Analysis of Genetic Algorithm on
Bearings-Only Target Motion Analysis

If we know our initial position $X_{obs}(0)$ and $Y_{obs}(0)$, target’s initial bearing $B_{obsd}(0)$, initial range from observer $R_{tar}(0)$; then we can extract target’s initial position $X_{tar}(0)$ and $Y_{tar}(0)$ from Eq. 1.

If we know our target’s course angle $C_{tar}$ and velocity $S_{tar}$; then we can extract target’s following positions $X_{tar}(t+\Delta t)$ and $Y_{tar}(t+\Delta t)$, which is changing in $\Delta t$ equal time, from Eq. 2.

When we get the positions of target $X_{target_{EA}}(t)$ and $Y_{target_{EA}}(t)$ using evolutionary strategies, we can get estimated bearing angles using Eq. 3.

The cost function for target motion analysis depends on Euclidean distance between estimated bearing angles $B_{estd}(t)$ and observed ones $B_{obsd}(t)$ is given by Eq. 4. This cost function has more calculation complexity but more feasible than the equidistant line distance difference cost function [11].

A solution individual is a chromosome bit array represents target motion. Solution individual’s structure is given in Eq. 5. $X_{0}$ and $Y_{0}$ are initial positions of target on $X$ and $Y$ axes respectively. $C$ is the course angle and $S$ is the velocity of target.

If we set a parameter bit-length for one of the four parameters above, we must know maximum $Max$ and minimum $Min$ value that parameter can get at limits. Then we use Eq. 6 with the degree of precision $p$, and the number of bits $m$. IEEE Standard for Floating-Point Arithmetic [12] says that for single precision floating point p parameter must be $log_{10}(2^{24})\cong 7.225$. If we set precision 7 and $Max=20000$, $Min=0$ meters for $X_{0}$ and $Y_{0}$ parameters of chromosome, due to Eq. 6 our bit width will be 38. For $C$ and $S$ bit-widths are 32 and 28 due to limits [360 0] degrees and [25 0] meters/seconds respectively. Total 136 bits were used; but if we decide to use single precision floating point, we would use only 128 bits for representation.

Instead of representing position by known bearing $B_{obsd}$ and range $R$; the authors chose to use two parameters $X_{0}$ and $Y_{0}$. These values are correlated by the corresponding bearing. Then using two parameter; creates correlation problems, increases search dimension so enlarges search space causing more time to obtain correct solution. So ideal chromosome must look like Eq. 7, this form uses 96 bits single precision floating point numbers.

In summary, genetic algorithm is a memory exhaustive and slow method to use in 2009, real valued evolutionary strategies uses real values instead of bit strings, which is more suitable for high level computing, and uses less memory.

Due to TMAGA, search space between estimation and limit values could be reduced 20$\%$ using 200 generations with 50 population size and 20 runs. If we think that $\sigma=0.3$ in case of degrees, which is a quite high noise, generation and population size are quite low to converge to the ideal solution. Even if TMAGA says the parameter space’s being narrowed into an infeasible area is a very unlikely case, it seems like very likely case, because genetic algorithm does not care the numerical value of the individual chromosome and does not use a feedback mechanism to narrow the area except the cost function. Because of this genetic algorithm results are expected to converge a different point in every different 20 runs.

If route is perpendicular to observer, noise becomes more destructive because determining the breaking point becomes more critical and it changes whole track even with a minimal change. In this case it is impossible to narrow the area to get correct result, because even with clean bearing it is hard to track target.

TMAGA uses a non-metric cost function called Total Deviation and shown in Eq. 8 where ${\Theta_{G}}_{i}$ and ${\Theta_{M}}_{i}$ are ground truth and calculated bearing angles of target at $i^{th}$ sample point. So this could cause divergence instead of convergence.

The ideal cost function is must be like in Eq. 9 to obtain Euclidean distance properties and become a metric distance measure.

Genetic algorithm’s mutation and crossover operators are somewhat weaker than real valued evolutionary strategy’s ones. Because bitwise crossover and mutation options are limited and binary numbers’ digit value is not meaningful by itself to these operators.

TMAGA uses the simplest selection method; individuals are weighted by their fitnesses obtained from cost function, then randomly picked from population using these weights. The highest weighted individual has the highest pick chance, but the lowest weighted one has being picked chance too. But this straight method is weak; because the highest two individual’s offspring won’t create a good individual every time, sometimes two bad individual could create the best individual; so stochastic universal sampling and elitism could help converge faster than used method.

TMAGA uses two piece crossover and randomly picked bitwise mutation methods. These methods are old fashioned for 2009 and their performances are not satisfying anymore. In real value measure non-uniform mutation and n-point real valued weighted averaging crossover methods could give much more better performance and easy to implement.

TMAGA’s main flow include two layered structure. Outer layer creates population, narrows search space and keeps track of best solution. Inner layer controls genetic algorithm operations. The flow can be seen in Fig. 2

Outer and inner layers have Monte Carlo runs separately; so Monte Carlo run increases exponentially. TMAGA uses 20 inner, 20 outer Monte Carlo runs. Inner Monte Carlo is used for narrowing space and genetic algorithm run, outer one is used for statistical analysis.

At the end of process, outer layer calculates weighted average of the best results from Monte Carlo runs and labels it as the best result.

TMAGA uses 50 population size, 200 generations for space narrowing, 500 generations for genetic algorithm, 20 statistical, 20 in-system Monte Carlo counts. If we calculate function evaluation count with these informations; $50*700*20=700000$ function evaluation will be done in a single Monte Carlo of TMAGA, if 20 statistical Monte Carlo runs completed it will be $14000000$. 14 million function evaluation is too much for $limits(X_{0})=[20000,0]$ meters, $limits(Y_{0})=[20000,0]$ meters, $limits(C)=[360,0]$ degrees and $limits(S)=[25,0]$ meters/seconds.

If we chose $(R_{0}|C|S)$ form Eq. 7 and $limits(R)=[28000,0]$ meters, $limits(C)=[360,0]$ degrees and $limits(S)=[25,0]$ meters/seconds; then search the space with 100 unit precision for $R$, 0.5 unit precision for $C$ and 0.1 unit precision for $S$ using brute force; $280*720*250=50400000$ function evaluation needed. Results for brute force searching is given in Table II with ground truth values in Table I. Even with $50400000$ function evaluations, we cannot get correct results.

From Table II we can see that even brute force cannot find a clear solution for that high noise levels. Thus we can say noise is disrupting useful information in bearing data.

We said that TMAGA has not guaranteed results before; because in the paper [4] there is no information about variance of Monte Carlo runs and statistics are only given for means or the best results. Thus we cannot determine the real performance of the TMAGA. Higher performances on higher error standart deviations like $\sigma_{noise}=0.5$ and $\sigma_{noise}=1.0$ in degree measure are suspicious. Additionally, perpendicular movement of target to observer makes tracking difficult; but TMAGA says it is tracking target very accurate with these error variances. Even we look at Table II, brute force couldn’t get the precise results.

If we look at Table 3 which is given by [4], we could see Monte Carlo is done two times and mean of the second Monte Carlo is given as the average result. There is no information about variance or standard deviation; so this type of reporting is not acceptable. Solutions are obtained by weighted averages of the first Monte Carlo’s results. There is no statistical relation between best solution and fitness in extra noisy data condition.

CMA-ES is a kind of evolutionary strategy that uses statistical feedback and covariance matrix for capturing search space shape. CMA-ES uses cost weighted recombination instead of crossover; random number generation using Gaussian distribution instead of mutation.

$m_{i}$ is the favorite solution at the start of each generation and could be obtained using Eq. 10. $w_{i:\lambda}$ is the weight obtained from the cost of the parent $i$ for the $\lambda^{th}$ offspring cluster, $x_{i}$ is the $i^{th}$ parent.

Covariance matrix $C$ determines the shape of the random number generation and step size $\sigma$ determines the width of probability density function. Initially we use $\sigma$ predefined around like a number 0.3 and $C$ unit diagonal matrix. We learn them each generation via the methods described by Hansen[9]. Then we use Eq. 11 to get new population individuals $x_{i}$ around the $m_{i}$ favorite individual.

CMA-ES narrows the search space using statistical information learned by updating $C$ and $\sigma$. This narrowing helps CMA-ES converge with less calculations than classical evolutionary strategies. Fig. 3 shows how CMA works and converges to the result by the help of $C$ and $\sigma$.