Multi-exponential Fitting
Multi-exponential fitting means fitting of data points by a sum of (decaying) exponential functions, with or without a constant term.
mexpfit(x, y, p0, w = NULL, const = TRUE, options = list())
x, y |
x-, y-coordinates of data points to be fitted. |
p0 |
starting values for the exponentials alone; can be positive or negative, but not zero. |
w |
weight vector; not used in this version. |
const |
logical; shall an absolute term be included. |
options |
list of options for |
The multi-exponential fitting problem is solved here with with a separable nonlinear least-squares approach. If the following function is to be fitted,
y = a_0 + a_1 e^{b_1 x} + … + a_n e^{b_n x}
it will be looked at as a nonlinear optimization problem of the coefficients b_i alone. Given the b_i, coefficients a_i are uniquely determined as solution of an (overdetermined) system of linear equations.
This approach reduces the dimension of the search space by half and improves numerical stability and accuracy. As a convex problem, the solution is unique and global.
To solve the nonlinear part, the function lsqnonlin that uses the
Levenberg-Marquard algorithm will be applied.
mexpfit returns a list with the following elements:
a0: the absolute term, 0 if const is false.
a: linear coefficients.
b: coefficient in the exponential functions.
ssq: the sum of squares for the final fitting.
iter: number of iterations resp. function calls.
errmess: an error or info message.
As the Jacobian for this expression is known, a more specialized approch
would be possible, without using lsqnonlin;
see the immoptibox of H. B. Nielsen, Techn. University of Denmark.
HwB email: <hwborchers@googlemail.com>
Madsen, K., and H. B. Nielsen (2010). Introduction to Optimization and Data Fitting. Technical University of Denmark, Intitute of Computer Science and Mathematical Modelling.
Nielsen, H. B. (2000). Separable Nonlinear Least Squares. IMM, DTU, Report IMM-REP-2000-01.
# 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(-0.3, -5.5, -7.6)
mexpfit(x, y, p0, const = FALSE)
## $a0
## [1] 0
## $a
## [1] 0.09510431 0.86071171 1.55758398
## $b
## [1] -1.000022 -3.000028 -5.000009
## $ssq
## [1] 1.936163e-16
## $iter
## [1] 26
## $errmess
## [1] "Stopped by small gradient."Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.