Context
Last week, I was discussing about how to use nls() for a specific
model with one of my colleague and I ended creating a piece of code to
show what I was talking about! Even though there are many posts exploring nls()
in more depth that I did (for instance this post on datascienceplus by Lionel Herzog),
I thought I could share these lines of command here!
Basically, we were talking about a model where the temperature (\(T\)) follows
a saturation curve starting from 10°C at t=0 (so T(0) = 10) and plateauing at \(T_{\inf}\).
$$T(t) = T_{\inf} - (T_{\inf} - T_0)\exp(-kt)$$
Goal and data
The goal here is to use nls() (Nonlinear Least Squares) to find \(k\) and \(T_{inf}\).
For the sack of clarity, I simulate the data, i.e. I use the saturation curve
with known parameter values, then I add some noise (here a white noise):
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library(magrittr)
# Parameters
## known
T0 = 10
## the ones we are looking for
k = 0.2
Tinf = 20
# Simulate data
## time
seqt <- seq(0, 50, .25)
## create a data frame
simdata <- cbind(
seqt = seqt,
sim = Tinf - (Tinf- T0) * exp(-k * seqt) + .5 * rnorm(length(seqt))
) %>% as.data.frame
head(simdata)
#R> seqt sim
#R> 1 0.00 9.303692
#R> 2 0.25 10.745711
#R> 3 0.50 11.112195
#R> 4 0.75 12.195986
#R> 5 1.00 11.064262
#R> 6 1.25 11.510062
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Use nls()
Now I call nls() to fit the data:
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res <- nls(sim ~ Tinf - (Tinf - 10)*exp(-k*seqt), simdata, list(Tinf = 1, k = .1))
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All the information needed are stored in res and display via the print method:
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res
#R> Nonlinear regression model
#R> model: sim ~ Tinf - (Tinf - 10) * exp(-k * seqt)
#R> data: simdata
#R> Tinf k
#R> 20.0709 0.1938
#R> residual sum-of-squares: 46.45
#R>
#R> Number of iterations to convergence: 8
#R> Achieved convergence tolerance: 2.562e-07
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Let’s draw a quick plot:
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## get the coefficients values
cr <- coef(res)
fitC <- function(x) cr[1] - (cr[1] - 10)*exp(-cr[2]*x)
plot(simdata[,1], simdata[,2], xlab = "Time (min)", ylab = "Temperature (°C)")
curve(fitC, 0, 50, add = TRUE, col = "#2f85a4", lwd = 3)
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That’s all folks!
Display information relative to the R session used to render this post.
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sessionInfo()
#R> R version 4.5.1 (2025-06-13)
#R> Platform: x86_64-pc-linux-gnu
#R> Running under: Ubuntu 24.04.2 LTS
#R>
#R> Matrix products: default
#R> BLAS: /usr/lib/x86_64-linux-gnu/openblas-pthread/libblas.so.3
#R> LAPACK: /usr/lib/x86_64-linux-gnu/openblas-pthread/libopenblasp-r0.3.26.so; LAPACK version 3.12.0
#R>
#R> locale:
#R> [1] LC_CTYPE=C.UTF-8 LC_NUMERIC=C LC_TIME=C.UTF-8 LC_COLLATE=C.UTF-8
#R> [5] LC_MONETARY=C.UTF-8 LC_MESSAGES=C.UTF-8 LC_PAPER=C.UTF-8 LC_NAME=C
#R> [9] LC_ADDRESS=C LC_TELEPHONE=C LC_MEASUREMENT=C.UTF-8 LC_IDENTIFICATION=C
#R>
#R> time zone: UTC
#R> tzcode source: system (glibc)
#R>
#R> attached base packages:
#R> [1] stats graphics grDevices utils datasets methods base
#R>
#R> other attached packages:
#R> [1] magrittr_2.0.3 inSilecoRef_0.1.1
#R>
#R> loaded via a namespace (and not attached):
#R> [1] sass_0.4.10 generics_0.1.4 xml2_1.3.8 blogdown_1.21 stringi_1.8.7
#R> [6] httpcode_0.3.0 digest_0.6.37 evaluate_1.0.4 bookdown_0.43 fastmap_1.2.0
#R> [11] plyr_1.8.9 jsonlite_2.0.0 backports_1.5.0 crul_1.5.0 promises_1.3.3
#R> [16] bibtex_0.5.1 jquerylib_0.1.4 cli_3.6.5 shiny_1.11.1 rlang_1.1.6
#R> [21] cachem_1.1.0 yaml_2.3.10 tools_4.5.1 dplyr_1.1.4 httpuv_1.6.16
#R> [26] DT_0.33 rcrossref_1.2.0 curl_6.4.0 vctrs_0.6.5 R6_2.6.1
#R> [31] mime_0.13 lifecycle_1.0.4 stringr_1.5.1 fs_1.6.6 htmlwidgets_1.6.4
#R> [36] miniUI_0.1.2 pkgconfig_2.0.3 pillar_1.11.0 bslib_0.9.0 later_1.4.2
#R> [41] glue_1.8.0 Rcpp_1.1.0 xfun_0.52 tibble_3.3.0 tidyselect_1.2.1
#R> [46] knitr_1.50 xtable_1.8-4 htmltools_0.5.8.1 rmarkdown_2.29 compiler_4.5.1
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