Evaluating Optimization Algorithms in MATLAB, Python, and R

As those of you who read my last post know, I’m at the NIMBioS-CAMBAM workshop on linking mathematical models to biological data here at UT Knoxville. Day 1 (today) was on parameter estimation and model identifiability. Specifically, we (quickly) covered maximum likelihood estimation and how to program our own estimation procedures in MATLAB.

If you’ve read this blog in the past, you know I’m an open-source kind of person. Naturally, I re-programmed all of the MATLAB examples in R. When I did, I noticed something funny. The optimization procedures in MATLAB gave different estimates than those in R. I asked a post-doc there, who seemed equally stumped but did mention that R’s optimization procedures are little funky. So, I took the next logical step and programmed the ML optimization routine into Python, using Scipy and Numpy, just to double check.

The model is an SIR epidemiological model that predicts the number of Susceptible, Infected, and Recovering people with, in this case, cholera. It relies on four parameters, Bi, Bw, e, and k. I won’t give the model here, you’ll see the formula in the code below. When optimizing, I made sure that MATLAB, R, and Python all used Nelder-Mead algorithms and, when possible, equivalent ODE solvers (ode45 in MATLAB and R).

I won’t post the MATLAB code here, because I didn’t write it and it’s multiple files etc etc, but I’ve gone over it line by line to make sure it’s identical to my R and Python code. It is.
MATLAB outputs estimates for Bi (0.2896), Bw (1.0629), e (0.0066) and k (0.0001). If you want the MATLAB files, or the data, I’ll send them to you.

In R, first import the data (see the Python code below for the actual data). Then, run the ode solver and optimization code:

## Main code for the SIR model example
library(deSolve) # Load the library to solve the ode

## Set initial parameter values
Bi <- 0.75
Bw <- 0.75
e <- 0.01
k <- 1/89193.18

## Combine parameters into a vector
params <- c(Bi, Bw, e, k)
names(params) <- c('Bi', 'Bw', 'e', 'k')

# Make a function for running the ODE
SIRode <- function(t, x, params){
S <- x[1]
I <- x[2]
W <- x[3]
R <- x[4]

Bi <- params[1]
Bw <- params[2]
e <- params[3]
k <- params[4]

dS <- -Bi*S*I - Bw*S*W
dI <- Bi*S*I + Bw*S*W - 0.25*I
dW <- e*(I - W)
dR <- 0.25*I

output <- c(dS, dI, dW, dR)
list(output)
}

# Set initial conditions
I0 <- data[1]*k
R0 <- 0
S0 <- (1-I0)
W0 <- 0

initCond <- c(S0, I0, W0, R0)

# Simulate the model using our initial parameter guesses
initSim <- ode(initCond, tspan, SIRode, params, method='ode45')

plot(tspan, initSim[,3]/k, type='l')
points(tspan, data)

# Make a function for optimzation of the parameters
LLode <- function(params){

k <- params[4]

I0 <- data[1]*k
R0 <- 0
S0 <- 1 - I0
W0 <- 0

initCond <- c(S0, I0, W0, R0)

# Run the ODE
odeOut <- ode(initCond, tspan, SIRode, params, method='ode45')

# Measurement variable
y <- odeOut[,3]/k

diff <- (y - data)
LL <- t(diff) %*% diff

return(LL)
}

# Run the optimization procedure
MLresults <- optim(params, LLode, method='Nelder-Mead')

## Resimulate the ODE
estParms <- MLresults$par
estParms
MLresults$value

I0est <- data[1]*estParms[4]
S0est <- 1 - I0est
R0 <- 0
W0 <- 0
initCond <- c(S0est, I0est, W0, R0)

odeEstOut <- ode(initCond, tspan, SIRode, estParms, method='ode45')
estY <- odeEstOut[,3]/estParms[4]

plot(tspan, data, pch=16, xlab='Time', ylab='Number Infected')
lines(tspan, estY)

Running this code gives me estimates of Bi (0.30), Bw (1.05), e (0.0057), and k (0.0001).

R_OptimizeI rounded the values, but the unrounded values are even closer to MATLAB. So far, I’m fairly satisfied.

We can run the same procedure in Python:


from scipy.optimize import minimize
from scipy import integrate
import pylab as py

## load data

tspan = np.array([0, 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, 91, 98, 105, 112, 119, 126, 133, 140, 147, 154, 161])
data = np.array([ 113, 60, 70, 140, 385, 2900, 4600, 5400, 5300, 6350, 5350, 4400, 3570, 2300, 1900, 2200, 1700, 1170, 830, 750, 770, 520, 550, 380 ])

## define ODE equations
def SIRode(N, t, Bi, Bw, e, k):
    return(
    -Bi*N[0]*N[1] - Bw*N[0]*N[2] ,
    +Bi*N[0]*N[1] + Bw*N[0]*N[2] - 0.25*N[1] ,
    +e*(N[1] - N[2]) ,
    +0.25*N[1]
    )

# Set parameter values
Bi = 0.75
Bw = 0.75
e = 0.01
k = 1/89193.18

# Set initial conditions
I0 = data[0]*k
S0 = 1 - I0
N = [S0, I0, 0]

# Run the ode
Nt = integrate.odeint(SIRode, N, tspan, args=(Bi, Bw, e, k))

# Get the second column of data corresponding to I
INt = [row[1] for row in Nt]
INt = np.divide(INt, k)

py.clf()
py.plot(tspan, data, 'o')
py.plot(tspan, INt)
py.show()

def LLode(x):
    Bi = x[0]
    Bw = x[1]
    e = x[2]
    k = x[3]

    I0 = data[0]*k
    S0 = 1-I0
    N0 = [S0, I0, k]

    Nt = integrate.odeint(SIRode, N0, tspan, args=(Bi, Bw, e, k))

    INt = [row[1] for row in Nt]
    INt = np.divide(INt, k)

    difference = data - INt

    LL = np.dot(difference, difference)

    return LL

x0 = [Bi, Bw, e, k]

results = minimize(LLode, x0, method='nelder-mead')

print results.x

estParams = results.x

Bi = estParams[0]
Bw = estParams[1]
e = estParams[2]
k = estParams[3]

I0 = data[0]*k
S0 = 1 - I0
N = [S0, I0, 0]

Nt = integrate.odeint(SIRode, N, tspan, args=(Bi, Bw, e, k))

INt = [row[1] for row in Nt]
INt = np.divide(INt, k)

py.clf()
py.plot(tspan, data, 'o')
py.plot(tspan, INt)
py.show()

Python gave me estimates of Bi (0.297), Bw (1.106), e (0.0057), and k (0.00001).

Python_optimizeOverall, I have to say I’m pretty satisfied with the performance of both R and Python. I also didn’t find programming these sorts of optimization procedures in R or Python to be any more difficult than in MATLAB (discounting for the fact that I’m not terribly familiar with MATLAB, but I’m also only somewhat familiar with Python, so they’re roughly equivalent here). I initially wrote this post because I could not get R to sync up with MATLAB, so I tried it in Python and got the same results as in R. I then found out that something was wrong with the MATLAB code, so that all three matched up pretty well.

I’ll say it again, R + Python = awesome.

UPDATE:

I’ve had a couple of requests for time comparisons between MATLAB, R, and Python. I don’t have those for MATLAB, I don’t know MATLAB well.

For Python, if I wrap the mimize function in:

t0 = time.time()
miminize....# run the optimizer
t1 = time.time()
print t1 - t0

I get 3.17 seconds.

In R, if I use system.time( ) to time the optim( ) function, I get about 39 seconds. That pretty much matches my feeling that R is just laboriously slow compared with how quickly Python evaluates the function.

About these ads

12 thoughts on “Evaluating Optimization Algorithms in MATLAB, Python, and R

  1. Could you give us optimization time in R and Python?
    By the way: some estimates differ substantially.
    R: Bi (0.30), Bw (1.5), e (0.0057), and k (0.0001).
    P: Bi (0.297), Bw (1.106), e (0.0057), and k (0.00001).

    • That’s a typo. R actually gives Bw(1.05), which is much closer to 1.106. Also, if I’m remembering correctly, the model has identifiability issues, so the parameters may not be estimated accurately (that was part of the exercise).

      I don’t have precise optimization times, but I can tell you that MATLAB is by far the fastest, followed closely by Python, while R was much slower than either. Of course, MATLAB uses parallel processing (I think), taking advantage of all four cores on my processes, while R does not. I’m not sure about Python. When doing heavy computations, I use the ‘snowfall’ package in R to use multi-core processing, which speeds up the computation substantially. So base MATLAB to base R: advantage MATLAB. base MATLAB to multi-core R is unknown. I haven’t done that.

    • I cut out the line where I load in a separate data file that defines the data and tspan because I have a separate R script that I source in with that information. You can get that info from the Python code and modify it from R.

    • See my response above, but I don’t have actual computation times. Eyeballing it, it seems that MATLAB ~ Python << R, but again R doesn't take advantage of multi-core processing like MATLAB and, possibly, Python. R was brutally slow compared to Python and MATLAB. Python might be faster than MATLAB… but I'm not good at MATLAB so I don't know how to get computational times (or in Python, for that matter).

    • I did that to standardize across MATLAB, R, and Python. Like I said, I didn’t write the MATLAB code and I’m not overly familiar with it. I know the optimize function that was used in MATLAB used Nelder-Mead, so I used that method in R and Python for equivalence.

  2. Pingback: R vs Python: Practical Data Analysis (Nonlinear Regression) | Climate Change Ecology

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s