Fits \(y = f(x, \theta) + \epsilon\) by least squares, the model
being written as a formula such as y ~ a * (1 - exp(-b * x))
whose parameters are the names in start. Gauss-Newton follows
the nls algorithm of Bates and Chambers (1992): increments from the QR
decomposition of the gradient, step halving with a step factor that
doubles back after each success, and the relative-offset convergence
criterion of Bates and Watts (1988). Levenberg-Marquardt is available
for poor starting values. The Jacobian is analytic (via
stats::deriv) when possible, otherwise by central differences.
Usage
morie_nls(
formula,
data,
start,
algorithm = c("gauss-newton", "levenberg-marquardt"),
control = list()
)
# S3 method for class 'morie_nls'
print(x, ...)
# S3 method for class 'morie_nls'
summary(object, ...)
# S3 method for class 'summary.morie_nls'
print(x, ...)
# S3 method for class 'morie_nls'
coef(object, ...)
# S3 method for class 'morie_nls'
vcov(object, ...)
# S3 method for class 'morie_nls'
fitted(object, ...)
# S3 method for class 'morie_nls'
residuals(object, ...)
# S3 method for class 'morie_nls'
deviance(object, ...)
# S3 method for class 'morie_nls'
df.residual(object, ...)
# S3 method for class 'morie_nls'
logLik(object, ...)
# S3 method for class 'morie_nls'
predict(object, newdata = NULL, ...)Arguments
- formula
Model formula,
response ~ expression; a one-sided~ expressionminimises the sum of squares of the expression. Functions from base R andpnorm,dnorm,qnorm,plogis,dlogis,qlogismay be used.- data
Data frame (or named list) holding the variables.
- start
Named list or vector of starting values; vector-valued entries give indexed parameters (
b[1],b[2], namedb1,b2in the output).- algorithm
"gauss-newton"(default) or"levenberg-marquardt".- control
List of settings:
maxiter(50),tol(1e-5, on the relative offset),minFactor(1/1024),scaleOffset(0),warnOnly(FALSE: non-convergence is an error),jacobian("auto", "analytic" or "central"), and for Levenberg-Marquardtlambda(initial damping, 1e-3) andmaxlambda(1e16).- x, object
A
"morie_nls"fit.- ...
Unused.
- newdata
Optional data frame (or list) of predictor values.
Value
An object of class "morie_nls": coefficients,
coef_table (Estimate, Std. Error, t value, Pr(>|t|)),
sigma (residual standard error), df (p, n - p),
cov.unscaled, vcov, fitted.values,
residuals, deviance, convInfo (isConv,
finIter, finTol, stopCode, stopMessage), algorithm,
jacobian, formula, call.
References
Bates, D. M. and Watts, D. G. (1988). Nonlinear Regression Analysis and Its Applications. Wiley.
Bates, D. M. and Chambers, J. M. (1992). Nonlinear models. In Chambers, J. M. and Hastie, T. J. (eds), Statistical Models in S. Wadsworth & Brooks/Cole.
Marquardt, D. W. (1963). An algorithm for least-squares estimation of nonlinear parameters. SIAM Journal on Applied Mathematics 11, 431-441.
More, J. J. (1978). The Levenberg-Marquardt algorithm: implementation and theory. Lecture Notes in Mathematics 630, 105-116.
Examples
set.seed(1)
d <- data.frame(x = seq(0, 5, length.out = 40))
d$y <- 3 * (1 - exp(-0.8 * d$x)) + rnorm(40, sd = 0.05)
fit <- morie_nls(y ~ a * (1 - exp(-b * x)), d, start = list(a = 1, b = 1))
summary(fit)
#>
#> Formula: y ~ a * (1 - exp(-b * x))
#>
#> Parameters:
#> Estimate Std. Error t value Pr(>|t|)
#> a 2.994317 0.015653 191.296 < 2.2e-16 ***
#> b 0.811495 0.013871 58.501 < 2.2e-16 ***
#> ---
#> Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#>
#> Residual standard error: 0.04456 on 38 degrees of freedom
#>
#> Number of iterations to convergence: 5
#> Achieved convergence tolerance: 3.706773e-07
#> (gauss-newton, analytic Jacobian)
predict(fit, data.frame(x = c(1, 2)))
#> [1] 1.664261 2.403515