R Shiny Application: Nonlinear Exponential Fit

Introduction

• Exponential decay is a simple mathematical model yet very effective in modeling real-world signal behavior in many areas of Natural Science such as Radioactivity, Vibrations, Chemical Reactions, Geophysics, and etc.

• This simple Shiny App was developed for users who want to learn about the exponential decay model, its key parameters, and how the key parameters alter the signal.

• Additionally, the Shiny App enables the users to show prediction and its 95% confidence interval estimated from the nonlinear regression model.

Exponential Decay Model

Normalized signal equation is written in the following form:

\[S(t) = exp(-\frac{t}{\lambda}) + \epsilon\]

where \(S(t)\) is the normalized signal intensity (a.u.), \(t\) is time (s), \(\lambda\) is the Decaying Time Constant (s), and \(\epsilon\) is random noise.

Simulated Data

set.seed(1234567); t<-seq(1,300); lambda<-50 # true lambda
s<-exp(-t/lambda)+0.1*rnorm(length(t)); par(mar=c(4,4,0,0))
plot(t,s, xlab="Time (s)", ylab = "Signal Intensity (a.u.)")
legend("topright", legend=c("Data"), pch = c(1))

Non-Linear Exponential Fitting

df<-data.frame(t = t, s = s); par(mar=c(4,4,0,0))
fit<-nls(s ~ exp(-t/lambda), data = df,start = list(lambda=35.0))
sfit<-predict(fit,list(t=df$t))
plot(t,s, xlab="Time (s)", ylab = "Signal Intensity (a.u.)", col="black")
lines(t,sfit,lwd=3,col="red")
legend("topright", legend=c("Data", "Nonlinear Fit"), col=c("black", "red"), pch=c(1,-1), lty=c(0,1))

Prediction and Relative Error

paste("True lambda: ",lambda," (s)")

## [1] "True lambda:  50  (s)"

paste("Predicted lambda: ",round(coef(fit)[[1]],digits=2))

## [1] "Predicted lambda:  48.64"

paste("Relative Error: ",round(100*(coef(fit)[[1]]-lambda)/lambda),"%")

## [1] "Relative Error:  -3 %"

A Simple R Shiny Apps

Here is the weblink to a very simple R shiny app that performs the nonlinear exponential fitting.

Thank you!