V_GAUSSMIXD marginal and conditional Gaussian mixture densities
Usage: (1) [mz,vz,wz]=v_gaussmixd(y,m,v,w); % If {m,v,w} has dimension p, then y specifies
% elements 1:q and {mz,vz,wz} is a GMM for elements q+1:p
(2) [mz,vz,wz]=v_gaussmixd(y,m,v,w,[],[1 3 4]); % y specifies elements 1,3,4 and {mz,vz,wz}
% is a GMM for the remaining elements
Inputs: Input mixture: k mixtures, p dimensions
Output mixture: k mixtures, r dimensions
Conditioning data: n data values, q dimensions
Y(n,q) = conditioning input data: x*a'+ b'= y
M(k,p) = mixture means for x(p)
V(k,p) or V(p,p,k) variances (diagonal or full)
W(k,1) = mixture weights
A(q,p) = conditioning transformation: y=x*A'+ B' (where y and x are row vectors).
B(q,1) If A is omitted or null, y=x*I(B,:)' where I is the identity matrix.
If B is also omitted or null, y=x*I(1:q,:)'.
F(r,p) = output transformation z = x*F'+G' (where z and x are row vectors).
G(r,1) If F is omitted or null, z = x*f' where I is the identity matrix.
If G is also omitted or null, z=x*I(q+1:p,:)' or, if A was also null,
the complement of y.
Outputs (if 1 or 2 outputs specified):
MZ(n,r) = Global mean of z for each y
VZ(r,r,n) Global full covariances of z (now dependent on y)
Outputs (if 3 outputs specified):
MZ(k,r,n) = mixture means of z for each y
VZ(k,r) or VZ(r,r,k) variances of z (diagonal or full); surprisingly it is independent of y
WZ(n,k)
The output mixture covariances will be diagonal if either r=1 or else the following three
conditions are all true:
(a) the input mixture covariances are diagonal and
(b) matrix A has at most one non-zero element in any row or column and
(c) matrix F has at most one non-zero element in any column
Several of the output variables can be squeezed if r=1 but this is not done automatically.
This routine can be used for inference: if you train a GMM on p-dimensional data
then, if y(n,q) contains observations of the first q components, then z=v_gaussmixd(y,m,v,w)
will return the estimated values of the remaining p-q components.
See also: v_gaussmix, v_gaussmixg, v_gaussmixp, v_randvec0001 function [mz,vz,wz]=v_gaussmixd(y,m,v,w,a,b,f,g) 0002 %V_GAUSSMIXD marginal and conditional Gaussian mixture densities 0003 % 0004 % Usage: (1) [mz,vz,wz]=v_gaussmixd(y,m,v,w); % If {m,v,w} has dimension p, then y specifies 0005 % % elements 1:q and {mz,vz,wz} is a GMM for elements q+1:p 0006 % (2) [mz,vz,wz]=v_gaussmixd(y,m,v,w,[],[1 3 4]); % y specifies elements 1,3,4 and {mz,vz,wz} 0007 % % is a GMM for the remaining elements 0008 % 0009 % Inputs: Input mixture: k mixtures, p dimensions 0010 % Output mixture: k mixtures, r dimensions 0011 % Conditioning data: n data values, q dimensions 0012 % 0013 % Y(n,q) = conditioning input data: x*a'+ b'= y 0014 % M(k,p) = mixture means for x(p) 0015 % V(k,p) or V(p,p,k) variances (diagonal or full) 0016 % W(k,1) = mixture weights 0017 % A(q,p) = conditioning transformation: y=x*A'+ B' (where y and x are row vectors). 0018 % B(q,1) If A is omitted or null, y=x*I(B,:)' where I is the identity matrix. 0019 % If B is also omitted or null, y=x*I(1:q,:)'. 0020 % F(r,p) = output transformation z = x*F'+G' (where z and x are row vectors). 0021 % G(r,1) If F is omitted or null, z = x*f' where I is the identity matrix. 0022 % If G is also omitted or null, z=x*I(q+1:p,:)' or, if A was also null, 0023 % the complement of y. 0024 % 0025 % Outputs (if 1 or 2 outputs specified): 0026 % 0027 % MZ(n,r) = Global mean of z for each y 0028 % VZ(r,r,n) Global full covariances of z (now dependent on y) 0029 % 0030 % Outputs (if 3 outputs specified): 0031 % 0032 % MZ(k,r,n) = mixture means of z for each y 0033 % VZ(k,r) or VZ(r,r,k) variances of z (diagonal or full); surprisingly it is independent of y 0034 % WZ(n,k) 0035 % 0036 % The output mixture covariances will be diagonal if either r=1 or else the following three 0037 % conditions are all true: 0038 % (a) the input mixture covariances are diagonal and 0039 % (b) matrix A has at most one non-zero element in any row or column and 0040 % (c) matrix F has at most one non-zero element in any column 0041 % 0042 % Several of the output variables can be squeezed if r=1 but this is not done automatically. 0043 % 0044 % This routine can be used for inference: if you train a GMM on p-dimensional data 0045 % then, if y(n,q) contains observations of the first q components, then z=v_gaussmixd(y,m,v,w) 0046 % will return the estimated values of the remaining p-q components. 0047 % 0048 % See also: v_gaussmix, v_gaussmixg, v_gaussmixp, v_randvec 0049 0050 % Copyright (C) Mike Brookes 2000-2012 0051 % Version: $Id: v_gaussmixd.m 10865 2018-09-21 17:22:45Z dmb $ 0052 % 0053 % VOICEBOX is a MATLAB toolbox for speech processing. 0054 % Home page: http://www.ee.ic.ac.uk/hp/staff/dmb/voicebox/voicebox.html 0055 % 0056 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 0057 % This program is free software; you can redistribute it and/or modify 0058 % it under the terms of the GNU General Public License as published by 0059 % the Free Software Foundation; either version 2 of the License, or 0060 % (at your option) any later version. 0061 % 0062 % This program is distributed in the hope that it will be useful, 0063 % but WITHOUT ANY WARRANTY; without even the implied warranty of 0064 % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 0065 % GNU General Public License for more details. 0066 % 0067 % You can obtain a copy of the GNU General Public License from 0068 % http://www.gnu.org/copyleft/gpl.html or by writing to 0069 % Free Software Foundation, Inc.,675 Mass Ave, Cambridge, MA 02139, USA. 0070 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 0071 0072 [n,q]=size(y); 0073 [k,p]=size(m); 0074 % We set fv=1 if input V contains full covariance matrices with dimensions V(r,r,k). 0075 % This is true either if V has >2 dimensions or V=V(r,r) and r>k 0076 fv=ndims(v)>2 || size(v,1)>k; % full covariance matrix is supplied 0077 anull=nargin<5 || isempty(a); % no A matrix specified 0078 if anull 0079 a=eye(p); 0080 if nargin<6 || isempty(b) 0081 b=1:q; 0082 end 0083 a=a(b,:); 0084 b0=b; % save row selection 0085 b=zeros(q,1); 0086 elseif nargin<6 || isempty(b) 0087 b=zeros(q,1); 0088 end 0089 if nargin<7 || isempty(f) 0090 f=eye(p); 0091 if nargin<8 || isempty(g) 0092 if anull 0093 f(b0,:)=[]; 0094 else 0095 f=f(q+1:end,:); 0096 end 0097 else 0098 f=f(g,:); % G selects the output variables 0099 end 0100 r=size(f,1); 0101 g=zeros(r,1); 0102 elseif nargin<8 || isempty(g) 0103 r=size(f,1); 0104 g=zeros(r,1); 0105 else 0106 r=size(f,1); 0107 end 0108 0109 yb=y-repmat(b',n,1); % remove the b term 0110 [lp,wz]=v_gaussmixp(yb,m,v,w,a); % find mixture weights 0111 mz=zeros(n,r,k); 0112 ma=a~=0; % check for sparse a and f matrices (common case) 0113 mf=f~=0; 0114 % We set dvo=1 if the output mixture covariances are structurally 0115 % diagonal. This is the case if either: 0116 % (1) r=1 since in this case they are scalar values, or else 0117 % (2) the following three conditions are all true: 0118 % (a) the input mixture covariances are diagonal and 0119 % (b) matrix A has at most one non-zero element in any row or column and 0120 % (c) matrix F has at most one non-zero element in any column 0121 dvo=r==1 || (~fv && all(sum(ma,1)<=1) && all(sum(ma,2)<=1) && all(sum(mf,1)<=1)); 0122 if dvo % structurally diagonal output covariances 0123 vz=zeros(k,r); % diagonal output variances (one row per mixture) independent of y 0124 for i=1:k % loop for each mixture 0125 if fv 0126 vi=v(:,:,i); 0127 else 0128 vi=diag(v(i,:)); % convert to full covariance matrices 0129 end 0130 hi=vi*a'/(a*vi*a'); % regression coefficient matrix (p#q] 0131 vz(i,:)=diag(f*(vi-hi*a*vi)*f')'; % variance of z (independent of y) 0132 mi=m(i,:); % input mean for mixure i 0133 m0=(mi-mi*a'*hi')*f'+g'; % y-independent part of mean 0134 mz(:,:,i)=(repmat(m0,n,1)+yb*hi'*f'); % mean for each y 0135 end 0136 else 0137 vz=zeros(r,r,k); 0138 for i=1:k % loop for each mixture 0139 if fv 0140 vi=v(:,:,i); 0141 else 0142 vi=diag(v(i,:)); % convert to full covariance matrices 0143 end 0144 hi=vi*a'/(a*vi*a'); % regression coefficient matrix (p#q] 0145 vz(:,:,i)=f*(vi-hi*a*vi)*f'; % variance of z (independent of y) 0146 mi=m(i,:); % input mean for mixure i 0147 m0=(mi-mi*a'*hi')*f'+g'; % y-independent part of mean 0148 mz(:,:,i)=(repmat(m0,n,1)+yb*hi'*f'); % mean for each y 0149 end 0150 end 0151 if nargout<3 0152 mt=reshape(sum(reshape(mz,n*r,k).*repmat(wz,r,1),2),n,r); % global mean 0153 if nargout>1 % need to calculate global variance as well 0154 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 0155 % We calculate the global covariance by adding up the weighted mixture 0156 % covariances corrected for the fact that the mixture mean does not equal 0157 % the global mean. 0158 % To save calculations, we calculate only the lower triangle of the 0159 % symmetric covariance matrix and then expand it at the end into a full matrix. 0160 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 0161 rl=r*(r+1)/2; % number of elements in lower triangular covariance matrix 0162 lix=1:r^2; 0163 cix=repmat(1:r,r,1); 0164 rix=cix'; 0165 lix(cix>rix)=[]; % index of lower triangular elements 0166 rlix=rix(lix); 0167 clix=cix(lix); 0168 lixi=zeros(r,r); 0169 lixi(lix)=1:rl; 0170 lixi=lixi'; 0171 lixi(lix)=1:rl; % reverse index to build full matrices 0172 0173 vt=zeros(n,rl); % reserve space for lower triangular output covariances 0174 for i=1:k 0175 if dvo 0176 vi=diag(vz(i,:)); 0177 else 0178 vi=vz(:,:,i); 0179 end 0180 mzt=mz(:,:,i)-mt; 0181 vt=vt+repmat(wz(:,i),1,r^2).*(repmat(vi(lix),n,1)+mzt(:,rlix).*mzt(:,clix)); 0182 end 0183 vz=permute(reshape(vt(:,lixi),[n,r,r]),[2 3 1]); 0184 end 0185 mz=mt; 0186 else 0187 mz=permute(mz,3:-1:1); 0188 end