Constrained Fitting of a Model to Data

Description

Fitting a model to data, with lower and/or upper bounds

Usage

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modFit(f, p, ..., lower = -Inf, upper = Inf,
  method = c("Marq", "Port", "Newton",
           "Nelder-Mead", "BFGS", "CG", "L-BFGS-B", "SANN",
           "Pseudo", "bobyqa"), jac = NULL,
  control = list(), hessian = TRUE)

## S3 method for class 'modFit'
summary(object, cov=TRUE,...)

## S3 method for class 'modFit'
deviance(object,  ...)

## S3 method for class 'modFit'
coef(object, ...)

## S3 method for class 'modFit'
residuals(object, ...)

## S3 method for class 'modFit'
df.residual(object, ...)

## S3 method for class 'modFit'
plot(x, ask = NULL, ...)

## S3 method for class 'summary.modFit'
print(x, digits = max(3, getOption("digits") - 3),
               ...)

Arguments

f

a function to be minimized, with first argument the vector of parameters over which minimization is to take place. It should return either a vector of residuals (of model versus data) or an element of class modCost (as returned by a call to modCost.

p

initial values for the parameters to be optimized over.

...

additional arguments passed to function f (modFit) or passed to the methods.

lower

lower bounds on the parameters; if unbounded set equal to -Inf.

upper

upper bounds on the parameters; if unbounded set equal to Inf.

method

The method to be used, one of "Marq", "Port", "Newton", "Nelder-Mead", "BFGS", "CG", "L-BFGS-B", "SANN", "Pseudo", "bobyqa" - see details.

jac

A function that calculates the Jacobian; it should be called as jac(x, ...) and return the matrix with derivatives of the model residuals as a function of the parameters. Supplying the Jacobian can substantially improve performance; see last example.

hessian

TRUE if Hessian is to be estimated. Note that, if set to FALSE, then a summary cannot be estimated.

control

additional control arguments passed to the optimisation routine - see details of nls.lm ("Marq"), nlminb ("Port"), optim ("Nelder-Mead", "BFGS", "CG", "L-BFGS-B", "SANN"), nlm ("Newton") or pseudoOptim("Pseudo").

object

an object of class modFit.

x

an object of class modFit.

digits

number of significant digits in printout.

cov

when TRUE also calculates the parameter covariances.

ask

logical; if TRUE, the user is asked before each plot, if NULL the user is only asked if more than one page of plots is necessary and the current graphics device is set interactive, see par(ask=.) and dev.interactive.

Details

Note that arguments after ... must be matched exactly.

The method to be used is specified by argument method which can be one of the methods from function optim:

  • "Nelder-Mead", the default from optim,

  • "BFGS", a quasi-Newton method,

  • "CG", a conjugate-gradient method,

  • "L-BFGS-B", constrained quasi-Newton method,

  • "SANN", method of simulated annealing.

Or one of the following:

  • "Marq", the Levenberg-Marquardt algorithm (nls.lm from package minpack) - the default. Note that this method is the only least squares method.

  • "Newton", a Newton-type algorithm (see nlm),

  • "Port", the Port algorithm (see nlminb),

  • "Pseudo", a pseudorandom-search algorithm (see pseudoOptim),

  • "bobyqa", derivative-free optimization by quadratic approximation from package minqa.

For difficult problems it may be efficient to perform some iterations with Pseudo, which will bring the algorithm near the vicinity of a (the) minimum, after which the default algorithm (Marq) is used to locate the minimum more precisely.

The implementation for the routines from optim differs from constrOptim which implements an adaptive barrier algorithm and which allows a more flexible implementation of linear constraints.

For all methods except L-BFGS-B, Port, Pseudo, and bobyqa that handle box constraints internally, bounds on parameters are imposed by a transformation of the parameters to be fitted.

In case both lower and upper bounds are specified, this is achieved by a tangens and arc tangens transformation.

This is, parameter values, p', generated by the optimisation routine, and which are located in the range [-Inf, Inf] are transformed, before they are passed to f as:

p = 1/2 * (upper + lower) + (upper - lower) * arctan(p')/pi

.

which maps them into the interval [lower, upper].

Before the optimisation routine is called, the original parameter values, as given by argument p are mapped from [lower,upper] to [-Inf, Inf] by:

p' = tan(pi/2 * (2 * p - upper - lower) / (upper - lower))

In case only lower or upper bounds are specified, this is achieved by a log transformation and a corresponding exponential back transformation.

In case parameters are transformed (all methods) or in case the method Port, Pseudo, Marq or bobyqa is selected, the Hessian is approximated as 2 * J^T * J, where J is the Jacobian, estimated by finite differences.

This ignores the second derivative terms, but this is reasonable if the method has truly converged to the minimum.

Note that finite differences are not extremely precise.

In case the Levenberg-Marquard method (Marq) is used, and parameters are not transformed, 0.5 times the Hessian of the least squares problem is returned by nls.lm, the original Marquardt algorithm. To make it compatible, this value is multiplied with 2 and the TRUE Hessian is thus returned by modFit.

Value

a list of class modFit containing the results as returned from the called optimisation routines.

This includes the following:

par

the best set of parameters found.

ssr

the sum of squared residuals, evaluated at the best set of parameters.

Hessian

A symmetric matrix giving an estimate of the Hessian at the solution found - see note.

residuals

the result of the last f evaluation; that is, the residuals.

ms

the mean squared residuals, i.e. ssr/length(residuals).

var_ms

the weighted and scaled variable mean squared residuals, one value per observed variable; only when f returns an element of class modCost; NA otherwise.

var_ms_unscaled

the weighted, but not scaled variable mean squared residuals

var_ms_unweighted

the raw variable mean squared residuals, unscaled and unweighted.

...

any other arguments returned by the called optimisation routine.

Note: this means that some return arguments of the original optimisation functions are renamed.

More specifically, "objective" and "counts" from routine nlminb (method = "Port") are renamed; "value" and "counts"; "niter" and "minimum" from routine nls.lm (method=Marq) are renamed; "counts" and "value"; "minimum" and "estimate" from routine nlm (method = "Newton") are renamed.

The list returned by modFit has a method for the summary, deviance, coef, residuals, df.residual and print.summary – see note.

Note

The summary method is based on an estimate of the parameter covariance matrix. In computing the covariance matrix of the fitted parameters, the problem is treated as if it were a linear least squares problem, linearizing around the parameter values that minimize Chi^2.

The covariance matrix is estimated as 1/(0.5*Hessian).

This computation relies on several things, i.e.:

  1. the parameter values are located at the minimum (i.e. the fitting algorithm has converged).

  2. the observations y_j are subject to independent errors whose variances are well estimated by 1 / (n - p) times the residual sum of squares (where n = number of data points, p = number of parameters).

  3. the model is not too nonlinear.

This means that the estimated covariance (correlation) matrix and the confidence intervals derived from it may be worthless if the assumptions behind the covariance computation are invalid.

If in doubt about the validity of the summary computations, use Monte Carlo fitting instead, or run a modMCMC.

Other methods included are:

  • deviance, which returns the model deviance,

  • coef, which extracts the values of the fitted parameters,

  • residuals,which extracts the model residuals,

  • df.residual which returns the residual degrees of freedom

  • print.summary, producing a nice printout of the summary.

Specifying a function to estimate the Jacobian matrix via argument jac may increase speed. The Jacobian is used in the methods "Marq", "BFGS", "CG", "L-BFGS", "Port", and is also used at the end, to estimate the Hessian at the optimum.

Specification of the gradient in routines "BFGS", "CG", "L-BFGS" from optim and "port" from nlminb is not allowed here. Within modFit, the gradient is rather estimated from the Jacobian jac and the function f.

Author(s)

Karline Soetaert <karline.soetaert@nioz.nl>,

Thomas Petzoldt <thomas.petzoldt@tu-dresden.de>

References

Bates, D., Mullen, K. D. Nash, J. C. and Varadhan, R. 2014. minqa: Derivative-free optimization algorithms by quadratic approximation. R package. https://cran.r-project.org/package=minqa

Gay, D. M., 1990. Usage Summary for Selected Optimization Routines. Computing Science Technical Report No. 153. AT&T Bell Laboratories, Murray Hill, NJ 07974.

Powell, M. J. D. (2009). The BOBYQA algorithm for bound constrained optimization without derivatives. Report No. DAMTP 2009/NA06, Centre for Mathematical Sciences, University of Cambridge, UK. http://www.damtp.cam.ac.uk/user/na/NA_papers/NA2009_06.pdf

Press, W. H., Teukolsky, S. A., Vetterling, W. T. and Flannery, B. P., 2007. Numerical Recipes in C. Cambridge University Press.

Price, W.L., 1977. A Controlled Random Search Procedure for Global Optimisation. The Computer Journal, 20: 367-370.

Soetaert, K. and Petzoldt, T., 2010. Inverse Modelling, Sensitivity and Monte Carlo Analysis in R Using Package FME. Journal of Statistical Software 33(3) 1–28. http://www.jstatsoft.org/v33/i03

Please see also additional publications on the help pages of the individual algorithms.

See Also

constrOptim for constrained optimization.

Examples

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## =======================================================================
## logistic growth model
## =======================================================================
TT    <- seq(1, 60, 5)
N0    <- 0.1
r     <- 0.5
K     <- 100

## perturbed analytical solution
Data <- data.frame(
  time = TT,
     N = K / (1+(K/N0-1) * exp(-r*TT)) * (1 + rnorm(length(TT), sd = 0.01))
)

plot(TT, Data[,"N"], ylim = c(0, 120), pch = 16, col = "red",
     main = "logistic growth", xlab = "time", ylab = "N")


##===================================
## Fitted with analytical solution  #
##===================================

## initial "guess"
parms <- c(r = 2, K = 10, N0 = 5)

## analytical solution
model <- function(parms,time)
  with (as.list(parms), return(K/(1+(K/N0-1)*exp(-r*time))))

## run the model with initial guess and plot results
lines(TT, model(parms, TT), lwd = 2, col = "green")

## FITTING algorithm 1
ModelCost <- function(P) {
 out <- model(P, TT)
 return(Data$N-out)  # residuals
}

(Fita <- modFit(f = ModelCost, p = parms))

times <- 0:60
lines(times, model(Fita$par, times), lwd = 2, col = "blue")
summary(Fita)

##===================================
##  Fitted with numerical solution  #
##===================================

## numeric solution
logist <- function(t, x, parms) {
  with(as.list(parms), {
    dx <- r * x[1] * (1 - x[1]/K)
    list(dx)
  })
}

## model cost,
ModelCost2 <- function(P) {
 out <- ode(y = c(N = P[["N0"]]), func = logist, parms = P, times = c(0, TT))
 return(modCost(out, Data))  # object of class modCost
}

Fit <- modFit(f = ModelCost2, p = parms, lower = rep(0, 3),
              upper = c(5, 150, 10))

out <- ode(y = c(N = Fit$par[["N0"]]), func = logist, parms = Fit$par,
           times = times)

lines(out, col = "red", lty = 2)
legend("right", c("data", "original", "fitted analytical", "fitted numerical"),
       lty = c(NA, 1, 1, 2), lwd = c(NA, 2, 2, 1),
       col = c("red", "green", "blue", "red"), pch = c(16, NA, NA, NA))
summary(Fit)
plot(residuals(Fit))

## =======================================================================
## the use of the Jacobian
## =======================================================================

## We use modFit to solve a set of linear equations
A <- matrix(nr = 30, nc = 30, runif(900))
X <- runif(30)
B <- A %*% X

## try to find vector "X"; the Jacobian is matrix A

## Function that returns the vector of residuals
FUN <- function(x)
  as.vector(A %*% x - B)

## Function that returns the Jacobian
JAC <- function(x) A

## The port algorithm
print(system.time(
  mf <- modFit(f = FUN, p = runif(30), method = "Port")
))
print(system.time(
  mf <- modFit(f = FUN, p = runif(30), method = "Port", jac = JAC)
))
max(abs(mf$par - X))  # should be very small

## BFGS
print(system.time(
  mf <- modFit(f = FUN, p = runif(30), method = "BFGS")
))
print(system.time(
  mf <- modFit(f = FUN, p = runif(30), method = "BFGS", jac = JAC)
))
max(abs(mf$par - X))

## Levenberg-Marquardt
print(system.time(
  mf <- modFit(f = FUN, p = runif(30), jac = JAC)
))
max(abs(mf$par - X))