lsqnonlin: Nonlinear Least-Squares Fitting

Description

lsqnonlin solves nonlinear least-squares problems, including nonlinear data-fitting problems, through the Levenberg-Marquardt approach.

lsqnonneg solve nonnegative least-squares constraints problem.

Usage

lsqnonlin(fun, x0, options = list(), ...)
lsqnonneg(C, d)

Arguments

fun

User-defined, vector-valued function.

starting point.

...

additional parameters passed to the function.

options

list of options, for details see below.

C, d

matrix and vector such that C x - d will be minimized with x >= 0.

Value

Returns a list with the following elements:
- x: the point with least sum of squares value.
- ssq: the sum of squares.
- ng: norm of last gradient.
- nh: norm of last step used.
- mu: damping parameter of Levenberg-Marquardt.
- neval: number of function evaluations.
- errno: error number, corresponds to error message.
- errmess: error message, i.e. reason for stopping.

Details

lsqnonlin computes the sum-of-squares of the vector-valued function fun, that is if $f(x) = (f_1(x), \ldots ,f_n(x))$ then

m i n | | f (x) | |_{2}^{2} = m i n (f_{1} (x)^{2} + \dots + f_{n} (x)^{2})

will be minimized.

x=lsqnonlin(fun,x0) starts at point x0 and finds a minimum of the sum of squares of the functions described in fun. fun shall return a vector of values and not the sum of squares of the values. (The algorithm implicitly sums and squares fun(x).)

options is a list with the following components and defaults:

tau: used in starting value for Marquardt parameter.
tolx: stopping parameter for step length.
tolg: stopping parameter for gradient.
maxevalthe maximum number of function evaluations.

Typical values for tau are from 1e-6...1e-3...1 with small values for good starting points and larger values for not so good or known bad starting points.

lsqnonneg solves the linear least-squares problem C x - d, x nonnegative, transforming it with the `trick' x --> exp(x) into a nonlinear one and solving it applying lsqnonlin.

References

Madsen, K., and H. B.Nielsen (2010).Introduction to Optimization and Data Fitting. Technical University of Denmark, Intitute of Computer Science and Mathematical Modelling.

Fletcher, R., (1971). A Modified Marquardt Subroutine for Nonlinear Least Squares. Report AERE-R 6799, Harwell.

Examples

Run this code

##  Rosenberg function as least-squares problem
x0  <- c(0, 0)
fun <- function(x) c(10*(x[2]-x[1]^2), 1-x[1])
lsqnonlin(fun, x0)

##  Example from R-help
y <- c(5.5199668,  1.5234525,  3.3557000,  6.7211704,  7.4237955,  1.9703127,
       4.3939336, -1.4380091,  3.2650180,  3.5760906,  0.2947972,  1.0569417)
x <- c(1,   0,   0,   4,   3,   5,  12,  10,  12, 100, 100, 100)
# Define target function as difference
f <- function(b)
     b[1] * (exp((b[2] - x)/b[3]) * (1/b[3]))/(1 + exp((b[2] - x)/b[3]))^2 - y
x0 <- c(21.16322, 8.83669, 2.957765)
lsqnonlin(f, x0)        # ssq 50.50144 at c(36.133144, 2.572373, 1.079811)

# nls() will break down
# nls(Y ~ a*(exp((b-X)/c)*(1/c))/(1 + exp((b-X)/c))^2,
#     start=list(a=21.16322, b=8.83669, c=2.957765), algorithm = "plinear")
# Error: step factor 0.000488281 reduced below 'minFactor' of 0.000976563

## Least-squares data fitting
# Define fun(p, x)
lsqcurvefit <- function(fun, p0, xdata, ydata) {
    fn <- function(p, x) fun(p, xdata) - ydata
    lsqnonlin(fn, p0)    
}

##  Lanczos1 data (artificial data)
#   f(x) = 0.0951*exp(-x) + 0.8607*exp(-3*x) + 1.5576*exp(-5*x)
x <- linspace(0, 1.15, 24)
y <- c(2.51340000, 2.04433337, 1.66840444, 1.36641802, 1.12323249, 0.92688972,
       0.76793386, 0.63887755, 0.53378353, 0.44793636, 0.37758479, 0.31973932,
       0.27201308, 0.23249655, 0.19965895, 0.17227041, 0.14934057, 0.13007002,
       0.11381193, 0.10004156, 0.08833209, 0.07833544, 0.06976694, 0.06239313)

p0 <- c(1.2, 0.3, 5.6, 5.5, 6.5, 7.6)
fp <- function(p, x) p[1]*exp(-p[2]*x) + p[3]*exp(-p[4]*x) + p[5]*exp(-p[6]*x)
lsqcurvefit(fp, p0, x, y)

##  Example for lsqnonneg()
C <- matrix(c(0.0372, 0.2868,
              0.6861, 0.7071,
              0.6233, 0.6245,
              0.6344, 0.6170), nrow = 4, ncol = 2, byrow = TRUE)
d <- c(0.8587, 0.1781, 0.0747, 0.8405)

sol <- lsqnonneg(C, d)
cbind(qr.solve(C, d), sol$x)
# -2.563884  5.515869e-08
#  3.111911  6.929003e-01

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