#
Do Random and Chaotic Sequences Really Cause Different PSO Performance?

Paul Moritz Nörenberg and Hendrik Richter
HTWK Leipzig University of Applied Sciences
Faculty of Engineering
Postfach 301166, D–04251 Leipzig, Germany.
Email<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
###### Abstract

Our topic is performance differences between using random and chaos for
particle swarm optimization (PSO). We take random sequences with different
probability distributions and compare them to chaotic sequences with different
but also with same density functions. This enables us to differentiate between
differences in the origin of the sequences (random number generator or chaotic
nonlinear system) and statistical differences expressed by the underlying
distributions. Our findings (obtained by evaluating the PSO performance for
various benchmark problems using statistical hypothesis testing) cast
considerable doubt on previous results which compared random to chaos and
suggested that the choice leads to intrinsic differences in performance.

##  1 Introduction

If and under what circumstances chaotic sequences give different performance
results than random sequences in evolutionary computation is a topic of
intensive debate [3, 2, 4, 13, 9, 5, 12, 10, 15]. This debate has mainly been
conducted by taking a chaotic map and comparing for a set of benchmark
problems the performance results obtained by sequences from the map with the
results involving pseudo random number generators (PRNGs). In most of these
works we find the claim that chaos somehow improves the performance as
compared to random but different results have also been reported [1, 2, 5, 12,
10, 15, 6, 11, 14]. With the focus on particle swarm optimization (PSO), this
paper adds to the discussion but proposes a different approach.

Suppose there are differences in PSO performance between using chaotic and
random sequences and given the PSO implementations and parameters are the same
except the sequences used. Then, these differences in performance must
correspond to differences between the sequences, and a question worth asking
is how these sequences could possibly differ. Apart from the difference in
origin of both types of sequences, the main reason which could make chaotic
sequences different from random ones concerns the sequences as a whole and
involves differences in the underlying distribution or density.

Our addition to the discussion about performance differences between chaotic
and random sequences is to disentangle the effect of the underlying
distribution and effects connected to how the sequences are generated (PRNGs
or chaotic maps). It is well-known that PRNGs can produce random sequences
associated with different probability distributions, thus offering to study
the effect that different distributions have on PSO performance, see Figure
1(a) for the distributions considered in this study. A first observation is
that chaotic sequences obtained from a certain variety of nonlinear maps also
have different invariant densities, thus enabling to study the same effect,
see Figure 1(b) for the densities considered here. Another key observation is
that certain chaotic sequences have a density which has the same algebraic
description as the distribution of known random sequences. For instance, the
time evolution of the Logistic map with $r=4$ has the density
$\varrho(z)=\left(\pi\sqrt{z(1-z)}\right)^{-1}$ which is the same function as
the probability density of the Beta distribution $\mathcal{B}(\alpha/\beta)$
with $\alpha=\beta=0.5$. In other words, for certain parameter settings
chaotic sequences from the Logistic map and random sequences from the Beta
distribution are statistically equivalent as they can be described by one and
the same distribution. These observations suggest the following experimental
setup. We take chaotic and random sequences which have either the same
distribution or different ones. We test these sequences in otherwise identical
PSO implementations. In this way, we can differentiate the effect that
different or same distributions have on performance for either random or
chaotic sequences. Our main result obtained by null hypothesis testing with
the Wilcoxon rank-sum test is that the underlying distribution is the key
factor in performance, while the origin (either chaotic or random) is
secondary.

##  2 Experimental setup

Particle swarm optimization (PSO) is a generational population-based algorithm
used for calculating optima of single- and multidimensional functions $f(x)$.
A PSO particle holds 4 variables for each generation $t$: its position
$x(t)\in\mathbb{R}^{D}$ and velocity $v(t)\in\mathbb{R}^{D}$ in search space,
its own best position (local best position) $p^{l}(t)$ and the best position
of any particle of the population that occurred during all generations (global
best position) $p^{g}(t)$. The particle movement is described by

$\displaystyle x(t+1)=$ $\displaystyle x(t)+v(t+1)$ (1) $\displaystyle
v(t+1)=$ $\displaystyle
wv(t)+c_{1}r_{1}(p^{l}(t)-x(t))+c_{2}r_{2}(p^{g}(t)-x(t)).$ (2)

The parameters $w$, $c_{1}$ and $c_{2}$ are the inertial, cognitive and social
weights. The random variables $r_{1},r_{2}$ are taken from the sequences
produced by either the PRNGs or chaotic maps. At $t=0$, each particle $i$,
$i=1,2,\ldots I$, of swarm size $I$ is initialized randomly at position $x(0)$
with velocity $v(0)$. The fitness function is evaluated for the position of
every particle. The position with the highest fitness that occurred in this
evaluation is stored as global best position in each particle.

In the simulation we take standard values of PSO parameters frequently used:
$w=0.79$, $c_{1}=1.49$, $c_{2}=1.49$, $200$ generations per run, and $1000$
runs per sample. The performance of PSO runs is quantified by the mean
distance error, which is the mean of the absolute difference in search space
between the found best fitness value and the global optimum of the function
$f(x)$. A good performance means a low mean distance between the found best
result and the actual optimum of the fitness function.

(a) (b)

Figure 1: Distribution of sequences used in the study (a) Natural invariant
densities of chaotic maps. (b) Probability density functions of random
variables.

To compare the impact of chaotic and random sequences on PSO the following
test functions $f(x)$ from the IEEE CEC 2013 test suite [7] are employed:

  * •

1 Equal Maxima (1 dimension)

  * •

2 Uneven Decreasing Maxima (1 dimension)

  * •

3 Himmelblau (2 dimension)

  * •

4 Six-Hump Camel Back (2 dimension)

  * •

5 Shubert (2 dimension)

  * •

6 Vincent (2 dimension)

  * •

7-9 Rastrigin (10, 20 and 30 dimensions)

  * •

10-12 Rosenbrook (10, 20 and 30 dimensions)

  * •

13-15 Sphere (10, 20 and 30 dimensions).

Random sequences with the following distributions are used:

  * •

$\mathcal{B}(0.5/0.5)$ Beta distribution

  * •

$\mathcal{B}(1/5)$ Beta distribution

  * •

$\mathcal{B}(5/1)$ Beta distribution

  * •

$\mathcal{N}(0.5/0.1)$ normal distribution.

Chaotic sequences from the following chaotic maps are taken:

  * •

Logistic map, $z(k+1)=rz(k)(1-z(k))$, $r=4$

  * •

Cubic map, $z(k+1)=rz(k)(1-z(k)^{2})$, $r=2.62$

  * •

Bellows map, $z(k+1)=rz(k)(1+z(k))^{-6}$, $r=2.0$.

Table 1: Comparison between chaotic and random sequences for the test function
according to Section LABEL:sec:test and the distributions in Section
LABEL:sec:dist. Mean and standard deviation of the PSO performance over 1000
runs each. Bold numbers indicate the best performance for each of the 15 test
functions.

Test |  |  |  |  |  |  |  |  |  |  |  |  |  |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
function | Logistic Map | Cubic Map | Bellows Map | $\mathcal{B}(0.5/0.5)$ | $\mathcal{N}(0.5/0.1)$ | $\mathcal{B}(1/5)$ | $\mathcal{B}(5/1)$
| $\mu$ | $\sigma$ | $\mu$ | $\sigma$ | $\mu$ | $\sigma$ | $\mu$ | $\sigma$ | $\mu$ | $\sigma$ | $\mu$ | $\sigma$ | $\mu$ | $\sigma$
1 | 0.44e-4 | 0.16e-3 | 0.32e-4 | 0.18e-3 | 0.03e-4 | 0.04e-3 | 0.41e-4 | 0.16e-3 | 0.00e-4 | 0.00e-3 | 0.02e-4 | 0.03e-3 | 0.00e-4 | 0.00e-3
2 | 0.10e-2 | 1.05e-2 | 0.26e-2 | 1.95e-2 | 0.15e-2 | 1.38e-2 | 0.16e-2 | 1.48e-2 | 1.12e-2 | 4.12e-2 | 1.13e-2 | 4.24e-2 | 1.19e-2 | 4.24e-2
3 | 1.58e-1 | 0.69e-2 | 1.58e-1 | 0.70e-2 | 1.60e-1 | 0.94e-2 | 1.58e-1 | 0.68e-2 | 1.63e-1 | 1.17e-2 | 1.59e-1 | 0.87e-2 | 1.59e-1 | 0.86e-2
4 | 1.66e-1 | 2.61e-2 | 1.66e-1 | 2.59e-2 | 1.66e-1 | 2.62e-2 | 1.66e-1 | 2.60e-2 | 1.66e-1 | 2.61e-2 | 1.68e-1 | 2.54e-2 | 1.66e-1 | 2.61e-2
5 | 5.16e-2 | 3.26e-2 | 5.22e-2 | 3.06e-2 | 5.23e-2 | 3.19e-2 | 5.22e-2 | 3.21e-2 | 4.93e-2 | 3.09e-2 | 4.82e-2 | 2.88e-2 | 4.98e-2 | 3.07e-2
6 | 3.62e-6 | 0.22e-5 | 3.50e-6 | 0.22e-5 | 4.11e-6 | 1.41e-5 | 3.68e-6 | 0.22e-5 | 3.44e-6 | 0.22e-5 | 3.49e-6 | 0.22e-5 | 3.45e-6 | 0.22e-5
7 | 7.74e-2 | 2.11e-2 | 7.27e-2 | 1.94e-2 | 7.73e-2 | 2.11e-2 | 7.67e-2 | 2.22e-2 | 7.39e-2 | 1.92e-2 | 7.37e-2 | 1.88e-2 | 7.51e-2 | 2.02e-2
8 | 8.03e-2 | 1.75e-2 | 7.72e-2 | 1.77e-2 | 8.09e-2 | 1.99e-2 | 7.95e-2 | 1.79e-2 | 7.65e-2 | 1.74e-2 | 7.60e-2 | 1.54e-2 | 7.84e-2 | 1.85e-2
9 | 8.38e-2 | 1.78e-2 | 8.02e-2 | 1.73e-2 | 8.47e-2 | 1.98e-2 | 8.35e-2 | 1.77e-2 | 8.02e-2 | 1.84e-2 | 7.75e-2 | 1.44e-2 | 8.39e-2 | 2.11e-2
10 | 7.28e-2 | 1.73e-2 | 7.29e-2 | 1.79e-2 | 8.05e-2 | 1.58e-2 | 7.20e-2 | 1.80e-2 | 8.93e-2 | 1.28e-2 | 8.87e-2 | 1.21e-2 | 8.65e-2 | 1.35e-2
11 | 9.14e-2 | 0.74e-2 | 9.18e-2 | 0.69e-2 | 9.07e-2 | 0.78e-2 | 9.12e-2 | 0.74e-2 | 9.05e-2 | 0.77e-2 | 9.06e-2 | 0.70e-2 | 9.04e-2 | 0.76e-2
12 | 9.06e-2 | 0.63e-2 | 9.05e-2 | 0.66e-2 | 9.10e-2 | 0.66e-2 | 9.12e-2 | 0.61e-2 | 9.03e-2 | 0.67e-2 | 9.08e-2 | 0.64e-2 | 9.04e-2 | 0.70e-2
13 | 0.01e-4 | 0.02e-4 | 0.03e-4 | 0.05e-4 | 0.05e-4 | 0.38e-4 | 0.01e-4 | 0.02e-4 | 0.36e-4 | 1.73e-4 | 0.36e-4 | 1.72e-4 | 1.36e-4 | 2.26e-4
14 | 0.64e-2 | 0.28e-2 | 0.51e-2 | 0.27e-2 | 0.81e-2 | 0.33e-2 | 0.68e-2 | 0.29e-2 | 0.73e-2 | 0.30e-2 | 0.86e-2 | 0.33e-2 | 0.94e-2 | 0.35e-2
15 | 1.67e-2 | 0.41e-2 | 1.55e-2 | 0.40e-2 | 1.83e-2 | 0.43e-2 | 1.71e-2 | 0.42e-2 | 1.67e-2 | 0.43e-2 | 1.91e-2 | 0.44e-2 | 1.97e-2 | 0.44e-2

##  3 Results

Table 1 presents a comparison between chaotic and random sequences for the
test functions and the distributions in Section 2. For each combination of
test function and distribution the mean performance and the standard deviation
of the performance are given for 1000 PSO runs each. Bold numbers indicate the
best performance for each of the 15 test functions. The results vary not much
for a given test function; there are several test functions where more than
one distribution scores best results, but there are also subtle differences in
performance. For instance, PSO driven by sequences associated with the Cubic
map performs best for 5 out of the 15 test functions, but Bellows map only
scores 1 out of 15 and the Logistic map has 4 out of 15. For the random
sequences the results are a little more balanced with PSO driven by sequences
from $\mathcal{B}(0.5/0.5)$ and $\mathcal{N}(0.5/0.1)$ performing best 4 times
out of 15, while $\mathcal{B}(1/5)$ and $\mathcal{B}(5/1)$ come up with 3 best
results. At least for the selection considered, every distribution is best for
at least one test function. Moreover, if we were to compare chaotic sequences
from the Cubic map and random sequences from $\mathcal{B}(1/5)$ (or
$\mathcal{B}(5/1)$), we might be inclined to conclude that chaos and random
give different performances with chaos somewhat being better. In the
following, we analyse the performance results and argue that these differences
are rather caused by differences in the underlying distribution and to a
lesser degree by the fact that the sequences come from either PRNGs or chaotic
maps.

A first step is by displaying the distribution of the 1000 PSO runs as
boxplots to evaluate not only mean and standard deviation of the results, but
also quartiles and outliers. With the two dimensional Shubert function we take
an example showing some systematic differences of the PSO performance which
are typical for all test functions considered, see Figure 2. In addition to
the data in Table 1, we also give the result of a re-run with
$\mathcal{B}(0.5/0.5)$, for which we get the same mean and standard deviation,
but a slight difference in the lower whisker. Although the means and standard
deviations and also the median, first and third quartiles for all
distributions are almost equal, for PSO run samples that were driven by
$\mathcal{B}(0.5/0.5)$, the Logistic map and Bellows map, the lower quartile
of the performance data is significantly narrower than their counterparts
resulting from $\mathcal{B}(1/5)$, $\mathcal{B}(5/1)$, $\mathcal{N}(0.5/0.1)$
and the Cubic map. In other words, we may conclude that PSO performs slightly
worse when using $\mathcal{B}(0.5/0.5)$, Logistic map or Bellows map,
respectively, as for sequences from the other 3 distributions very good
results fall into the quartile group 1. Moreover, for these 3 distributions
the performance is almost the same. Furthermore, $\mathcal{B}(0.5/0.5)$ in
both runs and the Logistic map are most similar.

Figure 2: Box plot for PSO performance data for the Shubert function. Table 2:
Performance comparison between chaotic and random sequences by Wilcoxon rank-
sum test for all 15 test function according to Section LABEL:sec:test and the
7 distributions in Section LABEL:sec:dist. For each cell the tuple
$[+,-,\thickapprox]$ indicates for $+$ that the sequences from the
distribution in the row performs better, for $-$ that the sequences from the
distribution in the column performs better, while for $\thickapprox$ there is
no difference in performance. Bold numbers indicate comparisons between
chaotic and random sequences.

| $\mathcal{B}(0.5/0.5)$ |  cubic map |  Bellows map | $\mathcal{N}(0.5/0.1)$ | $\mathcal{B}(1/5)$ | $\mathcal{B}(5/1)$ | chaotic vs. random
---|---|---|---|---|---|---|---
$\mathcal{B}(0.5/0.5)$ | [1,0,14] |  |  | [4,7,4] | [5,6,4] | [5,4,6] |
logistic map | [2,1,12] | [1,5,9] | [5,2,8] | [3,6,6] | [5,6,4] | [4,5,6] | 14 ($+$), 18 ($-$), 28 ($\thickapprox$)
cubic map | [5,2,8] |  | [9,1,4] | [5,3,7] | [5,4,6] | [7,2,6] | 22 ($+$), 11 ($-$), 27 ($\thickapprox$)
Bellows map | [2,4,9] |  |  | [3,7,5] | [5,6,4] | [4,5,6] | 14 ($+$), 22 ($-$), 24 ($\thickapprox$)
$\mathcal{N}(0.5/0.1)$ |  |  |  |  | [2,3,10] | [5,3,7] |
$\mathcal{B}(1/5)$ |  |  |  |  |  | [5,2,8] |
random vs. chaotic | 7 ($+$), 9 ($-$), 29 ($\thickapprox$) |  |  | 16 ($+$), 11 ($-$), 18 ($\thickapprox$) | 16 ($+$), 15 ($-$), 14 ($\thickapprox$) | 12 ($+$), 15 ($-$), 18 ($\thickapprox$) |

Our second analysis step is to compare the performance with a nonparametric
statistical test of the null hypothesis. For this task, the Wilcoxon rank-sum
test is widely used to verify performance differences between variants of
metaheuristic algorithms. It is here taken to evaluate performance differences
between chaotic and random sequences. The two sided test evaluates the null
hypothesis that two given data samples are samples from continuous
distributions with the same median. We use a 0.05 significance level. The
Wilcoxon rank-sum test is used on every combination of PSO performance
samples. Table 2 states for each comparison between differently driven PSO the
following tuple $[+,-,\thickapprox]$ with:

  * •

$+$ …number of test functions where the PSO driven by the sequence in the row
performs better

  * •

$-$ …number of test functions where the PSO driven by the sequence in the
column performs better

  * •

$\thickapprox$ …number of test functions where the PSO performances are not
distinguishable by the Wilcoxon rank-sum test.

The results show that performance differences primarily occur when the used
sequences differ in the underlying probability distribution. For instance,
compare the results for the Logistic map with $\mathcal{B}(0.5/0.5)$, which
gives 5 superior, 6 inferior and 4 equal performances, or the results for
$\mathcal{B}(0.5/0.5)$ vs. $\mathcal{B}(1/5)$ with 4 superior, 5 inferior and
6 equal. In contrast the PSO using sequences from $\mathcal{B}(0.5/0.5)$ and
the Logistic map have very small deviations in performance with 1 superior, 2
inferior and 12 equal. This is almost on a par with a re-run with another set
of realizations of $\mathcal{B}(0.5/0.5)$, which gives 1 inferior and 14
equal. Another interesting comparison is the overall performance of chaotic
sequences vs. random sequences, see the bold numbers in Table 2 and the
summation in the last row and column. Here, we find for chaos vs. random that
the Cubic map gives 22 superior, 11 inferior and 27 equal performances, while
the Logistic and Bellows map score 14 superior vs. 18 (and 22) inferior and 28
(and 24) equal performances. Similar results can be recorded from the
perspective of random sequences vs. chaotic sequences. For instance,
$\mathcal{B}(0.5/0.5)$ has 7 superior, 9 inferior and 29 equal performances,
but $\mathcal{N}(0.5/0.1)$ scores 16 superior, 11 inferior and 18 equal. To
summarize some chaotic sequences perform better than random, but there are
also some random sequences which perform better than chaos, which agrees with
previously reported result [1, 2, 5, 12, 10, 15, 6, 11, 14]. In view of the
results presented here, however, it does not appear plausible to assume the
presence of general and systematic differences in performance between chaotic
and random sequences irrespective and independent of taking into account the
underlying distribution. Given the fact that for random and chaotic sequences
with the same distribution, Logistic map and $\mathcal{B}(0.5/0.5)$, we get
the smallest difference in performance, while for all other combinations of
different sources, we get different performances, the results rather support
the conclusion that the underlying distribution rather than the origin is the
main influential factor in PSO performance.

##  4 Discussion

There is an ongoing debate about using random and chaotic sequences for
driving population-based metaheuristic search algorithms and the effect on
behaviour and performance such a selection has [3, 2, 4, 13, 9, 5, 12, 10,
15]. This paper adds to this discussion with the focus on PSO and proposed the
following approach. We not only use random sequences associated with different
probability distributions, but also compare to chaotic sequences with
different density distributions and complete the experimental setup by random
and chaotic sequences with the same distribution. This enables us to
differentiate between differences in the origin of the sequences (PRNGs or
chaotic maps) and statistical differences expressed by the underlying
distributions.

These differences in statistical properties can potentially have profound
influences on PSO search dynamics, and thus on behaviour and performance. The
particle movement directly depends on the sequences as successive particle
positions are calculated via cognitive and social components weighted by the
elements from the sequences. The distribution associated with the sequences
defines global, long-term search preferences of the particle swarm. If, for
instance, the distribution is rather bell shaped, as a normal distribution,
then the search is more likely centered to the interior of the search space
region where the swarm is currently situated. If, on the other hand, the
distribution is rather U-shaped, as Beta distributions with $\alpha=\beta<1$,
the search more likely concentrates on the exterior of the current region. How
such a varying search bias interacts with unimodal or multimodal functions of
the optimization problem causes different types of swarm diversity decline. By
this mechanism does the distribution of the numbers in the random and chaotic
sequences influence the ratio of exploration and exploitation during the PSO
run and thus behavior and performance.

In the experimental setup, we considered 3 sources of chaotic sequences,
Logistic, Cubic and Bellows map, and 4 sources of random sequences, a normal
distribution $\mathcal{N}(0.5/0.1)$ and 3 Beta distributions,
$\mathcal{B}(0.5/0.5)$, $\mathcal{B}(1/5)$ and $\mathcal{B}(5/1)$. With this
selection, we took distributions which are not constant but have shape over
the sample space. This is in contrast to an uniform distribution, which is
constant and produces random sequences which are equally spread on a fixed
interval. For such a type of random distribution results similar to those
discussed here are known. The invariant density of the tent map equals an
uniform distribution and differences in performance between chaotic sequences
generated by the tent map as compared to sequences generated by the Logistic
map have been shown [6, 11], which is consistent with and supplements our
results.

##  5 Conclusions

We considered PSO and address the question whether or not using chaotic or
random sequences really causes different performance. Thus, our discussion is
not so much about what sequence performs best, but rather about identifying
reasons why some sequences are better than others. Our findings cast
considerable doubt on previous results which compared random to chaos and
suggested that there are intrinsic differences in PSO performance,
irrespective and independent of the associated distributions. We may choose
either random or chaotic sequences to drive PSO and observe differences in
performance. But most likely these differences occur because the underlying
distributions of the chosen sequences are different. In other words, if
chaotic sequences for a certain optimization problem (or also for a certain
collection of optimization problems) lead to better performance then this is
primary the case because the distribution is more fitting to the problem.

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