Particle Swarm Optimization (PSO)
PSO is a population-based optimization algorithm that mimics the collective behavior of birds or fish. It operates with a group of particles, each representing a possible solution to an optimization problem. These particles move through a solution space, continuously adjusting their positions and velocities. Each particle keeps track of its personal best solution and learns from the global best solution found by any particle in the population.
By balancing personal experience and global knowledge, PSO efficiently explores the solution space, gradually converging towards an optimal solution. The algorithm's performance relies on relevant parameters like the number of particles, maximum velocity, and learning factors.
As with most topics, there is a wealth of extensive and valuable information about PSO available on Google, and this post does not duplicate it. Instead, I focus on applying the psoptim() function from the R package pso to solve practical problems.
R code
The following R code estimates the Nelson-Siegel parameters by using the PSO.
#========================================================# # Quantitative Financial Econometrics & Derivatives # ML/DL using R, Python, Tensorflow by Sang-Heon Lee # # https://shleeai.blogspot.com #--------------------------------------------------------# # Estimating the Nelson-Siegel model # using Particle Swarm Optimizer (PSO) #========================================================# graphics.off(); rm(list = ls()) library(pso) # psoptim #----------------------------------------------- # Objective function #----------------------------------------------- objfunc <- function(para, y, m) { beta <- para[1:3]; la <- para[4] C <- cbind(rep(1,length(m)), (1-exp(-la*m))/(la*m), (1-exp(-la*m))/(la*m)-exp(-la*m)) return(sum((y - C%*%beta)^2)) } #======================================================= # 1. Read data #======================================================= # b1, b2, b3, lambda for comparisons ns_reg_para_rmse1 <- c( 4.26219396, -4.08609206, -4.90893865, 0.02722607, 0.04883786) ns_reg_para_rmse2 <- c( 4.97628654, -4.75365297, -6.40263059, 0.05046789, 0.04157326) str.zero <- " mat rate1 rate2 3 0.0781 0.0591 6 0.1192 0.0931 9 0.1579 0.1270 12 0.1893 0.1654 24 0.2669 0.3919 36 0.3831 0.8192 48 0.5489 1.3242 60 0.7371 1.7623 72 0.9523 2.1495 84 1.1936 2.4994 96 1.4275 2.7740 108 1.6424 2.9798 120 1.8326 3.1662 144 2.1715 3.4829 180 2.5489 3.7827 240 2.8093 3.9696" df <- read.table(text = str.zero, header=TRUE) m <- df$mat y1 <- df$rate1; y2 <- df$rate2 #============================================================== # 2. Particle Swarm Optimizer # : PSO Optimization with constraints #============================================================== # NS estimation with 1st data y <- y1 x_init <- c(y[16], y[1]-y[16], 2*y[6]-y[1]-y[16], 0.0609) set.seed(90) psopt<-psoptim(x_init, fn = objfunc, lower = c(0, -20, -20, 0.015), upper = c(20, 20, 20, 0.075), y = y, m = m) nsptim_out1 <- c(psopt$par, sqrt(psopt$value/length(m))) # NS estimation with 2nd data y <- y2 x_init <- c(y[16], y[1]-y[16], 2*y[6]-y[1]-y[16], 0.0609) set.seed(90) psopt <- psoptim(x_init, fn = objfunc, y = y, m = m, lower = c(0, -20, -20, 0.015), upper = c(20, 20, 20, 0.075), control = list(trace=1, REPORT=50, reltol=1e-4, abstol=1e-8, hybrid=TRUE, hybrid.control=list(maxit=10))) nsptim_out2 <- c(psopt$par, sqrt(psopt$value/length(m))) #======================================================= # 3. Results and Comparisons #======================================================= ns_reg_para_rmse1 nsptim_out1 ns_reg_para_rmse2 nsptim_out2 | cs |
As can be seen in the following results, these simple examples yield the same outcomes with the given estimates.
Some research has applied hybrid particle swarm optimization and achieved promising results in financial applications such as asset allocation and yield curve fitting, among others, indicating the need for further exploration and investigation. Therefore, this post serves as a small step and starting point for guiding its usage. \(\blacksquare\)
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