Package 'nlsr'

Title: Functions for Nonlinear Least Squares Solutions - Updated 2022
Description: Provides tools for working with nonlinear least squares problems. For the estimation of models reliable and robust tools than nls(), where the the Gauss-Newton method frequently stops with 'singular gradient' messages. This is accomplished by using, where possible, analytic derivatives to compute the matrix of derivatives and a stabilization of the solution of the estimation equations. Tools for approximate or externally supplied derivative matrices are included. Bounds and masks on parameters are handled properly.
Authors: John C Nash [aut, cre], Duncan Murdoch [aut], Fernando Miguez [ctb], Arkajyoti Bhattacharjee [ctb]
Maintainer: John C Nash <[email protected]>
License: GPL-2
Version: 2023.8.31
Built: 2024-06-07 05:46:50 UTC
Source: CRAN

Help Index


coef.nlsr

Description

prepare and display result of nlsr computations

Usage

## S3 method for class 'nlsr'
coef(object, ...)

Arguments

object

an object of class nlsr

...

additional data needed to evaluate the modeling functions Default FALSE

Details

The set of possible controls to set is as follows

Author(s)

J C Nash 2014-7-16 nashjc _at_ uottawa.ca


fitted.nlsr

Description

prepare and display fits of nlsr computations

Usage

## S3 method for class 'nlsr'
fitted(object = NULL, data = parent.frame(), ...)

Arguments

object

an object of class nlsr

data

a data frame with the data for which fits are wanted.

...

additional data needed to evaluate the modeling functions Default FALSE

Author(s)

J C Nash 2014-7-16 revised 2022-11-22 nashjc _at_ uottawa.ca


jaback

Description

approximate Jacobian via forward differences

Usage

jaback(pars, resfn = NULL, bdmsk = NULL, resbest = NULL, ndstep = 1e-07, ...)

Arguments

pars

a named numeric vector of parameters to the model

resfn

a function to compute a vector of residuals

bdmsk

Vector defining bounds and masks. Default is NULL

resbest

If supplied, a vector of the residuals at the parameters pars to save re-evaluation.

ndstep

A tolerance used to alter parameters to compute numerical approximations to derivatives. Default 1e-7.

...

Extra information needed to compute the residuals

Author(s)

J C Nash 2014-7-16 nashjc _at_ uottawa.ca


jacentral

Description

Approximate Jacobian via central differences. Note this needs two evaluations per parameter, but generally gives much better approximation of the derivatives.

Usage

jacentral(
  pars,
  resfn = NULL,
  bdmsk = NULL,
  resbest = NULL,
  ndstep = 1e-07,
  ...
)

Arguments

pars

a named numeric vector of parameters to the model

resfn

a function to compute a vector of residuals

bdmsk

Vector defining bounds and masks. Default is NULL

resbest

If supplied, a vector of the residuals at the parameters pars to save re-evaluation.

ndstep

A tolerance used to alter parameters to compute numerical approximations to derivatives. Default 1e-7.

...

Extra information needed to compute the residuals

Author(s)

J C Nash 2014-7-16 revised 2022-11-22 nashjc _at_ uottawa.ca


jafwd

Description

approximate Jacobian via forward differences

Usage

jafwd(pars, resfn = NULL, bdmsk = NULL, resbest = NULL, ndstep = 1e-07, ...)

Arguments

pars

a named numeric vector of parameters to the model

resfn

a function to compute a vector of residuals

bdmsk

Vector defining bounds and masks. Default is NULL

resbest

If supplied, a vector of the residuals at the parameters pars to save re-evaluation.

ndstep

A tolerance used to alter parameters to compute numerical approximations to derivatives. Default 1e-7.

...

Extra information needed to compute the residuals

Author(s)

J C Nash 2014-7-16 nashjc _at_ uottawa.ca


jand

Description

approximate Jacobian via numDeriv::jacobian

Usage

jand(pars, resfn = NULL, bdmsk = NULL, resbest = NULL, ndstep = 1e-07, ...)

Arguments

pars

a named numeric vector of parameters to the model

resfn

a function to compute a vector of residuals

bdmsk

Vector defining bounds and masks. Default is NULL

resbest

If supplied, a vector of the residuals at the parameters pars to save re-evaluation.

ndstep

A tolerance used to alter parameters to compute numerical approximations to derivatives. Default 1e-7.

...

Extra information needed to compute the residuals

Author(s)

J C Nash 2014-7-16 nashjc _at_ uottawa.ca


model2rjfun

Description

These functions create functions to evaluate residuals or sums of squares at particular parameter locations.

Usage

model2rjfun(modelformula, pvec, data = NULL, jacobian = TRUE, testresult = TRUE, ...)
SSmod2rjfun(modelformula, pvec, data = NULL, jacobian = TRUE, testresult = TRUE, ...)
model2ssgrfun(modelformula, pvec, data = NULL, gradient = TRUE, 
              testresult = TRUE, ...)
modelexpr(fun)

Arguments

modelformula

A formula describing a nonlinear regression model.

pvec

A vector of parameters.

data

A dataframe, list or environment holding data used in the calculation.

jacobian

Whether to compute the Jacobian matrix.

testresult

Whether to test the function by evaluating it at pvec.

gradient

Whether to compute the gradient vector.

fun

A function produced by one of model2rjfun or model2ssgrfun.

...

Dot arguments, that is, arguments that may be supplied by name = value to supply information needed to compute specific quantities in the model.

Details

If pvec does not have names, the parameters will have names generated in the form ‘⁠p_<n>⁠’, e.g. p_1, p_2. Names that appear in pvec will be taken to be parameters of the model.

The data argument may be a dataframe, list or environment, or NULL. If it is not an environment, one will be constructed using the components of data with parent environment set to be the environment of modelformula.

SSmod2rjfun returns a function with header function(prm), which evaluates the residuals (and if jacobian is TRUE the Jacobian matrix) of the selfStart model (the rhs is used) at prm. The residuals are defined to be the right hand side of modelformula minus the left hand side. Note that the selfStart model used in the model formula must be available (i.e., loaded). If this function is called from nlxb() then the control element japprox must be set to value SSJac.

Value

model2rjfun returns a function with header function(prm), which evaluates the residuals (and if jacobian is TRUE the Jacobian matrix) of the model at prm. The residuals are defined to be the right hand side of modelformula minus the left hand side.

model2ssgrfun returns a function with header function(prm), which evaluates the sum of squared residuals (and if gradient is TRUE the gradient vector) of the model at prm.

modelexpr returns the expression used to calculate the vector of residuals (and possibly the Jacobian) used in the previous functions.

Author(s)

John Nash and Duncan Murdoch

See Also

nls

Examples

# We do not appear to have an example for modelexpr. See nlsr-devdoc.Rmd for one.

y <- c(5.308, 7.24, 9.638, 12.866, 17.069, 23.192, 31.443, 38.558,
       50.156, 62.948, 75.995, 91.972)

tt <- seq_along(y)  # for testing

mydata <- data.frame(y = y, tt = tt)
f <- y ~ b1/(1 + b2 * exp(-1 * b3 * tt))
p <- c(b1 = 1, b2 = 1, b3 = 1)
rjfn <- model2rjfun(f, p, data = mydata)
rjfn(p)
rjfnnoj <- model2rjfun(f, p, data = mydata, jacobian=FALSE)
rjfnnoj(p)

myexp <- modelexpr(rjfn)
cat("myexp:"); print(myexp)

ssgrfn <- model2ssgrfun(f, p, data = mydata)
ssgrfn(p)

ssgrfnnoj <- model2ssgrfun(f, p, data = mydata, gradient=FALSE)
ssgrfnnoj(p)

nlfb: nonlinear least squares modeling by functions

Description

A simplified and hopefully robust alternative to finding the nonlinear least squares minimizer that causes 'formula' to give a minimal residual sum of squares.

Usage

nlfb(
  start,
  resfn,
  jacfn = NULL,
  trace = FALSE,
  lower = -Inf,
  upper = Inf,
  weights = NULL,
  data = NULL,
  ctrlcopy = FALSE,
  control = list(),
  ...
)

Arguments

start

a numeric vector with all elements present e.g., start=c(b1=200, b2=50, b3=0.3) The start vector for this nlfb, unlike nlxb, does not need to be named.

resfn

A function that evaluates the residual vector for computing the elements of the sum of squares function at the set of parameters start. Where this function is created by actions on a formula or expression in nlxb, this residual vector will be created by evaluation of the 'model - data', rather than the conventional 'data - model' approach. The sum of squares is the same.

jacfn

A function that evaluates the Jacobian of the sum of squares function, that is, the matrix of partial derivatives of the residuals with respect to each of the parameters. If NULL (default), uses an approximation.

The Jacobian MUST be returned as the attribute "gradient" of this function, allowing jacfn to have the same name and be the same code block as resfn, which may permit some efficiencies of computation.

trace

TRUE for console output during execution

lower

a vector of lower bounds on the parameters. If a single number, this will be applied to all. Default -Inf.

upper

a vector of upper bounds on the parameters. If a single number, this will be applied to all parameters. Default Inf.

weights

A vector of fixed weights or a function producing one. See the Details below.

data

a data frame of variables used by resfn and jacfn to compute the required residuals and Jacobian.

ctrlcopy

If TRUE use control supplied as is. This avoids reprocessing controls.

control

a list of control parameters. See nlsr.control().

...

additional data needed to evaluate the modeling functions

Details

nlfb is particularly intended to allow for the resolution of very ill-conditioned or else near zero-residual problems for which the regular nls() function is ill-suited.

This variant uses a qr solution without forming the sum of squares and cross products t(J)

Neither this function nor nlxb have provision for parameter scaling (as in the parscale control of optim and package optimx). This would be more tedious than difficult to introduce, but does not seem to be a priority feature to add.

The weights argument can be a vector of fixed weights, in which case the objective function that will be minimized is the sum of squares where each residual is multiplied by the square root of the corresponding weight. Default NULL implies unit weights. weights may alternatively be a function with header function(parms, resids) to compute such a vector.

Value

A list of the following items:

coefficients

A named vector giving the parameter values at the supposed solution.

ssquares

The sum of squared residuals at this set of parameters.

resid

The weighted residual vector at the returned parameters.

jacobian

The jacobian matrix (partial derivatives of residuals w.r.t. the parameters) at the returned parameters.

feval

The number of residual evaluations (sum of squares computations) used.

jeval

The number of Jacobian evaluations used.

weights0

The weights argument as specified.

weights

The weights vector at the final fit.

Author(s)

J C Nash 2014-7-16 nashjc _at_ uottawa.ca

Examples

library(nlsr)
# Scaled Hobbs problem
shobbs.res  <-  function(x){ # scaled Hobbs weeds problem -- residual
  # This variant uses looping
  if(length(x) != 3) stop("shobbs.res -- parameter vector n!=3")
  y  <-  c(5.308, 7.24, 9.638, 12.866, 17.069, 23.192, 31.443,
           38.558, 50.156, 62.948, 75.995, 91.972)
  tt  <-  1:12
  res  <-  100.0*x[1]/(1+x[2]*10.*exp(-0.1*x[3]*tt)) - y
}
shobbs.jac  <-  function(x) { # scaled Hobbs weeds problem -- Jacobian
  jj  <-  matrix(0.0, 12, 3)
  tt  <-  1:12
  yy  <-  exp(-0.1*x[3]*tt)
  zz  <-  100.0/(1+10.*x[2]*yy)
  jj[tt,1]   <-   zz
  jj[tt,2]   <-   -0.1*x[1]*zz*zz*yy
  jj[tt,3]   <-   0.01*x[1]*zz*zz*yy*x[2]*tt
  attr(jj, "gradient") <- jj
  jj
}
st <- c(b1=2, b2=1, b3=1) # a default starting vector (named!)
# Default controls, standard Nash-Marquardt algorithm
anlf0 <- nlfb(start=st, resfn=shobbs.res, jacfn=shobbs.jac,
               trace=TRUE, control=list(prtlvl=1))
anlf0

# Hartley with step reduction factor of .2
anlf0h <- nlfb(start=st, resfn=shobbs.res, jacfn=shobbs.jac,
              trace=TRUE, control=list(prtlvl=1, lamda=0, laminc=1.0,
              lamdec=1.0, phi=0, stepredn=0.2))
anlf0h

anlf1bm <- nlfb(start=st, resfn=shobbs.res, jacfn=shobbs.jac, lower=c(2,0,0),
                upper=c(2,6,3), trace=TRUE, control=list(prtlvl=1))
anlf1bm
cat("backtrack using stepredn=0.2\n")
anlf1bmbt <- nlfb(start=st, resfn=shobbs.res, jacfn=shobbs.jac, lower=c(2,0,0),
                upper=c(2,6,3), trace=TRUE, control=list(stepredn=0.2, prtlvl=1))
anlf1bmbt
## Short output
pshort(anlf1bm)
anlf2bm <- nlfb(start=st, resfn=shobbs.res, jacfn=shobbs.jac, lower=c(2,0,0),
                upper=c(2,6,9), trace=TRUE, control=list(prtlvl=1))
anlf2bm
cat("backtrack using stepredn=0.2\n")

anlf2bmbt <- nlfb(start=st, resfn=shobbs.res, jacfn=shobbs.jac, lower=c(2,0,0),
                upper=c(2,6,9), trace=TRUE, control=list(stepredn=0.2, prtlvl=1))
anlf2bmbt
## Short output
pshort(anlf2bm)

nlsDeriv Functions to take symbolic derivatives.

Description

Compute derivatives of simple expressions symbolically, allowing user-specified derivatives.

Usage

nlsDeriv(expr, name, derivEnv = sysDerivs, do_substitute = FALSE, verbose = FALSE, ...)

codeDeriv(expr, namevec, hessian = FALSE, derivEnv = sysDerivs, 
          do_substitute = FALSE, verbose = FALSE, ...)

fnDeriv(expr, namevec, args = all.vars(expr), env = environment(expr), 
          do_substitute = FALSE, verbose = FALSE, ...)

Arguments

expr

An expression represented in a variety of ways. See Details.

name

The name of the variable with respect to which the derivative will be computed.

derivEnv

The environment in which derivatives are stored.

do_substitute

If TRUE, use substitute to get the expression passed as expr, otherwise evaluate it.

verbose

If TRUE, then diagnostic output will be printed as derivatives and simplifications are recognized.

...

Additional parameters which will be passed to codeDeriv from fnDeriv, and to nlsSimplify from nlsDeriv and codeDeriv.

namevec

Character vector giving the variable names with respect to which the derivatives will be taken.

hessian

Logical indicator of whether the 2nd derivatives should also be computed.

args

Desired arguments for the function. See Details below.

env

The environment to be attached to the created function. If NULL, the caller's frame is used.

Details

Functions nlsDeriv and codeDeriv are designed as replacements for the stats package functions D and deriv respectively, though the argument lists do not match exactly.

The nlsDeriv function computes a symbolic derivative of an expression or language object. Known derivatives are stored in derivEnv; the default sysDerivs contains expressions for all of the derivatives recognized by deriv, but in addition allows differentiation with respect to any parameter where it makes sense. It also allows the derivative of abs and sign, using an arbitrary choice of 0 at the discontinuities.

The codeDeriv function computes an expression for efficient calculation of the expression value together with its gradient and optionally the Hessian matrix.

The fnDeriv function wraps the codeDeriv result in a function. If the args are given as a character vector (the default), the arguments will have those names, with no default values. Alternatively, a custom argument list with default values can be created using alist; see the example below.

The expr argument will be converted to a language object using dex (but note the different default for do_substitute). Normally it should be a formula with no left hand side, e.g. ~ x^2 , or an expression vector e.g. expression(x, x^2, x^3) , or a language object e.g. quote(x^2). In codeDeriv and fnDeriv the expression vector must be of length 1.

The newDeriv function is used to define a new derivative. The expr argument should match the header of the function as a call to it (e.g. as in the help pages), and the deriv argument should be an expression giving the derivative, including calls to D(arg), which will not be evaluated, but will be substituted with partial derivatives of that argument with respect to name. See the examples below.

If expr or deriv is missing in a call to newDeriv(), it will return the currently saved derivative record from derivEnv. If name is missing in a call to nlsDeriv with a function call, it will print a message describing the derivative formula and return NULL.

To handle functions which act differently if a parameter is missing, code the default value of that parameter to .MissingVal, and give a derivative that is conditional on missing() applied to that parameter. See the derivatives of "-" and "+" in the file derivs.R for an example.

Value

If expr is an expression vector, nlsDeriv and nlsSimplify return expression vectors containing the response. For formulas or language objects, a language object is returned.

codeDeriv always returns a language object.

fnDeriv returns a closure (i.e. a function).

nlsDeriv returns the symbolic derivative of the expression.

newDeriv with expr and deriv specified is called for the side effect of recording the derivative in derivEnv. If expr is missing, it will return the list of names of functions for which derivatives are recorded. If deriv is missing, it will return its record for the specified function.

Note

newDeriv(expr, deriv, ...) will issue a warning if a different definition for the derivative exists in the derivative table.

Author(s)

Duncan Murdoch

See Also

deriv

Examples

nlsDeriv(~ sin(x+y), "x")

 f <- function(x) x^2
 newDeriv(f(x), 2*x*D(x))
 nlsDeriv(~ f(abs(x)), "x")
 
 nlsDeriv(~ pnorm(x, sd=2, log = TRUE), "x")
 fnDeriv(~ pnorm(x, sd = sd, log = TRUE), "x")
 f <- fnDeriv(~ pnorm(x, sd = sd, log = TRUE), "x", args = alist(x =, sd = 2))
 f
 f(1)
 100*(f(1.01) - f(1))  # Should be close to the gradient
 
       # The attached gradient attribute (from f(1.01)) is
       # meaningless after the subtraction.
       
 # Multiple point example
 xvals <- c(1, 3, 4.123)
 print(f(xvals))
 # Getting a hessian matrix
 f2 <- ~ (x-2)^3*y - y^2
 mydf2 <- fnDeriv(f2, c("x","y"), hessian=TRUE)
 # display the resulting function
 print(mydf2)
 x <- c(1, 2)
 y <- c(0.5, 0.1)
 evalmydf2 <- mydf2(x, y)
 print(evalmydf2)
 # the first index of the hessian attribute is the point at which we want the hessian
 hmat1 <- as.matrix(attr(evalmydf2,"hessian")[1,,])
 print(hmat1)
 hmat2 <- as.matrix(attr(evalmydf2,"hessian")[2,,])
 print(hmat2)

nlsr function

Description

Provides class nls solution to a nonlinear least squares solution using the Nash Marquardt tools.

Usage

nlsr(formula = NULL,  data = NULL,  start = NULL,  control = NULL,
    trace = FALSE, subset = NULL, lower = -Inf, upper = Inf,  weights = NULL,
    ...)

Arguments

formula

The modeling formula. Looks like 'y~b1/(1+b2*exp(-b3*T))'

data

a data frame containing data for variables used in the formula that are NOT the parameters. This data may also be defined in the parent frame i.e., 'global' to this function

start

MUST be a named vector with all elements present e.g., start=c(b1=200, b2=50, b3=0.3)

control

a list of control parameters. See nlsr.control().

trace

TRUE for console output during execution (default FALSE)

subset

an optional vector specifying a subset of observations to be used in the fitting process. NOT used currently by nlxb() or nlfb() and will throw an error if present and not NULL.

lower

a vector of lower bounds on the parameters. If a single number, this will be applied to all parameters Default -Inf.

upper

a vector of upper bounds on the parameters. If a single number, this will be applied to all parameters. Default Inf.

weights

A vector of fixed weights. The objective function that will be minimized is the sum of squares where each residual is multiplied by the square root of the corresponding weight. Default NULL implies unit weights.

...

additional data needed to evaluate the modeling functions

Value

A solution object of type nls


nlsr.control

Description

Set and provide defaults of controls for package nlsr

Usage

nlsr.control(control)

Arguments

control

A list of controls. If missing, the defaults are provided. See below. If a named control is provided, e.g., via a call ctrllist<- nlsr.control(japprox="jand"), then that value is substituted for the default of the control in the FULL list of controls that is returned.

NOTE: at 2022-6-17 there is NO CHECK FOR VALIDITY

The set of possible controls to set is as follows, and is returned.

Value

femax

INTEGER limit on the number of evaluations of residual function Default 10000.

japprox

CHARACTER name of the Jacobian approximation to use Default NULL, since we try to use analytic gradient

jemax

INTEGER limit on the number of evaluations of the Jacobian Default 5000

lamda

REAL initial value of the Marquardt parameter Default 0.0001 Note: mis-spelling as in JNMWS, kept as historical serendipity.

lamdec

REAL multiplier used to REDUCE lambda (0 < lamdec < laminc) Default 4, so lamda <- lamda * (lamdec/laminc)

laminc

REAL multiplier to INCREASE lambda (1 < laminc Default 10

nbtlim

if stepredn > 0, then maximum number of backtrack loops (in addition to default evaluation); Default 6

ndstep

REAL stepsize for numerical Jacobian approximation Default 1e-7

offset

REAL A value used to test for numerical equality, i.e. a and b are taken equal if (a + offset) == (b + offset) Default 100.

phi

REAL Factor used to add unit Marquardt stabilization matrix in Nash modification of LM method. Default 1

prtlvl

INTEGER The higher the value, the more intermediate output is provided. Default 0

psi

REAL Factor used to add scaled Marquardt stabilization matrix in Nash modification of LM method. Default 0

rofftest

LOGICAL If TRUE, perform (safeguarded) relative offset convergence test Default TRUE

scaleOffset

a positive constant to be added to the denominator sum-of-squares in the relative offset convergence criteria. Default 0

smallsstest

LOGICAL. If TRUE tests sum of squares and terminates if very small. Default TRUE

stepredn

REAL Factor used to reduce the stepsize in a Gauss-Newton algorithm (Hartley's method). 0 means NO backtrack. Default 0

watch

LOGICAL to provide a pause at the end of each iteration for user to monitor progress. Default FALSE

Author(s)

J C Nash 2014-7-16 revised 2022-11-22 nashjc _at_ uottawa.ca


nlsr-package Tools for solving nonlinear least squares problems The package provides some tools related to using the Nash variant of Marquardt's algorithm for nonlinear least squares. Jacobians can usually be developed by automatic or symbolic derivatives.

Description

nlsr-package

Tools for solving nonlinear least squares problems

The package provides some tools related to using the Nash variant of Marquardt's algorithm for nonlinear least squares. Jacobians can usually be developed by automatic or symbolic derivatives.

Usage

nlsr.package()

Details

This package includes methods for solving nonlinear least squares problems specified by a modeling expression and given a starting vector of named paramters. Note: You must provide an expression of the form lhs ~ rhsexpression so that the residual expression rhsexpression - lhs can be computed. The expression can be enclosed in quotes, and this seems to give fewer difficulties with R. Data variables must already be defined, either within the parent environment or else in the dot-arguments. Other symbolic elements in the modeling expression must be standard functions or else parameters that are named in the start vector.

The main functions in nlsr are:

nlfb Nash variant of the Marquardt procedure for nonlinear least squares, with bounds constraints, using a residual and optionally Jacobian described as R functions.

nlxb Nash variant of the Marquardt procedure for nonlinear least squares, with bounds constraints, using an expression to describe the residual via an R modeling expression. The Jacobian is computed via symbolic differentiation.

wrapnlsr Uses nlxb to solve nonlinear least squares then calls nls() to create an object of type nls. nlsr is an alias for wrapnlsr

model2rjfun returns a function with header function(prm), which evaluates the residuals (and if jacobian is TRUE the Jacobian matrix) of the model at prm. The residuals are defined to be the right hand side of modelformula minus the left hand side.

model2ssgrfun returns a function with header function(prm), which evaluates the sum of squared residuals (and if gradient is TRUE the gradient vector) of the model at prm.

modelexpr returns the expression used to calculate the vector of residuals (and possibly the Jacobian) used in the previous functions.

Author(s)

John C Nash and Duncan Murdoch

References

Nash, J. C. (1979, 1990) _Compact Numerical Methods for Computers. Linear Algebra and Function Minimisation._ Adam Hilger./Institute of Physics Publications

Nash, J. C. (2014) _Nonlinear Parameter Optimization Using R Tools._ Wiley


nlsrSS - solve selfStart nonlinear least squares with nlsr package

Description

This function uses the getInitial() function to estimate starting parameters for a Gauss-Newton iteration, then calls nlsr::nlxb() appropriately to find a solution to the required nonlinear least squares problem.

Usage

nlsrSS(formula, data)

Arguments

formula

a model formula incorporating a selfStart function in the right hand side

data

a data frame with named columns that allow evaluation of the formula

Value

A solution object of class nlsr.

List of solution elements

resid

weighted residuals at the proposed solution

jacobian

Jacobian matrix at the proposed solution

feval

residual function evaluations used to reach solution from starting parameters

jeval

Jacobian function (or approximation) evaluations used

coefficients

a named vector of proposed solution parameters

ssquares

weighted sum of squared residuals (often the deviance)

lower

lower bounds on parameters

upper

upper bounds on parameters

maskidx

vector if indices of fixed (masked) parameters

weights

specified weights on observations

formula

the modeling formula

resfn

the residual function (unweighted) based on the formula

Author(s)

J C Nash 2022-9-14 nashjc _at_ uottawa.ca


nlxb: nonlinear least squares modeling by formula

Description

A simplified and hopefully robust alternative to finding the nonlinear least squares minimizer that causes 'formula' to give a minimal residual sum of squares.

Usage

nlxb(
  formula,
  data = parent.frame(),
  start,
  trace = FALSE,
  lower = NULL,
  upper = NULL,
  weights = NULL,
  control = list(),
  ...
)

Arguments

formula

The modeling formula. Looks like 'y~b1/(1+b2*exp(-b3*T))'

data

a data frame containing data for variables used in the formula that are NOT the parameters. This data may also be defined in the parent frame i.e., 'global' to this function

start

MUST be a named vector with all elements present e.g., start=c(b1=200, b2=50, b3=0.3)

trace

TRUE for console output during execution

lower

a vector of lower bounds on the parameters. If a single number, this will be applied to all parameters Default NULL.

upper

a vector of upper bounds on the parameters. If a single number, this will be applied to all parameters. Default NULL.

weights

A vector of fixed weights or a function or formula producing one. See the Details below.

control

a list of control parameters. See nlsr.control().

...

additional data needed to evaluate the modeling functions

Details

nlxb is particularly intended to allow for the resolution of very ill-conditioned or else near zero-residual problems for which the regular nls() function is ill-suited.

This variant uses a qr solution without forming the sum of squares and cross products t(J)

Neither this function nor nlfb have provision for parameter scaling (as in the parscale control of optim and package optimx). This would be more tedious than difficult to introduce, but does not seem to be a priority feature to add.

There are many controls, and some of them are important for nlxb. In particular, if the derivatives needed for developing the Jacobian are NOT in the derivatives table, then we must supply code elsewhere as specified by the control japprox. This was originally just for numerical approximations, with the character strings "jafwd", "jaback", "jacentral" and "jand" leading to the use of a forward, backward, central or package numDeriv approximation. However, it is also possible to use code embedded in the residual function created using the formula. This is particularly useful for selfStart models, and we use the character string "SSJac" to point to such Jacobian code. Note how the starting parameter vector is found using the getInitial function from the stats package as in an example below.

The weights argument can be a vector of fixed weights, in which case the objective function that will be minimized is the sum of squares where each residual is multiplied by the square root of the corresponding weight. Default NULL implies unit weights.

weights may alternatively be a function with header function(parms, resids) to compute such a vector, or a formula whose right hand side gives an expression for the weights. Variables in the expression may include the following:

A variable named resid

The current residuals.

A variable named fitted

The right hand side of the model formula.

Parameters

The parameters of the model.

Data

Values from data.

Vars

Variables in the environment of the formula.

Value

list of solution elements

resid

weighted residuals at the proposed solution

jacobian

Jacobian matrix at the proposed solution

feval

residual function evaluations used to reach solution from starting parameters

jeval

Jacobian function (or approximation) evaluations used

coefficients

a named vector of proposed solution parameters

ssquares

weighted sum of squared residuals (often the deviance)

lower

lower bounds on parameters

upper

upper bounds on parameters

maskidx

vector if indices of fixed (masked) parameters

weights0

weights specified in function call

weights

weights at the final solution

formula

the modeling formula

resfn

the residual function (unweighted) based on the formula

Author(s)

J C Nash 2014-7-16 nashjc _at_ uottawa.ca

Examples

library(nlsr)
weed <- c(5.308, 7.24, 9.638, 12.866, 17.069, 23.192, 31.443,
        38.558, 50.156, 62.948, 75.995, 91.972)
tt <- 1:12
weeddf <- data.frame(tt, weed)
frm <- 
wmodu <- weed ~ b1/(1+b2*exp(-b3*tt)) # Unscaled
## nls from unit start FAILS
start1<-c(b1=1, b2=1, b3=1)
hunls1 <- try(nls(wmodu, data=weeddf, start=start1, trace=FALSE))
if (! inherits(hunls1, "try-error")) print(hunls1) ## else cat("Failure -- try-error\n")
## nlxb from unit start
hunlx1 <- try(nlxb(wmodu, data=weeddf, start=c(b1=1, b2=1, b3=1), trace=FALSE))
if (! inherits(hunlx1, "try-error")) print(hunlx1)

st2h<-c(b1=185, b2=10, b3=.3)
#' hunls2 <- try(nls(wmodu, data=weeddf, start=st2h, trace=FALSE))
if (! inherits(hunls1, "try-error")) print(hunls1) ## else cat("Failure -- try-error\n")
## nlxb from unit start
hunlx1 <- try(nlxb(wmodu, data=weeddf, start=st2h, trace=FALSE))
if (! inherits(hunlx1, "try-error")) print(hunlx1)

# Functional models need to use a Jacobian approximation or external calculation.
# For example, the SSlogis() selfStart model from \code{stats} package.

# nls() needs NO starting value
hSSnls<-try(nls(weed~SSlogis(tt, Asym, xmid, scal), data=weeddf))
summary(hSSnls)
# We need to get the start for nlxb explicitly
stSS <- getInitial(weed ~ SSlogis(tt, Asym, xmid, scal), data=weeddf)
hSSnlx<-try(nlxb(weed~SSlogis(tt, Asym, xmid, scal), data=weeddf, start=stSS))
hSSnlx

# nls() can only bound parameters with algorithm="port"
#   and minpack.lm is unreliable in imposing bounds, but nlsr copes fine.
lo<-c(0, 0, 0)
up<-c(190, 10, 2) # Note: start must be admissible.
bnls0<-try(nls(wmodu, data=weeddf, start=st2h,
         lower=lo, upper=up)) # should complain and fail
 
bnls<-try(nls(wmodu, data=weeddf, start=st2h,
         lower=lo, upper=up, algorith="port"))
summary(bnls)
bnlx<-try(nlxb(wmodu, data=weeddf, start=st2h, lower=lo, upper=up))
bnlx

# nlxb() can also MASK (fix) parameters. The mechanism of maskidx from nls
# is NO LONGER USED. Instead we set upper and lower parameters equal for
# the masked parameters. The start value MUST be equal to this fixed value.
lo<-c(190, 0, 0) # mask first parameter
up<-c(190, 10, 2)
strt <- c(b1=190, b2=1, b3=1)
mnlx<-try(nlxb(wmodu, start=strt, data=weeddf, 
         lower=lo, upper=up))
mnlx
mnls<-try(nls(wmodu, data=weeddf, start=strt,
         lower=lo, upper=up, algorith="port"))
summary(mnls)

# Try first parameter masked and see if we get SEs 
lo<-c(200, 0, 0) # mask first parameter
up<-c(100, 10, 5)
strt <- c(b1=200, b2=1, b3=1)
mnlx<-try(nlxb(wmodu, start=strt, data=weeddf, 
         lower=lo, upper=up))
mnlx
mnls<-try(nls(wmodu, data=weeddf, start=strt,
         lower=lo, upper=up, algorith="port"))
summary(mnls) 

# Try with weights on the observations
mnlx<-try(nlxb(wmodu, start=strt, data=weeddf, 
               weights = ~ 1/weed))
mnlx

numericDerivR: numerically evaluates the gradient of an expression. All in R

Description

This version is all in R to replace the C version in package stats

Usage

numericDerivR(
  expr,
  theta,
  rho = parent.frame(),
  dir = 1,
  eps = .Machine$double.eps^(1/if (central) 3 else 2),
  central = FALSE
)

Arguments

expr

expression or call to be differentiated. Should evaluate to a numeric vector.

theta

character vector of names of numeric variables used in expr.

rho

environment containing all the variables needed to evaluate expr.

dir

numeric vector of directions, typically with values in -1, 1 to use for the finite differences; will be recycled to the length of theta.

eps

a positive number, to be used as unit step size hh for the approximate numerical derivative (f(x+h)-f(x))/h (f(x+h)-f(x))/h or the central version, see central.

central

logical indicating if central divided differences should be computed, i.e., (f(x+h) - f(x-h)) / 2h (f(x+h)-f(x-h))/2h. These are typically more accurate but need more evaluations of f()f().

Value

The value of eval(expr, envir = rho) plus a matrix attribute "gradient". The columns of this matrix are the derivatives of the value with respect to the variables listed in theta.

Examples

ex <- expression(a/(1+b*exp(-tt*c)) - weed)
weed <- c(5.308, 7.24, 9.638, 12.866, 17.069, 23.192, 31.443,
              38.558, 50.156, 62.948, 75.995, 91.972)
tt <- 1:12
a <- 200; b <- 50; c <- 0.3
dhobb <- numericDerivR(ex, theta=c("a", "b", "c"))
print(dhobb)
# exf <- ~ a/(1+b*exp(-tt*c)) - weed
# Note that a formula doesn't work
# dh1 <-  try(numericDerivR(exf, theta=c("a", "b", "c")))

nvec

Description

Compact display of a specified named vector

Usage

nvec(vec)

Arguments

vec

a named vector of parameters

Value

none (Note that we may want to change this.)

Author(s)

J C Nash 2014-7-16 nashjc _at_ uottawa.ca


pctrl

Description

Compact display of specified control vector for package nlsr.

Usage

pctrl(control)

Arguments

control

a control list

Value

none

Author(s)

J C Nash 2014-7-16 nashjc _at_ uottawa.ca


pnls

Description

Compact display of specified nls object x

Usage

pnls(x)

Arguments

x

an nls() result object from nls() or nlsLM()

Value

none

Author(s)

J C Nash 2014-7-16, 2023-5-8 nashjc _at_ uottawa.ca


pnlslm

Description

Compact display of specified nls.lm object x. This code returns the iteration count in a different variable from that of nls objects.

Usage

pnlslm(x)

Arguments

x

an nls() result object from minpack.lm::nls.lm()

Value

none

Author(s)

J C Nash 2014-7-16, 2023-5-8 nashjc _at_ uottawa.ca


predict.nlsr

Description

prepare and display predictions from an nlsr model

Usage

## S3 method for class 'nlsr'
predict(object = NULL, newdata = list(), ...)

Arguments

object

an object of class nlsr

newdata

a dataframe containing columns that match the original dataframe used to estimate the nonlinear model in the nlsr object

...

additional data needed to evaluate the modeling functions Default FALSE

Author(s)

J C Nash 2014-7-16 nashjc _at_ uottawa.ca


print.nlsr

Description

prepare and display result of nlsr computations

Usage

## S3 method for class 'nlsr'
print(x, ...)

Arguments

x

an object of class nlsr

...

additional data needed to evaluate the modeling functions Default FALSE

Details

The set of possible controls to set is as follows

Author(s)

J C Nash 2014-7-16 nashjc _at_ uottawa.ca


prt

Description

To display the calling name of x and print the object with the print.nlsr() function.

Usage

prt(x)

Arguments

x

an object of class nlsr

Author(s)

J C Nash 2022-11-22 nashjc _at_ uottawa.ca


pshort

Description

To provide a 1-line summary of an object of class nlsr.

Usage

pshort(x)

Arguments

x

an object of class nlsr

Author(s)

J C Nash 2022-11-22 nashjc _at_ uottawa.ca


rawres

Description

Prepare and display raw residuals of nlsr computations NOTE: we use model - data form i.e., rhs - lhs

Usage

rawres(object = NULL, data = parent.frame(), ...)

Arguments

object

an object of class nlsr

data

a data frame with the data for which fits are wanted

...

additional data needed to evaluate the modeling functions

Value

A vector of the raw residuals

Author(s)

J C Nash 2014-7-16 revised 2022-11-22 nashjc _at_ uottawa.ca


resgr

Description

Computes the gradient of the sum of squares function for nonlinear least squares where resfn and jacfn supply the residuals and Jacobian

Usage

resgr(prm, resfn, jacfn, ...)

Arguments

prm

a numeric vector of parameters to the model

resfn

a function to compute a vector of residuals

jacfn

a function to compute the Jacobian of the sum of squares. If the value is quoted, then the function is assumed to be a numerical approximation. Currently one of "jafwd", "jaback", "jacentral", or "jand".

...

Extra information needed to compute the residuals

Details

Does NOT (yet) handle calling of code built into selfStart models. That is, codes that in nlxb employ control japprox="SSJac".

Value

The main object returned is the numeric vector of residuals computed at prm by means of the function resfn. There are Jacobian and gradient attributes giving the Jacobian (matrix of 1st partial derivatives whose row i contains the partial derivative of the i'th residual w.r.t. each of the parameters) and the gradient of the sum of squared residuals w.r.t. each of the parameters. Moreover, the Jacobian is repeated within the gradient attribute of the Jacobian. This somewhat bizarre structure is present for compatibility with nls() and some other legacy functions, as well as to simplify the call to nlfb().

Author(s)

J C Nash 2014-7-16 revised 2022-11-22 nashjc _at_ uottawa.ca


resid.nlsr

Description

prepare and display result of nlsr computations

Usage

## S3 method for class 'nlsr'
resid(object, ...)

Arguments

object

an object of class nlsr

...

additional data needed to evaluate the modeling functions

Author(s)

J C Nash nashjc _at_ uottawa.ca

### remove _at_export to try to overcome NAMESPACE issue


residuals.nlsr

Description

prepare and display result of nlsr computations

Usage

## S3 method for class 'nlsr'
residuals(object, ...)

Arguments

object

an object of class nlsr

...

additional data needed to evaluate the modeling functions

Author(s)

J C Nash nashjc _at_ uottawa.ca


resss

Description

compute the sum of squares from resfn at parameters prm

Usage

resss(prm, resfn, ...)

Arguments

prm

a named numeric vector of parameters to the model

resfn

a function to compute a vector of residuals

...

Extra information needed to compute the residuals

Author(s)

J C Nash 2014-7-16 nashjc _at_ uottawa.ca


Alternative self start for three-parameter logistic function SSlogis

Description

Self starter for a 3-parameter logistic function.

The equation for this function is:

f(input)=Asym/(1+exp((xmidinput)/scal))f(input) = Asym/(1 + exp((xmid - input)/scal))

The approach of the function SSlogis() in base R uses a different algorithm and returns the actual solution rather than starting parameters, so runs a complete set of iterations, only to try to repeat from the solution with the standard algorithm.

Usage

SSlogisJN(input, Asym, xmid, scal)

Arguments

input

input vector (input)

Asym

asymptotic value for large values of x

xmid

a numeric parameter representing the x value at the inflection point of the curve. The value of SSlogisJN will be Asym/2 at xmid.

scal

numeric scale parameter on the input axis

References

Ratkowsky, David A. (1983) Nonlinear Regression Modeling, A Unified Practical Approach, Dekker: New York, section 8.3.2

Examples

{
## require(ggplot2)
require(nlsr)
set.seed(1234)
x <- seq(0, 20, .2)
y <- SSlogisJN(x, 5, 10, .5) + rnorm(length(x), 0, 0.15)
frm<-y ~ SSlogisJN(x, Asym, xmid, scal)
dat <- data.frame(x = x, y = y)
strt<-getInitial(frm, dat)
cat("Proposed start:\n"); print(strt)
fit <- nlxb(frm, strt, data = dat, control=list(japprox="SSJac"))
print(fit)
## plot
## ggplot(data = dat, aes(x = x, y = y)) + 
##   geom_point() + 
##  geom_line(aes(y = fitted(fit)))
}

summary.nlsr

Description

prepare display result of nlsr computations - NOT compact output

Usage

## S3 method for class 'nlsr'
summary(object, ...)

Arguments

object

an object of class nlsr

...

additional data needed to evaluate the modeling functions

Details

The set of possible controls to set is as follows

Author(s)

J C Nash 2014-7-16 nashjc _at_ uottawa.ca


wrapnlsr

Description

Provides class nls solution to a nonlinear least squares solution using the Nash Marquardt tools.

Usage

wrapnlsr(formula = NULL,  data = NULL,  start = NULL,  control = NULL,
    trace = FALSE, subset = NULL, lower = -Inf, upper = Inf,  weights = NULL,
    ...)

Arguments

formula

The modeling formula. Looks like 'y~b1/(1+b2*exp(-b3*T))'

data

a data frame containing data for variables used in the formula that are NOT the parameters. This data may also be defined in the parent frame i.e., 'global' to this function

start

MUST be a named vector with all elements present e.g., start=c(b1=200, b2=50, b3=0.3)

control

a list of control parameters. See nlsr.control().

trace

TRUE for console output during execution (default FALSE)

subset

an optional vector specifying a subset of observations to be used in the fitting process. NOT used currently by nlxb() or nlfb() and will throw an error if present and not NULL.

lower

a vector of lower bounds on the parameters. If a single number, this will be applied to all parameters Default -Inf.

upper

a vector of upper bounds on the parameters. If a single number, this will be applied to all parameters. Default Inf.

weights

A vector of (usually fixed) weights. The objective function that will be minimized is the sum of squares where each residual is multiplied by the square root of the corresponding weight. Default NULL implies unit weights.

...

additional data needed to evaluate the modeling functions

Value

A solution object of type nls

Examples

library(nlsr)
cat("kvanderpoel.R test of wrapnlsr\n")
x<-c(1,3,5,7)
y<-c(37.98,11.68,3.65,3.93)
pks28<-data.frame(x=x,y=y)
fit0<-try(nls(y~(a+b*exp(1)^(-c*x)), data=pks28, start=c(a=0,b=1,c=1), 
              trace=TRUE))
print(fit0)
fit1<-nlxb(y~(a+b*exp(-c*x)), data=pks28, start=c(a=0,b=1,c=1), trace = TRUE)
print(fit1) 
cat("\n\n or better\n")
fit2<-wrapnlsr(y~(a+b*exp(-c*x)), data=pks28, start=c(a=0,b=1,c=1), 
               lower=-Inf, upper=Inf, trace = TRUE)
fit2

weed <- c(5.308, 7.24, 9.638, 12.866, 17.069, 23.192, 31.443,
          38.558, 50.156, 62.948, 75.995, 91.972)
tt <- 1:12
weeddf <- data.frame(tt, weed)
hobbsu <- weed ~ b1/(1+b2*exp(-b3*tt))
st2 <- c(b1=200, b2=50, b3=0.3)
wts <- 0.5^tt # a straight scaling comes via wts <- rep(0.01, 12)
lo <- c(200, 0, 0)
up <- c(1000, 1000, 1000)
whuw2 <-  try(wrapnlsr(start=st2, formula=hobbsu, data=weeddf, subset=2:11,
                  weights=wts, trace=TRUE, lower=lo, upper=up))
summary(whuw2)
deviance(whuw2)
whuw2a <-  try(nlsr(start=st2, formula=hobbsu, data=weeddf, subset=2:11,
                  weights=wts, trace=TRUE, lower=lo, upper=up))
summary(whuw2a)
deviance(whuw2a)