Documentation
¶
Overview ¶
Package numbers provides a variety of functions related to statistics
Index ¶
- Constants
- func AbsInt(x int) int
- func AdaptiveSimpsons(f func(float64) float64, a float64, b float64, errorThreshold float64, ...) float64
- func AdaptiveSimpsonsLog(f func(float64) float64, a float64, b float64, errorThreshold float64, ...) float64
- func AddBinomMapEntry(n int) []float64
- func ApproxEqual(a, b, epsilon float64) bool
- func AverageFloat64(f []float64) float64
- func BetaClosure(alpha float64, beta float64) func(float64) float64
- func BetaDist(x float64, alpha float64, beta float64) float64
- func BetaFunc(x float64, y float64) float64
- func BetaIntegral(left float64, right float64, alpha float64, beta float64) float64
- func BetaLeftIntegral(x float64, alpha float64, beta float64) float64
- func BetaRightIntegral(x float64, alpha float64, beta float64) float64
- func BetaSampler(a float64, b float64) func() (float64, float64)
- func BinomCoefficient(n int, k int) int
- func BinomCoefficientLog(n int, k int) float64
- func BinomialDist(n int, k int, p float64, logOutput bool) (float64, bool)
- func BinomialDistLog(n int, k int, p float64) float64
- func BinomialDistLogSlice(n int, k int, p float64, binomCache [][]float64) float64
- func BinomialExpressionLog(n int, k int, p float64) float64
- func BinomialLeftSummation(n int, k int, p float64, logOutput bool) float64
- func BinomialRightSummation(n int, k int, p float64, logOutput bool) float64
- func BinomialSum(left int, right int, n int, p float64, logOutput bool) float64
- func BoundedRejectionSample(boundingSampler func() (float64, float64), f func(float64) float64, ...) (float64, float64)
- func DefiniteIntegral(f func(float64) float64, start float64, end float64) float64
- func DefiniteSmallIntegral(f func(float64) float64, start float64, end float64) float64
- func DigitsBaseTen(x int) int
- func DiscardBurnIn(t McmcTrace, burnIn int)
- func ExpDist(x float64) float64
- func Factorial(n int) int
- func FastRejectionSampler(xLeft float64, xRight float64, f func(float64) float64, bins int, ...) []float64
- func FisherExact(a, b, c, d int, aSmall bool) float64
- func GammaClosure(alpha float64, beta float64) func(float64) float64
- func GammaDist(x float64, alpha float64, beta float64) float64
- func GammaIntegral(left float64, right float64, alpha float64, beta float64) float64
- func GammaLeftIntegral(x float64, alpha float64, beta float64) float64
- func GammaRightIntegral(x float64, alpha float64, beta float64) float64
- func GammaSampler(a float64, b float64) func() (float64, float64)
- func GeometricDist(k int, p float64) float64
- func GoldenSectionMaxSearch(f func(float64) float64, a float64, b float64, epsilon float64) float64
- func GoldenSectionMinSearch(f func(float64) float64, a float64, b float64, epsilon float64) float64
- func HighestDensityInterval(t McmcTrace, proportion float64) (float64, float64)
- func InitializeFastRejectionSampler(xLeft float64, xRight float64, f func(float64) float64, bins int) ([]float64, float64)
- func LogIntegrate(f func(float64) float64, a float64, b float64, n int) float64
- func LogIntegrateIterative(f func(float64) float64, a float64, b float64, maxIter int, ...) float64
- func Max[E constraints.Ordered](a, b E) E
- func MaxMany[E constraints.Ordered](s ...E) E
- func MeanMcmcTrace(t McmcTrace) float64
- func Min[E constraints.Ordered](a, b E) E
- func MinMany[E constraints.Ordered](s ...E) E
- func NegativeBinomialCdf(x float64, R float64, P float64) float64
- func NegativeBinomialDist(k int, r float64, p float64, logOutput bool) (float64, bool)
- func NormalAdaptiveIntegral(left string, right string, mu float64, sigma float64) float64
- func NormalClosure(mu float64, sigma float64) func(float64) float64
- func NormalDist(x float64, mu float64, sigma float64) float64
- func NormalLeftIntegral(x float64, mu float64, sigma float64) float64
- func NormalRightIntegral(x float64, mu float64, sigma float64) float64
- func Pearson(a []float64, b []float64) float64
- func Plot(f func(float64) float64, left float64, right float64, bins int, outFile string)
- func PlotBinomCoefficient(n int, outFile string)
- func PoissonDist(k int, lambda float64) float64
- func PoissonLeftSummation(k int, lambda float64) float64
- func PoissonRightSummation(k int, lambda float64) float64
- func PoissonSum(left int, right int, lambda float64) float64
- func RandBeta(a float64, b float64) float64
- func RandBinomial(alias BinomialAlias) int
- func RandExp() (float64, float64)
- func RandFloat64InRange(x float64, y float64) float64
- func RandGamma(a float64, b float64) (float64, float64)
- func RandGeometric(p float64) int
- func RandInt64InRange(x int64, y int64) int64
- func RandIntInRange(x int, y int) int
- func RegularizedIncompleteBeta(a float64, b float64, x float64) float64
- func RejectionSample(xLeft float64, xRight float64, yMax float64, f func(float64) float64, ...) float64
- func RejectionSampleChooseBin(xLeft float64, xRight float64, stepSize float64, f func(float64) float64, ...) float64
- func SampleInverseNormal(mu float64, sigma float64) float64
- func ScaledBetaSampler(a float64, b float64, multiplier float64) func() (float64, float64)
- func StandardDeviationFloat64(f []float64) float64
- func StandardNormalDist(x float64) float64
- func VarianceFloat64(f []float64) float64
- type BinomialAlias
- type McmcTrace
Constants ¶
const Ln4 float64 = 1.38629436112
const MaxInt = 1<<(bits.UintSize-1) - 1
const MaxIterations int = 200
const MaxUint = 1<<bits.UintSize - 1
const MinInt = -MaxInt - 1
const RelError float64 = 1.0e-8
const Small float64 = 1.0e-30
Variables ¶
This section is empty.
Functions ¶
func AdaptiveSimpsons ¶
func AdaptiveSimpsons(f func(float64) float64, a float64, b float64, errorThreshold float64, maxDepth int) float64
AdaptiveSimpsons returns the integral from a to b of function f The error in the calculation should be less than or equal to errorThreshold. If this can not be achieved within maxDepth number recursions, then the function aborts.
func AdaptiveSimpsonsLog ¶
func AdaptiveSimpsonsLog(f func(float64) float64, a float64, b float64, errorThreshold float64, maxDepth int) float64
AdaptiveSimpsons returns the log of the integral from a to b of g(x), where f(x) = log(g(x)) The error in the calculation should be less than or equal to errorThreshold. If this can not be achieved within maxDepth number recursions, then the function aborts.
func AddBinomMapEntry ¶
AddBinomMapEntry adds an entry to a binomMap containing a slice of binomial coefficients in logSpace for a particular n value.
func ApproxEqual ¶ added in v1.0.1
ApproxEqual determines if two floating-point numbers are equal within a specified tolerance level.
func AverageFloat64 ¶
AverageFloat64 returns the mean value of type float64 from a slice of type float64.
func BetaClosure ¶
BetaClosure returns an instantiation of a Beta Distribution with fixed alpha and beta parameters.
func BetaDist ¶
BetaDist returns the probability density of a beta distribution with parameters alpha and beta at position x.
func BetaFunc ¶
BetaFunc returns B(x, y), where B is the Beta Function, also known as the Euler integral of the first kind.
func BetaIntegral ¶
BetaIntegral calculates the integral under a beta distribution with parameters alpha and beta between a specified left and right bound.
func BetaLeftIntegral ¶
BetaLeftIntegral returns the integral to the left of an input point x from a beta distribution with parameters alpha and beta.
func BetaRightIntegral ¶
BetaRightIntegral returns the integral to the right of an input point x from a beta distribution with parameters alpha and beta.
func BetaSampler ¶
BetaSampler returns an instantiation of RandBeta for a specified a and b parameter.
func BinomCoefficient ¶
BinomCoefficient calculates the Binomial Coefficient, which is also called the "Choose" Function. The answer returned is "n choose k" or n!/(n-k)!k!
func BinomCoefficientLog ¶
BinomCoefficientLog returns log(n choose k), where log is the natural logarithm. Ideal for large numbers as this raises the overflow ceiling considerably.
func BinomialDist ¶
BinomialDist returns the probability mass from a binomial distribution with k successes out of n observations with success probability p. The second return is false if no overflow/underflow was detected. If underflow was detected, the program returns 0 and true. If logOutput is true, answer will be returned as log(answer).
func BinomialDistLog ¶
BinomialDistLog returns log(BinomialDist), where log is the natural logarithm. This is ideal for very small probabilities to avoid underflow.
func BinomialDistLogSlice ¶
BinomialDistLogSlice returns log(BinomialDist), where log is the natural logarithm. This function is similar to BinomialDistLog but passes in a map[int][]float64, where the int key refers to n and the []float64 map values are the corresponding binomial coefficients for index k. Useful to not recalculate the binomial coefficient each time when binomial densities must be constantly evaluated in logSpace, like in MCMC.
func BinomialExpressionLog ¶
BinomialExpressionLog returns p^n * (1 - p)^n-k, which is also referred to as the binomial expression. The answer is provided in logSpace (.
func BinomialLeftSummation ¶
BinomialLeftSummation calculates the sum of binomial probabilities to the left of k successes for a binomial distribution with n experiments and a success probability of p, inclusive.
func BinomialRightSummation ¶
BinomialRightSummation calculates the sum of binomial probabilities to the right of k successes for a binomial distribution with n experiments and a success probability of p, inclusive.
func BinomialSum ¶
BinomialSum calculates the sum of probabilities in a binomial distribution with n experiments and success probability p between two input k values. Inclusive on both ends.
func BoundedRejectionSample ¶
func BoundedRejectionSample(boundingSampler func() (float64, float64), f func(float64) float64, xLeft float64, xRight float64, maxIteration int) (float64, float64)
BoundedRejectionSample returns a rejection sample of a function f using a bounding function boundingSampler between a specified left and right bound.
func DefiniteIntegral ¶
DefiniteIntegral computes the definite integral of f(x) dx from start to end.
func DefiniteSmallIntegral ¶
DefiniteSmallIntegral is like DefiniteIntegral with absolute error set to zero, so only relative error defines convergence conditions. slower than DefiniteIntegral, but more accurate for small values.
func DiscardBurnIn ¶
DiscardBurnIn will remove the the first i values in an McmcTrace, where i is equal to the input value burnIn.
func FastRejectionSampler ¶
func FastRejectionSampler(xLeft float64, xRight float64, f func(float64) float64, bins int, maxSampleDepth int, samples int) []float64
FastRejectionSampler returns simulated values from an a func(float64) float64 between a left and right value using an optimized rejection sampler that divides the function support into discrete bins with optimized sampling heights. maxSampleDepth triggers the log.Fatalf in the RejectionSample func, and samples is the number of values to be returned.
func FisherExact ¶
FisherExact computes a one-sided Fisher's Exact test on the 2x2 table provided The test is for the matrix: [a b] [c d] aSmall being true tests for the ratio of a to b being small, given the ratio of c to d aSmall being false tests for the ratio of a to b being large, given the ratio of c to d.
func GammaClosure ¶
GamaClosure returns an instantiation of a Gamma Distribution with fixed alpha and beta parameters.
func GammaDist ¶
GammaDist returns the probability density of a gamma distribution with parameters alpha and beta at position x. alpha is the shape parameter and beta is the rate parameter.
func GammaIntegral ¶
GammaIntegral calculates the integral between an input left and right bound of a gamma distribution with parameters alpha and beta.
func GammaLeftIntegral ¶
GammaLeftIntegral calculates the integral to the left of an input position on a gamma distribution with parameters alpha and beta.
func GammaRightIntegral ¶
GammaRightIntegral calculates the integral to the right of an input position on a gamma distribution with parameters alpha and beta.
func GammaSampler ¶
GammaSampler returns an instantiation of RandGamma for specified a and b parameters.
func GeometricDist ¶
GeometricDist returns the density of the geometric for k failures with success probability p. Note that this is the version of the geometric distribution with support from 0 to +INF.
func GoldenSectionMaxSearch ¶
GoldenSectionMaxSearch returns returns a local maximum for the input range a to b from an input function f with a user specified error epsilon. For unimodal functions, we are guaranteed to find the maximum if it is contained within the interval a to b. For polymodal functions, this function will converge on one of its modes contained in the interval a to b.
func GoldenSectionMinSearch ¶
GoldenSectionMinSearch returns a local minimum for the input range a to b from an input function f with a user specified error epsilon. For a unimodal function, we are guaranteed to find the minimum if it is contained within the interval a to b. For polymodal functions, this function will converge on one of its modes contained in the interval a to b.
func HighestDensityInterval ¶
HighestDensityInterval returns the HDI credible interval for an input McmcTrace struct. Proportion is the proportion of iterations in the credible interval (ex. 0.95 for a 95% credible interval).
func InitializeFastRejectionSampler ¶
func InitializeFastRejectionSampler(xLeft float64, xRight float64, f func(float64) float64, bins int) ([]float64, float64)
InitializeFastRejectionSampler takes in the parameters of a rejection sampler and returns the binHeights and sumHeights variables.
func LogIntegrate ¶
LogIntegrate evaluates log(int_a^b f(x)dx) in cases where f returns log(f(x)). Uses the rectangle rule.
func LogIntegrateIterative ¶
func LogIntegrateIterative(f func(float64) float64, a float64, b float64, maxIter int, relativeError float64) float64
LogIntegrateIterative repeatedly calls LogIntegrate with progressively more bins until the relative error between iterations is less than the relativeError inut variable. Each interation uses 10x more bins.
func Max ¶
func Max[E constraints.Ordered](a, b E) E
Max returns the maximum of two input values of an ordered type.
func MaxMany ¶
func MaxMany[E constraints.Ordered](s ...E) E
MaxMany returns the maximum of any number of input values of an ordered type.
func MeanMcmcTrace ¶
MeanMcmcTrace returns the mean value of the posterior distribution estimated by an McmcTrace.
func Min ¶
func Min[E constraints.Ordered](a, b E) E
Min returns the minimum of two input values of an ordered type.
func MinMany ¶
func MinMany[E constraints.Ordered](s ...E) E
MinMany returns the minimum of any number of input values of an ordered type.
func NegativeBinomialCdf ¶ added in v1.0.1
NegativeBinomialCdf returns the CDF of the negative binomial distribution for an input score x for a distribution defined by parameters R and P.
func NegativeBinomialDist ¶ added in v1.0.1
NegativeBinomialDist returns the probability mass of a negative binomial distribution with shape parameter r and success probability p at k. Pr(X = k) the second return is true if overflow/underflow was detected.
func NormalAdaptiveIntegral ¶
NormalAdaptiveIntegral returns the integral under a normal probability distribution with mean mu and standard deviation sigma from a specified left and right bound.
func NormalClosure ¶
NormalClosure returns an instantiation of a normal distribution for a particular mean mu and standard deviation sigma.
func NormalDist ¶
NormalDist returns the normal distribution value x for a distribution with mean mu and standard deviation sigma.
func NormalLeftIntegral ¶
NormalLeftIntegral returns the area under the curve of an input normal probability distribution defined by mean (mu) and standard deviation (sigma) to the left of an input point x.
func NormalRightIntegral ¶
NormalLeftIntegral returns the area under the curve of an input normal probability distribution defined by mean (mu) and standard deviation (sigma) to the right of an input point x.
func Pearson ¶
Pearson calculates the Pearson Correlation Coefficient between two slices of float64.
func Plot ¶
Plot returns the values of a function(float64) float64 between a left and right bound for a specified number of bins. the answer is written to a file as a CSV for subsequent visualization. Half-closed interval [left, right).
func PlotBinomCoefficient ¶
PlotBinomCoefficient writes binomial coefficients (n choose k) from k=1 to k=n-1 to an output file for downstream visualization.
func PoissonDist ¶
PoissonDist returns the probability density of a poisson distribution with parameter lambda at the integer value k.
func PoissonLeftSummation ¶
PoissonLeftSummation calculates the sum of probabilities to the left of an integer k, inclusive.
func PoissonRightSummation ¶
PoissonRightSummation calculates the sum of probabilities to the right of an integer k, inclusive.
func PoissonSum ¶
PoissonSum calculates the sum of probabilities between an input left and right bound, inclusive on both ends.
func RandBeta ¶
RandBeta is the basic beta variate generator from Cheng 1978. Uses the BA algorithm, which is less optimized, but still runs effectively for my use case. More optimized algorithms (with greater programming complexity) are described in the Cheng paper, and should be implemented (TODO) if required.
func RandBinomial ¶
func RandBinomial(alias BinomialAlias) int
RandBinomial generates binomial distributed variates from a pre-generated BinomialAlias struct, which can be made for a specified binomial distribution with 'MakeBinomialAlias'.
func RandExp ¶
RandExp Returns a random variable as a float64 from a standard exponential distribution. f(x)=e**-x. Algorithm from Ahrens, J.H. and Dieter, U. (1972). Computer methods for sampling from the exponential and normal distributions. Comm. ACM, 15, 873-882.
func RandFloat64InRange ¶
RandFloat64InRange returns a pseudorandom value of type int between x and y.
func RandGamma ¶
RandGamma returns a random x,y point drawn from a gamma distribution with parameters alpha and beta. y corresponds to the function density at that x value. a > 1 uses the method from Marsaglia and Tsang 2000. Written for k, theta parameters, so the first step converts b to 1 / b to evaluate gamma in terms of alpha and beta parameters. a < 1 uses the method from Ahrens, J.H. and Dieter, U. (1974). Computer methods for sampling from gamma, beta, poisson and binomial distributions. Computing, 12, 223-246.
func RandGeometric ¶
RandGeometric returns a geometrically-distributed random variate using the inverse transform. Note the geometric distribution has CDF: F(x) = 1 - (1-p)^x. The inverse transform is derived as (1-p)^x = 1 - F(x) -> x*log(1-p) = log(1 - F(x)) -> x = log(1 - F(x)) / log(1-p). Note that this is the version of the geometric distribution with support from 0 to +INF.
func RandInt64InRange ¶
RandInt64InRange returns a pseudorandom value of type int64 between x and y. Output includes x, but not y.
func RandIntInRange ¶
RandIntInRange returns a pseudorandom value of type int between x and y. Output includes x, but not y.
func RegularizedIncompleteBeta ¶ added in v1.0.1
RegularizedIncompleteBeta computes the Regularized Incomplete Beta Function for parameters A and B at query point X. Translation of an implementation at Copyright (c) 2016, 2017 Lewis Van Winkle, zlib license. https://codeplea.com/incomplete-beta-function-c
func RejectionSample ¶
func RejectionSample(xLeft float64, xRight float64, yMax float64, f func(float64) float64, maxIteration int) float64
RejectionSample returns simulated values from an arbitrary function between a specified left and right bound using a simple rejection sampling method.
func RejectionSampleChooseBin ¶
func RejectionSampleChooseBin(xLeft float64, xRight float64, stepSize float64, f func(float64) float64, maxIteration int, sumHeights float64, binHeights []float64) float64
RejectionSampleChooseBin is a helper function of FAstRejectionSampler.
func SampleInverseNormal ¶
SampleInverseNormal returns a simulated value from a normal distribution.
func ScaledBetaSampler ¶
ScaledBetaSampler returns an instatiation of RandBeta where the returned density has been scaled by the input variable 'multiplier'.
func StandardDeviationFloat64 ¶
StandardDeviationFloat64 returns the standard deviation (type float64) from a slice of type float64.
func StandardNormalDist ¶
StandardNormalDist returns the probability density for an input x value on a standard normal distribution (mu=0, sigma=1).
func VarianceFloat64 ¶
VarianceFloat64 returns the variance (type float64) from a slice of type float64.
Types ¶
type BinomialAlias ¶
BinomialAlias contains alias table information used to generate binomial distributed random variates for pre-specified parameters. Setting up the BinomialAlias struct runs in O(n) time. However, once the alias is set up for a fixed cost, random variates can be generated from the distribution in O(1) time. More information on the alias method can be found here: https://en.wikipedia.org/wiki/Alias_method
func MakeBinomialAlias ¶
func MakeBinomialAlias(n int, p float64) BinomialAlias
MakeBinomialAlias generates a BinomialAlias struct for a specified binomial distribution of n trials with success probability p. Note if the probability of a binomial outcome is less than math.MinFloat64, it will never be generated. This implementation therefore draws variates from an approximation of the binomial distribution that is truncated to zero when its tail crosses below the float underflow threshold.
type McmcTrace ¶
type McmcTrace struct {
Parameter []float64 //Parameter state, where Parameter[i] is the value of Parameter in the ith iteration.
}
McmcTrace is a general struct for Mcmc trace output. Used for discarding burn-in and calculating the mean and credible interval.
func ReadMcmcTrace ¶
ReadMcmcTrace takes a filename of the trace output file and a parameter name of interest. The function returns the values of that parameter across the Mcmc run.
Source Files
Directories