imm_filter.m

%IMM_FILTER  Interacting Multiple Model (IMM) Filter prediction and update steps
%
% Syntax:
%   [X_i,P_i,MU,X,P] = IMM_FILTER(X_ip,P_ip,MU_ip,p_ij,ind,dims,A,Q,Y,H,R)
%
% In:
%   X_ip  - Cell array containing N^j x 1 mean state estimate vector for
%           each model j after update step of previous time step
%   P_ip  - Cell array containing N^j x N^j state covariance matrix for 
%           each model j after update step of previous time step
%   MU_ip - Vector containing the model probabilities at previous time step
%   p_ij  - Model transition matrix
%   ind   - Indices of state components for each model as a cell array
%   dims  - Total number of different state components in the combined system
%   A     - State transition matrices for each model as a cell array.
%   Q     - Process noise matrices for each model as a cell array.
%   Y    - Dx1 measurement vector.
%   H    - Measurement matrices for each model as a cell array.
%   R    - Measurement noise covariances for each model as a cell array.
%
%
% Out:
%   X_p  - Updated state mean for each model as a cell array
%   P_p  - Updated state covariance for each model as a cell array
%   MU   - Model probabilities as vector
%   X    - Combined state mean estimate
%   P    - Combined state covariance estimate
%   
% Description:
%   IMM filter prediction and update steps. Use this instead
%   of separate prediction and update functions, if you don't need
%   the prediction estimates.
%
% See also:
%   IMM_UPDATE, IMM_SMOOTH, IMM_FILTER

% History:
%   01.11.2007 JH The first official version.
%
% Copyright (C) 2007 Jouni Hartikainen
%
% $Id: imm_update.m 111 2007-11-01 12:09:23Z jmjharti $
%
% This software is distributed under the GNU General Public 
% Licence (version 2 or later); please refer to the file 
% Licence.txt, included with the software, for details.

function [X_i,P_i,MU,X,P] = imm_filter(X_ip,P_ip,MU_ip,p_ij,ind,dims,A,Q,Y,H,R)
    % Number of models
    m = length(X_ip);

    % Default values for state mean and covariance
    MM_def = zeros(dims,1);
    PP_def = diag(20*ones(dims,1));

    % Normalizing factors for mixing probabilities
    c_j = zeros(1,m);
    for j = 1:m
        for i = 1:m
            c_j(j) = c_j(j) + p_ij(i,j).*MU_ip(i);
        end
    end
    
    % Mixing probabilities
    MU_ij = zeros(m,m);
    for i = 1:m
        for j = 1:m
            MU_ij(i,j) = p_ij(i,j) * MU_ip(i) / c_j(j);
        end
    end
    
    % Calculate the mixed state mean for each filter   
    X_0j = cell(1,m);
    for j = 1:m
        X_0j{j} = zeros(dims,1);
        for i = 1:m
            X_0j{j}(ind{i}) = X_0j{j}(ind{i}) + X_ip{i}*MU_ij(i,j);
        end
    end
    
    % Calculate the mixed state covariance for each filter    
    P_0j = cell(1,m);
    for j = 1:m
        P_0j{j} = zeros(dims,dims);
        for i = 1:m
            P_0j{j}(ind{i},ind{i}) = P_0j{j}(ind{i},ind{i}) + MU_ij(i,j)*(P_ip{i} + (X_ip{i}-X_0j{j}(ind{i}))*(X_ip{i}-X_0j{j}(ind{i}))');
        end
    end

    % Space for estimates
    X_p = cell(1,m);
    P_p = cell(1,m);
    X_i = cell(1,m);
    P_i = cell(1,m);
    lambda = zeros(1,m);

    % Filter the estimates for each model
    for i = 1:m
        % Predict the estimates
        [X_p{i}, P_p{i}] = kf_predict(X_0j{i}(ind{i}),P_0j{i}(ind{i},ind{i}),A{i},Q{i});
        % Update the estimates
        [X_i{i}, P_i{i}, K, IM, IS] = kf_update(X_p{i},P_p{i},Y,H{i},R{i});
        
        % Calculate likelihoods
        lambda(i) = kf_lhood(X_p{i},P_p{i},Y,H{i},R{i});
    end
    
     % Calculate the model probabilities   
    MU = zeros(1,m); 
    c = sum(lambda.*c_j);
    MU = c_j.*lambda/c;

        
    % Output the combined updated state mean and covariance, if wanted.
    if nargout > 3
        % Space for estimates    
        X = zeros(dims,1);
        P = zeros(dims,dims);
        % Updated state mean        
        for i = 1:m
            X(ind{i}) = X(ind{i}) + MU(i)*X_i{i};
        end
        % Updated state covariance
        for i = 1:m
            P(ind{i},ind{i}) = P(ind{i},ind{i}) + MU(i)*(P_i{i} + (X_i{i}-X(ind{i}))*(X_i{i}-X(ind{i}))');
        end
    end