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ToleranceFactor

ToleranceFactor

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16 May 2009 (Updated )

ToleranceFactor computes the exact tolerance factor for the two-sided tolerance interval

ToleranceFactor(n,coverage,confidence,options)
function [k,options] = ToleranceFactor(n,coverage,confidence,options)
%ToleranceFactor computes the exact tolerance factor k for the two-sided
%(optionally also for the one-sided) p-content and gamma-confidence
%tolerance interval
%   TI = [Xmean - k * S, Xmean + k * S],
%where Xmean = mean(X), S = std(X), X = [X_1,...,X_n] is a random sample
%from the distribution N(mu,sig2) with unknown mean mu and variance sig2.
%
%The value of the tolerance factor k is determined such that the tolerance
%intervals with the confidence gamma cover at least the fraction p
%('coverage') of the distribution N(mu,sigma^2), i.e.
%   Prob[ Prob( Xmean - k * S < X < Xmean + k * S ) >= p ]= gamma,
%for X ~ N(mu,sig2) which is independent with Xmean and S. For more details
%see e.g. Krishnamoorthy and Mathew (2009).
%
%Syntax:
%k = ToleranceFactor(n,coverage,confidence)
%[k,options] = ToleranceFactor(n,coverage,confidence)
%or
%k = ToleranceFactor(n,coverage,confidence,options)
%
%Input:
% - n: samlpe size
% - coverage:
%    Prob( Xmean - k * S < X < Xmean + k * S ) >= coverage,
% - confidence:
%    Prob[ Prob( Xmean - k * S < X < Xmean + k * S ) >= p ] = confidence.
% - options: structure with further optional parameters:

% - options.DegreesOfFreedom:
%    set the degrees of freedom (different from nu = n-1),
%    for one sample, with n observations, set nu = n - 1,
%    for m samples, each with n observations, set nu = m * (n - 1).
% - options.Onesided:
%    logical flag for calculating the factor of (upper) one-sided
%    tolerance interval; default value: onesided = false;
% - options.NormalizingConstant
%    normalizing constant c; default value: c = 1/n. Alternatively, for
%    computing the nonsimultaneous tolerance factors in linear regression
%    use use e.g. c = x'*inv(X'*X)*x, here X is the regression design
%    matrix and x is the vector of predictors.
%
%Output:
% - k, the calculated tolerance factor for tolerance interval
%
%Example:
%Calculate the tolerance factor k for the two-sided statistical p-content
%and gamma-coverage tolerance interval, with p = 0.80, and gamma = 0.95.
%
%Generate the random sample X of size n = 7 from a normal distribution
%N(mu,sig2) with mu = 5, sig2 = 0.5^2 and estimate the tolerance interval,
%based on the estimated sample mean and the sample standard deviation.
%
%Check the quality of estimatd TI: Generate a new random sample Z of size
%N = 1000 from N(mu,sig2) and calculate proportion of the generated
%observations that are covered by the estimated TI.
%
%n = 7;
%p = 0.80;
%gamma = 0.95;
%k = ToleranceFactor(n,p,gamma)
%
%mu = 5; sig = 0.5; X = mu + sig * randn(1,n);
%Xmean = mean(X);
%S = std(X);
%TI = [Xmean - k * S, Xmean + k * S]
%
%N = 1000;
%Z = mu + sig * randn(1,N);
%prop = sum(TI(1) < Z & Z < TI(2))/N
%
%Dependence:
%stats\norminv.m, stats\chi2inv.m, stats\nctinv.m
%
%See also:
%MATLAB\fzero.m
%
%References:
%
%Krishnamoorthy K., Mathew T.: Statistical Tolerance Regions: Theory,
%Applications, and Computation. Wiley, ISBN: 978-0-470-38026-0, 512 pages,
%May 2009.
%
%ISO 16269-6:2005: Statistical interpretation of data - Part 6:
%Determination of statistical tolerance intervals.
%
%Janiga I., Garaj I.: Two-sided tolerance limits of normal distributions
%with unknown means and unknown common variability. MEASUREMENT SCIENCE
%REVIEW, Volume 3, Section 1, 2003, 75-78.
%
%Cite this algorithm as:
%Viktor Witkovsky: ToleranceFactor - The Matlab algorithm for computing the
%exact tolerance factors of the two-sided non-simultaneous tolerance
%intervals for normal distribution. MATLAB Central File Exchange, May 2009.
%http://www.mathworks.com/matlabcentral/fileexchange/ .
%
%Viktor Witkovsky
%Institute of Mesaurement Science
%Slovak Academy of Sciences
%Dubravska cesta 9
%84104 Bratislava
%Slovak Republic
%E-mail: witkovsky@savba.sk
%http://www.um.sav.sk/en/department-03/viktor-witkovsky.html

%(c) Viktor Witkovsky, 2009, (witkovsky@savba.sk)
%Ver.: 20-Aug-2009 12:02:16

error(nargchk(1, 4, nargin));

if nargin < 2
    coverage = 0.95;
end
if nargin < 3
    confidence = 0.95;
end
if nargin < 4
    options = struct('DegreesOfFreedom',n-1,...
        'NormalizingConstant',1/n,'Onesided',false);
    % Gauss-Hermite nodes and weights (order 499)
    options.GHweights = [2.335254455909132e-020  8.727520180512132e-020 ...
        3.192492986472047e-019  1.143079966323183e-018  4.006410689872739e-018 ...
        1.374645207412527e-017  4.617487133659739e-017  1.518531475276486e-016 ...
        4.889535045861178e-016  1.541555393524301e-015  4.759049063256765e-015 ...
        1.438705492841146e-014  4.259268561632164e-014  1.234894400387112e-013 ...
        3.506518269122293e-013  9.751989883410978e-013  2.656440833077607e-012 ...
        7.087862727356068e-012  1.852493672185278e-011  4.742879166739566e-011 ...
        1.189565769188374e-010  2.922891421620524e-010  7.036084851819923e-010 ...
        1.659432282079332e-009  3.834542792182359e-009  8.681771070540109e-009 ...
        1.926009618049048e-008  4.186761739458309e-008  8.918290399352374e-008 ...
        1.861581441292868e-007  3.807962364485096e-007  7.633542883937283e-007 ...
        1.499667160723453e-006  2.887425542736802e-006  5.448606677924250e-006 ...
        1.007697238150531e-005  1.826650426945954e-005  3.245428369842251e-005 ...
        5.651840553955225e-005  9.647588492209113e-005  1.614239536456450e-004 ...
        2.647564252726149e-004  4.256601318096663e-004  6.708503311910412e-004 ...
        1.036435786038859e-003  1.569714986526230e-003  2.330601619290612e-003 ...
        3.392274252602869e-003  4.840564691851943e-003  6.771570420208750e-003 ...
        9.287030186384078e-003  1.248716483984685e-002  1.646097132612199e-002 ...
        2.127435734221991e-002  2.695697939802246e-002  3.348912916697763e-002 ...
        4.079040867403259e-002  4.871214063099133e-002  5.703538422726187e-002 ...
        6.547601341067191e-002  7.369756694384425e-002  8.133157079705838e-002 ...
        8.800390531735061e-002  9.336473180595961e-002  9.711870844600029e-002 ...
        9.905188660428430e-002]';
    options.GHnodes = [6.551018453076543e+000  6.449557694700710e+000 ...
        6.348166092579258e+000  6.246842375771083e+000  6.145585283625421e+000 ...
        6.044393565536637e+000  5.943265980703555e+000  5.842201297893355e+000 ...
        5.741198295210179e+000  5.640255759867880e+000  5.539372487966921e+000 ...
        5.438547284275569e+000  5.337778962014790e+000  5.237066342647089e+000 ...
        5.136408255669010e+000  5.035803538407177e+000  4.935251035817865e+000 ...
        4.834749600289886e+000  4.734298091450734e+000  4.633895375975809e+000 ...
        4.533540327400811e+000  4.433231825937007e+000  4.332968758289295e+000 ...
        4.232750017477094e+000  4.132574502657917e+000  4.032441118953432e+000 ...
        3.932348777278103e+000  3.832296394170188e+000  3.732282891625054e+000 ...
        3.632307196930786e+000  3.532368242505885e+000  3.432464965739177e+000 ...
        3.332596308831606e+000  3.232761218640027e+000  3.132958646522919e+000 ...
        3.033187548187789e+000  2.933446883540434e+000  2.833735616535785e+000 ...
        2.734052715030375e+000  2.634397150636366e+000  2.534767898577027e+000 ...
        2.435163937543635e+000  2.335584249553732e+000  2.236027819810664e+000 ...
        2.136493636564379e+000  2.036980690973368e+000  1.937487976967762e+000 ...
        1.838014491113478e+000  1.738559232477368e+000  1.639121202493338e+000 ...
        1.539699404829372e+000  1.440292845255389e+000  1.340900531511917e+000 ...
        1.241521473179509e+000  1.142154681548851e+000  1.042799169491520e+000 ...
        9.434539513313326e-001  8.441180427162521e-001  7.447904604907797e-001 ...
        6.454702225688125e-001  5.461563478068849e-001  4.468478558777696e-001 ...
        3.475437671443922e-001  2.482431025339954e-001  1.489448834125141e-001 ...
        4.964813145912062e-002]';
end

% Set the optional parameters
c = options.NormalizingConstant;
onesided = options.Onesided;
nu = options.DegreesOfFreedom;
GHweights = options.GHweights;
GHnodes = options.GHnodes;

% Set values for limit cases
if confidence == 0
    k = NaN;
    return
elseif confidence == 1
    k = Inf;
    return
elseif coverage == 1
    k = Inf;
    return
elseif coverage == 0
    k =  0;
    return
end

if nu == Inf
    k = norminv(1-(1-coverage)/2);
    return
elseif nu <= 0
    error('VW:ToleranceFactor','Degrees of freedom should be positive ...')
end

% Get the starting point: Howe's approximation
if nu < 2^20,
    kH = sqrt( (nu * (1+c) * norminv((1-coverage)/2)^2) ...
        / chi2inv(1-confidence,nu));
else
    kH = sqrt(((1+c) * norminv((1-coverage)/2)^2));
end
k0 = kH;

% Compute the tolerance factor
if onesided
    k = sqrt(c)*nctinv(confidence,nu,norminv(coverage)/sqrt(c));
else
    x = sqrt(2 * c) * GHnodes;
    root = FindRoot(x,coverage);
    ncx2points = nu * root.^2;
    k = fzero(@(k) Integral(k,nu,GHweights,ncx2points)- confidence,k0);
end
%% Auxiliary function Integral
function int = Integral(k,nu,GHweights,ncx2points)
%Auxiliary function Integral for computing two sided tolerance intervals.
%Integral evaluates the integral defined by eq. (1.2.4) in Krishnamoorthy
%and Mathew: Statistical Tolerance Regions, 2009, p.7, by the Gauss-Hermite
%quadrature.

%(c) Viktor Witkovsky (witkovsky@savba.sk)
%Ver.: 12-Aug-2009 11:31:14

x = ncx2points / k^2;
fun = gammainc(x/2,nu/2);
int = 1.0 - 1.128379167095513 * GHweights' * fun;
%% Auxiliry function FindRoot
function r = FindRoot(x,coverage,tol,maxiter)
%ROOTFIDER numerically finds the root(s) r, the solution to the equation
%normcdf(x+r) - normcdf(x-r) = coverage, 
%given x and coverage by the Halley's method

%(c) Viktor Witkovsky (witkovsky@savba.sk)
%Ver.: 20-Aug-2009 12:49:42

if nargin < 4
    maxiter = 100;
end
if nargin < 3
    tol = 3 * eps;
end

r = x;
iter = 0;
while true
    iter = iter + 1;
    [fun,funD1,funD2] = Coverage(x,r,coverage);
    r  = r - 2 * fun .* funD1 ./ (2 * funD1.^2 - fun .* funD2);
    if iter > maxiter
        break
    end
    if all(abs(fun) < tol)
        break
    end
end
%% Auxiliary function Coverage
function [fun,funD1,funD2] = Coverage(x,r,p)
%Auxiliary function Coverage: fun = normcdf(x+r) - normcdf(x-r);

%(c) Viktor Witkovsky (witkovsky@savba.sk)
%Ver.: 20-Aug-2009 12:49:42

if nargin < 2
    p = 0;
end

sqrt2 = 1.414213562373095;
sqrt2pi = 2.506628274631;

fun = 0.5 * ( erfc(-(x+r)/sqrt2) - erfc(-(x-r)/sqrt2) ) - p;
aux1 = exp(-0.5 * (x + r).^2);
aux2 = exp(-0.5 * (x - r).^2);
funD1 = (aux1 + aux2)/sqrt2pi;
funD2 = ((x - r) .* aux2 - (x + r) .* aux1)/sqrt2pi;
%%

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