mml

Class Simplex

java.lang.Object

la.la.Value

la.la.Function

la.la.Function.Native

mml.UPModel

mml.ByPdf

mml.R_D

mml.Simplex

All Implemented Interfaces:

java.lang.Comparable<Value>

Direct Known Subclasses:

Dirichlet, Simplex.Uniform

public abstract class Simplex
extends R_D

For Models of data s : K_Simplex; a standard K_Simplex is the space of (K+1)-Vectors, {[s₀, ..., s_K]} (NB. 0..K) where 0 ≤ s_i ≤ 1 and ∑s_i = 1. (An alternative, equivalent definition is K-Vectors {[s[0], ..., s[K-1]]}, 0 ≤ s[i] ≤ 1 where ∑s[i] ≤ 1.) The present formulation fits conveniently with MultiState.M.
The standard K_Simplex has K+1 vertices in R^K+1, {(1,0,...,0), (0,1,...,0), ..., (0,0,...,1)}. The length of a side, the distance between any two vertices, is √2. The length of the standard 1_Simplex in R² is √2. The area of the standard 2_Simplex in R³ is (√3)/2. The volume of the standard 3_Simplex in R⁴ is 1/3 = (√4)/3!. The K-volume of the standard K_Simplex in R^K+1 is (√(K+1))/K!. -- [wikip]
Also see Direction for another K-dimensional subspace of R^K+1.

Nested Class Summary

Nested Classes

Modifier and Type	Class and Description
static class	Simplex.Uniform The UnParameterised Uniform Model over a K_Simplex data-space (Mdl is fully parameterised).

Nested classes/interfaces inherited from class mml.R_D

R_D.Forest, R_D.ForestSearch, R_D.Independent, R_D.M, R_D.NrmDir, R_D.Transform

Nested classes/interfaces inherited from class mml.UPModel

UPModel.Est

Nested classes/interfaces inherited from class la.la.Function.Native

Function.Native.WithInverse

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

Function.Cts2Cts, Function.Cts2Cts2Cts, Function.CtsD2CtsD, Function.HasInverse, Function.Native, Function.Native2, Function.Native3

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 inherited from class mml.UPModel

dp

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
Simplex(Value dp)

Method Summary

Modifier and Type	Method and Description
void	check(Value v) Perform a validity check on datum v — that it is L1-normalised.
int	D() D, the dimension of [0, 1]D which contains the K-Simplex; D = K()+1.
abstract int	K() K, the dimension (degrees of freedom) of the data-space which is a sub-space of [0, 1]K+1.
double	logArea() The area of the standard K-Simplex is (√(K+1))/K!, so log of that.

Methods inherited from class mml.R_D

ds2L1stats, ds2NorL1stats, ds2Nstats, sp2Model, transform

Methods inherited from class mml.UPModel

apply, defnParams, estimator, main, stats, stats, stats, stats, toString, transform

Methods inherited from class la.la.Function

bOp, fromBop, fromUop, type, uOp

Methods inherited from class la.la.Value

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

Methods inherited from class java.lang.Object

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

Constructor Detail

Simplex

public Simplex(Value dp)

Method Detail

D

public int D()

D, the dimension of [0, 1]^D which contains the K-Simplex; D = K()+1. Also see R_D.D().

Specified by:

D in class R_D

K

public abstract int K()

K, the dimension (degrees of freedom) of the data-space which is a sub-space of [0, 1]^K+1. K = D-1. Also see Direction.K().

logArea

public double logArea()

The area of the standard K-Simplex is (√(K+1))/K!, so log of that. E.g., (i) Bool ~ 1-Simplex, vol=√2; (ii) {R|G|B} ~ 2-Simplex, vol=(√3)/2!; (iii) DNA ~ 3-Simplex, vol=1/3 = (√(K+1))/K!; ... etc..

check

public void check(Value v)

Perform a validity check on datum v — that it is L₁-normalised.