PBPSO Algorithm Comparisons

Comparison of initial and final swarms for 3 different parameter based algorithms

Synthetic data sets are used to test the degree of swarm convergence for the following algorithms:

PSO with "tempest" boundary conditions

Velocity weighting factor: w=0.
Individuals leaving the solution domain are destroyed a new one is randomly created within the solution domain.

PSO with "pong" boundary conditions

Velocity weighting factor: w=0.
Individuals leaving the solution domain bounce back in a distance equal to their "out of bounds" distance x.
If x is greater than the bounding box length L, they bounce back in a distance equal to the fraction part of x/L.

VPSO with "pong" boundary conditions

Velocity weighting factor: 0.0 ≤ w ≤ 0.4.
Weighting factors decrease as the sum of the fractional change in global parameters decreases to <1.0×10^-5.
Individuals leaving the solution domain bounce back in a distance equal to their "out of bounds" distance x.
If x is greater than the bounding box length L, they bounce back in a distance equal to the fraction part of x/L.

Each graph contains parameter values for 10 initial swarms and the corresponding 10 final swarms after the algorithm has locked in on a solution. Each swarm consists of 200 individuals, so each graph depicts 2000 initial and final individuals' positions. The degree of convergence of individuals towards the optimal solution is not a measure of which algorithm retrieves the size distribution that is consistently closest to the target solution, but instead reflects the time it takes for the algorithm to reach the performance goal of all global parameters having a fractional change of less than 1.0×10^–5 for 100 consecutive iterations. It should be noted that "tempest" boundary conditions generally result in more individuals leaving the solution domain. Since their replacements are randomly generated, this appears in the graphs as less overall convergence. It was initially thought that this behavior would allow the algorithm to explore more of the solution domain, but after reviewing the results the faster convergence using "pong" boundary conditions seems preferable.

Average parameter values with uncertainties appearing in the results tables below using the method described by DuPaul (AJUR, submitted 02/14/2017) to reduce the set of ten parameter values based on the reduced chi-squared for aerosol optical depths calculated from the size distributions.

Junge Size Distribution

n_J(r) = dN/dr = N₀βr^–(α+1)

Graphics:Junge Initial and Final Swarms - PSO Tempest BCs
● target solution ● initial swarms ● final swarms

Comparison of Results
Solution N₀×β (cm^–3) α

Target 7.5×10⁵ 3

PSO - tempest (7.499919±0.4)×10⁵ 3

PSO - pong (7.499922±0.5)×10⁵ 3

VPSO - pong (7.499915±0.6)×10⁵ 3

Comparison of Results
Solution	N₀×β (cm^–3)	α
Target	7.5×10⁵	3
PSO - tempest	(7.499919±0.4)×10⁵	3
PSO - pong	(7.499922±0.5)×10⁵	3
VPSO - pong	(7.499915±0.6)×10⁵	3

Since N₀ and β are not independent, the swarm convergence appears to be weak with many individuals unable to converge before the performance criteria are met. However, the factor N₀×β is what is important for this distribution and all algorithms yield the same result.

Modified Gamma Size Distribution

n_G(r) = dN/dr = N₀r^βe^–αr

Graphics:Modified Gamma Initial and Final Swarms - PSO Tempest BCs
● target solution ● initial swarms ● final swarms

Comparison of Results
Solution N₀ (cm^–(3+β)) α (μm^–1) β

Target 2.5×10⁹ 0.1 0.35

PSO - tempest (2.6±0.8)×10⁹ 0.097±0.007 0.33±0.04

PSO - pong (2.5±0.7)×10⁹ 0.098±0.005 0.34±0.03

VPSO - pong (4.0±0.7)×10⁹ 0.100±0.004 0.40±0.02

Comparison of Results
Solution	N₀ (cm^–(3+β))	α (μm^–1)	β
Target	2.5×10⁹	0.1	0.35
PSO - tempest	(2.6±0.8)×10⁹	0.097±0.007	0.33±0.04
PSO - pong	(2.5±0.7)×10⁹	0.098±0.005	0.34±0.03
VPSO - pong	(4.0±0.7)×10⁹	0.100±0.004	0.40±0.02

The peak of this modified gamma distribution occurs at r_p = β/α. β affects the small r behavior while α affects r_p and hence the large r behavior. In a log-log plot of n_G(r), β is the slope for r < 1 while the drop of for r > r_p is affected by both α and β. The individuals tend to converge to a surface within the solution domain. This means that there are multiple size distributions that can yield the same set of aerosol optical depths within instrument accuracy. The possibility of having multiple valid size distributions is reflected in the uncertainties associated with the distribution parameters. The results indicate that PSO-pong is the most accurate algorithm.

Lognormal Size Distribution

n_L(r) = dN/dr = (2π)^–1/2N₀(βr)^–1exp[–ln(r/α)²/2β²]

Graphics:Lognormal Initial and Final Swarms - PSO Tempest BCs
● target solution ● initial swarms ● final swarms

Comparison of Results
Solution N₀ (cm^–2) α (μm) β r_mode (μm) r_mean (μm)

Target 1.0×10⁷ 0.500 0.307 0.455 0.524

PSO - tempest 1.0×10⁷ 0.500 0.307 0.455 0.524

PSO - pong 1.0×10⁷ 0.500 0.307 0.455 0.524

VPSO - pong 1.0×10⁷ 0.500 0.307 0.455 0.524

Comparison of Results
Solution	N₀ (cm^–2)	α (μm)	β	r_mode (μm)	r_mean (μm)
Target	1.0×10⁷	0.500	0.307	0.455	0.524
PSO - tempest	1.0×10⁷	0.500	0.307	0.455	0.524
PSO - pong	1.0×10⁷	0.500	0.307	0.455	0.524
VPSO - pong	1.0×10⁷	0.500	0.307	0.455	0.524

All algorithms are equally good at resolving the target solution. Individuals tend to converge to a point within the solution domain, with a few straggling behind unable to converge before the performance criteria are met.

Bimodal Size Distribution

n_B(r) = dN/dr = (2π)^–1/2N₀(β₀r)^–1exp[–ln(r/α₀)²/2β₀²] + (2π)^–1/2N₁(β₁r)^–1exp[–ln(r/α₁)²/2β₁²]

Aitken Mode
Graphics:Bimodal Mode 1 Initial and Final Swarms - PSO Tempest BCs
● target solution ● initial swarms ● final swarms

Accumulation Mode
Graphics:Bimodal Mode 2 Initial and Final Swarms - PSO Tempest BCs
● target solution ● initial swarms ● final swarms

Comparison of Results
Solution N₀ (cm^–2) α₀ (μm) β₀ N₁ (cm^–2) α₁ (μm) β₁ r_0,mode (μm) r_0,mean (μm) r_1,mode (μm) r_1,mean (μm)

Target 1.0×10⁷ 0.100 0.307 1.0×10⁶ 2.00 0.307 0.091 0.105 1.820 2.097

PSO - tempest (2.1±1.1)×10⁷ 0.102±0.017 0.28±0.04 (8.1±0.4)×10⁶ 2.28±0.09 0.28±0.03 0.095±0.016 0.107±0.018 2.10±0.09 2.37±0.10

PSO - pong (1.0±0.5)×10⁷ 0.118±0.011 0.27±0.05 (1.15±0.27)×10⁶ 1.94±0.24 0.32±0.05 0.110±0.011 0.122±0.012 1.75±0.23 2.05±0.26

VPSO - pong (2.4±1.1)×10⁷ 0.118±0.027 0.30±0.05 (1.08±0.22)×10⁶ 2.12±0.25 0.29±0.05 0.107±0.025 0.123±0.028 1.95±0.23 2.21±0.26

Comparison of Results
Solution	N₀ (cm^–2)	α₀ (μm)	β₀	N₁ (cm^–2)	α₁ (μm)	β₁	r_0,mode (μm)	r_0,mean (μm)	r_1,mode (μm)	r_1,mean (μm)
Target	1.0×10⁷	0.100	0.307	1.0×10⁶	2.00	0.307	0.091	0.105	1.820	2.097
PSO - tempest	(2.1±1.1)×10⁷	0.102±0.017	0.28±0.04	(8.1±0.4)×10⁶	2.28±0.09	0.28±0.03	0.095±0.016	0.107±0.018	2.10±0.09	2.37±0.10
PSO - pong	(1.0±0.5)×10⁷	0.118±0.011	0.27±0.05	(1.15±0.27)×10⁶	1.94±0.24	0.32±0.05	0.110±0.011	0.122±0.012	1.75±0.23	2.05±0.26
VPSO - pong	(2.4±1.1)×10⁷	0.118±0.027	0.30±0.05	(1.08±0.22)×10⁶	2.12±0.25	0.29±0.05	0.107±0.025	0.123±0.028	1.95±0.23	2.21±0.26

Individuals tend to converge to a point within the solution domain. Trade-offs between the Aitken mode and accumulation mode parameters lead to multiple valid size distributions as indicated by the uncertainties in the size distribution parameters. The results indicate that PSO-pong is the best overal algorithm when considering both accuracy and precision.