MODEL FITTING

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MODEL FITTING

• jiangyushan

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Introduction
The goal of model fitting is to choose values for
the parameters in a function to best describe a
set of data. There are many possible reason to do
this. If a specific meaningful form for the
function with a small number of free parameters
is known in advance, this is called parametric
fitting, and finding the parameter values
themselves may be the goad (for example,
measuring an exponential decay constant to
determine a reaction rate).
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Visual Model Fitting with the Original Data
Suppose we want to fit the model
to the data shown in Fig2.1. How might we choose
the constants and to determine the line
that best fits the data? Generally, when more
than two data points exist, all of them cannot be
expected to lie exactly along a single straight
line, even if such a line accurately models the
relationship between the two variables
.
Fig2.1
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Interpolation

• The most straightforward way to fit a curve to
  the data D is to find a function f that passes
  through each of the n data points or samples.
  That is , find a function f such that

Once a function is determined that satisfies
(2.1) ,the value of y at other points can be
estimated. In this case, the process of
estimating y is referred to as interpolation.
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Main Method for Interpolation and Curfitting
Piecewise-Linear Interpolation Polynomial
Interpolation Lagrange Interpolating
Polynomials Newtonss Difference Formula Cubic
Splines Chebyshev Approximation Criterion Least
squares curvefiting
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Matlab Functions

• poly,interp1,spline, pchip, polyfit, polyval
• Lsqnonlin, lsqcurvefit, regress, nlinfit

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Piecewise-Linear Interpolation
interp1 1-D data interpolation (table
lookup) Syntax yi interp1(x,Y,xi) yi
interp1(Y,xi) yi interp1(x,Y,xi,method) yi
interp1(x,Y,xi,method,'extrap') yi
interp1(x,Y,xi,method,extrapval) pp
interp1(x,Y,method,'pp') Description yi
interp1(x,Y,xi) interpolates to find yi, the
values of the underlying function Y at the points
in the vector or array xi. x must be a vector. Y
can be a scalar, a vector, or an array of any
dimension.
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Visual Model Fitting with the Original Data
Ordinarily, there will be some vertical
discrepancy between a few of the data points and
any particular line under consideration. We refer
to these vertical discrepancies as absolute
deviations (Fig 2.2). For the best-fitting line,
we might try to minimize the sum of these
absolute deviations, leading to the model
depicted in Fig2.2. Although success may be
achieved in minimizing the sum of the absolute
deviations, the absolute deviation from
individual points may be quite large. For
example, consider point D in Fig2.2. If the
modeler has confidence in the accuracy of this
data point, there would be concern for the
predictions made from the fitted line near the
point. As an alternative, suppose a line is
selected that minimizes the largest deviation
from any point. Applying this criterion to the
data points might give the line shown in Fig2.3.
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Visual Model Fitting with the Original Data
Absolute deviation
Fig2.2 minimizing the sum of these absolute
deviations from the fitted line
Fig2.3 minimizing the largest absolute deviations
from the fitted line
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lsqcurvefit

• x lsqcurvefit(fun,x0,xdata,ydata)
• Solve nonlinear curve-fitting (data-fitting)
  problems in least-squares sense
• EquationFind coefficients x that best fit the
  equation
• given input data xdata, and the observed output
  ydata, where xdata and ydata are matrices or
  vectors of length n, and F (xdata) is a
  matrix-valued or vector-valued function.

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Objectives

• Understand the difference between interpolation
  and curvefitting
• Know how to express piecewise-linear
  interpolation parametrically
• Know why interpolation with high-degree
  polynomials is undesirable
• Known why finding the interpolating polynomial by
  solving a linear algebraic system can be
  inaccurate.

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Objectives

• Know how to construct and use Largrange
  interpolating polynomials to perform
  interpolation
• Be able to fit curves to the underlying trend of
  data using least-squares techniques.
• Understand how curve fitting techniques can be
  used to solve practical engineering problems.
• Understand the relative strengths and weaknesses
  of each computational method and know which are
  most applicable for a given problem.

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The growth of a population of fruit flies

• The following data represent the growth of a
  population of fruit flies over a 6-week period.
  Test the following models by plotting an
  appropriate set of data. Estimate the parameters
  of the models

T (days) 7 14 21 28 35 42
P (number of observed flies) 8 41 133 250 280 297