIntegralRatio(double x, double y, double r, double sqrtomr2)

Parameters

Type	Name	Description
Double	x	First upper limit.
Double	y	Second upper limit.
Double	r	Correlation coefficient.
Double	sqrtomr2	sqrt(1-r*r)

Returns

Type	Description
Double

NormalCdfInv(Double)

Computes the inverse of the cumulative Gaussian distribution, i.e. NormalCdf(NormalCdfInv(p)) == p. For example, NormalCdfInv(0.5) == 0. This is also known as the Gaussian quantile function.

Declaration

public static double NormalCdfInv(double p)

Parameters

Type	Name	Description
Double	p	A real number in [0,1].

Returns

Type	Description
Double	A number x such that NormalCdf(x) == p.

NormalCdfLn(Double)

The natural logarithm of the cumulative Gaussian distribution.

Declaration

public static double NormalCdfLn(double x)

Parameters

Type	Name	Description
Double	x	Any real number.

Returns

Type	Description
Double	ln(NormalCdf(x)).

Remarks

This function provides higher accuracy than Math.Log(NormalCdf(x)), which can fail for x < -7.

NormalCdfLn(Double, Double, Double)

Computes the natural logarithm of the cumulative bivariate normal distribution.

Declaration

public static double NormalCdfLn(double x, double y, double r)

Parameters

Type	Name	Description
Double	x	First upper limit.
Double	y	Second upper limit.
Double	r	Correlation coefficient.

Returns

Type	Description
Double	ln(phi(x,y,r))

NormalCdfLogit(Double)

The log-odds of the cumulative Gaussian distribution.

Declaration

public static double NormalCdfLogit(double x)

Parameters

Type	Name	Description
Double	x	Any real number.

Returns

Type	Description
Double	ln(NormalCdf(x)/(1-NormalCdf(x))).

NormalCdfMomentRatio(Int32, Double)

Computes int_0^infinity t^n N(t;x,1) dt / (n! N(x;0,1))

Declaration

public static double NormalCdfMomentRatio(int n, double x)

Parameters

Type	Name	Description
Int32	n	The exponent
Double	x	Any real number

Returns

Type	Description
Double

NormalCdfMomentRatioDiff(Int32, Double, Double, Int32)

Computes NormalCdfMomentRatio(n, x+delta)-NormalCdfMomentRatio(n, x)

Declaration

public static double NormalCdfMomentRatioDiff(int n, double x, double delta, int startingIndex = 1)

Parameters

Type	Name	Description
Int32	n	A non-negative integer
Double	x	Any real number
Double	delta	A finite real number with absolute value less than 9.9, or less than 70% of the absolute value of x.
Int32	startingIndex	The first moment to use in the power series. Used to skip leading terms. For example, 2 will skip NormalCdfMomentRatio(n+1, x).

Returns

Type	Description
Double

NormalCdfRatio(Double)

Computes NormalCdf(x)/N(x;0,1) to high accuracy.

Declaration

public static double NormalCdfRatio(double x)

Parameters

Type	Name	Description
Double	x	Any real number.

Returns

Type	Description
Double

NormalCdfRatioDiff(Double, Double, Int32)

Computes NormalCdfRatio(x+delta)-NormalCdfRatio(x)

Declaration

public static double NormalCdfRatioDiff(double x, double delta, int startingIndex = 1)

Parameters

Type	Name	Description
Double	x	A real number less than 37
Double	delta	A finite real number with absolute value less than 9.9, or less than 70% of the absolute value of x.
Int32	startingIndex	The first moment to use in the power series. Used to skip leading terms. For example, 2 will skip NormalCdfMomentRatio(1, x).

Returns

Type	Description
Double

NormalCdfRatioLn(Double)

Computes ln(NormalCdf(x)/N(x;0,1)) to high accuracy.

Declaration

public static double NormalCdfRatioLn(double x)

Parameters

Type	Name	Description
Double	x	Any real number.

Returns

Type	Description
Double

NormalCdfRatioLn(Double, Double, Double, Double)

Computes the natural logarithm of the cumulative bivariate normal distribution, minus the log-density of the bivariate normal distribution at x and y, plus 0.5log(1-rr).

Declaration

public static double NormalCdfRatioLn(double x, double y, double r, double sqrtomr2)

Parameters

Type	Name	Description
Double	x	First upper limit. Must be finite.
Double	y	Second upper limit. Must be finite.
Double	r	Correlation coefficient.
Double	sqrtomr2	sqrt(1-r*r)

Returns

Type	Description
Double	ln(phi(x,y,r)/N([x;y],[0;0],[1 r; r 1])

OneMinusSqrtOneMinus(Double)

Computes 1-sqrt(1-x) to high accuracy.

Declaration

public static double OneMinusSqrtOneMinus(double x)

Parameters

Type	Name	Description
Double	x	A real number between 0 and 1

Returns

Type	Description
Double

PreviousDouble(Double)

Returns the largest double precision number less than value, if one exists. Otherwise returns value.

Declaration

public static double PreviousDouble(double value)

Parameters

Type	Name	Description
Double	value

Returns

Type	Description
Double

PreviousDoubleWithPositiveDifference(Double)

Returns the biggest double precision number such that when it is subtracted from value, the result is strictly greater than zero, if one exists. Otherwise returns value.

Declaration

public static double PreviousDoubleWithPositiveDifference(double value)

Parameters

Type	Name	Description
Double	value

Returns

Type	Description
Double

ReciprocalFactorialMinus1(Double)

Computes 1/Gamma(x+1) - 1 to high accuracy

Declaration

public static double ReciprocalFactorialMinus1(double x)

Parameters

Type	Name	Description
Double	x	A real number >= 0

Returns

Type	Description
Double

RemoveNaNs(Double[], Int32, Int32)

Given an array, returns a new array with all NANs removed.

Declaration

public static double[] RemoveNaNs(double[] array, int start, int length)

Parameters

Type	Name	Description
Double[]	array	The source array
Int32	start	The start index in the source array
Int32	length	How many items to look at in the source array

Returns

Type	Description
Double[]

RisingFactorialLnOverN(Double, Double)

Computes the logarithm of the Pochhammer function, divided by n: (GammaLn(x + n) - GammaLn(x))/n

Declaration

public static double RisingFactorialLnOverN(double x, double n)

Parameters

Type	Name	Description
Double	x	A real number > 0
Double	n	If zero, result is 0.

Returns

Type	Description
Double

Softmax(IList<Double>)

Exponentiate array elements and normalize to sum to 1.

Declaration

public static Vector Softmax(IList<double> x)

Parameters

Type	Name	Description
IList<Double>	x	May be +/-infinity

Returns

Type	Description
Vector	A Vector p where p[k] = exp(x[k])/sum_j exp(x[j])

Remarks

Sparse lists and vectors are handled efficiently

Tetragamma(Double)

Evaluates Tetragamma, the forth derivative of logGamma(x)

Declaration

public static double Tetragamma(double x)

Parameters

Type	Name	Description
Double	x

Returns

Type	Description
Double

ToStringExact(Double)

Returns a decimal string that exactly equals a double-precision number, unlike double.ToString which always returns a rounded result.

Declaration

public static string ToStringExact(double x)

Parameters

Type	Name	Description
Double	x

Returns

Type	Description
String

Trigamma(Double)

Evaluates Trigamma(x), the derivative of Digamma(x).

Declaration

public static double Trigamma(double x)

Parameters

Type	Name	Description
Double	x	Any real value.

Returns

Type	Description
Double	Trigamma(x).

Trigamma(Double, Double)

Second derivative of the natural logarithm of the multivariate Gamma function.

Declaration

public static double Trigamma(double x, double d)

Parameters

Type	Name	Description
Double	x	A real value >= 0
Double	d	The dimension, an integer > 0

Returns

Type	Description
Double	trigamma_d(x)

Ulp(Double)

Returns the positive distance between a value and the next representable value that is larger in magnitude.

Declaration

public static double Ulp(double value)

Parameters

Type	Name	Description
Double	value	Any double-precision value.

Returns

Type	Description
Double

WeightedAverage(Double, Double, Double, Double)

Returns (weight1value1 + weight2value2)/(weight1 + weight2), avoiding underflow and overflow.
The result is guaranteed to be between value1 and value2, monotonic in value1 and value2, and equal to Average(value1, value2) when weight1 == weight2.

Declaration

public static double WeightedAverage(double weight1, double value1, double weight2, double value2)

Parameters

Type	Name	Description
Double	weight1	Any number >=0
Double	value1	Any number
Double	weight2	Any number >=0
Double	value2	Any number

Returns

Type	Description
Double