| Title: | Nonlinear least squares examples from NIST |
|---|---|
| Description: | Datasets for testing nonlinear regression routines. |
| Authors: | original from National Institutes for Standards and Technology (NIST) http://www.itl.nist.gov/div898/strd/nls/nls_main.shtml R port by Douglas Bates <[email protected]> |
| Maintainer: | Douglas Bates <[email protected]> |
| License: | GPL (>= 2) |
| Version: | 0.9-13 |
| Built: | 2024-11-16 06:23:34 UTC |
| Source: | CRAN |
The Bennett5 data frame has 154 rows and 2 columns of data from a
magnetism study
This data frame contains the following columns:
A numeric vector of magnetism values.
A numeric vector of log(time).
These data are the result of a NIST study involving superconductivity magnetization modeling. The response variable is magnetism, and the predictor variable is the log of time in minutes.
Bennett, L., L. Swartzendruber, and H. Brown, NIST (1994). Superconductivity Magnetization Modeling.
Try <- function(expr) if (!inherits(val <- try(expr), "try-error")) val plot(y ~ x, data = Bennett5) Try(fm1 <- nls(y ~ b1*(b2+x)**(-1/b3), data = Bennett5, start = c(b1 = -2000, b2 = 50, b3 = 0.8), trace = TRUE)) Try(fm1a <- nls(y ~ b1*(b2+x)**(-1/b3), data = Bennett5, start = c(b1 = -2000, b2 = 50, b3 = 0.8), trace = TRUE, alg = "port")) Try(fm2 <- nls(y ~ b1*(b2+x)**(-1/b3), data = Bennett5, start = c(b1 = -1500, b2 = 45, b3 = 0.85), trace = TRUE)) Try(fm2a <- nls(y ~ b1*(b2+x)**(-1/b3), data = Bennett5, start = c(b1 = -1500, b2 = 45, b3 = 0.85), trace = TRUE, alg = "port")) Try(fm3 <- nls(y ~ (b2+x)**(-1/b3), data = Bennett5, alg = "plinear", start = c( b2 = 50, b3 = 0.8), trace = TRUE)) Try(fm4 <- nls(y ~ (b2+x)**(-1/b3), data = Bennett5, alg = "plinear", start = c( b2 = 45, b3 = 0.8), trace = TRUE))Try <- function(expr) if (!inherits(val <- try(expr), "try-error")) val plot(y ~ x, data = Bennett5) Try(fm1 <- nls(y ~ b1*(b2+x)**(-1/b3), data = Bennett5, start = c(b1 = -2000, b2 = 50, b3 = 0.8), trace = TRUE)) Try(fm1a <- nls(y ~ b1*(b2+x)**(-1/b3), data = Bennett5, start = c(b1 = -2000, b2 = 50, b3 = 0.8), trace = TRUE, alg = "port")) Try(fm2 <- nls(y ~ b1*(b2+x)**(-1/b3), data = Bennett5, start = c(b1 = -1500, b2 = 45, b3 = 0.85), trace = TRUE)) Try(fm2a <- nls(y ~ b1*(b2+x)**(-1/b3), data = Bennett5, start = c(b1 = -1500, b2 = 45, b3 = 0.85), trace = TRUE, alg = "port")) Try(fm3 <- nls(y ~ (b2+x)**(-1/b3), data = Bennett5, alg = "plinear", start = c( b2 = 50, b3 = 0.8), trace = TRUE)) Try(fm4 <- nls(y ~ (b2+x)**(-1/b3), data = Bennett5, alg = "plinear", start = c( b2 = 45, b3 = 0.8), trace = TRUE))
The Chwirut1 data frame has 214 rows and 2 columns giving
This data frame contains the following columns:
A numeric vector of ultrasonic response values
A numeric vector or metal distance values
These data are the result of a NIST study involving ultrasonic calibration. The response variable is ultrasonic response, and the predictor variable is metal distance.
Chwirut, D., NIST (197?). Ultrasonic Reference Block Study.
Try <- function(expr) if (!inherits(val <- try(expr), "try-error")) val plot(y ~ x, data = Chwirut1) Try(fm1 <- nls(y ~ exp(-b1*x)/(b2+b3*x), data = Chwirut1, trace = TRUE, start = c(b1 = 0.1, b2 = 0.01, b3 = 0.02))) Try(fm1a <- nls(y ~ exp(-b1*x)/(b2+b3*x), data = Chwirut1, trace = TRUE, start = c(b1 = 0.1, b2 = 0.01, b3 = 0.02), alg = "port")) Try(fm2 <- nls(y ~ exp(-b1*x)/(b2+b3*x), data = Chwirut1, trace = TRUE, start = c(b1 = 0.15, b2 = 0.008, b3 = 0.010))) Try(fm2a <- nls(y ~ exp(-b1*x)/(b2+b3*x), data = Chwirut1, trace = TRUE, start = c(b1 = 0.15, b2 = 0.008, b3 = 0.010), alg = "port")) Try(fm3 <- nls(y ~ exp(-b1*x)/(1+p3*x), data = Chwirut1, trace = TRUE, start = c(b1 = 0.1, p3 = 0.02/0.01), algorithm = "plinear")) Try(fm4 <- nls(y ~ exp(-b1*x)/(1+p3*x), data = Chwirut1, trace = TRUE, start = c(b1 = 0.15, p3 = 0.01/0.008), algorithm = "plinear"))Try <- function(expr) if (!inherits(val <- try(expr), "try-error")) val plot(y ~ x, data = Chwirut1) Try(fm1 <- nls(y ~ exp(-b1*x)/(b2+b3*x), data = Chwirut1, trace = TRUE, start = c(b1 = 0.1, b2 = 0.01, b3 = 0.02))) Try(fm1a <- nls(y ~ exp(-b1*x)/(b2+b3*x), data = Chwirut1, trace = TRUE, start = c(b1 = 0.1, b2 = 0.01, b3 = 0.02), alg = "port")) Try(fm2 <- nls(y ~ exp(-b1*x)/(b2+b3*x), data = Chwirut1, trace = TRUE, start = c(b1 = 0.15, b2 = 0.008, b3 = 0.010))) Try(fm2a <- nls(y ~ exp(-b1*x)/(b2+b3*x), data = Chwirut1, trace = TRUE, start = c(b1 = 0.15, b2 = 0.008, b3 = 0.010), alg = "port")) Try(fm3 <- nls(y ~ exp(-b1*x)/(1+p3*x), data = Chwirut1, trace = TRUE, start = c(b1 = 0.1, p3 = 0.02/0.01), algorithm = "plinear")) Try(fm4 <- nls(y ~ exp(-b1*x)/(1+p3*x), data = Chwirut1, trace = TRUE, start = c(b1 = 0.15, p3 = 0.01/0.008), algorithm = "plinear"))
The Chwirut2 data frame has nr rows and nc columns giving
This data frame contains the following columns:
A numeric vector of ultrasonic response values.
A numeric vector of metal distance values.
These data are the result of a NIST study involving ultrasonic calibration. The response variable is ultrasonic response, and the predictor variable is metal distance.
Chwirut, D., NIST (197?). Ultrasonic Reference Block Study.
Try <- function(expr) if (!inherits(val <- try(expr), "try-error")) val plot(y ~ x, data = Chwirut2) Try(fm1 <- nls(y ~ exp(-b1*x)/(b2+b3*x), data = Chwirut2, trace = TRUE, start = c(b1 = 0.1 , b2 = 0.01, b3 = 0.02))) Try(fm1a <- nls(y ~ exp(-b1*x)/(b2+b3*x), data = Chwirut2, trace = TRUE, start = c(b1 = 0.1 , b2 = 0.01, b3 = 0.02), alg = "port")) Try(fm2 <- nls(y ~ exp(-b1*x)/(b2+b3*x), data = Chwirut2, trace = TRUE, start = c(b1 = 0.15 , b2 = 0.008, b3 = 0.01))) Try(fm2a <- nls(y ~ exp(-b1*x)/(b2+b3*x), data = Chwirut2, trace = TRUE, start = c(b1 = 0.15 , b2 = 0.008, b3 = 0.01), alg = "port")) Try(fm3 <- nls(y ~ exp(-b1*x)/(1+p3*x), data = Chwirut2, trace = TRUE, start = c(b1 = 0.1, p3 = 2.), alg = "plinear")) Try(fm4 <- nls(y ~ exp(-b1*x)/(1+p3*x), data = Chwirut2, trace = TRUE, start = c(b1 = 0.15, p3 = 0.01/0.008), alg = "plinear"))Try <- function(expr) if (!inherits(val <- try(expr), "try-error")) val plot(y ~ x, data = Chwirut2) Try(fm1 <- nls(y ~ exp(-b1*x)/(b2+b3*x), data = Chwirut2, trace = TRUE, start = c(b1 = 0.1 , b2 = 0.01, b3 = 0.02))) Try(fm1a <- nls(y ~ exp(-b1*x)/(b2+b3*x), data = Chwirut2, trace = TRUE, start = c(b1 = 0.1 , b2 = 0.01, b3 = 0.02), alg = "port")) Try(fm2 <- nls(y ~ exp(-b1*x)/(b2+b3*x), data = Chwirut2, trace = TRUE, start = c(b1 = 0.15 , b2 = 0.008, b3 = 0.01))) Try(fm2a <- nls(y ~ exp(-b1*x)/(b2+b3*x), data = Chwirut2, trace = TRUE, start = c(b1 = 0.15 , b2 = 0.008, b3 = 0.01), alg = "port")) Try(fm3 <- nls(y ~ exp(-b1*x)/(1+p3*x), data = Chwirut2, trace = TRUE, start = c(b1 = 0.1, p3 = 2.), alg = "plinear")) Try(fm4 <- nls(y ~ exp(-b1*x)/(1+p3*x), data = Chwirut2, trace = TRUE, start = c(b1 = 0.15, p3 = 0.01/0.008), alg = "plinear"))
The DanielWood data frame has 6 rows and 2 columns giving the
energy radiated from a carbon filament versus the absolute temperature
of the filament.
This data frame contains the following columns:
A numeric vector of the energy radiated from a carbon filament lamp.
A numeric vector of the temperature of the filament (1000 K).
These data and model are described in Daniel and Wood (1980), and originally published in E.S.Keeping, "Introduction to Statistical Inference," Van Nostrand Company, Princeton, NJ, 1962, p. 354. The response variable is energy radiated from a carbon filament lamp per cm**2 per second, and the predictor variable is the absolute temperature of the filament in 1000 degrees Kelvin.
Daniel, C. and F. S. Wood (1980). Fitting Equations to Data, Second Edition. New York, NY: John Wiley and Sons, pp. 428-431.
Try <- function(expr) if (!inherits(val <- try(expr), "try-error")) val plot(y ~ x, data = DanielWood) Try(fm1 <- nls(y ~ b1*x**b2, data = DanielWood, trace = TRUE, start = c(b1 = 1, b2 = 5))) Try(fm1a <- nls(y ~ b1*x**b2, data = DanielWood, trace = TRUE, start = c(b1 = 1, b2 = 5), alg = "port")) Try(fm2 <- nls(y ~ b1*x**b2, data = DanielWood, trace = TRUE, start = c(b1 = 0.7, b2 = 4))) Try(fm2a <- nls(y ~ b1*x**b2, data = DanielWood, trace = TRUE, start = c(b1 = 0.7, b2 = 4), alg = "port")) Try(fm3 <- nls(y ~ x**b2, data = DanielWood, trace = TRUE, start = c(b2 = 5), algorithm = "plinear")) Try(fm4 <- nls(y ~ x**b2, data = DanielWood, trace = TRUE, start = c(b2 = 4), algorithm = "plinear"))Try <- function(expr) if (!inherits(val <- try(expr), "try-error")) val plot(y ~ x, data = DanielWood) Try(fm1 <- nls(y ~ b1*x**b2, data = DanielWood, trace = TRUE, start = c(b1 = 1, b2 = 5))) Try(fm1a <- nls(y ~ b1*x**b2, data = DanielWood, trace = TRUE, start = c(b1 = 1, b2 = 5), alg = "port")) Try(fm2 <- nls(y ~ b1*x**b2, data = DanielWood, trace = TRUE, start = c(b1 = 0.7, b2 = 4))) Try(fm2a <- nls(y ~ b1*x**b2, data = DanielWood, trace = TRUE, start = c(b1 = 0.7, b2 = 4), alg = "port")) Try(fm3 <- nls(y ~ x**b2, data = DanielWood, trace = TRUE, start = c(b2 = 5), algorithm = "plinear")) Try(fm4 <- nls(y ~ x**b2, data = DanielWood, trace = TRUE, start = c(b2 = 4), algorithm = "plinear"))
The Eckerle4 data frame has 35 rows and 2 columns giving
transmittance as a function of wavelength.
This data frame contains the following columns:
A numeric vector of transmittance values.
A numeric vector of wavelengths.
These data are the result of a NIST study involving circular interference transmittance. The response variable is transmittance, and the predictor variable is wavelength.
Eckerle, K., NIST (197?). Circular Interference Transmittance Study.
Try <- function(expr) if (!inherits(val <- try(expr), "try-error")) val plot(y ~ x, data = Eckerle4) ## should fail - ridiculous starting value for b3 Try(fm1 <- nls(y ~ (b1/b2) * exp(-0.5*((x-b3)/b2)**2), Eckerle4, trace = TRUE, start = c(b1 = 1, b2 = 10, b3 = 500))) Try(fm1a <- nls(y ~ (b1/b2) * exp(-0.5*((x-b3)/b2)**2), Eckerle4, trace = TRUE, alg = "port", start = c(b1 = 1, b2 = 10, b3 = 500))) Try(fm2 <- nls(y ~ (b1/b2) * exp(-0.5*((x-b3)/b2)**2), Eckerle4, trace = TRUE, start = c(b1 = 1.5, b2 = 5, b3 = 450))) Try(fm2a <- nls(y ~ (b1/b2) * exp(-0.5*((x-b3)/b2)**2), Eckerle4, trace = TRUE, alg = "port", start = c(b1 = 1.5, b2 = 5, b3 = 450))) ## should fail - ridiculous starting value for b3 Try(fm3 <- nls(y ~ (1/b2) * exp(-0.5*((x-b3)/b2)**2), Eckerle4, trace = TRUE, start = c(b2 = 10, b3 = 500), algorithm = "plinear")) Try(fm4 <- nls(y ~ (1/b2) * exp(-0.5*((x-b3)/b2)**2), Eckerle4, trace = TRUE, start = c(b2 = 5, b3 = 450), algorithm = "plinear"))Try <- function(expr) if (!inherits(val <- try(expr), "try-error")) val plot(y ~ x, data = Eckerle4) ## should fail - ridiculous starting value for b3 Try(fm1 <- nls(y ~ (b1/b2) * exp(-0.5*((x-b3)/b2)**2), Eckerle4, trace = TRUE, start = c(b1 = 1, b2 = 10, b3 = 500))) Try(fm1a <- nls(y ~ (b1/b2) * exp(-0.5*((x-b3)/b2)**2), Eckerle4, trace = TRUE, alg = "port", start = c(b1 = 1, b2 = 10, b3 = 500))) Try(fm2 <- nls(y ~ (b1/b2) * exp(-0.5*((x-b3)/b2)**2), Eckerle4, trace = TRUE, start = c(b1 = 1.5, b2 = 5, b3 = 450))) Try(fm2a <- nls(y ~ (b1/b2) * exp(-0.5*((x-b3)/b2)**2), Eckerle4, trace = TRUE, alg = "port", start = c(b1 = 1.5, b2 = 5, b3 = 450))) ## should fail - ridiculous starting value for b3 Try(fm3 <- nls(y ~ (1/b2) * exp(-0.5*((x-b3)/b2)**2), Eckerle4, trace = TRUE, start = c(b2 = 10, b3 = 500), algorithm = "plinear")) Try(fm4 <- nls(y ~ (1/b2) * exp(-0.5*((x-b3)/b2)**2), Eckerle4, trace = TRUE, start = c(b2 = 5, b3 = 450), algorithm = "plinear"))
The ENSO data frame has 168 rows and 2 columns giving atmospheric
pressure differences over time.
This data frame contains the following columns:
A numeric vector of monthly averaged atmospheric pressure differences between Easter Island and Darwin, Australia.
A numeric vector of time values.
The data are monthly averaged atmospheric pressure differences between Easter Island and Darwin, Australia. This difference drives the trade winds in the southern hemisphere. Fourier analysis of the data reveals 3 significant cycles. The annual cycle is the strongest, but cycles with periods of approximately 44 and 26 months are also present. These cycles correspond to the El Nino and the Southern Oscillation. Arguments to the SIN and COS functions are in radians.
Kahaner, D., C. Moler, and S. Nash, (1989). Numerical Methods and Software. Englewood Cliffs, NJ: Prentice Hall, pp. 441-445.
Try <- function(expr) if (!inherits(val <- try(expr), "try-error")) val plot(y ~ x, data = ENSO) plot(y ~ x, data = ENSO, type = "l") # to see the pattern more clearly Try(fm1 <- nls(y ~ b1 + b2*cos( 2*pi*x/12 ) + b3*sin( 2*pi*x/12 ) + b5*cos( 2*pi*x/b4 ) + b6*sin( 2*pi*x/b4 ) + b8*cos( 2*pi*x/b7 ) + b9*sin( 2*pi*x/b7 ), data = ENSO, trace = TRUE, start = c(b1 = 11.0, b2 = 3.0, b3 = 0.5, b4 = 40.0, b5 = -0.7, b6 = -1.3, b7 = 25.0, b8 = -0.3, b9 = 1.4))) Try(fm1a <- nls(y ~ b1 + b2*cos( 2*pi*x/12 ) + b3*sin( 2*pi*x/12 ) + b5*cos( 2*pi*x/b4 ) + b6*sin( 2*pi*x/b4 ) + b8*cos( 2*pi*x/b7 ) + b9*sin( 2*pi*x/b7 ), data = ENSO, trace = TRUE, alg = "port", start = c(b1 = 11.0, b2 = 3.0, b3 = 0.5, b4 = 40.0, b5 = -0.7, b6 = -1.3, b7 = 25.0, b8 = -0.3, b9 = 1.4))) Try(fm2 <- nls(y ~ b1 + b2*cos( 2*pi*x/12 ) + b3*sin( 2*pi*x/12 ) + b5*cos( 2*pi*x/b4 ) + b6*sin( 2*pi*x/b4 ) + b8*cos( 2*pi*x/b7 ) + b9*sin( 2*pi*x/b7 ), data = ENSO, trace = TRUE, start = c(b1 = 10.0, b2 = 3.0, b3 = 0.5, b4 = 44.0, b5 = -1.5, b6 = 0.5, b7 = 26.0, b8 = -0.1, b9 = 1.5))) Try(fm2a <- nls(y ~ b1 + b2*cos( 2*pi*x/12 ) + b3*sin( 2*pi*x/12 ) + b5*cos( 2*pi*x/b4 ) + b6*sin( 2*pi*x/b4 ) + b8*cos( 2*pi*x/b7 ) + b9*sin( 2*pi*x/b7 ), data = ENSO, trace = TRUE, alg = "port", start = c(b1 = 10.0, b2 = 3.0, b3 = 0.5, b4 = 44.0, b5 = -1.5, b6 = 0.5, b7 = 26.0, b8 = -0.1, b9 = 1.5))) Try(fm3 <- nls(y ~ cbind(1, cos( 2*pi*x/12 ), sin( 2*pi*x/12 ), cos( 2*pi*x/b4 ), sin( 2*pi*x/b4 ), cos( 2*pi*x/b7 ), sin( 2*pi*x/b7 )), data = ENSO, trace = TRUE, start = c(b4 = 40.0, b7 = 25.0), algorithm = "plinear")) Try(fm4 <- nls(y ~ cbind(1, cos( 2*pi*x/12 ), sin( 2*pi*x/12 ), cos( 2*pi*x/b4 ), sin( 2*pi*x/b4 ), cos( 2*pi*x/b7 ), sin( 2*pi*x/b7 )), data = ENSO, trace = TRUE, start = c(b4 = 44.0, b7 = 26.0), algorithm = "plinear"))Try <- function(expr) if (!inherits(val <- try(expr), "try-error")) val plot(y ~ x, data = ENSO) plot(y ~ x, data = ENSO, type = "l") # to see the pattern more clearly Try(fm1 <- nls(y ~ b1 + b2*cos( 2*pi*x/12 ) + b3*sin( 2*pi*x/12 ) + b5*cos( 2*pi*x/b4 ) + b6*sin( 2*pi*x/b4 ) + b8*cos( 2*pi*x/b7 ) + b9*sin( 2*pi*x/b7 ), data = ENSO, trace = TRUE, start = c(b1 = 11.0, b2 = 3.0, b3 = 0.5, b4 = 40.0, b5 = -0.7, b6 = -1.3, b7 = 25.0, b8 = -0.3, b9 = 1.4))) Try(fm1a <- nls(y ~ b1 + b2*cos( 2*pi*x/12 ) + b3*sin( 2*pi*x/12 ) + b5*cos( 2*pi*x/b4 ) + b6*sin( 2*pi*x/b4 ) + b8*cos( 2*pi*x/b7 ) + b9*sin( 2*pi*x/b7 ), data = ENSO, trace = TRUE, alg = "port", start = c(b1 = 11.0, b2 = 3.0, b3 = 0.5, b4 = 40.0, b5 = -0.7, b6 = -1.3, b7 = 25.0, b8 = -0.3, b9 = 1.4))) Try(fm2 <- nls(y ~ b1 + b2*cos( 2*pi*x/12 ) + b3*sin( 2*pi*x/12 ) + b5*cos( 2*pi*x/b4 ) + b6*sin( 2*pi*x/b4 ) + b8*cos( 2*pi*x/b7 ) + b9*sin( 2*pi*x/b7 ), data = ENSO, trace = TRUE, start = c(b1 = 10.0, b2 = 3.0, b3 = 0.5, b4 = 44.0, b5 = -1.5, b6 = 0.5, b7 = 26.0, b8 = -0.1, b9 = 1.5))) Try(fm2a <- nls(y ~ b1 + b2*cos( 2*pi*x/12 ) + b3*sin( 2*pi*x/12 ) + b5*cos( 2*pi*x/b4 ) + b6*sin( 2*pi*x/b4 ) + b8*cos( 2*pi*x/b7 ) + b9*sin( 2*pi*x/b7 ), data = ENSO, trace = TRUE, alg = "port", start = c(b1 = 10.0, b2 = 3.0, b3 = 0.5, b4 = 44.0, b5 = -1.5, b6 = 0.5, b7 = 26.0, b8 = -0.1, b9 = 1.5))) Try(fm3 <- nls(y ~ cbind(1, cos( 2*pi*x/12 ), sin( 2*pi*x/12 ), cos( 2*pi*x/b4 ), sin( 2*pi*x/b4 ), cos( 2*pi*x/b7 ), sin( 2*pi*x/b7 )), data = ENSO, trace = TRUE, start = c(b4 = 40.0, b7 = 25.0), algorithm = "plinear")) Try(fm4 <- nls(y ~ cbind(1, cos( 2*pi*x/12 ), sin( 2*pi*x/12 ), cos( 2*pi*x/b4 ), sin( 2*pi*x/b4 ), cos( 2*pi*x/b7 ), sin( 2*pi*x/b7 )), data = ENSO, trace = TRUE, start = c(b4 = 44.0, b7 = 26.0), algorithm = "plinear"))
The Gauss1 data frame has 250 rows and 2 columns of generated data.
This data frame contains the following columns:
A numeric vector of generated responses.
A numeric vector of generated input values.
The data are generated data with two well-separated Gaussians on a decaying exponential baseline plus normally distributed zero-mean noise with variance = 6.25.
Rust, B., NIST (1996).
Try <- function(expr) if (!inherits(val <- try(expr), "try-error")) val plot(y ~ x, data = Gauss1) Try(fm1 <- nls(y ~ b1*exp( -b2*x ) + b3*exp( -(x-b4)**2 / b5**2 ) + b6*exp( -(x-b7)**2 / b8**2 ), data = Gauss1, trace = TRUE, start = c(b1 = 97.0, b2 = 0.009, b3 = 100.0, b4 = 65.0, b5 = 20.0, b6 = 70.0, b7 = 178., b8 = 16.5))) Try(fm1a <- nls(y ~ b1*exp( -b2*x ) + b3*exp( -(x-b4)**2 / b5**2 ) + b6*exp( -(x-b7)**2 / b8**2 ), data = Gauss1, trace = TRUE, start = c(b1 = 97.0, b2 = 0.009, b3 = 100.0, b4 = 65.0, b5 = 20.0, b6 = 70.0, b7 = 178., b8 = 16.5), alg = "port")) Try(fm2 <- nls(y ~ b1*exp( -b2*x ) + b3*exp( -(x-b4)**2 / b5**2 ) + b6*exp( -(x-b7)**2 / b8**2 ), data = Gauss1, trace = TRUE, start = c(b1 = 94.0, b2 = 0.0105, b3 = 99.0, b4 = 63.0, b5 = 25.0, b6 = 71.0, b7 = 180.0, b8 = 20.0))) Try(fm2a <- nls(y ~ b1*exp( -b2*x ) + b3*exp( -(x-b4)**2 / b5**2 ) + b6*exp( -(x-b7)**2 / b8**2 ), data = Gauss1, trace = TRUE, start = c(b1 = 94.0, b2 = 0.0105, b3 = 99.0, b4 = 63.0, b5 = 25.0, b6 = 71.0, b7 = 180.0, b8 = 20.0), alg = "port")) Try(fm3 <- nls(y ~ cbind(exp(-b2*x), exp(-(x-b4)**2/b5**2), exp(-(x-b7)**2/b8**2)), data = Gauss1, trace = TRUE, start = c( b2 = 0.009, b4 = 65.0, b5 = 20.0, b7 = 178., b8 = 16.5), algorithm = "plinear")) Try(fm4 <- nls(y ~ cbind(exp(-b2*x), exp(-(x-b4)**2/b5**2), exp(-(x-b7)**2/b8**2)), data = Gauss1, trace = TRUE, start = c( b2 = 0.0105, b4 = 63.0, b5 = 25.0, b7 = 180., b8 = 20.0), algorithm = "plinear"))Try <- function(expr) if (!inherits(val <- try(expr), "try-error")) val plot(y ~ x, data = Gauss1) Try(fm1 <- nls(y ~ b1*exp( -b2*x ) + b3*exp( -(x-b4)**2 / b5**2 ) + b6*exp( -(x-b7)**2 / b8**2 ), data = Gauss1, trace = TRUE, start = c(b1 = 97.0, b2 = 0.009, b3 = 100.0, b4 = 65.0, b5 = 20.0, b6 = 70.0, b7 = 178., b8 = 16.5))) Try(fm1a <- nls(y ~ b1*exp( -b2*x ) + b3*exp( -(x-b4)**2 / b5**2 ) + b6*exp( -(x-b7)**2 / b8**2 ), data = Gauss1, trace = TRUE, start = c(b1 = 97.0, b2 = 0.009, b3 = 100.0, b4 = 65.0, b5 = 20.0, b6 = 70.0, b7 = 178., b8 = 16.5), alg = "port")) Try(fm2 <- nls(y ~ b1*exp( -b2*x ) + b3*exp( -(x-b4)**2 / b5**2 ) + b6*exp( -(x-b7)**2 / b8**2 ), data = Gauss1, trace = TRUE, start = c(b1 = 94.0, b2 = 0.0105, b3 = 99.0, b4 = 63.0, b5 = 25.0, b6 = 71.0, b7 = 180.0, b8 = 20.0))) Try(fm2a <- nls(y ~ b1*exp( -b2*x ) + b3*exp( -(x-b4)**2 / b5**2 ) + b6*exp( -(x-b7)**2 / b8**2 ), data = Gauss1, trace = TRUE, start = c(b1 = 94.0, b2 = 0.0105, b3 = 99.0, b4 = 63.0, b5 = 25.0, b6 = 71.0, b7 = 180.0, b8 = 20.0), alg = "port")) Try(fm3 <- nls(y ~ cbind(exp(-b2*x), exp(-(x-b4)**2/b5**2), exp(-(x-b7)**2/b8**2)), data = Gauss1, trace = TRUE, start = c( b2 = 0.009, b4 = 65.0, b5 = 20.0, b7 = 178., b8 = 16.5), algorithm = "plinear")) Try(fm4 <- nls(y ~ cbind(exp(-b2*x), exp(-(x-b4)**2/b5**2), exp(-(x-b7)**2/b8**2)), data = Gauss1, trace = TRUE, start = c( b2 = 0.0105, b4 = 63.0, b5 = 25.0, b7 = 180., b8 = 20.0), algorithm = "plinear"))
The Gauss2 data frame has 250 rows and 2 columns giving
This data frame contains the following columns:
A numeric vector of generated response values.
A numeric vector of generated input values.
The data are two slightly-blended Gaussians on a decaying exponential baseline plus normally distributed zero-mean noise with variance = 6.25.
Rust, B., NIST (1996)
Try <- function(expr) if (!inherits(val <- try(expr), "try-error")) val plot(y ~ x, data = Gauss2) Try(fm1 <- nls(y ~ b1*exp( -b2*x ) + b3*exp( -(x-b4)**2 / b5**2 ) + b6*exp( -(x-b7)**2 / b8**2 ), data = Gauss2, trace = TRUE, start = c(b1 = 96, b2 = 0.009, b3 = 103, b4 = 106, b5 = 18, b6 = 72, b7 = 151, b8 = 18))) Try(fm1a <- nls(y ~ b1*exp( -b2*x ) + b3*exp( -(x-b4)**2 / b5**2 ) + b6*exp( -(x-b7)**2 / b8**2 ), data = Gauss2, trace = TRUE, start = c(b1 = 96, b2 = 0.009, b3 = 103, b4 = 106, b5 = 18, b6 = 72, b7 = 151, b8 = 18), alg = "port")) Try(fm2 <- nls(y ~ b1*exp( -b2*x ) + b3*exp( -(x-b4)**2 / b5**2 ) + b6*exp( -(x-b7)**2 / b8**2 ), data = Gauss2, trace = TRUE, start = c(b1 = 98, b2 = 0.0105, b3 = 103, b4 = 105, b5 = 20, b6 = 73, b7 = 150, b8 = 20))) Try(fm2a <- nls(y ~ b1*exp( -b2*x ) + b3*exp( -(x-b4)**2 / b5**2 ) + b6*exp( -(x-b7)**2 / b8**2 ), data = Gauss2, trace = TRUE, start = c(b1 = 98, b2 = 0.0105, b3 = 103, b4 = 105, b5 = 20, b6 = 73, b7 = 150, b8 = 20), alg = "port")) Try(fm3 <- nls(y ~ cbind(exp(-b2*x), exp(-(x-b4)**2/b5**2), exp(-(x-b7)**2/b8**2)), data = Gauss2, trace = TRUE, start = c(b2 = 0.009, b4 = 106, b5 = 18, b7 = 151, b8 = 18), algorithm = "plinear")) Try(fm4 <- nls(y ~ cbind(exp(-b2*x), exp(-(x-b4)**2/b5**2), exp(-(x-b7)**2/b8**2)), data = Gauss2, trace = TRUE, start = c(b2 = 0.0105, b4 = 105, b5 = 20, b7 = 150, b8 = 20), algorithm = "plinear"))Try <- function(expr) if (!inherits(val <- try(expr), "try-error")) val plot(y ~ x, data = Gauss2) Try(fm1 <- nls(y ~ b1*exp( -b2*x ) + b3*exp( -(x-b4)**2 / b5**2 ) + b6*exp( -(x-b7)**2 / b8**2 ), data = Gauss2, trace = TRUE, start = c(b1 = 96, b2 = 0.009, b3 = 103, b4 = 106, b5 = 18, b6 = 72, b7 = 151, b8 = 18))) Try(fm1a <- nls(y ~ b1*exp( -b2*x ) + b3*exp( -(x-b4)**2 / b5**2 ) + b6*exp( -(x-b7)**2 / b8**2 ), data = Gauss2, trace = TRUE, start = c(b1 = 96, b2 = 0.009, b3 = 103, b4 = 106, b5 = 18, b6 = 72, b7 = 151, b8 = 18), alg = "port")) Try(fm2 <- nls(y ~ b1*exp( -b2*x ) + b3*exp( -(x-b4)**2 / b5**2 ) + b6*exp( -(x-b7)**2 / b8**2 ), data = Gauss2, trace = TRUE, start = c(b1 = 98, b2 = 0.0105, b3 = 103, b4 = 105, b5 = 20, b6 = 73, b7 = 150, b8 = 20))) Try(fm2a <- nls(y ~ b1*exp( -b2*x ) + b3*exp( -(x-b4)**2 / b5**2 ) + b6*exp( -(x-b7)**2 / b8**2 ), data = Gauss2, trace = TRUE, start = c(b1 = 98, b2 = 0.0105, b3 = 103, b4 = 105, b5 = 20, b6 = 73, b7 = 150, b8 = 20), alg = "port")) Try(fm3 <- nls(y ~ cbind(exp(-b2*x), exp(-(x-b4)**2/b5**2), exp(-(x-b7)**2/b8**2)), data = Gauss2, trace = TRUE, start = c(b2 = 0.009, b4 = 106, b5 = 18, b7 = 151, b8 = 18), algorithm = "plinear")) Try(fm4 <- nls(y ~ cbind(exp(-b2*x), exp(-(x-b4)**2/b5**2), exp(-(x-b7)**2/b8**2)), data = Gauss2, trace = TRUE, start = c(b2 = 0.0105, b4 = 105, b5 = 20, b7 = 150, b8 = 20), algorithm = "plinear"))
The Gauss3 data frame has 250 rows and 2 columns giving generated
data of Gaussian peaks with a decaying exponential background.
This data frame contains the following columns:
A numeric vector of generated responses.
A numeric vector of generated inputs.
The data are two strongly-blended Gaussians on a decaying exponential baseline plus normally distributed zero-mean noise with variance = 6.25.
Rust, B., NIST (1996).
Try <- function(expr) if (!inherits(val <- try(expr), "try-error")) val plot(y ~ x, data = Gauss3) Try(fm1 <- nls(y ~ b1*exp( -b2*x ) + b3*exp( -(x-b4)**2 / b5**2 ) + b6*exp( -(x-b7)**2 / b8**2 ), data = Gauss3, trace = TRUE, start = c(b1 = 94.9, b2 = 0.009, b3 = 90.1, b4 = 113, b5 = 20, b6 = 73.8, b7 = 140, b8 = 20))) Try(fm1a <- nls(y ~ b1*exp( -b2*x ) + b3*exp( -(x-b4)**2 / b5**2 ) + b6*exp( -(x-b7)**2 / b8**2 ), data = Gauss3, trace = TRUE, start = c(b1 = 94.9, b2 = 0.009, b3 = 90.1, b4 = 113, b5 = 20, b6 = 73.8, b7 = 140, b8 = 20), alg = "port")) Try(fm2 <- nls(y ~ b1*exp( -b2*x ) + b3*exp( -(x-b4)**2 / b5**2 ) + b6*exp( -(x-b7)**2 / b8**2 ), data = Gauss3, trace = TRUE, start = c(b1 = 96, b2 = 0.0096, b3 = 80, b4 = 110, b5 = 25, b6 = 74, b7 = 139, b8 = 25))) Try(fm2a <- nls(y ~ b1*exp( -b2*x ) + b3*exp( -(x-b4)**2 / b5**2 ) + b6*exp( -(x-b7)**2 / b8**2 ), data = Gauss3, trace = TRUE, start = c(b1 = 96, b2 = 0.0096, b3 = 80, b4 = 110, b5 = 25, b6 = 74, b7 = 139, b8 = 25), alg = "port")) Try(fm3 <- nls(y ~ cbind(exp(-b2*x), exp(-(x-b4)**2/b5**2), exp(-(x-b7)**2/b8**2)), data = Gauss3, trace = TRUE, start = c(b2 = 0.009, b4 = 113, b5 = 20, b7 = 140, b8 = 20), algorithm = "plinear")) Try(fm4 <- nls(y ~ cbind(exp(-b2*x), exp(-(x-b4)**2/b5**2), exp(-(x-b7)**2/b8**2)), data = Gauss3, trace = TRUE, start = c(b2 = 0.0096, b4 = 110, b5 = 25, b7 = 139, b8 = 25), algorithm = "plinear"))Try <- function(expr) if (!inherits(val <- try(expr), "try-error")) val plot(y ~ x, data = Gauss3) Try(fm1 <- nls(y ~ b1*exp( -b2*x ) + b3*exp( -(x-b4)**2 / b5**2 ) + b6*exp( -(x-b7)**2 / b8**2 ), data = Gauss3, trace = TRUE, start = c(b1 = 94.9, b2 = 0.009, b3 = 90.1, b4 = 113, b5 = 20, b6 = 73.8, b7 = 140, b8 = 20))) Try(fm1a <- nls(y ~ b1*exp( -b2*x ) + b3*exp( -(x-b4)**2 / b5**2 ) + b6*exp( -(x-b7)**2 / b8**2 ), data = Gauss3, trace = TRUE, start = c(b1 = 94.9, b2 = 0.009, b3 = 90.1, b4 = 113, b5 = 20, b6 = 73.8, b7 = 140, b8 = 20), alg = "port")) Try(fm2 <- nls(y ~ b1*exp( -b2*x ) + b3*exp( -(x-b4)**2 / b5**2 ) + b6*exp( -(x-b7)**2 / b8**2 ), data = Gauss3, trace = TRUE, start = c(b1 = 96, b2 = 0.0096, b3 = 80, b4 = 110, b5 = 25, b6 = 74, b7 = 139, b8 = 25))) Try(fm2a <- nls(y ~ b1*exp( -b2*x ) + b3*exp( -(x-b4)**2 / b5**2 ) + b6*exp( -(x-b7)**2 / b8**2 ), data = Gauss3, trace = TRUE, start = c(b1 = 96, b2 = 0.0096, b3 = 80, b4 = 110, b5 = 25, b6 = 74, b7 = 139, b8 = 25), alg = "port")) Try(fm3 <- nls(y ~ cbind(exp(-b2*x), exp(-(x-b4)**2/b5**2), exp(-(x-b7)**2/b8**2)), data = Gauss3, trace = TRUE, start = c(b2 = 0.009, b4 = 113, b5 = 20, b7 = 140, b8 = 20), algorithm = "plinear")) Try(fm4 <- nls(y ~ cbind(exp(-b2*x), exp(-(x-b4)**2/b5**2), exp(-(x-b7)**2/b8**2)), data = Gauss3, trace = TRUE, start = c(b2 = 0.0096, b4 = 110, b5 = 25, b7 = 139, b8 = 25), algorithm = "plinear"))
The Hahn1 data frame has 236 rows and 2 columns of data from a
study on the thermal expansion of copper.
This data frame contains the following columns:
A numeric vector of values of the coefficient of thermal expansion.
A numeric vector of temperatures (K).
These data are the result of a NIST study involving the thermal expansion of copper. The response variable is the coefficient of thermal expansion, and the predictor variable is temperature in degrees Kelvin.
Hahn, T., NIST (197?). Copper Thermal Expansion Study.
Try <- function(expr) if (!inherits(val <- try(expr), "try-error")) val plot(y ~ x, data = Hahn1) Try(fm1 <- nls(y ~ (b1+b2*x+b3*x**2+b4*x**3) / (1+b5*x+b6*x**2+b7*x**3), data = Hahn1, trace = TRUE, start = c(b1 = 10, b2 = -1, b3 = 0.05, b4 = -0.00001, b5 = -0.05, b6 = 0.001, b7 = -0.000001))) Try(fm1a <- nls(y ~ (b1+b2*x+b3*x**2+b4*x**3) / (1+b5*x+b6*x**2+b7*x**3), data = Hahn1, trace = TRUE, alg = "port", start = c(b1 = 10, b2 = -1, b3 = 0.05, b4 = -0.00001, b5 = -0.05, b6 = 0.001, b7 = -0.000001))) Try(fm2 <- nls(y ~ (b1+b2*x+b3*x**2+b4*x**3) / (1+b5*x+b6*x**2+b7*x**3), data = Hahn1, trace = TRUE, start = c(b1 = 1, b2 = -0.1, b3 = 0.005, b4 = -0.000001, b5 = -0.005, b6 = 0.0001, b7 = -0.0000001))) Try(fm2a <- nls(y ~ (b1+b2*x+b3*x**2+b4*x**3) / (1+b5*x+b6*x**2+b7*x**3), data = Hahn1, trace = TRUE, alg = "port", start = c(b1 = 1, b2 = -0.1, b3 = 0.005, b4 = -0.000001, b5 = -0.005, b6 = 0.0001, b7 = -0.0000001))) Try(fm3 <- nls(y ~ cbind(1, x, x^2, x^3)/(1+x*(b5+x*(b6+x*b7))), data = Hahn1, trace = TRUE, algorithm = "plinear", start = c(b5 = -0.05, b6 = 0.001, b7 = -0.000001))) Try(fm4 <- nls(y ~ cbind(1, x, x^2, x^3)/(1+x*(b5+x*(b6+x*b7))), data = Hahn1, trace = TRUE, algorithm = "plinear", start = c(b5 = -0.005, b6 = 0.0001, b7 = -0.0000001)))Try <- function(expr) if (!inherits(val <- try(expr), "try-error")) val plot(y ~ x, data = Hahn1) Try(fm1 <- nls(y ~ (b1+b2*x+b3*x**2+b4*x**3) / (1+b5*x+b6*x**2+b7*x**3), data = Hahn1, trace = TRUE, start = c(b1 = 10, b2 = -1, b3 = 0.05, b4 = -0.00001, b5 = -0.05, b6 = 0.001, b7 = -0.000001))) Try(fm1a <- nls(y ~ (b1+b2*x+b3*x**2+b4*x**3) / (1+b5*x+b6*x**2+b7*x**3), data = Hahn1, trace = TRUE, alg = "port", start = c(b1 = 10, b2 = -1, b3 = 0.05, b4 = -0.00001, b5 = -0.05, b6 = 0.001, b7 = -0.000001))) Try(fm2 <- nls(y ~ (b1+b2*x+b3*x**2+b4*x**3) / (1+b5*x+b6*x**2+b7*x**3), data = Hahn1, trace = TRUE, start = c(b1 = 1, b2 = -0.1, b3 = 0.005, b4 = -0.000001, b5 = -0.005, b6 = 0.0001, b7 = -0.0000001))) Try(fm2a <- nls(y ~ (b1+b2*x+b3*x**2+b4*x**3) / (1+b5*x+b6*x**2+b7*x**3), data = Hahn1, trace = TRUE, alg = "port", start = c(b1 = 1, b2 = -0.1, b3 = 0.005, b4 = -0.000001, b5 = -0.005, b6 = 0.0001, b7 = -0.0000001))) Try(fm3 <- nls(y ~ cbind(1, x, x^2, x^3)/(1+x*(b5+x*(b6+x*b7))), data = Hahn1, trace = TRUE, algorithm = "plinear", start = c(b5 = -0.05, b6 = 0.001, b7 = -0.000001))) Try(fm4 <- nls(y ~ cbind(1, x, x^2, x^3)/(1+x*(b5+x*(b6+x*b7))), data = Hahn1, trace = TRUE, algorithm = "plinear", start = c(b5 = -0.005, b6 = 0.0001, b7 = -0.0000001)))
The Kirby2 data frame has 151 rows and 2 columns of data from an
NIST study on scanning electron microscope line width standards.
This data frame contains the following columns:
A numeric vector of response values.
A numeric vector of input values.
These data are the result of a NIST study involving scanning electron microscope line with standards.
Kirby, R., NIST (197?). Scanning electron microscope line width standards.
Try <- function(expr) if (!inherits(val <- try(expr), "try-error")) val plot(y ~ x, data = Kirby2) Try(fm1 <- nls(y ~ (b1 + b2*x + b3*x**2) / (1 + b4*x + b5*x**2), data = Kirby2, trace = TRUE, start = c(b1 = 2, b2 = -0.1, b3 = 0.003, b4 = -0.001, b5 = 0.00001))) Try(fm1a <- nls(y ~ (b1 + b2*x + b3*x**2) / (1 + b4*x + b5*x**2), data = Kirby2, trace = TRUE, alg = "port", start = c(b1 = 2, b2 = -0.1, b3 = 0.003, b4 = -0.001, b5 = 0.00001))) Try(fm2 <- nls(y ~ (b1 + b2*x + b3*x**2) / (1 + b4*x + b5*x**2), data = Kirby2, trace = TRUE, start = c(b1 = 1.5, b2 = -0.15, b3 = 0.0025, b4 = -0.0015, b5 = 0.00002))) Try(fm2a <- nls(y ~ (b1 + b2*x + b3*x**2) / (1 + b4*x + b5*x**2), data = Kirby2, trace = TRUE, alg = "port", start = c(b1 = 1.5, b2 = -0.15, b3 = 0.0025, b4 = -0.0015, b5 = 0.00002))) Try(fm3 <- nls(y ~ cbind(1, x, x**2)/(1 + x*(b4 + b5*x)), data = Kirby2, trace = TRUE, algorithm = "plinear", start = c(b4 = -0.001, b5 = 0.00001))) Try(fm4 <- nls(y ~ cbind(1, x, x**2)/(1 + x*(b4 + b5*x)), data = Kirby2, trace = TRUE, algorithm = "plinear", start = c(b4 = -0.0015, b5 = 0.00002)))Try <- function(expr) if (!inherits(val <- try(expr), "try-error")) val plot(y ~ x, data = Kirby2) Try(fm1 <- nls(y ~ (b1 + b2*x + b3*x**2) / (1 + b4*x + b5*x**2), data = Kirby2, trace = TRUE, start = c(b1 = 2, b2 = -0.1, b3 = 0.003, b4 = -0.001, b5 = 0.00001))) Try(fm1a <- nls(y ~ (b1 + b2*x + b3*x**2) / (1 + b4*x + b5*x**2), data = Kirby2, trace = TRUE, alg = "port", start = c(b1 = 2, b2 = -0.1, b3 = 0.003, b4 = -0.001, b5 = 0.00001))) Try(fm2 <- nls(y ~ (b1 + b2*x + b3*x**2) / (1 + b4*x + b5*x**2), data = Kirby2, trace = TRUE, start = c(b1 = 1.5, b2 = -0.15, b3 = 0.0025, b4 = -0.0015, b5 = 0.00002))) Try(fm2a <- nls(y ~ (b1 + b2*x + b3*x**2) / (1 + b4*x + b5*x**2), data = Kirby2, trace = TRUE, alg = "port", start = c(b1 = 1.5, b2 = -0.15, b3 = 0.0025, b4 = -0.0015, b5 = 0.00002))) Try(fm3 <- nls(y ~ cbind(1, x, x**2)/(1 + x*(b4 + b5*x)), data = Kirby2, trace = TRUE, algorithm = "plinear", start = c(b4 = -0.001, b5 = 0.00001))) Try(fm4 <- nls(y ~ cbind(1, x, x**2)/(1 + x*(b4 + b5*x)), data = Kirby2, trace = TRUE, algorithm = "plinear", start = c(b4 = -0.0015, b5 = 0.00002)))
The Lanczos1 data frame has 24 rows and 2 columns of generated
data.
This data frame contains the following columns:
A numeric vector of generated responses.
A numeric vector of generated input values
These data are taken from an example discussed in
Lanczos (1956). The data were generated to 14-digits
of accuracy using
f(x) = 0.0951*exp(-x) + 0.8607*exp(-3*x) + 1.5576*exp(-5*x).
Lanczos, C. (1956). Applied Analysis. Englewood Cliffs, NJ: Prentice Hall, pp. 272-280.
Try <- function(expr) if (!inherits(val <- try(expr), "try-error")) val plot(y ~ x, data = Lanczos1) ## plot on log scale to see the apparent number of exponential terms plot(y ~ x, data = Lanczos1, log = "y") ## data are an exact fit so the convergence criterion fails Try(fm1 <- nls(y ~ b1*exp(-b2*x) + b3*exp(-b4*x) + b5*exp(-b6*x), data = Lanczos1, trace = TRUE, start = c(b1 = 1.2, b2 = 0.3, b3 = 5.6, b4 = 5.5, b5 = 6.5, b6 = 7.6))) Try(fm1a <- nls(y ~ b1*exp(-b2*x) + b3*exp(-b4*x) + b5*exp(-b6*x), data = Lanczos1, trace = TRUE, alg = "port", start = c(b1 = 1.2, b2 = 0.3, b3 = 5.6, b4 = 5.5, b5 = 6.5, b6 = 7.6))) ## data are an exact fit so the convergence criterion fails Try(fm2 <- nls(y ~ b1*exp(-b2*x) + b3*exp(-b4*x) + b5*exp(-b6*x), data = Lanczos1, trace = TRUE, start = c(b1 = 0.5, b2 = 0.7, b3 = 3.6, b4 = 4.2, b5 = 4, b6 = 6.3))) Try(fm2a <- nls(y ~ b1*exp(-b2*x) + b3*exp(-b4*x) + b5*exp(-b6*x), data = Lanczos1, trace = TRUE, alg = "port", start = c(b1 = 0.5, b2 = 0.7, b3 = 3.6, b4 = 4.2, b5 = 4, b6 = 6.3))) ## data are an exact fit so the convergence criterion fails Try(fm3 <- nls(y ~ exp(outer(x,-c(b2, b4, b6))), data = Lanczos1, trace = TRUE, algorithm = "plinear", start = c(b2 = 0.3, b4 = 5.5, b6 = 7.6))) ## data are an exact fit so the convergence criterion fails Try(fm4 <- nls(y ~ exp(outer(x,-c(b2, b4, b6))), data = Lanczos1, trace = TRUE, algorithm = "plinear", start = c(b2 = 0.7, b4 = 4.2, b6 = 6.3)))Try <- function(expr) if (!inherits(val <- try(expr), "try-error")) val plot(y ~ x, data = Lanczos1) ## plot on log scale to see the apparent number of exponential terms plot(y ~ x, data = Lanczos1, log = "y") ## data are an exact fit so the convergence criterion fails Try(fm1 <- nls(y ~ b1*exp(-b2*x) + b3*exp(-b4*x) + b5*exp(-b6*x), data = Lanczos1, trace = TRUE, start = c(b1 = 1.2, b2 = 0.3, b3 = 5.6, b4 = 5.5, b5 = 6.5, b6 = 7.6))) Try(fm1a <- nls(y ~ b1*exp(-b2*x) + b3*exp(-b4*x) + b5*exp(-b6*x), data = Lanczos1, trace = TRUE, alg = "port", start = c(b1 = 1.2, b2 = 0.3, b3 = 5.6, b4 = 5.5, b5 = 6.5, b6 = 7.6))) ## data are an exact fit so the convergence criterion fails Try(fm2 <- nls(y ~ b1*exp(-b2*x) + b3*exp(-b4*x) + b5*exp(-b6*x), data = Lanczos1, trace = TRUE, start = c(b1 = 0.5, b2 = 0.7, b3 = 3.6, b4 = 4.2, b5 = 4, b6 = 6.3))) Try(fm2a <- nls(y ~ b1*exp(-b2*x) + b3*exp(-b4*x) + b5*exp(-b6*x), data = Lanczos1, trace = TRUE, alg = "port", start = c(b1 = 0.5, b2 = 0.7, b3 = 3.6, b4 = 4.2, b5 = 4, b6 = 6.3))) ## data are an exact fit so the convergence criterion fails Try(fm3 <- nls(y ~ exp(outer(x,-c(b2, b4, b6))), data = Lanczos1, trace = TRUE, algorithm = "plinear", start = c(b2 = 0.3, b4 = 5.5, b6 = 7.6))) ## data are an exact fit so the convergence criterion fails Try(fm4 <- nls(y ~ exp(outer(x,-c(b2, b4, b6))), data = Lanczos1, trace = TRUE, algorithm = "plinear", start = c(b2 = 0.7, b4 = 4.2, b6 = 6.3)))
The Lanczos2 data frame has 24 rows and 2 columns of generated data.
This data frame contains the following columns:
A numeric vector of generated responses.
A numeric vector of generated input values.
These data are taken from an example discussed in
Lanczos (1956). The data were generated to 6-digits
of accuracy using
f(x) = 0.0951*exp(-x) + 0.8607*exp(-3*x) + 1.5576*exp(-5*x).
Lanczos, C. (1956). Applied Analysis. Englewood Cliffs, NJ: Prentice Hall, pp. 272-280.
Try <- function(expr) if (!inherits(val <- try(expr), "try-error")) val plot(y ~ x, data = Lanczos2) ## plot log response to see the number of exponential terms plot(y ~ x, data = Lanczos2, log = "y") ## Numerical derivatives do not produce sufficient accuracy to converge Try(fm1 <- nls(y ~ b1*exp(-b2*x) + b3*exp(-b4*x) + b5*exp(-b6*x), data = Lanczos2, trace = TRUE, start = c(b1 = 1.2, b2 = 0.3, b3 = 5.6, b4 = 5.5, b5 = 6.5, b6 = 7.6))) Try(fm1a <- nls(y ~ b1*exp(-b2*x) + b3*exp(-b4*x) + b5*exp(-b6*x), data = Lanczos2, trace = TRUE, alg = "port", start = c(b1 = 1.2, b2 = 0.3, b3 = 5.6, b4 = 5.5, b5 = 6.5, b6 = 7.6))) ## Numerical derivatives do not produce sufficient accuracy to converge Try(fm2 <- nls(y ~ b1*exp(-b2*x) + b3*exp(-b4*x) + b5*exp(-b6*x), data = Lanczos2, trace = TRUE, start = c(b1 = 0.5, b2 = 0.7, b3 = 3.6, b4 = 4.2, b5 = 4, b6 = 6.3))) Try(fm2a <- nls(y ~ b1*exp(-b2*x) + b3*exp(-b4*x) + b5*exp(-b6*x), data = Lanczos2, trace = TRUE, alg = "port", start = c(b1 = 0.5, b2 = 0.7, b3 = 3.6, b4 = 4.2, b5 = 4, b6 = 6.3))) ## Numerical derivatives do not produce sufficient accuracy to converge Try(fm3 <- nls(y ~ exp(outer(x,-c(b2, b4, b6))), data = Lanczos2, trace = TRUE, algorithm = "plinear", start = c(b2 = 0.3, b4 = 5.5, b6 = 7.6))) ## Numerical derivatives do not produce sufficient accuracy to converge Try(fm4 <- nls(y ~ exp(outer(x,-c(b2, b4, b6))), data = Lanczos2, trace = TRUE, algorithm = "plinear", start = c(b2 = 0.7, b4 = 4.2, b6 = 6.3))) ## Use analytic derivatives Lanczos <- deriv(~ b1*exp(-b2*x) + b3*exp(-b4*x) + b5*exp(-b6*x), paste("b", 1:6, sep = ""), function(x, b1, b2, b3, b4, b5, b6){}) Try(fm5 <- nls(y ~ Lanczos(x, b1, b2, b3, b4, b5, b6), data = Lanczos2, trace = TRUE, start = c(b1 = 1.2, b2 = 0.3, b3 = 5.6, b4 = 5.5, b5 = 6.5, b6 = 7.6))) Try(fm5a <- nls(y ~ Lanczos(x, b1, b2, b3, b4, b5, b6), data = Lanczos2, trace = TRUE, alg = "port", start = c(b1 = 1.2, b2 = 0.3, b3 = 5.6, b4 = 5.5, b5 = 6.5, b6 = 7.6))) Try(fm6 <- nls(y ~ Lanczos(x, b1, b2, b3, b4, b5, b6), data = Lanczos2, trace = TRUE, start = c(b1 = 0.5, b2 = 0.7, b3 = 3.6, b4 = 4.2, b5 = 4, b6 = 6.3))) Try(fm6a <- nls(y ~ Lanczos(x, b1, b2, b3, b4, b5, b6), data = Lanczos2, trace = TRUE, alg = "port", start = c(b1 = 0.5, b2 = 0.7, b3 = 3.6, b4 = 4.2, b5 = 4, b6 = 6.3)))Try <- function(expr) if (!inherits(val <- try(expr), "try-error")) val plot(y ~ x, data = Lanczos2) ## plot log response to see the number of exponential terms plot(y ~ x, data = Lanczos2, log = "y") ## Numerical derivatives do not produce sufficient accuracy to converge Try(fm1 <- nls(y ~ b1*exp(-b2*x) + b3*exp(-b4*x) + b5*exp(-b6*x), data = Lanczos2, trace = TRUE, start = c(b1 = 1.2, b2 = 0.3, b3 = 5.6, b4 = 5.5, b5 = 6.5, b6 = 7.6))) Try(fm1a <- nls(y ~ b1*exp(-b2*x) + b3*exp(-b4*x) + b5*exp(-b6*x), data = Lanczos2, trace = TRUE, alg = "port", start = c(b1 = 1.2, b2 = 0.3, b3 = 5.6, b4 = 5.5, b5 = 6.5, b6 = 7.6))) ## Numerical derivatives do not produce sufficient accuracy to converge Try(fm2 <- nls(y ~ b1*exp(-b2*x) + b3*exp(-b4*x) + b5*exp(-b6*x), data = Lanczos2, trace = TRUE, start = c(b1 = 0.5, b2 = 0.7, b3 = 3.6, b4 = 4.2, b5 = 4, b6 = 6.3))) Try(fm2a <- nls(y ~ b1*exp(-b2*x) + b3*exp(-b4*x) + b5*exp(-b6*x), data = Lanczos2, trace = TRUE, alg = "port", start = c(b1 = 0.5, b2 = 0.7, b3 = 3.6, b4 = 4.2, b5 = 4, b6 = 6.3))) ## Numerical derivatives do not produce sufficient accuracy to converge Try(fm3 <- nls(y ~ exp(outer(x,-c(b2, b4, b6))), data = Lanczos2, trace = TRUE, algorithm = "plinear", start = c(b2 = 0.3, b4 = 5.5, b6 = 7.6))) ## Numerical derivatives do not produce sufficient accuracy to converge Try(fm4 <- nls(y ~ exp(outer(x,-c(b2, b4, b6))), data = Lanczos2, trace = TRUE, algorithm = "plinear", start = c(b2 = 0.7, b4 = 4.2, b6 = 6.3))) ## Use analytic derivatives Lanczos <- deriv(~ b1*exp(-b2*x) + b3*exp(-b4*x) + b5*exp(-b6*x), paste("b", 1:6, sep = ""), function(x, b1, b2, b3, b4, b5, b6){}) Try(fm5 <- nls(y ~ Lanczos(x, b1, b2, b3, b4, b5, b6), data = Lanczos2, trace = TRUE, start = c(b1 = 1.2, b2 = 0.3, b3 = 5.6, b4 = 5.5, b5 = 6.5, b6 = 7.6))) Try(fm5a <- nls(y ~ Lanczos(x, b1, b2, b3, b4, b5, b6), data = Lanczos2, trace = TRUE, alg = "port", start = c(b1 = 1.2, b2 = 0.3, b3 = 5.6, b4 = 5.5, b5 = 6.5, b6 = 7.6))) Try(fm6 <- nls(y ~ Lanczos(x, b1, b2, b3, b4, b5, b6), data = Lanczos2, trace = TRUE, start = c(b1 = 0.5, b2 = 0.7, b3 = 3.6, b4 = 4.2, b5 = 4, b6 = 6.3))) Try(fm6a <- nls(y ~ Lanczos(x, b1, b2, b3, b4, b5, b6), data = Lanczos2, trace = TRUE, alg = "port", start = c(b1 = 0.5, b2 = 0.7, b3 = 3.6, b4 = 4.2, b5 = 4, b6 = 6.3)))
The Lanczos3 data frame has 24 rows and 2 columns of generated data.
This data frame contains the following columns:
A numeric vector of generated responses.
A numeric vector of generated input values.
These data are taken from an example discussed in
Lanczos (1956). The data were generated to 5-digits
of accuracy using
f(x) = 0.0951*exp(-x) + 0.8607*exp(-3*x) + 1.5576*exp(-5*x).
Lanczos, C. (1956). Applied Analysis. Englewood Cliffs, NJ: Prentice Hall, pp. 272-280.
Try <- function(expr) if (!inherits(val <- try(expr), "try-error")) val plot(y ~ x, data = Lanczos3) ## plot log response to see the number of exponential terms plot(y ~ x, data = Lanczos3, log = "y") Try(fm1 <- nls(y ~ b1*exp(-b2*x) + b3*exp(-b4*x) + b5*exp(-b6*x), data = Lanczos3, trace = TRUE, start = c(b1 = 1.2, b2 = 0.3, b3 = 5.6, b4 = 5.5, b5 = 6.5, b6 = 7.6))) Try(fm1a <- nls(y ~ b1*exp(-b2*x) + b3*exp(-b4*x) + b5*exp(-b6*x), data = Lanczos3, trace = TRUE, alg = "port", start = c(b1 = 1.2, b2 = 0.3, b3 = 5.6, b4 = 5.5, b5 = 6.5, b6 = 7.6))) Try(fm2 <- nls(y ~ b1*exp(-b2*x) + b3*exp(-b4*x) + b5*exp(-b6*x), data = Lanczos3, trace = TRUE, start = c(b1 = 0.5, b2 = 0.7, b3 = 3.6, b4 = 4.2, b5 = 4, b6 = 6.3))) Try(fm2a <- nls(y ~ b1*exp(-b2*x) + b3*exp(-b4*x) + b5*exp(-b6*x), data = Lanczos3, trace = TRUE, alg = "port", start = c(b1 = 0.5, b2 = 0.7, b3 = 3.6, b4 = 4.2, b5 = 4, b6 = 6.3))) Try(fm3 <- nls(y ~ exp(outer(x,-c(b2, b4, b6))), data = Lanczos3, trace = TRUE, algorithm = "plinear", start = c(b2 = 0.3, b4 = 5.5, b6 = 7.6))) Try(fm4 <- nls(y ~ exp(outer(x,-c(b2, b4, b6))), data = Lanczos3, trace = TRUE, algorithm = "plinear", start = c(b2 = 0.7, b4 = 4.2, b6 = 6.3))) ## Use analytic derivatives Lanczos <- deriv(~ b1*exp(-b2*x) + b3*exp(-b4*x) + b5*exp(-b6*x), paste("b", 1:6, sep = ""), function(x, b1, b2, b3, b4, b5, b6){}) Try(fm5 <- nls(y ~ Lanczos(x, b1, b2, b3, b4, b5, b6), data = Lanczos3, trace = TRUE, start = c(b1 = 1.2, b2 = 0.3, b3 = 5.6, b4 = 5.5, b5 = 6.5, b6 = 7.6))) Try(fm5a <- nls(y ~ Lanczos(x, b1, b2, b3, b4, b5, b6), data = Lanczos3, trace = TRUE, alg = "port", start = c(b1 = 1.2, b2 = 0.3, b3 = 5.6, b4 = 5.5, b5 = 6.5, b6 = 7.6))) Try(fm6 <- nls(y ~ Lanczos(x, b1, b2, b3, b4, b5, b6), data = Lanczos3, trace = TRUE, start = c(b1 = 0.5, b2 = 0.7, b3 = 3.6, b4 = 4.2, b5 = 4, b6 = 6.3))) Try(fm6a <- nls(y ~ Lanczos(x, b1, b2, b3, b4, b5, b6), data = Lanczos3, trace = TRUE, alg = "port", start = c(b1 = 0.5, b2 = 0.7, b3 = 3.6, b4 = 4.2, b5 = 4, b6 = 6.3)))Try <- function(expr) if (!inherits(val <- try(expr), "try-error")) val plot(y ~ x, data = Lanczos3) ## plot log response to see the number of exponential terms plot(y ~ x, data = Lanczos3, log = "y") Try(fm1 <- nls(y ~ b1*exp(-b2*x) + b3*exp(-b4*x) + b5*exp(-b6*x), data = Lanczos3, trace = TRUE, start = c(b1 = 1.2, b2 = 0.3, b3 = 5.6, b4 = 5.5, b5 = 6.5, b6 = 7.6))) Try(fm1a <- nls(y ~ b1*exp(-b2*x) + b3*exp(-b4*x) + b5*exp(-b6*x), data = Lanczos3, trace = TRUE, alg = "port", start = c(b1 = 1.2, b2 = 0.3, b3 = 5.6, b4 = 5.5, b5 = 6.5, b6 = 7.6))) Try(fm2 <- nls(y ~ b1*exp(-b2*x) + b3*exp(-b4*x) + b5*exp(-b6*x), data = Lanczos3, trace = TRUE, start = c(b1 = 0.5, b2 = 0.7, b3 = 3.6, b4 = 4.2, b5 = 4, b6 = 6.3))) Try(fm2a <- nls(y ~ b1*exp(-b2*x) + b3*exp(-b4*x) + b5*exp(-b6*x), data = Lanczos3, trace = TRUE, alg = "port", start = c(b1 = 0.5, b2 = 0.7, b3 = 3.6, b4 = 4.2, b5 = 4, b6 = 6.3))) Try(fm3 <- nls(y ~ exp(outer(x,-c(b2, b4, b6))), data = Lanczos3, trace = TRUE, algorithm = "plinear", start = c(b2 = 0.3, b4 = 5.5, b6 = 7.6))) Try(fm4 <- nls(y ~ exp(outer(x,-c(b2, b4, b6))), data = Lanczos3, trace = TRUE, algorithm = "plinear", start = c(b2 = 0.7, b4 = 4.2, b6 = 6.3))) ## Use analytic derivatives Lanczos <- deriv(~ b1*exp(-b2*x) + b3*exp(-b4*x) + b5*exp(-b6*x), paste("b", 1:6, sep = ""), function(x, b1, b2, b3, b4, b5, b6){}) Try(fm5 <- nls(y ~ Lanczos(x, b1, b2, b3, b4, b5, b6), data = Lanczos3, trace = TRUE, start = c(b1 = 1.2, b2 = 0.3, b3 = 5.6, b4 = 5.5, b5 = 6.5, b6 = 7.6))) Try(fm5a <- nls(y ~ Lanczos(x, b1, b2, b3, b4, b5, b6), data = Lanczos3, trace = TRUE, alg = "port", start = c(b1 = 1.2, b2 = 0.3, b3 = 5.6, b4 = 5.5, b5 = 6.5, b6 = 7.6))) Try(fm6 <- nls(y ~ Lanczos(x, b1, b2, b3, b4, b5, b6), data = Lanczos3, trace = TRUE, start = c(b1 = 0.5, b2 = 0.7, b3 = 3.6, b4 = 4.2, b5 = 4, b6 = 6.3))) Try(fm6a <- nls(y ~ Lanczos(x, b1, b2, b3, b4, b5, b6), data = Lanczos3, trace = TRUE, alg = "port", start = c(b1 = 0.5, b2 = 0.7, b3 = 3.6, b4 = 4.2, b5 = 4, b6 = 6.3)))
The MGH09 data frame has 11 rows and 2 columns giving
This data frame contains the following columns:
A numeric vector of response values.
A numeric vector of input values.
This problem was found to be difficult for some very good algorithms. There is a local minimum at (+inf, -14.07..., -inf, -inf) with final sum of squares 0.00102734....
See More, J. J., Garbow, B. S., and Hillstrom, K. E. (1981). Testing unconstrained optimization software. ACM Transactions on Mathematical Software. 7(1): pp. 17–41.
Kowalik, J.S., and M. R. Osborne, (1978). Methods for Unconstrained Optimization Problems. New York, NY: Elsevier North-Holland.
Try <- function(expr) if (!inherits(val <- try(expr), "try-error")) val plot(y ~ x, data = MGH09) ## starting values for this attempt are ridiculous Try(fm1 <- nls(y ~ b1*(x**2+x*b2) / (x**2+x*b3+b4), data = MGH09, trace = TRUE, start = c(b1 = 25, b2 = 39, b3 = 41.5, b4 = 39))) Try(fm1a <- nls(y ~ b1*(x**2+x*b2) / (x**2+x*b3+b4), data = MGH09, trace = TRUE, alg = "port", start = c(b1 = 25, b2 = 39, b3 = 41.5, b4 = 39))) Try(fm2 <- nls(y ~ b1*(x**2+x*b2) / (x**2+x*b3+b4), data = MGH09, trace = TRUE, start = c(b1 = 0.25, b2 = 0.39, b3 = 0.415, b4 = 0.39))) Try(fm2a <- nls(y ~ b1*(x**2+x*b2) / (x**2+x*b3+b4), data = MGH09, trace = TRUE, alg = "port", start = c(b1 = 0.25, b2 = 0.39, b3 = 0.415, b4 = 0.39))) Try(fm3 <- nls(y ~ cbind(x, x**2) / (x**2+x*b3+b4), data = MGH09, trace = TRUE, algorithm = "plinear", start = c(b3 = 41.5, b4 = 39))) Try(fm4 <- nls(y ~ cbind(x, x**2) / (x**2+x*b3+b4), data = MGH09, trace = TRUE, algorithm = "plinear", start = c(b3 = 0.415, b4 = 0.39)))Try <- function(expr) if (!inherits(val <- try(expr), "try-error")) val plot(y ~ x, data = MGH09) ## starting values for this attempt are ridiculous Try(fm1 <- nls(y ~ b1*(x**2+x*b2) / (x**2+x*b3+b4), data = MGH09, trace = TRUE, start = c(b1 = 25, b2 = 39, b3 = 41.5, b4 = 39))) Try(fm1a <- nls(y ~ b1*(x**2+x*b2) / (x**2+x*b3+b4), data = MGH09, trace = TRUE, alg = "port", start = c(b1 = 25, b2 = 39, b3 = 41.5, b4 = 39))) Try(fm2 <- nls(y ~ b1*(x**2+x*b2) / (x**2+x*b3+b4), data = MGH09, trace = TRUE, start = c(b1 = 0.25, b2 = 0.39, b3 = 0.415, b4 = 0.39))) Try(fm2a <- nls(y ~ b1*(x**2+x*b2) / (x**2+x*b3+b4), data = MGH09, trace = TRUE, alg = "port", start = c(b1 = 0.25, b2 = 0.39, b3 = 0.415, b4 = 0.39))) Try(fm3 <- nls(y ~ cbind(x, x**2) / (x**2+x*b3+b4), data = MGH09, trace = TRUE, algorithm = "plinear", start = c(b3 = 41.5, b4 = 39))) Try(fm4 <- nls(y ~ cbind(x, x**2) / (x**2+x*b3+b4), data = MGH09, trace = TRUE, algorithm = "plinear", start = c(b3 = 0.415, b4 = 0.39)))
The MGH10 data frame has 16 rows and 2 columns.
This data frame contains the following columns:
A numeric vector of response values.
A numeric vector of input values.
This problem was found to be difficult for some very good algorithms.
See More, J. J., Garbow, B. S., and Hillstrom, K. E. (1981). Testing unconstrained optimization software. ACM Transactions on Mathematical Software. 7(1): pp. 17-41.
Meyer, R. R. (1970). Theoretical and computational aspects of nonlinear regression. In Nonlinear Programming, Rosen, Mangasarian and Ritter (Eds). New York, NY: Academic Press, pp. 465-486.
Try <- function(expr) if (!inherits(val <- try(expr), "try-error")) val plot(y ~ x, data = MGH10) ## check plot on log scale for shape plot(y ~ x, data = MGH10, log = "y") ## starting values for this run are ridiculous Try(fm1 <- nls(y ~ b1 * exp(b2/(x+b3)), data = MGH10, trace = TRUE, start = c(b1 = 2, b2 = 400000, b3 = 25000))) Try(fm1a <- nls(y ~ b1 * exp(b2/(x+b3)), data = MGH10, trace = TRUE, alg = "port", start = c(b1 = 2, b2 = 400000, b3 = 25000))) Try(fm2 <- nls(y ~ b1 * exp(b2/(x+b3)), data = MGH10, trace = TRUE, start = c(b1 = 0.02, b2 = 4000, b3 = 250))) Try(fm2a <- nls(y ~ b1 * exp(b2/(x+b3)), data = MGH10, trace = TRUE, alg = "port", start = c(b1 = 0.02, b2 = 4000, b3 = 250))) Try(fm3 <- nls(y ~ exp(b2/(x+b3)), data = MGH10, trace = TRUE, start = c(b2 = 400000, b3 = 25000), algorithm = "plinear")) Try(fm4 <- nls(y ~ exp(b2/(x+b3)), data = MGH10, trace = TRUE, start = c(b2 = 4000, b3 = 250), algorithm = "plinear"))Try <- function(expr) if (!inherits(val <- try(expr), "try-error")) val plot(y ~ x, data = MGH10) ## check plot on log scale for shape plot(y ~ x, data = MGH10, log = "y") ## starting values for this run are ridiculous Try(fm1 <- nls(y ~ b1 * exp(b2/(x+b3)), data = MGH10, trace = TRUE, start = c(b1 = 2, b2 = 400000, b3 = 25000))) Try(fm1a <- nls(y ~ b1 * exp(b2/(x+b3)), data = MGH10, trace = TRUE, alg = "port", start = c(b1 = 2, b2 = 400000, b3 = 25000))) Try(fm2 <- nls(y ~ b1 * exp(b2/(x+b3)), data = MGH10, trace = TRUE, start = c(b1 = 0.02, b2 = 4000, b3 = 250))) Try(fm2a <- nls(y ~ b1 * exp(b2/(x+b3)), data = MGH10, trace = TRUE, alg = "port", start = c(b1 = 0.02, b2 = 4000, b3 = 250))) Try(fm3 <- nls(y ~ exp(b2/(x+b3)), data = MGH10, trace = TRUE, start = c(b2 = 400000, b3 = 25000), algorithm = "plinear")) Try(fm4 <- nls(y ~ exp(b2/(x+b3)), data = MGH10, trace = TRUE, start = c(b2 = 4000, b3 = 250), algorithm = "plinear"))
The MGH17 data frame has 33 rows and 2 columns
This data frame contains the following columns:
A numeric vector of response values.
A numeric vector of input values.
This problem was found to be difficult for some very good algorithms.
See More, J. J., Garbow, B. S., and Hillstrom, K. E. (1981). Testing unconstrained optimization software. ACM Transactions on Mathematical Software. 7(1): pp. 17-41.
Osborne, M. R. (1972). Some aspects of nonlinear least squares calculations. In Numerical Methods for Nonlinear Optimization, Lootsma (Ed). New York, NY: Academic Press, pp. 171-189.
Try <- function(expr) if (!inherits(val <- try(expr), "try-error")) val plot(y ~ x, data = MGH17) ## Starting values here are ridiculous Try(fm1 <- nls(y ~ b1 + b2*exp(-x*b4) + b3*exp(-x*b5), data = MGH17, trace = TRUE, start = c(b1 = 50, b2 = 150, b3 = -100, b4 = 1, b5 = 2))) Try(fm1a <- nls(y ~ b1 + b2*exp(-x*b4) + b3*exp(-x*b5), data = MGH17, trace = TRUE, alg = "port", start = c(b1 = 50, b2 = 150, b3 = -100, b4 = 1, b5 = 2))) Try(fm2 <- nls(y ~ b1 + b2*exp(-x*b4) + b3*exp(-x*b5), data = MGH17, trace = TRUE, start = c(b1 = 0.5, b2 = 1.5, b3 = -1, b4 = 0.01, b5 = 0.02))) Try(fm2a <- nls(y ~ b1 + b2*exp(-x*b4) + b3*exp(-x*b5), data = MGH17, trace = TRUE, alg = "port", start = c(b1 = 0.5, b2 = 1.5, b3 = -1, b4 = 0.01, b5 = 0.02))) Try(fm3 <- nls(y ~ cbind(1, exp(-x*b4), exp(-x*b5)), data = MGH17, trace = TRUE, algorithm = "plinear", start = c(b4 = 1, b5 = 2))) Try(fm4 <- nls(y ~ cbind(1, exp(-x*b4), exp(-x*b5)), data = MGH17, trace = TRUE, algorithm = "plinear", start = c(b4 = 0.01, b5 = 0.02)))Try <- function(expr) if (!inherits(val <- try(expr), "try-error")) val plot(y ~ x, data = MGH17) ## Starting values here are ridiculous Try(fm1 <- nls(y ~ b1 + b2*exp(-x*b4) + b3*exp(-x*b5), data = MGH17, trace = TRUE, start = c(b1 = 50, b2 = 150, b3 = -100, b4 = 1, b5 = 2))) Try(fm1a <- nls(y ~ b1 + b2*exp(-x*b4) + b3*exp(-x*b5), data = MGH17, trace = TRUE, alg = "port", start = c(b1 = 50, b2 = 150, b3 = -100, b4 = 1, b5 = 2))) Try(fm2 <- nls(y ~ b1 + b2*exp(-x*b4) + b3*exp(-x*b5), data = MGH17, trace = TRUE, start = c(b1 = 0.5, b2 = 1.5, b3 = -1, b4 = 0.01, b5 = 0.02))) Try(fm2a <- nls(y ~ b1 + b2*exp(-x*b4) + b3*exp(-x*b5), data = MGH17, trace = TRUE, alg = "port", start = c(b1 = 0.5, b2 = 1.5, b3 = -1, b4 = 0.01, b5 = 0.02))) Try(fm3 <- nls(y ~ cbind(1, exp(-x*b4), exp(-x*b5)), data = MGH17, trace = TRUE, algorithm = "plinear", start = c(b4 = 1, b5 = 2))) Try(fm4 <- nls(y ~ cbind(1, exp(-x*b4), exp(-x*b5)), data = MGH17, trace = TRUE, algorithm = "plinear", start = c(b4 = 0.01, b5 = 0.02)))
The Misra1a data frame has 14 rows and 2 columns.
This data frame contains the following columns:
A numeric vector of volume values.
A numeric vector of pressure values.
These data are the result of a NIST study regarding dental research in monomolecular adsorption. The response variable is volume, and the predictor variable is pressure.
Misra, D., NIST (1978). Dental Research Monomolecular Adsorption Study.
Try <- function(expr) if (!inherits(val <- try(expr), "try-error")) val plot(y ~ x, data = Misra1a) Try(fm1 <- nls(y ~ b1*(1-exp(-b2*x)), data = Misra1a, trace = TRUE, start = c(b1 = 500, b2 = 0.0001) )) Try(fm1a <- nls(y ~ b1*(1-exp(-b2*x)), data = Misra1a, trace = TRUE, alg = "port", start = c(b1 = 500, b2 = 0.0001) )) Try(fm2 <- nls(y ~ b1*(1-exp(-b2*x)), data = Misra1a, trace = TRUE, start = c(b1 = 250, b2 = 0.0005) )) Try(fm2a <- nls(y ~ b1*(1-exp(-b2*x)), data = Misra1a, trace = TRUE, alg = "port", start = c(b1 = 250, b2 = 0.0005) )) Try(fm3 <- nls(y ~ 1-exp(-b2*x), data = Misra1a, trace = TRUE, start = c(b2 = 0.0001), algorithm = "plinear" )) Try(fm4 <- nls(y ~ 1-exp(-b2*x), data = Misra1a, trace = TRUE, start = c(b2 = 0.0005), algorithm = "plinear" )) ## Using a self-starting model Try(fm5 <- nls(y ~ SSasympOrig(x, Asym, lrc), data = Misra1a))Try <- function(expr) if (!inherits(val <- try(expr), "try-error")) val plot(y ~ x, data = Misra1a) Try(fm1 <- nls(y ~ b1*(1-exp(-b2*x)), data = Misra1a, trace = TRUE, start = c(b1 = 500, b2 = 0.0001) )) Try(fm1a <- nls(y ~ b1*(1-exp(-b2*x)), data = Misra1a, trace = TRUE, alg = "port", start = c(b1 = 500, b2 = 0.0001) )) Try(fm2 <- nls(y ~ b1*(1-exp(-b2*x)), data = Misra1a, trace = TRUE, start = c(b1 = 250, b2 = 0.0005) )) Try(fm2a <- nls(y ~ b1*(1-exp(-b2*x)), data = Misra1a, trace = TRUE, alg = "port", start = c(b1 = 250, b2 = 0.0005) )) Try(fm3 <- nls(y ~ 1-exp(-b2*x), data = Misra1a, trace = TRUE, start = c(b2 = 0.0001), algorithm = "plinear" )) Try(fm4 <- nls(y ~ 1-exp(-b2*x), data = Misra1a, trace = TRUE, start = c(b2 = 0.0005), algorithm = "plinear" )) ## Using a self-starting model Try(fm5 <- nls(y ~ SSasympOrig(x, Asym, lrc), data = Misra1a))
The Misra1b data frame has 14 rows and 2 columns.
It is the same data as Misra1a but a different model is fit.
This data frame contains the following columns:
A numeric vector of volume values.
A numeric vector of pressure values.
These data are the result of a NIST study regarding dental research in monomolecular adsorption. The response variable is volume, and the predictor variable is pressure.
Misra, D., NIST (1978). Dental Research Monomolecular Adsorption Study.
Try <- function(expr) if (!inherits(val <- try(expr), "try-error")) val plot(y ~ x, data = Misra1b) Try(fm1 <- nls(y ~ b1 * (1-(1+b2*x/2)**(-2)), data = Misra1b, trace = TRUE, start = c(b1 = 500, b2 = 0.0001) )) Try(fm1a <- nls(y ~ b1 * (1-(1+b2*x/2)**(-2)), data = Misra1b, trace = TRUE, alg = "port", start = c(b1 = 500, b2 = 0.0001) )) Try(fm2 <- nls(y ~ b1 * (1-(1+b2*x/2)**(-2)), data = Misra1b, trace = TRUE, start = c(b1 = 300, b2 = 0.0002) )) Try(fm2a <- nls(y ~ b1 * (1-(1+b2*x/2)**(-2)), data = Misra1b, trace = TRUE, alg = "port", start = c(b1 = 300, b2 = 0.0002) )) Try(fm3 <- nls(y ~ 1-(1+b2*x/2)**(-2), data = Misra1b, trace = TRUE, start = c(b2 = 0.0001), algorithm = "plinear" )) Try(fm4 <- nls(y ~ 1-(1+b2*x/2)**(-2), data = Misra1b, trace = TRUE, start = c(b2 = 0.0005), algorithm = "plinear" ))Try <- function(expr) if (!inherits(val <- try(expr), "try-error")) val plot(y ~ x, data = Misra1b) Try(fm1 <- nls(y ~ b1 * (1-(1+b2*x/2)**(-2)), data = Misra1b, trace = TRUE, start = c(b1 = 500, b2 = 0.0001) )) Try(fm1a <- nls(y ~ b1 * (1-(1+b2*x/2)**(-2)), data = Misra1b, trace = TRUE, alg = "port", start = c(b1 = 500, b2 = 0.0001) )) Try(fm2 <- nls(y ~ b1 * (1-(1+b2*x/2)**(-2)), data = Misra1b, trace = TRUE, start = c(b1 = 300, b2 = 0.0002) )) Try(fm2a <- nls(y ~ b1 * (1-(1+b2*x/2)**(-2)), data = Misra1b, trace = TRUE, alg = "port", start = c(b1 = 300, b2 = 0.0002) )) Try(fm3 <- nls(y ~ 1-(1+b2*x/2)**(-2), data = Misra1b, trace = TRUE, start = c(b2 = 0.0001), algorithm = "plinear" )) Try(fm4 <- nls(y ~ 1-(1+b2*x/2)**(-2), data = Misra1b, trace = TRUE, start = c(b2 = 0.0005), algorithm = "plinear" ))
The Misra1c data frame has 14 rows and 2 columns.
This is the same data as Misra1a but a different model is fit.
This data frame contains the following columns:
A numeric vector of volume values.
A numeric vector of pressure values.
These data are the result of a NIST study regarding dental research in monomolecular adsorption. The response variable is volume, and the predictor variable is pressure.
Misra, D., NIST (1978). Dental Research Monomolecular Adsorption Study.
Try <- function(expr) if (!inherits(val <- try(expr), "try-error")) val plot(y ~ x, data = Misra1c) Try(fm1 <- nls(y ~ b1*(1-(1+2*b2*x)**(-.5)), data = Misra1c, trace = TRUE, start = c(b1 = 500, b2 = 0.0001) )) Try(fm1a <- nls(y ~ b1*(1-(1+2*b2*x)**(-.5)), data = Misra1c, trace = TRUE, alg = "port", start = c(b1 = 500, b2 = 0.0001) )) Try(fm2 <- nls(y ~ b1*(1-(1+2*b2*x)**(-.5)), data = Misra1c, trace = TRUE, start = c(b1 = 600, b2 = 0.0002) )) Try(fm2a <- nls(y ~ b1*(1-(1+2*b2*x)**(-.5)), data = Misra1c, trace = TRUE, alg = "port", start = c(b1 = 600, b2 = 0.0002) )) Try(fm3 <- nls(y ~ 1-(1+2*b2*x)**(-.5), data = Misra1c, trace = TRUE, start = c(b2 = 0.0001), algorithm = "plinear" )) Try(fm4 <- nls(y ~ 1-(1+2*b2*x)**(-.5), data = Misra1c, trace = TRUE, start = c(b2 = 0.0002), algorithm = "plinear" ))Try <- function(expr) if (!inherits(val <- try(expr), "try-error")) val plot(y ~ x, data = Misra1c) Try(fm1 <- nls(y ~ b1*(1-(1+2*b2*x)**(-.5)), data = Misra1c, trace = TRUE, start = c(b1 = 500, b2 = 0.0001) )) Try(fm1a <- nls(y ~ b1*(1-(1+2*b2*x)**(-.5)), data = Misra1c, trace = TRUE, alg = "port", start = c(b1 = 500, b2 = 0.0001) )) Try(fm2 <- nls(y ~ b1*(1-(1+2*b2*x)**(-.5)), data = Misra1c, trace = TRUE, start = c(b1 = 600, b2 = 0.0002) )) Try(fm2a <- nls(y ~ b1*(1-(1+2*b2*x)**(-.5)), data = Misra1c, trace = TRUE, alg = "port", start = c(b1 = 600, b2 = 0.0002) )) Try(fm3 <- nls(y ~ 1-(1+2*b2*x)**(-.5), data = Misra1c, trace = TRUE, start = c(b2 = 0.0001), algorithm = "plinear" )) Try(fm4 <- nls(y ~ 1-(1+2*b2*x)**(-.5), data = Misra1c, trace = TRUE, start = c(b2 = 0.0002), algorithm = "plinear" ))
The Misra1d data frame has 14 rows and 2 columns.
This is the same data as Misra1a but a different model is fit.
This data frame contains the following columns:
A numeric vector of volume values.
A numeric vector of pressure values.
These data are the result of a NIST study regarding dental research in monomolecular adsorption. The response variable is volume, and the predictor variable is pressure.
Misra, D., NIST (1978). Dental Research Monomolecular Adsorption Study.
Try <- function(expr) if (!inherits(val <- try(expr), "try-error")) val plot(y ~ x, data = Misra1d) Try(fm1 <- nls(y ~ b1*b2*x*((1+b2*x)**(-1)), data = Misra1d, trace = TRUE, start = c(b1 = 500, b2 = 0.0001) )) Try(fm1a <- nls(y ~ b1*b2*x*((1+b2*x)**(-1)), data = Misra1d, trace = TRUE, alg = "port", start = c(b1 = 500, b2 = 0.0001) )) Try(fm2 <- nls(y ~ b1*b2*x*((1+b2*x)**(-1)), data = Misra1d, trace = TRUE, start = c(b1 = 450, b2 = 0.0003) )) Try(fm2a <- nls(y ~ b1*b2*x*((1+b2*x)**(-1)), data = Misra1d, trace = TRUE, alg = "port", start = c(b1 = 450, b2 = 0.0003) )) Try(fm3 <- nls(y ~ b2*x*((1+b2*x)**(-1)), data = Misra1d, trace = TRUE, start = c(b2 = 0.0001), algorithm = "plinear" )) Try(fm4 <- nls(y ~ b2*x*((1+b2*x)**(-1)), data = Misra1d, trace = TRUE, start = c(b2 = 0.0005), algorithm = "plinear" ))Try <- function(expr) if (!inherits(val <- try(expr), "try-error")) val plot(y ~ x, data = Misra1d) Try(fm1 <- nls(y ~ b1*b2*x*((1+b2*x)**(-1)), data = Misra1d, trace = TRUE, start = c(b1 = 500, b2 = 0.0001) )) Try(fm1a <- nls(y ~ b1*b2*x*((1+b2*x)**(-1)), data = Misra1d, trace = TRUE, alg = "port", start = c(b1 = 500, b2 = 0.0001) )) Try(fm2 <- nls(y ~ b1*b2*x*((1+b2*x)**(-1)), data = Misra1d, trace = TRUE, start = c(b1 = 450, b2 = 0.0003) )) Try(fm2a <- nls(y ~ b1*b2*x*((1+b2*x)**(-1)), data = Misra1d, trace = TRUE, alg = "port", start = c(b1 = 450, b2 = 0.0003) )) Try(fm3 <- nls(y ~ b2*x*((1+b2*x)**(-1)), data = Misra1d, trace = TRUE, start = c(b2 = 0.0001), algorithm = "plinear" )) Try(fm4 <- nls(y ~ b2*x*((1+b2*x)**(-1)), data = Misra1d, trace = TRUE, start = c(b2 = 0.0005), algorithm = "plinear" ))
The Nelson data frame has 128 rows and 3 columns of data from an
accelerated test of dialectric breakdown.
This data frame contains the following columns:
A numeric vector of dialectric breakdown strength values.
A numeric vector of time values.
A numeric vector of temperature values.
These data are the result of a study involving the analysis of performance degradation data from accelerated tests, published in IEEE Transactions on Reliability. The response variable is dialectric breakdown strength in kilo-volts, and the predictor variables are time in weeks and temperature in degrees Celsius.
Nelson, W. (1981). Analysis of Performance-Degradation Data. IEEE Transactions on Reliability. Vol. 2, R-30, No. 2, pp. 149-155.
Try <- function(expr) if (!inherits(val <- try(expr), "try-error")) val plot(y ~ x1, data = Nelson, log = "y") plot(y ~ x2, data = Nelson, log = "y") coplot(y ~ x1 | x2, data = Nelson) coplot(y ~ x2 | x1, data = Nelson) Try(fm1 <- nls(log(y) ~ b1 - b2*x1 * exp(-b3*x2), data = Nelson, start = c(b1 = 2, b2 = 0.0001, b3 = -0.01), trace = TRUE)) Try(fm1a <- nls(log(y) ~ b1 - b2*x1 * exp(-b3*x2), data = Nelson, trace = TRUE, alg = "port", start = c(b1 = 2, b2 = 0.0001, b3 = -0.01))) Try(fm2 <- nls(log(y) ~ b1 - b2*x1 * exp(-b3*x2), data = Nelson, start = c(b1 = 2.5, b2 = 0.000000005, b3 = -0.05), trace = TRUE)) Try(fm2 <- nls(log(y) ~ b1 - b2*x1 * exp(-b3*x2), data = Nelson, trace = TRUE, alg = "port", start = c(b1 = 2.5, b2 = 0.000000005, b3 = -0.05))) Try(fm3 <- nls(log(y) ~ cbind(1, -x1 * exp(-b3*x2)), data = Nelson, start = c(b3 = -0.01), trace = TRUE, algorithm = "plinear")) Try(fm4 <- nls(log(y) ~ cbind(1, -x1 * exp(-b3*x2)), data = Nelson, start = c(b3 = -0.05), trace = TRUE, algorithm = "plinear"))Try <- function(expr) if (!inherits(val <- try(expr), "try-error")) val plot(y ~ x1, data = Nelson, log = "y") plot(y ~ x2, data = Nelson, log = "y") coplot(y ~ x1 | x2, data = Nelson) coplot(y ~ x2 | x1, data = Nelson) Try(fm1 <- nls(log(y) ~ b1 - b2*x1 * exp(-b3*x2), data = Nelson, start = c(b1 = 2, b2 = 0.0001, b3 = -0.01), trace = TRUE)) Try(fm1a <- nls(log(y) ~ b1 - b2*x1 * exp(-b3*x2), data = Nelson, trace = TRUE, alg = "port", start = c(b1 = 2, b2 = 0.0001, b3 = -0.01))) Try(fm2 <- nls(log(y) ~ b1 - b2*x1 * exp(-b3*x2), data = Nelson, start = c(b1 = 2.5, b2 = 0.000000005, b3 = -0.05), trace = TRUE)) Try(fm2 <- nls(log(y) ~ b1 - b2*x1 * exp(-b3*x2), data = Nelson, trace = TRUE, alg = "port", start = c(b1 = 2.5, b2 = 0.000000005, b3 = -0.05))) Try(fm3 <- nls(log(y) ~ cbind(1, -x1 * exp(-b3*x2)), data = Nelson, start = c(b3 = -0.01), trace = TRUE, algorithm = "plinear")) Try(fm4 <- nls(log(y) ~ cbind(1, -x1 * exp(-b3*x2)), data = Nelson, start = c(b3 = -0.05), trace = TRUE, algorithm = "plinear"))
The Ratkowsky2 data frame has 9 rows and 2 columns.
This data frame contains the following columns:
A numeric vector of pasture yields.
A numeric vector of growing times.
This model and data are an example of fitting sigmoidal growth curves taken from Ratkowsky (1983). The response variable is pasture yield, and the predictor variable is growing time.
Ratkowsky, D.A. (1983). Nonlinear Regression Modeling. New York, NY: Marcel Dekker, pp. 61 and 88.
Try <- function(expr) if (!inherits(val <- try(expr), "try-error")) val plot(y ~ x, data = Ratkowsky2) Try(fm1 <- nls(y ~ b1 / (1+exp(b2-b3*x)), data = Ratkowsky2, trace = TRUE, start = c(b1 = 100, b2 = 1, b3 = 0.1))) Try(fm1a <- nls(y ~ b1 / (1+exp(b2-b3*x)), data = Ratkowsky2, trace = TRUE, alg = "port", start = c(b1 = 100, b2 = 1, b3 = 0.1))) Try(fm2 <- nls(y ~ b1 / (1+exp(b2-b3*x)), data = Ratkowsky2, trace = TRUE, start = c(b1 = 75, b2 = 2.5, b3 = 0.07))) Try(fm2a <- nls(y ~ b1 / (1+exp(b2-b3*x)), data = Ratkowsky2, trace = TRUE, alg = "port", start = c(b1 = 75, b2 = 2.5, b3 = 0.07))) Try(fm3 <- nls(y ~ 1 / (1+exp(b2-b3*x)), data = Ratkowsky2, trace = TRUE, start = c(b2 = 1, b3 = 0.1), alg = "plinear")) Try(fm4 <- nls(y ~ 1 / (1+exp(b2-b3*x)), data = Ratkowsky2, trace = TRUE, start = c(b2 = 2.5, b3 = 0.07), alg = "plinear")) ## Using a self-starting model Try(fm5 <- nls(y ~ SSlogis(x, Asym, xmid, scal), data = Ratkowsky2)) summary(fm5)Try <- function(expr) if (!inherits(val <- try(expr), "try-error")) val plot(y ~ x, data = Ratkowsky2) Try(fm1 <- nls(y ~ b1 / (1+exp(b2-b3*x)), data = Ratkowsky2, trace = TRUE, start = c(b1 = 100, b2 = 1, b3 = 0.1))) Try(fm1a <- nls(y ~ b1 / (1+exp(b2-b3*x)), data = Ratkowsky2, trace = TRUE, alg = "port", start = c(b1 = 100, b2 = 1, b3 = 0.1))) Try(fm2 <- nls(y ~ b1 / (1+exp(b2-b3*x)), data = Ratkowsky2, trace = TRUE, start = c(b1 = 75, b2 = 2.5, b3 = 0.07))) Try(fm2a <- nls(y ~ b1 / (1+exp(b2-b3*x)), data = Ratkowsky2, trace = TRUE, alg = "port", start = c(b1 = 75, b2 = 2.5, b3 = 0.07))) Try(fm3 <- nls(y ~ 1 / (1+exp(b2-b3*x)), data = Ratkowsky2, trace = TRUE, start = c(b2 = 1, b3 = 0.1), alg = "plinear")) Try(fm4 <- nls(y ~ 1 / (1+exp(b2-b3*x)), data = Ratkowsky2, trace = TRUE, start = c(b2 = 2.5, b3 = 0.07), alg = "plinear")) ## Using a self-starting model Try(fm5 <- nls(y ~ SSlogis(x, Asym, xmid, scal), data = Ratkowsky2)) summary(fm5)
The Ratkowsky3 data frame has 15 rows and 2 columns.
This data frame contains the following columns:
A numeric vector of dry weights of onion bulbs and tops.
A numeric vector of growing times.
This model and data are an example of fitting sigmoidal growth curves taken from Ratkowsky (1983). The response variable is the dry weight of onion bulbs and tops, and the predictor variable is growing time.
Ratkowsky, D.A. (1983). Nonlinear Regression Modeling. New York, NY: Marcel Dekker, pp. 62 and 88.
Try <- function(expr) if (!inherits(val <- try(expr), "try-error")) val plot(y ~ x, data = Ratkowsky3) ## causes NA/NaN/Inf error Try(fm1 <- nls(y ~ b1 / ((1+exp(b2-b3*x))**(1/b4)), data = Ratkowsky3, start = c(b1 = 100, b2 = 10, b3 = 1, b4 = 1), trace = TRUE)) Try(fm1a <- nls(y ~ b1 / ((1+exp(b2-b3*x))**(1/b4)), data = Ratkowsky3, start = c(b1 = 100, b2 = 10, b3 = 1, b4 = 1), alg = "port", trace = TRUE)) Try(fm2 <- nls(y ~ b1 / ((1+exp(b2-b3*x))**(1/b4)), data = Ratkowsky3, start = c(b1 = 700, b2 = 5, b3 = 0.75, b4 = 1.3), trace = TRUE)) Try(fm2a <- nls(y ~ b1 / ((1+exp(b2-b3*x))**(1/b4)), data = Ratkowsky3, start = c(b1 = 700, b2 = 5, b3 = 0.75, b4 = 1.3), alg = "port", trace = TRUE)) Try(fm3 <- nls(y ~ 1 / ((1+exp(b2-b3*x))**(1/b4)), data = Ratkowsky3, start = c(b2 = 10, b3 = 1, b4 = 1), algorithm = "plinear", trace = TRUE)) Try(fm4 <- nls(y ~ 1 / ((1+exp(b2-b3*x))**(1/b4)), data = Ratkowsky3, start = c(b2 = 5, b3 = 0.75, b4 = 1.3), algorithm = "plinear", trace = TRUE))Try <- function(expr) if (!inherits(val <- try(expr), "try-error")) val plot(y ~ x, data = Ratkowsky3) ## causes NA/NaN/Inf error Try(fm1 <- nls(y ~ b1 / ((1+exp(b2-b3*x))**(1/b4)), data = Ratkowsky3, start = c(b1 = 100, b2 = 10, b3 = 1, b4 = 1), trace = TRUE)) Try(fm1a <- nls(y ~ b1 / ((1+exp(b2-b3*x))**(1/b4)), data = Ratkowsky3, start = c(b1 = 100, b2 = 10, b3 = 1, b4 = 1), alg = "port", trace = TRUE)) Try(fm2 <- nls(y ~ b1 / ((1+exp(b2-b3*x))**(1/b4)), data = Ratkowsky3, start = c(b1 = 700, b2 = 5, b3 = 0.75, b4 = 1.3), trace = TRUE)) Try(fm2a <- nls(y ~ b1 / ((1+exp(b2-b3*x))**(1/b4)), data = Ratkowsky3, start = c(b1 = 700, b2 = 5, b3 = 0.75, b4 = 1.3), alg = "port", trace = TRUE)) Try(fm3 <- nls(y ~ 1 / ((1+exp(b2-b3*x))**(1/b4)), data = Ratkowsky3, start = c(b2 = 10, b3 = 1, b4 = 1), algorithm = "plinear", trace = TRUE)) Try(fm4 <- nls(y ~ 1 / ((1+exp(b2-b3*x))**(1/b4)), data = Ratkowsky3, start = c(b2 = 5, b3 = 0.75, b4 = 1.3), algorithm = "plinear", trace = TRUE))
The Roszman1 data frame has 25 rows and 2 columns of data on the
number of quantum defects in iodine atoms at different energy states.
This data frame contains the following columns:
A numeric vector of number of quantum defects.
A numeric vector of the excited energy state.
These data are the result of a NIST study involving quantum defects in iodine atoms. The response variable is the number of quantum defects, and the predictor variable is the excited energy state. The argument to the ARCTAN function is in radians.
Roszman, L., NIST (19??). Quantum Defects for Sulfur I Atom.
Try <- function(expr) if (!inherits(val <- try(expr), "try-error")) val plot(y ~ x, data = Roszman1) Try(fm1 <- nls(y ~ b1 - b2*x - atan(b3/(x-b4))/pi, data = Roszman1, start = c(b1 = 0.1, b2 = -0.00001, b3 = 1000, b4 = -100), trace = TRUE)) Try(fm1a <- nls(y ~ b1 - b2*x - atan(b3/(x-b4))/pi, data = Roszman1, start = c(b1 = 0.1, b2 = -0.00001, b3 = 1000, b4 = -100), alg = "port", trace = TRUE)) Try(fm2 <- nls(y ~ b1 - b2*x - atan(b3/(x-b4))/pi, data = Roszman1, start = c(b1 = 0.2, b2 = -0.0000015, b3 = 1200, b4 = -150), trace = TRUE)) Try(fm2a <- nls(y ~ b1 - b2*x - atan(b3/(x-b4))/pi, data = Roszman1, start = c(b1 = 0.2, b2 = -0.0000015, b3 = 1200, b4 = -150), alg = "port", trace = TRUE))Try <- function(expr) if (!inherits(val <- try(expr), "try-error")) val plot(y ~ x, data = Roszman1) Try(fm1 <- nls(y ~ b1 - b2*x - atan(b3/(x-b4))/pi, data = Roszman1, start = c(b1 = 0.1, b2 = -0.00001, b3 = 1000, b4 = -100), trace = TRUE)) Try(fm1a <- nls(y ~ b1 - b2*x - atan(b3/(x-b4))/pi, data = Roszman1, start = c(b1 = 0.1, b2 = -0.00001, b3 = 1000, b4 = -100), alg = "port", trace = TRUE)) Try(fm2 <- nls(y ~ b1 - b2*x - atan(b3/(x-b4))/pi, data = Roszman1, start = c(b1 = 0.2, b2 = -0.0000015, b3 = 1200, b4 = -150), trace = TRUE)) Try(fm2a <- nls(y ~ b1 - b2*x - atan(b3/(x-b4))/pi, data = Roszman1, start = c(b1 = 0.2, b2 = -0.0000015, b3 = 1200, b4 = -150), alg = "port", trace = TRUE))
The Thurber data frame has 37 rows and 2 columns.
This data frame contains the following columns:
A numeric vector of electron mobility values.
A numeric vector of logs of electron density values.
These data are the result of a NIST study involving semiconductor electron mobility. The response variable is a measure of electron mobility, and the predictor variable is the natural log of the density.
Thurber, R., NIST (197?). Semiconductor electron mobility modeling.
Try <- function(expr) if (!inherits(val <- try(expr), "try-error")) val plot(y ~ x, data = Thurber) Try(fm1 <- nls(y ~ (b1+x*(b2+x*(b3+b4*x))) / (1+x*(b5+x*(b6+x*b7))), data = Thurber, trace = TRUE, start = c(b1 = 1000, b2 = 1000, b3 = 400, b4 = 40, b5 = 0.7, b6 = 0.3, b7 = 0.03))) Try(fm1a <- nls(y ~ (b1+x*(b2+x*(b3+b4*x))) / (1+x*(b5+x*(b6+x*b7))), data = Thurber, trace = TRUE, alg = "port", start = c(b1 = 1000, b2 = 1000, b3 = 400, b4 = 40, b5 = 0.7, b6 = 0.3, b7 = 0.03))) Try(fm2 <- nls(y ~ (b1+x*(b2+x*(b3+b4*x))) / (1+x*(b5+x*(b6+x*b7))), data = Thurber, trace = TRUE, start = c(b1 = 1300, b2 = 1500, b3 = 500, b4 = 75, b5 = 1, b6 = 0.4, b7 = 0.05))) Try(fm2a <- nls(y ~ (b1+x*(b2+x*(b3+b4*x))) / (1+x*(b5+x*(b6+x*b7))), data = Thurber, trace = TRUE, alg = "port", start = c(b1 = 1300, b2 = 1500, b3 = 500, b4 = 75, b5 = 1, b6 = 0.4, b7 = 0.05))) Try(fm3 <- nls(y ~ outer(x, 0:3, "^")/(1+x*(b5+x*(b6+x*b7))), data = Thurber, trace = TRUE, start = c(b5 = 0.7, b6 = 0.3, b7 = 0.03), alg = "plinear")) Try(fm4 <- nls(y ~ outer(x, 0:3, "^")/(1+x*(b5+x*(b6+x*b7))), data = Thurber, trace = TRUE, start = c(b5 = 1, b6 = 0.4, b7 = 0.05), alg = "plinear"))Try <- function(expr) if (!inherits(val <- try(expr), "try-error")) val plot(y ~ x, data = Thurber) Try(fm1 <- nls(y ~ (b1+x*(b2+x*(b3+b4*x))) / (1+x*(b5+x*(b6+x*b7))), data = Thurber, trace = TRUE, start = c(b1 = 1000, b2 = 1000, b3 = 400, b4 = 40, b5 = 0.7, b6 = 0.3, b7 = 0.03))) Try(fm1a <- nls(y ~ (b1+x*(b2+x*(b3+b4*x))) / (1+x*(b5+x*(b6+x*b7))), data = Thurber, trace = TRUE, alg = "port", start = c(b1 = 1000, b2 = 1000, b3 = 400, b4 = 40, b5 = 0.7, b6 = 0.3, b7 = 0.03))) Try(fm2 <- nls(y ~ (b1+x*(b2+x*(b3+b4*x))) / (1+x*(b5+x*(b6+x*b7))), data = Thurber, trace = TRUE, start = c(b1 = 1300, b2 = 1500, b3 = 500, b4 = 75, b5 = 1, b6 = 0.4, b7 = 0.05))) Try(fm2a <- nls(y ~ (b1+x*(b2+x*(b3+b4*x))) / (1+x*(b5+x*(b6+x*b7))), data = Thurber, trace = TRUE, alg = "port", start = c(b1 = 1300, b2 = 1500, b3 = 500, b4 = 75, b5 = 1, b6 = 0.4, b7 = 0.05))) Try(fm3 <- nls(y ~ outer(x, 0:3, "^")/(1+x*(b5+x*(b6+x*b7))), data = Thurber, trace = TRUE, start = c(b5 = 0.7, b6 = 0.3, b7 = 0.03), alg = "plinear")) Try(fm4 <- nls(y ~ outer(x, 0:3, "^")/(1+x*(b5+x*(b6+x*b7))), data = Thurber, trace = TRUE, start = c(b5 = 1, b6 = 0.4, b7 = 0.05), alg = "plinear"))