gtsam/matlab/unstable_examples/+imuSimulator/covarianceAnalysisBetween.m

import gtsam.*;

% Test GTSAM covariances on a graph with betweenFactors
% Authors: Luca Carlone, David Jensen
% Date: 2014/4/6

clc
clear all
close all

%% Create ground truth trajectory
trajectoryLength = 49;
unsmooth_DP = 0.5; % controls smoothness on translation norm
unsmooth_DR = 0.1; % controls smoothness on translation norm

% possibly create random trajectory as ground Truth
gtValues = Values;
gtGraph = NonlinearFactorGraph;

sigma_ang = 1e-2;
sigma_cart = 0.1;
noiseVector = [sigma_ang; sigma_ang; sigma_ang; sigma_cart; sigma_cart; sigma_cart];
noise = noiseModel.Diagonal.Sigmas(noiseVector);

currentPoseKey = symbol('x', 0);
currentPose = Pose3; % initial pose
gtValues.insert(currentPoseKey, currentPose);
gtGraph.add(PriorFactorPose3(currentPoseKey, currentPose, noise));

for i=1:trajectoryLength
  currentPoseKey = symbol('x', i);
  gtDeltaPosition = unsmooth_DP*randn(3,1) + [20;0;0]; % create random vector with mean = [1 0 0] and sigma = 0.5
  gtDeltaRotation = unsmooth_DR*randn(3,1) + [0;0;0]; % create random rotation with mean [0 0 0] and sigma = 0.1 (rad)
  gtDeltaMatrix(i,:) = [gtDeltaRotation; gtDeltaPosition];
  deltaPose = Pose3.Expmap(gtDeltaMatrix(i,:)');
  
  % "Deduce" ground truth measurements
  % deltaPose are the gt measurements - save them in some structure
  currentPose = currentPose.compose(deltaPose);
  gtValues.insert(currentPoseKey, currentPose);
  
  % Add the factors to the factor graph
  gtGraph.add(BetweenFactorPose3(currentPoseKey-1, currentPoseKey, deltaPose, noise));
end

figure(1)
hold on;
plot3DTrajectory(gtValues, '-r', [], 1, Marginals(gtGraph, gtValues));
axis equal

numMonteCarloRuns = 100;
for k=1:numMonteCarloRuns
  % create a new graph
  graph = NonlinearFactorGraph;
  
  % noisy prior
  currentPoseKey = symbol('x', 0);
  noisyDelta = noiseVector .* randn(6,1);
  initialPose = Pose3.Expmap(noisyDelta);
  graph.add(PriorFactorPose3(currentPoseKey, initialPose, noise));
  
  for i=1:trajectoryLength
    currentPoseKey = symbol('x', i);
    
    % for each measurement: add noise and add to graph
    noisyDelta = gtDeltaMatrix(i,:)' + (noiseVector .* randn(6,1));
    noisyDeltaPose = Pose3.Expmap(noisyDelta);
    
    % Add the factors to the factor graph
    graph.add(BetweenFactorPose3(currentPoseKey-1, currentPoseKey, noisyDeltaPose, noise));
  end
  
  % optimize
  optimizer = GaussNewtonOptimizer(graph, gtValues);
  estimate = optimizer.optimize();
 
  figure(1)
  plot3DTrajectory(estimate, '-b');
  
  marginals = Marginals(graph, estimate);
  
  % for each pose in the trajectory
  for i=1:trajectoryLength+1
    % compute estimation errors
    currentPoseKey = symbol('x', i-1);
    gtPosition  = gtValues.at(currentPoseKey).translation.vector;
    estPosition = estimate.at(currentPoseKey).translation.vector;
    errPosition = estPosition - gtPosition;
    
    % compute covariances:
    cov = marginals.marginalCovariance(currentPoseKey);
    covPosition = cov(4:6,4:6);
    
    % compute NEES using (estimationError = estimatedValues - gtValues) and estimated covariances
    NEES(k,i) = errPosition' * inv(covPosition) * errPosition; % distributed according to a Chi square with n = 3 dof
  end

  figure(2)
  hold on
  plot(NEES(k,:),'-b','LineWidth',1.5)
end
%%
ANEES = mean(NEES);
plot(ANEES,'-r','LineWidth',2)
plot(3*ones(size(ANEES,2),1),'k--'); % Expectation(ANEES) = number of dof 
box on
set(gca,'Fontsize',16)
title('NEES and ANEES');

%%
figure(1)
box on
set(gca,'Fontsize',16)
title('Ground truth and estimates for each MC runs');

%% Let us compute statistics on the overall NEES
n = 3; % position vector dimension
N = numMonteCarloRuns; % number of runs
alpha = 0.01; % confidence level

% mean_value = n*N; % mean value of the Chi-square distribution 
% (we divide by n * N and for this reason we expect ANEES around 1)
r1 = chi2inv(alpha, n * N)  / (n * N);
r2 = chi2inv(1-alpha, n * N)  / (n * N);

% output here
fprintf(1, 'r1 = %g\n', r1);
fprintf(1, 'r2 = %g\n', r2);

figure(3)
hold on
plot(ANEES/n,'-b','LineWidth',2)
plot(ones(size(ANEES,2),1),'r-');
plot(r1*ones(size(ANEES,2),1),'k-.');
plot(r2*ones(size(ANEES,2),1),'k-.');
box on
set(gca,'Fontsize',16)
title('NEES normalized by dof VS bounds');

%% NEES COMPUTATION (Bar-Shalom 2001, Section 5.4)
% the nees for a single experiment (i) is defined as 
%               NEES_i = xtilda' * inv(P) * xtilda, 
% where xtilda in R^n is the estimation
% error, and P is the covariance estimated by the approach we want to test
% 
% Average NEES. Given N Monte Carlo simulations, i=1,...,N, the average
% NEES is:
%                   ANEES = sum(NEES_i)/N
% The quantity N*ANEES is distributed according to a Chi-square
% distribution with N*n degrees of freedom.
%
% For the single run case, N=1, therefore NEES = ANEES is distributed 
% according to a chi-square distribution with n degrees of freedom (e.g. n=3 
% if we are testing a position estimate)
% Therefore its mean should be n (difficult to see from a single run)
% and, with probability alpha, it should hold:
%
% NEES in [r1, r2]
%
% where r1 and r2 are built from the Chi-square distribution