Statistics with R for Business Analysts – Nonlinear Least Square

(R Tutorials for Citizen Data Scientist)

Statistics with R for Business Analysts – Nonlinear Least Square

When modeling real world data for regression analysis, we observe that it is rarely the case that the equation of the model is a linear equation giving a linear graph. Most of the time, the equation of the model of real world data involves mathematical functions of higher degree like an exponent of 3 or a sin function. In such a scenario, the plot of the model gives a curve rather than a line. The goal of both linear and non-linear regression is to adjust the values of the model’s parameters to find the line or curve that comes closest to your data. On finding these values we will be able to estimate the response variable with good accuracy.

In Least Square regression, we establish a regression model in which the sum of the squares of the vertical distances of different points from the regression curve is minimized. We generally start with a defined model and assume some values for the coefficients. We then apply the nls() function of R to get the more accurate values along with the confidence intervals.


The basic syntax for creating a nonlinear least square test in R is −

nls(formula, data, start)

Following is the description of the parameters used −

  • formula is a nonlinear model formula including variables and parameters.
  • data is a data frame used to evaluate the variables in the formula.
  • start is a named list or named numeric vector of starting estimates.



We will consider a nonlinear model with assumption of initial values of its coefficients. Next we will see what is the confidence intervals of these assumed values so that we can judge how well these values fir into the model.

So let’s consider the below equation for this purpose −

a = b1*x^2+b2

Let’s assume the initial coefficients to be 1 and 3 and fit these values into nls() function.

xvalues <- c(1.6,2.1,2,2.23,3.71,3.25,3.4,3.86,1.19,2.21)
yvalues <- c(5.19,7.43,6.94,8.11,18.75,14.88,16.06,19.12,3.21,7.58)

# Give the chart file a name.
png(file = "nls.png")

# Plot these values.

# Take the assumed values and fit into the model.
model <- nls(yvalues ~ b1*xvalues^2+b2,start = list(b1 = 1,b2 = 3))

# Plot the chart with new data by fitting it to a prediction from 100 data points. <- data.frame(xvalues = seq(min(xvalues),max(xvalues),len = 100))
lines($xvalues,predict(model,newdata =

# Save the file.

# Get the sum of the squared residuals.

# Get the confidence intervals on the chosen values of the coefficients.

When we execute the above code, it produces the following result −

[1] 1.081935
Waiting for profiling to be done...
       2.5%    97.5%
b1 1.137708 1.253135
b2 1.497364 2.496484

Non Linear least square R

We can conclude that the value of b1 is more close to 1 while the value of b2 is more close to 2 and not 3.


Classification in R – partial least squares discriminant in R


Statistics with R for Business Analysts – Nonlinear Least Square

Personal Career & Learning Guide for Data Analyst, Data Engineer and Data Scientist

Applied Machine Learning & Data Science Projects and Coding Recipes for Beginners

A list of FREE programming examples together with eTutorials & eBooks @ SETScholars

95% Discount on “Projects & Recipes, tutorials, ebooks”

Projects and Coding Recipes, eTutorials and eBooks: The best All-in-One resources for Data Analyst, Data Scientist, Machine Learning Engineer and Software Developer

Topics included: Classification, Clustering, Regression, Forecasting, Algorithms, Data Structures, Data Analytics & Data Science, Deep Learning, Machine Learning, Programming Languages and Software Tools & Packages.
(Discount is valid for limited time only)

Disclaimer: The information and code presented within this recipe/tutorial is only for educational and coaching purposes for beginners and developers. Anyone can practice and apply the recipe/tutorial presented here, but the reader is taking full responsibility for his/her actions. The author (content curator) of this recipe (code / program) has made every effort to ensure the accuracy of the information was correct at time of publication. The author (content curator) does not assume and hereby disclaims any liability to any party for any loss, damage, or disruption caused by errors or omissions, whether such errors or omissions result from accident, negligence, or any other cause. The information presented here could also be found in public knowledge domains.

Learn by Coding: v-Tutorials on Applied Machine Learning and Data Science for Beginners

Please do not waste your valuable time by watching videos, rather use end-to-end (Python and R) recipes from Professional Data Scientists to practice coding, and land the most demandable jobs in the fields of Predictive analytics & AI (Machine Learning and Data Science).

The objective is to guide the developers & analysts to “Learn how to Code” for Applied AI using end-to-end coding solutions, and unlock the world of opportunities!