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In many cases we have clear correlations and yet we do not know how to quantify them because they are nonlinear.

Introduction

The Simple linear regressions Are specific cases that apply only if the data follows a logic Y = AX + B. But in many cases, we can have nonlinear relationships that mathematically describe.

In this type of case, it is necessary, depending on the type of relationship, to linearize the data X and/or Y.

1. Visually identify the type of relationship

First step if we think the regression is nonlinear, visually identify the type of relationship a priori. For this, we build the cloud of points, then we search among one of the 3 models below the one that best fits our situation:

NameFormulaChart

exponential

It has a vertical asymptote
logarithmic



It has a horizontal asymptote

Power



She has no asymptote

2. Linearize the data

Exponential linearization

In this case we transform the data Y. We calculate ln (y) and trace the cloud of Point ln (y) according to X. If our choice is right, the dots form a right.

Logarithmic linearization

In this case only the data of X is transformed. We compute ln (x) and trace the cloud of Point Ln (x) according to Y. If our choice is good the dots form a straight.

Power linearization

In this case the X and Y data are transformed. We compute ln (x) and ln (y) and then trace the cloud of Point Ln (x) according to ln (y). If our choice is good the dots form a straight.

3. Calculate the linear regression line

Since our data is now linear, we calculate the right of Linear regression Associated Y = A * X + B.

4. Deduce the correlation formula

Having modified the initial data, we will have to do the reverse approach to find the model corresponding to our real situation:

Exponential

Logarithmic

Power

Source

R. Rafiq (2013) – econometrics, single and multiple linear regression

S. Champely (2004) – Statistics really applied to sport

J. M. Azaïs, J. M. Bardot (2012) – The linear model by example

P. Moshood (2006) – Theoretical and applied statistics

W. Fox (1999) – Social statistics

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