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jointPDFBayes.m
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jointPDFBayes.m
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function [state_posteriors,sens_posteriors,histogram_out]=jointPDFBayes(dir_mesh,speed_mesh,dir_states,speed_states,sensor_model,dir_std,speed_std,dir_bin_size,speed_bin_size,wind_data)
%========================================================================
% jointPDFBayes
% version 1.0 - January 18th, 2017
%
% This script computes the joint PDFs of the airflow (wind speed/wind
% direction) at a given location. For the computations of the posteriors,
% the script uses equations 1 to 3 from the article "Probabilistic Air
% Flow Modelling Using Turbulent and Laminar Characteristics for Ground
% and Aerial Robots".
% Inputs:
% dir_mesh,speed_mesh 2D matrices with the wind speed/direction values
% for each state in Gamma (state space)
% dir_states, speed_states: vectors that contain the possible wind speed/wind
% direction values of the discrete joint PDFs
% sensor_model: object that defines the sensor model according to
% equation 2
% dir_std,speed_std: Parameters of the sensor model (equation 2)
% dir_bin_size,speed_bin_size: Discretization parameters for the state
% space.
% wind data: Nx2 Measurement matrix that contains values of wind direction
% (degrees) in the first columd and wind speed (m/s) in the second column
% Outputs:
% histogram_out: Simple 2D histogram computed with the raw input
% state_posteriors: Joint PDFs computed with the Bayesian filter
% sens_posteriors: Joint PDFs computed using the sensor model ONLY
%
% NOTE: This script uses the circStat toolbox developed by Philipp
% Berens and co-authors. The circStat toolbox used in this work was
% downloaded from:
% https://se.mathworks.com/matlabcentral/fileexchange/10676-circular-statistics-toolbox--directional-statistics-
%
%========================================================================
%---------------------------------------
% Data initialization
%--------------------------------------
n_data=length(wind_data);
[n_rows,n_cols]=size(dir_mesh);
state_posteriors=(1/(n_rows*n_cols)).*ones(n_rows,n_cols);
state_accummulator=zeros(n_rows,n_cols);
state_accummulator_sens=zeros(n_rows,n_cols);
centers={dir_states,speed_states};
[N,C]=hist3(wind_data,centers);
histogram_out=N'./sum(N(:));
PX=ones(n_rows,n_cols).*(1/(n_rows*n_cols));
transition_matrices=zeros(n_rows,n_cols,n_rows,n_cols);
dir_vector=reshape(dir_mesh,[n_rows*n_cols,1]);
speed_vector=reshape(speed_mesh,[n_rows*n_cols,1]);
for i=1:n_data
%---------------------------------------
% Updates the transition matrix
%--------------------------------------
if i==1
prev_dir_val=wind_data(1,1);
prev_speed_val=wind_data(1,2);
distance_dir_vector=abs(rad2deg(circ_dist(deg2rad(dir_states),deg2rad(prev_dir_val))));
distance_speed_vector=abs(prev_speed_val-speed_states);
dir_idx=find(distance_dir_vector==min(distance_dir_vector));
previous_dir_idx=dir_idx(1);
speed_idx=find(distance_speed_vector==min(distance_speed_vector));
previous_speed_idx=speed_idx(1);
else
dir_val=wind_data(i,1);
speed_val=wind_data(i,2);
distance_vector_dir=abs(rad2deg(circ_dist(deg2rad(dir_vector),deg2rad(dir_val))));
distance_vector_speed=abs(speed_val-speed_vector);
distance_matrix_dir=reshape(distance_vector_dir,n_rows,n_cols);
distance_matrix_speed=reshape(distance_vector_speed,n_rows,n_cols);
idx_dir_mat=distance_matrix_dir<=3*dir_std;
idx_speed_mat=distance_matrix_speed<=3*speed_std;
idx_mat=idx_dir_mat.*idx_speed_mat;
[idx_r, idx_c]=find(idx_mat>0);
for k=1:length(idx_r)
i_r=idx_r(k);
i_c=idx_c(k);
bin_center_dir=dir_mesh(i_r,i_c);
bin_center_speed=speed_mesh(i_r,i_c);
dir_prior=sensor_model.dirPrior(dir_val,wrapTo360(bin_center_dir-dir_bin_size/2),wrapTo360(bin_center_dir+dir_bin_size/2));
speed_prior=sensor_model.speedPrior(speed_val,bin_center_speed-speed_bin_size/2,bin_center_speed+speed_bin_size/2);
transition_matrices(i_r,i_c,previous_speed_idx,previous_dir_idx)=transition_matrices(i_r,i_c,previous_speed_idx,previous_dir_idx)+dir_prior*speed_prior;
end
distance_dir_vector=abs(rad2deg(circ_dist(deg2rad(dir_states),deg2rad(dir_val))));
distance_speed_vector=abs(speed_val-speed_states);
dir_idx=find(distance_dir_vector==min(distance_dir_vector));
previous_dir_idx=dir_idx(1);
speed_idx=find(distance_speed_vector==min(distance_speed_vector));
previous_speed_idx=speed_idx(1);
end
%---------------------------------------
% Prediction Stage
%--------------------------------------
prediction_matrix=zeros(n_rows,n_cols);
for k=1:n_rows
for l=1:n_cols
trans_mat=transition_matrices(:,:,k,l);
trans_mat=squeeze(trans_mat);
prediction_matrix = prediction_matrix + PX(k,l).*trans_mat;
end
end
dir_val=wind_data(i,1);
speed_val=wind_data(i,2);
dir_vector=reshape(dir_mesh,[n_rows*n_cols,1]);
speed_vector=reshape(speed_mesh,[n_rows*n_cols,1]);
distance_vector_dir=abs(rad2deg(circ_dist(deg2rad(dir_vector),deg2rad(dir_val))));
distance_vector_speed=abs(speed_val-speed_vector);
distance_matrix_dir=reshape(distance_vector_dir,n_rows,n_cols);
distance_matrix_speed=reshape(distance_vector_speed,n_rows,n_cols);
idx_dir_mat=distance_matrix_dir<=3*dir_std;
idx_speed_mat=distance_matrix_speed<=3*speed_std;
idx_mat=idx_dir_mat.*idx_speed_mat+0;
[idx_r, idx_c]=find(idx_mat>0);
%---------------------------------------
% Correction Stage
%--------------------------------------
PX=zeros(n_rows,n_cols)*(1/(n_rows*n_cols));
for k=1:length(idx_r)
i_r=idx_r(k);
i_c=idx_c(k);
bin_center_dir=dir_mesh(i_r,i_c);
bin_center_speed=speed_mesh(i_r,i_c);
dir_prior=sensor_model.dirPrior(dir_val,wrapTo360(bin_center_dir-dir_bin_size/2),wrapTo360(bin_center_dir+dir_bin_size/2));
speed_prior=sensor_model.speedPrior(speed_val,bin_center_speed-speed_bin_size/2,bin_center_speed+speed_bin_size/2);
corrected_value=dir_prior*speed_prior*prediction_matrix(i_r,i_c);
state_accummulator(i_r,i_c)=state_accummulator(i_r,i_c)+corrected_value;
state_accummulator_sens(i_r,i_c)=state_accummulator_sens(i_r,i_c)+dir_prior*speed_prior;
PX(i_r,i_c)=corrected_value;
end
if (sum(PX(:))==0)
PX=(1/(n_rows*n_cols)).*ones(n_rows,n_cols);
else
PX=PX./sum(PX(:));
end
state_posteriors=state_accummulator/sum(state_accummulator(:));
sens_posteriors=state_accummulator_sens/sum(state_accummulator_sens(:));
end