lsqnonlin solves nonlinear least-squares problems, including
nonlinear data-fitting problems, through the Levenberg-Marquardt approach. lsqnonneg solve nonnegative least-squares constraints problem.
lsqnonlin(fun, x0, options = list(), ...)
lsqnonneg(C, d)lsqsep(flist, p0, xdata, ydata, const = TRUE)
lsqcurvefit(fun, p0, xdata, ydata)
C x - d will be
minimized with x >= 0.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.
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
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.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.
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.
nlm, nls## 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")Run the code above in your browser using DataLab