Nonlinear Regression

Introduction

Least squares problems fall into two categories: linear or ordinary least squares and nonlinear least squares, depending on whether or not the residuals are linear in all unknowns. The linear least-squares problem occurs in statistical regression analysis; it has a closed-form solution.

Non-linear least squares is the form of least squares analysis used to fit a set of m observations with a model that is non-linear in n unknown parameters (m ≥ n). It is used in some forms of nonlinear regression.

In statistics, nonlinear regression is a form of regression analysis in which observational data are modeled by a function which is a nonlinear combination of the model parameters and depends on one or more independent variables. The data are fitted by a method of successive approximations (iterations).

Using a spline Example

Using a Model/Function

Project #1

Create a bell curve data set (x,y). Add a small random amount to each y value. Fit a curve to the data. Plot the data points and the curve.

Equation for the X,Y Coordinates of a Bell Curve

Y = Ke-(X-M)2/(2σ2)

X,Yare the curve's x,y coordinates (used for plotting, etc.)
Kis the maximum Y coordinate; used to scale the Y coordinates
(height in Y units)
Mis the curve's mathematical mean (X coordinate of the mean)
σis the curve's standard deviation; determines how fat or skinny
the curve is (width in X units)
eis Euler's number; is a constant; is an irrational number
(defined in the Python numpy module and other libraries)
From: math.stackexchange.com

Links

Least squares (Wikipedia)

Non-linear least squares (Wikipedia)

Nonlinear Regression (Wikipedia)

Python for Data Analysis - Using scipy for data fitting

Is there a way to plot a curve of best fit without function? Python