pracma (version 1.7.0)

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)
lsqsep(flist, p0, xdata, ydata, const = TRUE)
lsqcurvefit(fun, p0, xdata, ydata)

Arguments

fun

User-defined, vector-valued function.

x0

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.

flist

list of (nonlinear) functions, depending on one extra parameter.

p0

starting parameters.

xdata, ydata

data points to be fitted.

const

logical; shall a constant term be included.

Value

• lsqnonlin 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.

  lsqnonneg returns a list of x the non-negative solition, and resid.norm the norm of the residual.

  lsqsep will return the coefficients sparately, a0 for the constant term (being 0 if const=FALSE) and the vectors a and b for the linear and nonlinear terms, respectively.

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 $$min || f(x) ||_2^2 = min(f_1(x)^2 + \ldots + f_n(x)^2)$$ 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 as 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, treating it through an active-set approach..

lsqsep solves the separable least-squares fitting problem

y = a0 + a1*f1(b1, x) + ... + an*fn(bn, x)

where fi are nonlinear functions each depending on a single extra paramater bi, and ai are additional linear parameters that can be separated out to solve a nonlinear problem in the bi alone.

lsqcurvefit is simply an application of lsqnonlin to fitting data points. fun(p, x) must be a function of two groups of variables such that p will be varied to minimize the least squares sum, see the example below.

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.

Lawson, C.L., and R.J. Hanson (1974). Solving Least-Squares Problems. Prentice-Hall, Chapter 23, p. 161.

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

See Also

nlm, nls

Examples

##  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

##  Example: Hougon function
x1 <- c(470, 285, 470, 470, 470, 100, 100, 470, 100, 100, 100, 285, 285)
x2 <- c(300,  80, 300,  80,  80, 190,  80, 190, 300, 300,  80, 300, 190)
x3 <- c( 10,  10, 120, 120,  10,  10,  65,  65,  54, 120, 120,  10, 120)
rate <- c(8.55,  3.79, 4.82, 0.02,  2.75, 14.39, 2.54,
          4.35, 13.00, 8.50, 0.05, 11.32,  3.13)
fun <- function(b)
        (b[1]*x2 - x3/b[5])/(1 + b[2]*x1 + b[3]*x2 + b[4]*x3) - rate
lsqnonlin(fun, rep(1, 5))
# $x    [1.25258502 0.06277577 0.04004772 0.11241472 1.19137819]
# $ssq  0.298901

##  Example for lsqnonneg()
C1 <- matrix( c(0.1210, 0.2319, 0.4398, 0.9342, 0.1370,
                0.4508, 0.2393, 0.3400, 0.2644, 0.8188,
                0.7159, 0.0498, 0.3142, 0.1603, 0.4302,
                0.8928, 0.0784, 0.3651, 0.8729, 0.8903,
                0.2731, 0.6408, 0.3932, 0.2379, 0.7349,
                0.2548, 0.1909, 0.5915, 0.6458, 0.6873,
                0.8656, 0.8439, 0.1197, 0.9669, 0.3461,
                0.2324, 0.1739, 0.0381, 0.6649, 0.1660,
                0.8049, 0.1708, 0.4586, 0.8704, 0.1556,
                0.9084, 0.9943, 0.8699, 0.0099, 0.1911), ncol = 5, byrow = TRUE)
C2 <- C1 - 0.5
d <- c(0.4225, 0.8560, 0.4902, 0.8159, 0.4608,
       0.4574, 0.4507, 0.4122, 0.9016, 0.0056)
( sol <- lsqnonneg(C1, d) )     #-> resid.norm   0.3694372
( sol <- lsqnonneg(C2, d) )     #-> $resid.norm  2.863979

##  Example for lsqcurvefit()
#   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 lsqsep()
f <- function(x) 0.5 + x^-0.5 + exp(-0.5*x)
set.seed(8237); n <- 15
x <- sort(0.5 + 9*runif(n))
y <- f(x)                       #y <- f(x) + 0.01*rnorm(n)

m <- 2
f1 <- function(b, x) x^b
f2 <- function(b, x) exp(b*x)
flist <- list(f1, f2)
start <- c(-0.25, -0.75)

sol <- lsqsep(flist, start, x, y, const = TRUE)
a0 <- sol$a0; a <- sol$a; b <- sol$b
fsol <- function(x) a0 + a[1]*f1(b[1], x) + a[2]*f2(b[2], x)

ezplot(f, 0.5, 9.5, col = "gray")
    points(x, y, col = "blue")
    xs <- linspace(0.5, 9.5, 51)
    ys <- fsol(xs)
    lines(xs, ys, col = "red")