Non Linear Equation

Green university of Bangladesh
Submitted by:
Name: Md. Al-Amin
ID: 172015031
Department of CSE
Non-Linear Equation
Submitted to:
Name: Prof. Dr. Md.Monirul Islam
Designation: Distinguished Professor
Department: CSE

 What is Non-linear regression ?
 Types of Non-linear regression
 Using different types of methods
 Example
 Difference between liner & non-linear regression
 Advantages
 Conclusion
Outline
1

What is Non-linear regression?
Nonlinear regression uses nonlinear regression equations, which take the
form Y = f(X,β) + ε
Where:
 X = a vector of p predictors,
 β = a vector of k parameters,
 f(-) = a known regression function,
 ε = an error term.
The formal definition is that if your regression equation looks like the one
above, it’s nonlinear regression. However, this is actually a lot more difficult
than it sounds. Take the following nonlinear regression equations:
 The Michaelis-Menten model: f(x,β) = (β1 x) / (β 2 + x).
 Y = β0 + (0.4 – β0)e-β
1
(x
i
-5) + εi.
2

Types of Non-linear regression
Nonlinear models can be classified into two categories
1. In the first category are models that are nonlinear in the variables, but still linear
in terms of the unknown parameters. This category includes models which are
made linear in the parameters via a transformation.
2. The second category of nonlinear models contains models which are nonlinear in
the parameters and which cannot be made linear in the parameters after a
transformation. For estimating models in this category the familiar least squares
technique is extended to an estimation procedure known as nonlinear least
squares.
3

Non-linear regression method
1. "Direct Computation" Method:
y = βo + β1*2-t/θ
2. "Derivative" Method:
η(β) ≈ ω(0) + Z(0)β
3. Self-Starting Method:
y = βo + β1 exp(-x/θ) + ε
4

When do we use a non-linear regression model?
 When at least one of the parameters is not
linear
 The general form of a non-linear
regression model is
y=η(x,β) + ε
where ε ~N(0, σ 2)
5

Example
Non-linear regression
graph. Where its in x axis
determine the Density and
y axis determine the
mobility. Finally we get
fitted line, which not
straight line.
Non-linear regression example 6

Types Linear Regression Non-linear Regression
Formula f = a1φ1(x) + …. + ap φp (x) any f(a,x)
Example f=a exp(-20x) f=exp(-ax)
Source Soft Hard
Choice Easy Difficult
Dimension Large Small
Multicolinearity Excess of parameters Lack of data
Interpretation Well-known Uncommon
Purpose Interpolation Extrapolation
Soft tools Many Few
Difference between Linear & Non-Linear Regression
7

 Apply differential weighting.
 Identify, and possibly exclude, outliers.
 Use a robust fitting method.
 Perform a normality test on the residuals.
 Inspect the correlation matrix and the dependency of each parameter.
 Compare the scatter of points from the line with the scatter among replicates
with the replicates test.
 Segmental linear regression.
 Use global nonlinear regression to fit one line to several data sets.
Advantages of Non-linear regression
8

 Use a non-linear regression model when at least one of the
parameters is not linear.
 Non-linear regression is an iterative procedure in which the number
of iterations depend on how quickly the parameters converge.
Three methods introduced:
 Direct Computation Method
 Derivative Method
 Self-Starting Method
Conclusion
9

Non Linear Equation

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Non Linear Equation