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<document url="fitting.html">

  <properties>
    <title>The Commons Math User Guide - Curve Fitting</title>
  </properties>

  <body>
    <section name="13 Curve Fitting">
      <subsection name="13.1 Overview" href="overview">
        <p>
          The fitting package deals with curve fitting for univariate real functions.
          When a univariate real function y = f(x) does depend on some unknown parameters
          p<sub>0</sub>, p<sub>1</sub> ... p<sub>n-1</sub>, curve fitting can be used to
          find these parameters. It does this by <em>fitting</em> the curve so it remains
          very close to a set of observed points (x<sub>0</sub>, y<sub>0</sub>),
          (x<sub>1</sub>, y<sub>1</sub>) ... (x<sub>k-1</sub>, y<sub>k-1</sub>). This
          fitting is done by finding the parameters values that minimizes the objective
          function Σ(y<sub>i</sub> - f(x<sub>i</sub>))<sup>2</sup>. This is actually a
          least-squares problem.
        </p>
        <p>
          For all provided curve fitters, the operating principle is the same.
          Users must
          <ul>
            <li>
              create an instance of the fitter using the <code>create</code> factory method of the
              appropriate class,
            </li>
            <li>
              call the <a href="../apidocs/org/apache/commons/math4/fitting/AbstractCurveFitter">fit</a>
              with a <code>Collection</code> of <a href="../apidocs/org/apache/commons/math4/fitting/WeightedObservedPoint.html">
                observed data points</a> as argument, which will return an array with the parameters that
              best fit the given data points.
            </li>
          </ul>

          The list of observed data points to be passed to <code>fit</code> can be built by incrementally
          adding instances to an instance of <a href="../apidocs/org/apache/commons/math4/fitting/WeightedObservedPoints.html">WeightedObservedPoints</a>,
          and then retrieve the list of <code>WeightedObservedPoint</code> by calling the <code>toList</code>
          method on that container.
          A weight can be associated with each observed point; it allows to take into account uncertainty
          on some points when they come from noisy measurements for example. If no such information exist and
          all points should be treated the same, it is safe to put 1.0 as the weight for all points (and this
          is the default when no weight is passed to the <code>add</code>.
        </p>

        <p>
          Some fitters require that initial values for the parameters are provided by the user,
          through the <code>withStartPoint</code> method, before attempting to perform the fit.
          When that's the case the fitter class usually defines a guessing procedure within a
          <code>ParameterGuesser</code> inner class, that attempts to provide appropriate initial
          values based on the user-supplied data.
          When initial values are required but are not provided, the <code>fit</code> method will
          internally call the guessing procedure.
        </p>

      </subsection>

      <subsection name="13.2 Implemented Functions" href="special">

        <p>
          Fitting of specific functions are provided through the following classes:
          <ul>
            <li>
              <a href="../apidocs/org/apache/commons/math4/fitting/PolynomialCurveFitter.html">
                PolynomialFitter</a> fits a
              <a href="../apidocs/org/apache/commons/math4/analysis/polynomials/PolynomialFunction.Parametric.html">
                polynomial</a> function.
            </li>
            <li>
              <a href="../apidocs/org/apache/commons/math4/fitting/HarmonicCurveFitter.html">
                HarmonicFitter</a> fits a
              <a href="../apidocs/org/apache/commons/math4/analysis/function/HarmonicOscillator.Parametric.html">
                harmonic</a> function.
              An instance of the inner class <a href="../apidocs/org/apache/commons/math4/fitting/HarmonicCurveFitter.ParameterGuesser.html">
                ParameterGuesser</a> can be used to retrieve initial values for the fitting procedure.
            </li>
            <li>
              <a href="../apidocs/org/apache/commons/math4/fitting/GaussianCurveFitter.html">
                GaussianFitter</a> fits a
              <a href="../apidocs/org/apache/commons/math4/analysis/function/Gaussian.Parametric.html">
                Gaussian</a> function.
              An instance of the inner class <a href="../apidocs/org/apache/commons/math4/fitting/GaussianCurveFitter.ParameterGuesser.html">
                ParameterGuesser</a> can be used to retrieve initial values for the fitting procedure.
            </li>
          </ul>
        </p>

        <p>
          The following example shows how to fit data with a polynomial function.
        </p>

        <source>// Collect data.
final WeightedObservedPoints obs = new WeightedObservedPoints();
obs.add(-1.00, 2.021170021833143);
obs.add(-0.99, 2.221135431136975);
obs.add(-0.98, 2.09985277659314);
obs.add(-0.97, 2.0211192647627025);
// ... Lots of lines omitted ...
obs.add(0.99, -2.4345814727089854);

// Instantiate a third-degree polynomial fitter.
final PolynomialCurveFitter fitter = PolynomialCurveFitter.create(3);

// Retrieve fitted parameters (coefficients of the polynomial function).
final double[] coeff = fitter.fit(obs.toList());
        </source>

      </subsection>

      <subsection name="13.3 General Case" href="general">
        <p>
          The <a href="../apidocs/org/apache/commons/math4/fitting/AbstractCurveFitter.html">
            AbstractCurveFitter</a> class provides the basic functionality for implementing other
          curve fitting classes.
          Users must provide their own implementation of the curve template as a class that implements
          the <a href="../apidocs/org/apache/commons/math4/analysis/ParametricUnivariateFunction.html">
            ParametricUnivariateFunction</a> interface.
        </p>
      </subsection>

     </section>
  </body>
</document>