Bayesian Estimation and Tracking: A Practical Guide

14.2 Root Mean Squared Errors

Using (3.16), at time t_n we can write the state covariance matrix as

(14.39)

When x_n is known, as it would be in a simulation, and is unbiased, the square root of the diagonal elements can be approximated by conditional root mean squared errors (or the square root of the mean of the squared errors).

As noted in Chapter 4, the first task of a simulation is to generate a truth state trajectory, {x_n, n = 0, 1, . . . , N}, for all times of interest {t_n, n = 0, 1, . . . , N}. A truth observation trajectory is then generated from the truth state trajectory and independent noise is added to each observation to create a set of simulated observations, . To compute the RMS errors, multiple sets of observations are created in a Monte Carlo fashion, so that their are M sets of independent observations, {z_n,m ; n = 0, 1, . . . , N ; m = 1, 2, . . . , M}. Note that the observation noise must be independent from time-sample to time-sample (z_i,m must be independent from z_j,m ∀ i ≠ j) and from Monte Carlo run to Monte Carlo run (z_n,l must be independent from z_n,k ∀ l ≠ ...