257 lines
8.7 KiB
Matlab
257 lines
8.7 KiB
Matlab
function [q,N] = quantile(X, p, dim, method, weights) % --*-- Unitary tests --*--
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% Quantiles of a sample via various methods.
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%
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% Q = QUANTILE2(X,P) returns quantiles of the values in X. P is a scalar
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% or a vector of cumulative probability values. When X is a vector, Q is
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% the same size as P, and Q(i) contains the P(i)-th quantile. When X is
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% a matrix, the i-th row of Q contains the P(i)-th quantiles of each
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% column of X. For N-D arrays, QUANTILE2 operates along the first
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% non-singleton dimension.
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%
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% Q = QUANTILE2(X,P,DIM) calculates quantiles along dimension DIM. The
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% DIM'th dimension of Q has length LENGTH(P).
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%
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% Q = QUANTILE2(X,P,DIM,METHOD) calculates quantiles using one of the
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% methods described in http://en.wikipedia.org/wiki/Quantile. The method
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% are designated 'R-1'...'R-9'; the default is R-8 as described in
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% http://bit.ly/1kX4NcT, whereas Matlab uses 'R-5'.
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%
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% Q = QUANTILE2(X,P,[],METHOD) uses the specified METHOD, but calculates
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% quantiles along the first non-singleton dimension.
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%
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% Q = QUANTILE2(X,P,[],METHOD,WEIGHTS) and QUANTILE2(X,P,[],[],WEIGHTS)
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% uses the array WEIGHTS to weight the values in X when calculating
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% quantiles. If no weighting is specified, the method determines the
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% real-valued index in to the data that is used to calculate the P(i)-th
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% quantile. When a weighting array WEIGHTS is specified (WEIGHTS should
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% be the same size as X), this index is mapped to the cumulative weights
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% (the weights are scaled to sum to N(i) - see below), and a new weighted
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% index is returned (using linear interpolation) for the point where the
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% cumulative weights equal the unweighted index. The weighted index is
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% used to calculate the P(i)-th quantile. If the values in WEIGHTS are
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% equal, then the weighted and unweighted index (and correpsonding
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% quantile) are identical. The default method R-8 is used if METHOD is
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% specified as an empty array ([]).
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%
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% [Q,N] = QUANTILE2(...) returns an array that is the same size as Q such
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% that N(i) is the number of points used to calculate Q(i).
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%
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% Further reading
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%
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% Hyndman, R.J.; Fan, Y. (November 1996). "Sample Quantiles in
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% Statistical Packages". The American Statistician 50 (4): 361-365.
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% Frigge, Michael; Hoaglin, David C.; Iglewicz, Boris (February 1989).
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% "Some Implementations of the Boxplot". The American Statistician 43
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% (1): 50-54.
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% Original file downloaded from:
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% http://fr.mathworks.com/matlabcentral/fileexchange/46555-quantile-calculation
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%
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% Copyright (C) 2014-2016 University of Surrey (Christopher Hummersone)
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% Copyright (C) 2016 Dynare Team
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%
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% This file is part of Dynare.
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%
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% Dynare is free software: you can redistribute it and/or modify
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% it under the terms of the GNU General Public License as published by
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% the Free Software Foundation, either version 3 of the License, or
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% (at your option) any later version.
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%
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% Dynare is distributed in the hope that it will be useful,
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% but WITHOUT ANY WARRANTY; without even the implied warranty of
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% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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% GNU General Public License for more details.
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%
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% You should have received a copy of the GNU General Public License
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% along with Dynare. If not, see <http://www.gnu.org/licenses/>.
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% Check input and make default assignments
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assert(isnumeric(X),'X must be a numeric');
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assert(isvector(p) & isnumeric(p),'P must be a numeric vector');
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assert(all(p>=0 & p<=1),'Values in P must be in the interval [0,1].')
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if nargin<2
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error('Not enough input arguments.')
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end
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dims = size(X);
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if nargin<3 || isempty(dim)
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dim = find(dims>1,1,'first'); % default dim
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else % validate input
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assert(isnumeric(dim) | isempty(dim),'DIM must be an integer or empty');
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assert(isint(dim) | isempty(dim),'DIM must be an integer or empty');
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assert(dim>0,'DIM must be greater than 0')
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end
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if nargin<4
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method = 'r-8'; % default method
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else % validate input
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if isempty(method)
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method = 'r-8'; % default method
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else
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assert(ischar(method),'METHOD must be a character array')
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end
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end
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if nargin<5
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weights = [];
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else
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assert(isequal(size(X),size(weights)) || isempty(weights),'WEIGHTS must be the same size as X');
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end
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% Choose a method
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% See http://en.wikipedia.org/wiki/Quantile#Estimating_the_quantiles_of_a_population
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switch lower(method)
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case 'r-1'
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min_con = @(N,p)(p==0);
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max_con = @(N,p)(false);
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h = @(N,p)((N*p)+.5);
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Qp = @(x,h)(x(ceil(h-.5)));
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case 'r-2'
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min_con = @(N,p)(p==0);
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max_con = @(N,p)(p==1);
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h = @(N,p)((N*p)+.5);
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Qp = @(x,h)((x(ceil(h-.5))+x(floor(h+.5)))/2);
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case 'r-3'
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min_con = @(N,p)(p<=(.5/N));
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max_con = @(N,p)(false);
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h = @(N,p)(N*p);
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Qp = @(x,h)(x(round(h)));
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case 'r-4'
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min_con = @(N,p)(p<(1/N));
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max_con = @(N,p)(p==1);
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h = @(N,p)(N*p);
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Qp = @(x,h)(x(floor(h)) + ((h-floor(h))*(x(floor(h)+1)-x(floor(h)))));
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case 'r-5'
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min_con = @(N,p)(p<(.5/N));
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max_con = @(N,p)(p>=((N-.5)/N));
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h = @(N,p)((N*p)+.5);
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Qp = @(x,h)(x(floor(h)) + ((h-floor(h))*(x(floor(h)+1)-x(floor(h)))));
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case 'r-6'
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min_con = @(N,p)(p<(1/(N+1)));
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max_con = @(N,p)(p>=(N/(N+1)));
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h = @(N,p)((N+1)*p);
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Qp = @(x,h)(x(floor(h)) + ((h-floor(h))*(x(floor(h)+1)-x(floor(h)))));
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case 'r-7'
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min_con = @(N,p)(false);
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max_con = @(N,p)(p==1);
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h = @(N,p)(((N-1)*p)+1);
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Qp = @(x,h)(x(floor(h)) + ((h-floor(h))*(x(floor(h)+1)-x(floor(h)))));
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case 'r-8'
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min_con = @(N,p)(p<((2/3)/(N+(1/3))));
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max_con = @(N,p)(p>=((N-(1/3))/(N+(1/3))));
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h = @(N,p)(((N+(1/3))*p)+(1/3));
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Qp = @(x,h)(x(floor(h)) + ((h-floor(h))*(x(floor(h)+1)-x(floor(h)))));
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case 'r-9'
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min_con = @(N,p)(p<((5/8)/(N+.25)));
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max_con = @(N,p)(p>=((N-(3/8))/(N+.25)));
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h = @(N,p)(((N+.25)*p)+(3/8));
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Qp = @(x,h)(x(floor(h)) + ((h-floor(h))*(x(floor(h)+1)-x(floor(h)))));
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otherwise
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error(['Method ''' method ''' does not exist'])
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end
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% calculate quartiles
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% reshape data so function works down columns
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order = mod(dim-1:dim+length(dims)-2,length(dims))+1;
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dims_shift = dims(order);
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x = rearrange(X,order,[dims_shift(1) prod(dims_shift(2:end))]);
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if ~isempty(weights)
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weights = rearrange(weights,order,[dims_shift(1) prod(dims_shift(2:end))]);
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cumwfunc = @accumulateWeights;
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wfunc = @weightedIndex;
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else
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cumwfunc = @(~,~,~,N) 1:N;
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wfunc = @(x,~) x;
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end
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% pre-allocate q
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q = zeros([length(p) prod(dims_shift(2:end))]);
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N = zeros([length(p) prod(dims_shift(2:end))]);
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for m = 1:length(p)
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for n = 1:numel(q)/length(p)
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[xSorted,ind] = sort(x(~isnan(x(:,n)),n)); % sort
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N(m,n) = length(xSorted); % sample size
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k = cumwfunc(weights,ind,n,N(m,n));
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switch N(m,n)
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case 0
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q(m,n) = NaN;
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case 1
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q(m,n) = xSorted;
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otherwise
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if min_con(N(m,n),p(m)) % at lower limit
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q(m,n) = xSorted(1);
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elseif max_con(N(m,n),p(m)) % at upper limit
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q(m,n) = xSorted(N(m,n));
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else % everything else
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huw = h(N(m,n),p(m)); % unweighted index
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hw = wfunc(huw,k);
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q(m,n) = Qp(xSorted,hw);
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end
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end
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end
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end
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% restore dims of q to equate to those of input
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q = irearrange(q,order,[length(p) dims_shift(2:end)]);
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N = irearrange(N,order,[length(p) dims_shift(2:end)]);
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% if q is a vector, make same shape as p
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if numel(p)==numel(q)
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q=reshape(q,size(p));
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N=reshape(N,size(p));
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end
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function cumweights = accumulateWeights(weights, ind, n, N)
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% ACCUMULATEWEIGHTS accumulate the weights
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wSorted = weights(ind,n); % sort weights
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wSorted = wSorted*N/sum(wSorted); % normalize weights to sum to N
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cumweights = cumsum(wSorted); % cumulative weights
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function hw = weightedIndex(huw, cumweights)
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% WEIGHTEDINDEX calculate index from cumulative weights
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ii = find(sign(cumweights-huw)<0,1,'last');
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jj = find(sign(cumweights-huw)>0,1,'first');
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if isempty(ii) || isempty(jj)
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hw = huw;
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else
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hw = ii + (huw-cumweights(ii))/(cumweights(jj)-cumweights(ii)); % weighted index
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end
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function y = isint(x)
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% ISINT check if input is whole number
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y = x==round(x);
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function y = rearrange(x,order,shape)
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%REARRANGE reshape and permute to make target dim column
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y = permute(x,order);
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y = reshape(y,shape);
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function y = irearrange(x,order,shape)
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%IREARRANGE reshape and permute to original size
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y = reshape(x,shape);
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y = ipermute(y,order);
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%@test:1
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%$ X = randn(10000000, 1);
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%$
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%$ try
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%$ q = quantile(X, [.25, .5, .75, .95 ]);
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%$ t(1) = true;
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%$ catch
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%$ t(1) = false;
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%$ end
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%$
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%$ e = [-0.674489750196082, 0, 0.674489750196082, 1.644853626951472];
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%$
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%$ if t(1)
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%$ for i=1:4
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%$ t(i+1) = abs(q(i)-e(i))<1e-3;
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%$ end
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%$ end
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%$
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%$ T = all(t);
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%@eof:1 |