mml

Class HeavyTail.Over_x1.M

java.lang.Object

la.la.Value

mml.Model

mml.UPModel.M

mml.ByPdf.M

mml.Continuous.M

mml.HeavyTail.Over_x1.M

All Implemented Interfaces:

java.lang.Comparable<Value>, HasPdf

Enclosing class:

HeavyTail.Over_x1

public class HeavyTail.Over_x1.M
extends Continuous.M

The fully parameterised Over_x1 Model; see pdf_x(x). The statistical parameter delta>1.

Nested Class Summary

Nested classes/interfaces inherited from class mml.Continuous.M

Continuous.M.Transform

Nested classes/interfaces inherited from class mml.Model

Model.Defaults

Nested classes/interfaces inherited from class la.la.Value

Value.Atomic, Value.Bool, Value.Char, Value.Chars, Value.Cts, Value.Defer, Value.Discrete, Value.Enum, Value.Inc_Or, Value.Int, Value.Lambda, Value.List, Value.Maybe, Value.Option, Value.Real, Value.Scannable, Value.Structured, Value.Triv, Value.Tuple

Field Summary

Fields

Modifier and Type	Field and Description
protected double	delta_x The double version of δ>1, and the normalising constant 'k'.
protected double	k The double version of δ>1, and the normalising constant 'k'.

Fields inherited from class mml.Model

msg1, msg2, sp

Fields inherited from class la.la.Value

C, comparator, CR, E, eight, ffalse, five, fiveR, four, fourR, half, negCR, negOne, negOneR, negTenR, nilVal, nine, None, one, oneR, PI, PIby2, point01, point1, point5, point9, quarter, seven, six, ten, tenR, three, threeR, triv, ttrue, two, twoR, zero, zeroR

Constructor Summary

Constructors

Constructor and Description
M(double msg1, double msg2, Value delta) Real-valued δ>1.

Method Summary

Modifier and Type	Method and Description
protected double	f(double x) f(x) = 1/(1+x)δ, where δ>1.
double	nlLH(Value ss) The negative log likelihood, nlLH(ss), of a data-set ds having statistics ss=stats(ds) (= ds itself).
double	nlPdf_x(double x) The negative log of pdf_x(x).
double	pdf_x(double x) A log-symmetric probability density function.

Methods inherited from class mml.Continuous.M

nlPdf, random_x, random_x, random, random, random, transform

Methods inherited from class mml.ByPdf.M

nlPr, nlPr, pdf

Methods inherited from class mml.UPModel.M

asGiven, asGiven, stats, stats, toString

Methods inherited from class mml.Model

asEstimator, asUPModel, m1m2sp, msg, msg1, msg1bits, msg2, msg2bits, msgBits, nl2LH, nl2Pr, pr, random, randomSeries, statParams, stats, stats, sumNlPr, transform, type, zeroTriv

Methods inherited from class la.la.Value

AoM, apply, bOp, compareTo, cons, cts, elt, errMsg, error, force, isTuple, just, main, n, nElts, nlAoM, pair, print, quad, real, RTE, triple, tuple, UOE, uOp, x

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, wait, wait, wait

Field Detail

delta_x

protected double delta_x

The double version of δ>1, and the normalising constant 'k'. See f(x) and pdf_x(x).

k

protected double k

The double version of δ>1, and the normalising constant 'k'. See f(x) and pdf_x(x).

Constructor Detail

M

public M(double msg1,
         double msg2,
         Value delta)

Real-valued δ>1. The closer δ is to 1.0, the closer the distribution is to (something like) 1/x. See f(x).

Method Detail

pdf_x

public double pdf_x(double x)

A log-symmetric probability density function. Now, pdf₁(x)=(δ−1)*f(x) is a valid pdf but is not log-symmetric. However, let y=1/x then dy/dx=−1/x², pdf₂(x)=pdf₁(1/x)/x². Then pdf(x)=(pdf₁(x)+pdf₂(x))/2 is log-symmetric: ∫_[a,b]pdf(x) ds = ∫_[1/b..1/a]pdf(1/x) dx. Also see f(x).

nlPdf_x

public double nlPdf_x(double x)

The negative log of pdf_x(x).

Specified by:

nlPdf_x in class Continuous.M

f

protected double f(double x)

f(x) = 1/(1+x)^δ, where δ>1. The integral, ∫_0..∞f(x) = [−1/((δ-1)(1+x)^δ-1)]_0..∞ = 1/(δ−1). f(x) is used by pdf_x(x).

nlLH

public double nlLH(Value ss)

The negative log likelihood, nlLH(ss), of a data-set ds having statistics ss=stats(ds) (= ds itself).

Specified by:

nlLH in class Model