Comparison of Heuristic Algorithms in Identification of Parameters of Anomalous Diffusion Model Based on Measurements from Sensors
Abstract
:1. Introduction
2. Anomalous Diffusion Model
3. Numerical Solution of Direct Problem
4. Inverse Problem and the Procedure for Its Solution
5. Meta-Heuristic Algorithms
5.1. ACO for Continuous Function Optimization
- Solution (pheromone) representation. Points from the search area ${\mathbb{R}}^{n}$ are identified as pheromone patches. In other words, the pheromone spot plays the role of a solution. Thus, k-th pheromone spot (or approximate solution) can be represented as ${\mathbf{x}}^{k}=({x}_{1}^{k},{x}_{2}^{k},\dots ,{x}_{n}^{k})$. Each solution (pheromone spot) has its quality calculated on the basis of fitness function $F\left({\mathbf{x}}^{k}\right)$. In each iteration of the algorithm, we store a fixed number of pheromone spots in the set of solutions (establish at the start of the algorithm).
- Transformation of the solution by the ant. The procedure of constructing a new solution, in the first place, consists in choosing one of the current solutions (pheromone spots) with a certain probability. The quality of the solution is a factor that determines the probability. The relationship here is as follows: with the increase in the quality of the solution, the probability of selection increases. In this paper, the following formula is adopted to calculate the probability (based on the rank) of the k-th solution:$${p}_{k}=\frac{{\omega}_{k}}{{\displaystyle \sum _{j=1}^{L}}{\omega}_{j}},$$$${\omega}_{k}=\frac{1}{qL\sqrt{2\pi}}{e}^{-\frac{{(rank\left(k\right)-1)}^{2}}{2{\left(qL\right)}^{2}}}.$$The symbol $rank\left(k\right)$ in the Equation (11) denotes the rank of the k-th solution in the set of solutions. The parameter q is a parameter that narrows the search area. In case of small value of q, the choice of the best solution is preferred. The greater q, the closer the probabilities of choosing each of the solutions. After choosing k-th solution, it is required to perform Gaussian sampling using the formula:$$g(x,\mu ,\sigma )=\frac{1}{\sigma \sqrt{2\pi}{e}^{-\frac{{(x-\mu )}^{2}}{2{\sigma}^{2}}}},$$
- Pheromone spots update. In each iteration of the ACO algorithm, M of new solutions is created (M denotes the number of ants). These solutions should be included in the solution set. In total, there are $L+M$ of pheromone spots in the set. Then the spots (solutions) are sorted by quality. The worst solutions in the M set are removed. Thus, the solution set always has a fixed number of elements equal to L.
Algorithm 1 Pseudocode of ACO algorithm. |
1: Initialization part. |
2: Configuration of ACO algorithm parameters. |
3: Initialization of starting population $\{{\mathbf{x}}^{1},{\mathbf{x}}^{2},\dots ,{\mathbf{x}}^{L}\}$ in a random way. |
4: Calculation value of the fitness function F for all pheromone spots and sorting them according to their rank (quality). |
5: Iterative part. |
6: for $\mathrm{iteration}i=1,2,\dots ,I$do |
7: Assignment of probability to pheromone spots according to the Equation (10). |
8: for $\mathrm{ant}m=1,2,\dots ,M$ do |
9: The ant chooses the k-th ($k=1,2,\dots ,L$) solution with probability ${p}_{k}$. |
10: for $\mathrm{coordinate}j=1,2,\dots ,n$ do |
11: Using the probability density function (12) in the sampling process, the ant changes the j-th coordinate of the k-th solution. |
12: end for |
13: end for |
14: Calculation the value of the fitness function F for M new solutions. |
15: Adding M new solutions to the set of archive of old, sorting the archive by quality and then rejection of the M worst solutions. |
16: end for |
17: return best solution ${\mathbf{x}}_{best}$. |
5.2. Dynamic Butterfly Optimization Algorithm
- Butterflies in the considered environment emit fragrances that differ in intensity, which results from the quality of the solution. Communication between these animals takes place through sensing the emitted fragrances.
- There are two ways of movement of a butterfly, namely: towards a more intense fragrance emitted by another butterfly and in a random direction.
- Global search is represented by:$${\mathbf{x}}^{new}={\mathbf{x}}^{old}+\left({r}^{2}\phantom{\rule{0.166667em}{0ex}}{\mathbf{x}}^{best}-{\mathbf{x}}^{old}\right)f,$$
- Local search move is formulated by:$${\mathbf{x}}^{new}={\mathbf{x}}^{old}+\left({r}^{2}\phantom{\rule{0.166667em}{0ex}}{\mathbf{x}}^{r1}-{\mathbf{x}}^{r2}\right)f,$$
Algorithm 2 Pseudocode of LSAM operator. |
1: ${\mathbf{x}}^{r}$—random solution among the top half best agents in population (obtained from BOA). |
2: $Fi{t}^{r}=F\left({\mathbf{x}}^{r}\right)$—value of the fitness function for ${\mathbf{x}}^{r}$. |
3: I—number of iterations, $\xi $—mutation rate. |
4: Iterative part. |
5: for $\mathrm{iteration}i=1,2,\dots ,I$do |
6: Calculate: $\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{\mathbf{x}}^{new}=$ Mutate(${\mathbf{x}}^{r},\xi $), $\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}Fi{t}^{new}=Fit\left({\mathbf{x}}^{new}\right)$. |
7: if $Fi{t}^{new}<Fi{t}^{r}$ then |
8: ${\mathbf{x}}^{r}={\mathbf{x}}^{new}$, $\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}Fi{t}^{r}=Fi{t}^{new}$. |
9: else |
10: Set a random solution ${\mathbf{x}}^{rnd}$ from the population, but not ${\mathbf{x}}^{r}$. |
11: Compute the fitness function $Fi{t}^{rnd}=Fit\left({\mathbf{x}}^{rnd}\right)$. |
12: if $Fi{t}^{new}<Fi{t}^{rnd}$ then |
13: ${\mathbf{x}}^{rnd}={\mathbf{x}}^{new}$ |
14: end if |
15: end if |
16: end for |
Algorithm 3 Pseudocode of DBOA. |
1: Initialization part. |
2: Determine parameters of BOA algorithm. N—number of butterfly in population, n—dimension, c—sensor modality and a, $\xi $, p parameters. |
3: Random generate starting population $\{{\mathbf{x}}^{1},{\mathbf{x}}^{2},\dots ,{\mathbf{x}}^{N}\}$. |
4: Calculate the value of the fitness function F (hence intensity of the stimulus $I=F$) for each butterfly ${\mathbf{x}}^{k}\phantom{\rule{4pt}{0ex}}(k=1,2,\dots ,N)$ in population. |
Iterative part. |
for $\mathrm{iteration}i=1,2,\dots ,I$do |
for $k=1,2,\dots ,N$ do |
Calculate value of fragnance for ${\mathbf{x}}^{k}$ with the use of Equation (13). |
5: end for |
Set the best agent ${\mathbf{x}}_{best}$ among the butterflies. |
for $k=1,2,\dots ,N$ do |
Set a random number r from range $[0,1]$. |
if $r<p$ then |
10: Convert solution ${\mathbf{x}}_{k}^{t}$ in accordance with the Equation (14). |
else |
Convert solution ${\mathbf{x}}_{k}^{t}$ in accordance with the Equation (15). |
end if |
end for |
15: Change value of the parameter a. |
Adopt the LSAM algorithm to convert the agents population with mutation rate $\xi $. |
end for |
return ${\mathbf{x}}^{best}$. |
5.3. Aquila Optimizer
- Expanded exploration. In the case that a predator is high in the air and wants to hunt other birds, it tilts vertically. After locating the victim from a height, Aquila begins nosediving with increasing speed. We can express this phenomenon with the use of the following equation:$${\mathbf{x}}^{new}=\left(1-\frac{i}{I}\right)\phantom{\rule{0.166667em}{0ex}}{\mathbf{x}}^{best}+\left({\mathbf{x}}^{mean}-rd\phantom{\rule{0.166667em}{0ex}}{\mathbf{x}}^{best}\right),$$$${\mathbf{x}}^{mean}=\frac{1}{N}\sum _{k=1}^{N}{\mathbf{x}}^{k}.$$
- Narrowed exploration. This technique involves circling the prey in flight and preparing to drop the earth and attack the prey. It is also known as short stroke contour flight. This is described in the algorithm by the equation:$${\mathbf{x}}^{new}=Lev{y}_{D}\phantom{\rule{0.166667em}{0ex}}{\mathbf{x}}^{best}+{\mathbf{x}}^{random}+rd\phantom{\rule{0.166667em}{0ex}}(rcos\varphi -rsin\varphi ),$$$$Lev{y}_{D}=\frac{s\phantom{\rule{0.166667em}{0ex}}u\phantom{\rule{0.166667em}{0ex}}\sigma}{{\left|v\right|}^{\frac{1}{\beta}}},$$$$\sigma =\frac{\mathsf{\Gamma}(1+\beta )sin\left(\frac{\pi \beta}{2}\right)}{\mathsf{\Gamma}\left(\frac{1+\beta}{2}\right){2}^{\frac{\beta -1}{2}}\beta}.$$In above equation $\mathsf{\Gamma}$ denotes gamma function. In order to determine the values of the parameters r and $\varphi $ the following formula is used:$$r={r}_{1}+V\phantom{\rule{0.166667em}{0ex}}{D}_{1},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\theta =-\xi \phantom{\rule{0.166667em}{0ex}}{D}_{1}+\frac{3\pi}{2},$$
- Expanded exploitation. This hunting technique begins with a vertical attack on a prey, which location is known within some approximation defining the search area. Thanks to this information, Aquila gets as close to its prey as possible. It can be described as follows:$${\mathbf{x}}^{neq}=\alpha \phantom{\rule{0.166667em}{0ex}}\left({\mathbf{x}}^{best}-{\mathbf{x}}^{mean}\right)-rd+\delta \phantom{\rule{0.166667em}{0ex}}\left(rd\phantom{\rule{0.166667em}{0ex}}(uB-lB)+lB\right),$$
- Narrowed exploitation. The characteristic feature of this technique are the stochastic movements of the bird, which attacks the prey in close proximity. It can be described by the formula:$${\mathbf{x}}^{new}=QF\phantom{\rule{0.166667em}{0ex}}{\mathbf{x}}^{best}-\left({G}_{1}\phantom{\rule{0.166667em}{0ex}}rd\phantom{\rule{0.166667em}{0ex}}{\mathbf{x}}^{mean}\right)-{G}_{2}\phantom{\rule{0.166667em}{0ex}}Lev{y}_{D}+{r}_{d}\phantom{\rule{0.166667em}{0ex}}{G}_{1},$$$$QF={i}^{\frac{2rd-1}{{(1-I)}^{2}}}.$$${G}_{1}$ and ${G}_{2}$ are described by:$${G}_{1}=2rd-1,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{G}_{2}=2{(t-T)}^{2}.$$We can adjust the algorithm with the above parameters.
Algorithm 4 Pseudocode of AO. |
1: Initialization part. |
2: Set up parameters of AO algorithm. |
3: Initialize population in a random way $\{{\mathbf{x}}^{1},{\mathbf{x}}^{2},\dots ,{\mathbf{x}}^{N}\}$. |
4: Iterative part. |
5: for $\mathrm{iteration}i=1,2,\dots ,I$do |
6: Determine values of the fitness function F for each agent in the population. |
7: Establish the best solution ${\mathbf{x}}^{best}$ in the population. |
8: for $k=1,2,\dots ,N$ do |
9: Calculate mean solution ${\mathbf{x}}^{mean}$ in the population. |
10: Improve parameters ${G}_{1},{G}_{2},QF$ of the algorithm. |
11: if $\mathrm{iteration}i\le \frac{2}{3}I$ then |
12: if $rd<0.5$ then |
13: Perform step expanded exploration (17) by updating solution ${\mathbf{x}}^{k}$. |
14: In the result solution ${\mathbf{x}}^{new,k}$ is obtained. |
15: if $F\left({\mathbf{x}}^{new,k}\right)<F\left({\mathbf{x}}^{k}\right)$ then make substitution ${\mathbf{x}}^{k}={\mathbf{x}}^{new,k}$ |
16: end if |
17: if $F\left({\mathbf{x}}^{new,k}\right)<F\left({\mathbf{x}}^{best}\right)$ then make substitution ${\mathbf{x}}^{best}={\mathbf{x}}^{new,k}$ |
18: end if |
19: else |
20: Perform step narrowed exploration (19) by updating solution ${\mathbf{x}}^{k}$. |
21: In the result solution ${\mathbf{x}}^{new,k}$ is obtained. |
22: if $F\left({\mathbf{x}}^{new,k}\right)<F\left({\mathbf{x}}^{k}\right)$ then make substitution ${\mathbf{x}}^{k}={\mathbf{x}}^{new,k}$. |
23: end if |
24: if $F\left({\mathbf{x}}^{new,k}\right)<F\left({\mathbf{x}}^{best}\right)$ then make substitution ${\mathbf{x}}^{best}={\mathbf{x}}^{new,k}$. |
25: end if |
26: end if |
27: else |
28: if $rd<0.5$ then |
29: Perform step Expanded exploitation (23) by updating solution ${\mathbf{x}}^{k}$. |
30: In the result solution ${\mathbf{x}}^{new,k}$ is obtained. |
31: if $F\left({\mathbf{x}}^{new,k}\right)<F\left({\mathbf{x}}^{k}\right)$ then make substitution ${\mathbf{x}}^{k}={\mathbf{x}}^{new,k}$. |
32: end if |
33: if $F\left({\mathbf{x}}^{new,k}\right)<F\left({\mathbf{x}}^{best}\right)$ then make substitution ${\mathbf{x}}^{best}={\mathbf{x}}^{new,k}$. |
34: end if |
35: else |
36: Perform step narrowed exploitation (24) by updating solution ${\mathbf{x}}^{k}$. |
37: In the result solution ${\mathbf{x}}^{new,k}$ is obtained. |
38: if $F\left({\mathbf{x}}^{new,k}\right)<F\left({\mathbf{x}}^{k}\right)$ then make substitution ${\mathbf{x}}^{k}={\mathbf{x}}^{new,k}$. |
39: end if |
40: if $F\left({\mathbf{x}}^{new,k}\right)<F\left({\mathbf{x}}^{best}\right)$ then make substitution ${\mathbf{x}}^{best}={\mathbf{x}}^{new,k}$. |
41: end if |
42: end if |
43: end if |
44: end for |
45: end for |
46: return ${\mathbf{x}}^{best}$. |
6. Numerical Example and Test of Algorithms
7. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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Algorithm | $\overline{\mathit{\lambda}}$ | ${\mathit{\delta}}_{\overline{\mathit{\lambda}}}[\%]$ | $\overline{\mathit{\beta}}$ | ${\mathit{\delta}}_{\overline{\mathit{\beta}}}[\%]$ | $\overline{\mathit{h}}\left(\mathit{t}\right)$ | ${\mathit{\delta}}_{\overline{\mathit{h}}}$ | F |
---|---|---|---|---|---|---|---|
ACO | $170.86$ | $7.14$ | $1.0838$ | $0.35$ | $2.27{t}^{2}+1.41t+10.71$ | $5.05$ | $272.95$ |
DBOA | $178.83$ | $2.81$ | $1.0818$ | $0.17$ | $2.42{t}^{2}-7.76t+94.46$ | $2.39$ | $0.45$ |
AO | $124.49$ | $32.34$ | $1.1021$ | $2.04$ | $2.11{t}^{2}+1.95t+20.01$ | $7.76$ | $482.39$ |
BOA | $194.27$ | $5.58$ | $1.0798$ | $0.02$ | $1.98{t}^{2}-5.42t+7.85$ | $21.55$ | $2501.21$ |
Algorithm | ${\mathsf{\Delta}}_{\mathbf{max}}$ | ${\mathsf{\Delta}}_{\mathbf{mean}}$ |
---|---|---|
ACO | $0.5111$ | $0.3575$ |
DBOA | $0.0261$ | $0.0131$ |
AO | $0.7471$ | $0.4361$ |
BOA | $2.7409$ | $0.9513$ |
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Brociek , R.; Wajda, A.; Słota, D. Comparison of Heuristic Algorithms in Identification of Parameters of Anomalous Diffusion Model Based on Measurements from Sensors. Sensors 2023, 23, 1722. https://doi.org/10.3390/s23031722
Brociek R, Wajda A, Słota D. Comparison of Heuristic Algorithms in Identification of Parameters of Anomalous Diffusion Model Based on Measurements from Sensors. Sensors. 2023; 23(3):1722. https://doi.org/10.3390/s23031722
Chicago/Turabian StyleBrociek , Rafał, Agata Wajda, and Damian Słota. 2023. "Comparison of Heuristic Algorithms in Identification of Parameters of Anomalous Diffusion Model Based on Measurements from Sensors" Sensors 23, no. 3: 1722. https://doi.org/10.3390/s23031722