From 225e5c703c7b55d802f3e9a6aeb92c7cd27ad386 Mon Sep 17 00:00:00 2001
From: Luc Maisonobe <luc@apache.org>
Date: Tue, 5 Feb 2008 18:09:06 +0000
Subject: [PATCH] completely rewrote estimation package documentation with
 downloadable example and explanation diagrams

git-svn-id: https://svn.apache.org/repos/asf/commons/proper/math/trunk@618726 13f79535-47bb-0310-9956-ffa450edef68
---
 .../TrajectoryDeterminationProblem.java       | 187 +++++++++
 .../userguide/estimation-class-diagram.png    | Bin 0 -> 20220 bytes
 .../userguide/estimation-sequence-diagram.png | Bin 0 -> 16190 bytes
 xdocs/userguide/estimation.xml                | 394 ++++++++++++++----
 4 files changed, 509 insertions(+), 72 deletions(-)
 create mode 100644 src/site/resources/userguide/TrajectoryDeterminationProblem.java
 create mode 100644 src/site/resources/userguide/estimation-class-diagram.png
 create mode 100644 src/site/resources/userguide/estimation-sequence-diagram.png

diff --git a/src/site/resources/userguide/TrajectoryDeterminationProblem.java b/src/site/resources/userguide/TrajectoryDeterminationProblem.java
new file mode 100644
index 000000000..32a0975f1
--- /dev/null
+++ b/src/site/resources/userguide/TrajectoryDeterminationProblem.java
@@ -0,0 +1,187 @@
+/*
+ * Licensed to the Apache Software Foundation (ASF) under one or more
+ * contributor license agreements.  See the NOTICE file distributed with
+ * this work for additional information regarding copyright ownership.
+ * The ASF licenses this file to You under the Apache License, Version 2.0
+ * (the "License"); you may not use this file except in compliance with
+ * the License.  You may obtain a copy of the License at
+ *
+ *      http://www.apache.org/licenses/LICENSE-2.0
+ *
+ * Unless required by applicable law or agreed to in writing, software
+ * distributed under the License is distributed on an "AS IS" BASIS,
+ * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+ * See the License for the specific language governing permissions and
+ * limitations under the License.
+ */
+
+import org.apache.commons.math.estimation.EstimationException;
+import org.apache.commons.math.estimation.EstimatedParameter;
+import org.apache.commons.math.estimation.EstimationProblem;
+import org.apache.commons.math.estimation.LevenbergMarquardtEstimator;
+import org.apache.commons.math.estimation.SimpleEstimationProblem;
+import org.apache.commons.math.estimation.WeightedMeasurement;
+
+public class TrajectoryDeterminationProblem extends SimpleEstimationProblem {
+
+    public static void main(String[] args) {
+        try {
+            TrajectoryDeterminationProblem problem =
+              new TrajectoryDeterminationProblem(0.0, 100.0, 800.0, 1.0, 0.0);
+
+            double[][] distances = {
+                    {   0.0, 806.5849 }, {  20.0, 796.8148 }, {  40.0, 791.0833 }, {  60.0, 789.6712 },
+                    {  80.0, 793.1334 }, { 100.0, 797.7248 }, { 120.0, 803.2785 }, { 140.0, 813.4939 },
+                    { 160.0, 826.9295 }, { 180.0, 844.0640 }, { 200.0, 863.3829 }, { 220.0, 883.3143 },
+                    { 240.0, 908.6867 }, { 260.0, 934.8561 }, { 280.0, 964.0730 }, { 300.0, 992.1033 },
+                    { 320.0, 1023.998 }, { 340.0, 1057.439 }, { 360.0, 1091.912 }, { 380.0, 1125.968 },
+                    { 400.0, 1162.789 }, { 420.0, 1201.517 }, { 440.0, 1239.176 }, { 460.0, 1279.347 } };
+            for (int i = 0; i < distances.length; ++i) {
+                problem.addDistanceMeasurement(1.0,  distances[i][0], distances[i][1]);
+            };
+
+            double[][] angles = {
+                    { 10.0, 1.415423 }, { 30.0, 1.352643 }, { 50.0, 1.289290 }, { 70.0, 1.225249 },
+                    { 90.0, 1.161203 }, {110.0, 1.098538 }, {130.0, 1.036263 }, {150.0, 0.976052 },
+                    {170.0, 0.917921 }, {190.0, 0.861830 }, {210.0, 0.808237 }, {230.0, 0.757043 },
+                    {250.0, 0.708650 }, {270.0, 0.662949 }, {290.0, 0.619903 }, {310.0, 0.579160 },
+                    {330.0, 0.541033 }, {350.0, 0.505590 }, {370.0, 0.471746 }, {390.0, 0.440155 },
+                    {410.0, 0.410522 }, {430.0, 0.382701 }, {450.0, 0.356957 }, {470.0, 0.332400 } };
+            for (int i = 0; i < angles.length; ++i) {
+                problem.addAngularMeasurement(3.0e7, angles[i][0], angles[i][1]);
+            };
+
+            LevenbergMarquardtEstimator estimator = new LevenbergMarquardtEstimator();
+            estimator.estimate(problem);
+            System.out.println("initial position: " + problem.getX0() + " " + problem.getY0());
+            System.out.println("velocity: " + problem.getVx0() + " " + problem.getVy0());
+
+        } catch (EstimationException ee) {
+          System.err.println(ee.getMessage());
+        }
+    }
+
+    public TrajectoryDeterminationProblem(double t0,
+                                          double  x0Guess, double  y0Guess,
+                                          double vx0Guess, double vy0Guess) {
+        this.t0 = t0;
+         x0 = new EstimatedParameter( "x0",  x0Guess);
+         y0 = new EstimatedParameter( "y0",  y0Guess);
+        vx0 = new EstimatedParameter("vx0", vx0Guess);
+        vy0 = new EstimatedParameter("vy0", vy0Guess);
+
+        // inform the base class about the parameters
+        addParameter(x0);
+        addParameter(y0);
+        addParameter(vx0);
+        addParameter(vy0);
+
+    }
+
+    public double getX0() {
+        return x0.getEstimate();
+    }
+
+    public double getY0() {
+        return y0.getEstimate();
+    }
+
+    public double getVx0() {
+        return vx0.getEstimate();
+    }
+
+    public double getVy0() {
+        return vy0.getEstimate();
+    }
+
+    public void addAngularMeasurement(double wi, double ti, double ai) {
+        // let the base class handle the measurement
+        addMeasurement(new AngularMeasurement(wi, ti, ai));
+    }
+
+    public void addDistanceMeasurement(double wi, double ti, double di) {
+        // let the base class handle the measurement
+        addMeasurement(new DistanceMeasurement(wi, ti, di));
+    }
+
+    public double x(double t) {
+        return x0.getEstimate() + (t - t0) * vx0.getEstimate();
+    }
+
+    public double y(double t) {
+        return y0.getEstimate() + (t - t0) * vy0.getEstimate();
+    }
+
+    private class AngularMeasurement extends WeightedMeasurement {
+
+        public AngularMeasurement(double weight, double t, double angle) {
+            super(weight, angle);
+            this.t = t;
+        }
+
+        public double getTheoreticalValue() {
+            return Math.atan2(y(t), x(t));
+        }
+
+        public double getPartial(EstimatedParameter parameter) {
+            double xt = x(t);
+            double yt = y(t);
+            double r  = Math.sqrt(xt * xt + yt * yt);
+            double u  = yt / (r + xt);
+            double c  = 2 * u / (1 + u * u);
+            if (parameter == x0) {
+                return -c;
+            } else if (parameter == vx0) {
+                return -c * t;
+            } else if (parameter == y0) {
+                return c * xt / yt;
+            } else {
+                return c * t * xt / yt;
+            }
+        }
+
+        private final double t;
+        private static final long serialVersionUID = -5990040582592763282L;
+
+    }
+
+    private class DistanceMeasurement extends WeightedMeasurement {
+
+        public DistanceMeasurement(double weight, double t, double angle) {
+            super(weight, angle);
+            this.t = t;
+        }
+
+        public double getTheoreticalValue() {
+            double xt = x(t);
+            double yt = y(t);
+            return Math.sqrt(xt * xt + yt * yt);
+        }
+
+        public double getPartial(EstimatedParameter parameter) {
+            double xt = x(t);
+            double yt = y(t);
+            double r  = Math.sqrt(xt * xt + yt * yt);
+            if (parameter == x0) {
+                return xt / r;
+            } else if (parameter == vx0) {
+                return xt * t / r;
+            } else if (parameter == y0) {
+                return yt / r;
+            } else {
+                return yt * t / r;
+            }
+        }
+
+        private final double t;
+        private static final long serialVersionUID = 3257286197740459503L;
+
+    }
+
+    private double t0;
+    private EstimatedParameter x0;
+    private EstimatedParameter y0;
+    private EstimatedParameter vx0;
+    private EstimatedParameter vy0;
+
+}
diff --git a/src/site/resources/userguide/estimation-class-diagram.png b/src/site/resources/userguide/estimation-class-diagram.png
new file mode 100644
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diff --git a/xdocs/userguide/estimation.xml b/xdocs/userguide/estimation.xml
index c214ab019..9dadf7368 100644
--- a/xdocs/userguide/estimation.xml
+++ b/xdocs/userguide/estimation.xml
@@ -1,22 +1,22 @@
 <?xml version="1.0"?>
 
 <!--
-   Licensed to the Apache Software Foundation (ASF) under one or more
+  Licensed to the Apache Software Foundation (ASF) under one or more
   contributor license agreements.  See the NOTICE file distributed with
   this work for additional information regarding copyright ownership.
   The ASF licenses this file to You under the Apache License, Version 2.0
   (the "License"); you may not use this file except in compliance with
   the License.  You may obtain a copy of the License at
-
-       http://www.apache.org/licenses/LICENSE-2.0
-
-   Unless required by applicable law or agreed to in writing, software
-   distributed under the License is distributed on an "AS IS" BASIS,
-   WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
-   See the License for the specific language governing permissions and
-   limitations under the License.
-  -->
   
+  http://www.apache.org/licenses/LICENSE-2.0
+  
+  Unless required by applicable law or agreed to in writing, software
+  distributed under the License is distributed on an "AS IS" BASIS,
+  WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+  See the License for the specific language governing permissions and
+  limitations under the License.
+-->
+
 <?xml-stylesheet type="text/xsl" href="./xdoc.xsl"?>
 <!-- $Revision: 480435 $ $Date: 2006-11-29 08:06:35 +0100 (mer., 29 nov. 2006) $ -->
 <document url="estimation.html">
@@ -29,86 +29,336 @@
     <section name="12 Parametric Estimation">
       <subsection name="12.1 Overview" href="overview">
         <p>
-        The estimation package provides classes to fit some non-linear model
-        to available observations depending on it. These problems are commonly
-        called estimation problems.
+          The estimation package provides classes to fit some non-linear
+          model to available observations depending on it. These
+          problems are commonly called estimation problems.
         </p>
         <p>
-        The estimation problems considered here are parametric problems where
-        a user-provided model depends on initially unknown scalar parameters and
-        several measurements made on values that depend on the model are available.
-        As examples, one can consider the center and radius of a circle given
-        points approximately lying on a ring, or a satellite orbit given range,
-        range-rate and angular measurements from various ground stations.
+          The estimation problems considered here are parametric
+          problems where a user-provided model depends on initially
+          unknown scalar parameters and several measurements made on
+          values that depend on the model are available. As examples,
+          one can consider the center and radius of a circle given
+          points approximately lying on a ring, or a satellite orbit
+          given range, range-rate and angular measurements from various
+          ground stations.
         </p>
         <p>
-        One important class of estimation problems is weighted least squares problems.
-        They basically consist in finding the values for some parameters p<sub>k</sub>
-        such that a cost function J = 
-        <!-- TODO: get entity for summation imported -->
-        sum(w<sub>i</sub> r<sub>i</sub><sup>2</sup>)
-        is minimized. The various r<sub>i</sub> terms represent the deviation
-        r<sub>i</sub> = mes<sub>i</sub> - mod<sub>i</sub> between the measurements and
-        the parameterized models. The w<sub>i</sub> factors are the measurements weights,
-        they are often chosen either all equal to 1.0 or proportional to the inverse of
-        the variance of the measurement type. The solver adjusts the values of the
-        estimated parameters p<sub>k</sub> which are not bound (i.e. the free parameters).
-        It does not touch the parameters which have been put in a bound state by the user.        
+          One important class of estimation problems is weighted least
+          squares problems. They basically consist in finding the values
+          for some parameters p<sub>k</sub> such that a cost function
+          J = sum(w<sub>i</sub>r<sub>i</sub><sup>2</sup>) is minimized.
+          The various r<sub>i</sub> terms represent the deviation
+          r<sub>i</sub> = mes<sub>i</sub> - mod<sub>i</sub>
+          between the measurements and the parameterized models. The
+          w<sub>i</sub> factors are the measurements weights, they are often
+          chosen either all equal to 1.0 or proportional to the inverse of the
+          variance of the measurement type. The solver adjusts the values of
+          the estimated parameters p<sub>k</sub> which are not bound (i.e. the
+          free parameters). It does not touch the parameters which have been
+          put in a bound state by the user.
         </p>
         <p>
-          The aim of this package is similar to the aim of the optimization package, but the
-          algorithms are entirely differents as:
+          The aim of this package is similar to the aim of the
+          optimization package, but the algorithms are entirely
+          different as:
           <ul>
             <li>
-              they need the partial derivatives of the measurements
-              with respect to the free parameters
+              they need the partial derivatives of the measurements with
+              respect to the free parameters
             </li>
             <li>
-              they are residuals based instead of generic cost functions based
+              they are residuals based instead of generic cost functions
+              based
             </li>
           </ul>
         </p>
-        
       </subsection>
-      <subsection name="12.2 Models" href="models">
-        The <a href="../apidocs/org/apache/commons/math/estimation/EstimationProblem.html">
-        org.apache.commons.math.estimation.EstimationProblem</a> interface is a very
-        simple container packing together parameters and measurements.
-      </subsection>
-      <subsection name="12.3 Parameters" href="parameters">
+
+      <subsection name="12.2 Problem modeling" href="problem">
         <p>
-        The <a href="../apidocs/org/apache/commons/math/estimation/EstimatedParameter.html">
-        org.apache.commons.math.estimation.EstimatedParameter</a> class to represent each
-        estimated parameter. The parameters are set up with a guessed value which will be
-        adjusted by the solver until a best fit is achieved. It is possible to change which
-        parameters are modified and which are preserved thanks to a bound property. Such
-        settings are often needed by expert users to handle contingency cases with very
-        low observability.
+          The problem modeling is the most important part for the
+          user. Understanding it is the key to proper use of the
+          package. One interface and two classes are provided for this
+          purpose: <a href="../apidocs/org/apache/commons/math/estimation/EstimationProblem.html">
+          EstimationProblem</a>, <a href="../apidocs/org/apache/commons/math/estimation/EstimatedParameter.html">
+          EstimatedParameter</a> and <a href="../apidocs/org/apache/commons/math/estimation/WeightedMeasurement.html">
+          WeightedMeasurement</a>.
         </p>
-      </subsection>
-      <subsection name="12.4 Measurements" href="measurements">
         <p>
-        The user extends the <a href="../apidocs/org/apache/commons/math/estimation/WeightedMeasurement.html">
-        org.apache.commons.math.estimation.WeightedMeasurement</a> abstract class to define its
-        own measurements. Each measurement types should have its own implementing class, for
-        example in the satellite example above , the user should define three classes, one
-        for range measurements, one for range-rates measurements and one for angular measurements.
-        Each measurement would correspond to an instance of the appropriate class, set up with
-        the date, a reference to the ground station, the weight and the measured value.
-         </p>
-      </subsection>
-      <subsection name="12.5 Solvers" href="solvers">
-        <p>
-        The package provides two common <a href="../apidocs/org/apache/commons/math/estimation/Estimator.html">
-        org.apache.commons.math.estimation.Estimator</a> implementations to solve weighted
-        least squares problems. The first one is based on the
-        <a href="../apidocs/org/apache/commons/math/estimation/GaussNewtonEstimator.html">Gauss-Newton</a> method.
-        The second one is based on the
-        <a href="../apidocs/org/apache/commons/math/estimation/LevenbergMarquardtEstimator.html">Levenberg-Marquardt</a>
-        method. The first one is best suited when a good approximation of the parameters is known while the second one
-        is more robust and can handle starting points far from the solution.
+          Consider the following example problem: we want to determine the
+          linear trajectory of a sailing ship by performing angular and
+          distance measurements from an observing  spot on the shore. The
+          problem model is represented by two equations:
         </p>
-      </subsection>
+        <p>
+          x(t) = x<sub>0</sub>+(t-t<sub>0</sub>)vx<sub>0</sub><br/>
+          y(t) = y<sub>0</sub>+(t-t<sub>0</sub>)vy<sub>0</sub>
+        </p>
+        <p>
+          These two equations depend on four parameters (x<sub>0</sub>, y<sub>0</sub>,
+          vx<sub>0</sub> and vy<sub>0</sub>). We want to determine these four parameters.
+        </p>
+        <p>
+          Assuming the observing spot is located at the origin of the coordinates
+          system and that the angular measurements correspond to the angle between
+          the x axis and the line of sight, the theoretical values of the angular
+          measurements at t<sub>i</sub> and of the distance measurements at
+          t<sub>j</sub> are modeled as follows:
+        </p>
+        <p>
+          angle<sub>i,theo</sub> = atan2(y(t<sub>i</sub>), x(t<sub>i</sub>))<br/>
+          distance<sub>j,theo</sub> = sqrt(x(t<sub>j</sub>)<sup>2</sup>+y(t<sub>j</sub>)<sup>2</sup>)   
+        </p>
+        <p>
+          The real observations generate a set of measurements values angle<sub>i,meas</sub>
+          and distance<sub>j,meas</sub>. 
+        </p>
+        <p>
+          The following class diagram shows one way to solve this problem using the
+          estimation package. The grey elements are already provided by the package
+          whereas the purple elements are developed by the user.
+        </p>
+        <img src="./estimation-class-diagram.png"/>
+        <p>
+          The <code>TrajectoryDeterminationProblem</code> class holds the linear model
+          equations x(t) and y(t). It delegate storage of the four parameters x<sub>0</sub>,
+          y<sub>0</sub>, vx<sub>0</sub> and vy<sub>0</sub> and of the various measurements
+          angle<sub>i,meas</sub> and distance<sub>j,meas</sub> to its base class
+          <code>SimpleEstimationProblem</code>. Since the theoretical values of the measurements
+          angle<sub>i,theo</sub> and distance<sub>j,theo</sub> depend on the linear model,
+          the two classes <code>AngularMeasurement</code> and <code>DistanceMeasurement</code>
+          are implemented as internal classes, thus having access to the equations of the
+          linear model and to the parameters.
+        </p>
+        <p>
+          Here are the various parts of the <code>TrajectoryDeterminationProblem.java</code>
+          source file. This example, with an additional <code>main</code> method is
+          available <a href="./TrajectoryDeterminationProblem.java">here</a>.
+        </p>
+        <dd>First, the general setup of the class: declarations, fields, constructor, setters and getters:
+          <source>
+public class TrajectoryDeterminationProblem extends SimpleEstimationProblem {
+  public TrajectoryDeterminationProblem(double t0,
+                                        double  x0Guess, double  y0Guess,
+                                        double vx0Guess, double vy0Guess) {
+    this.t0 = t0;
+     x0 = new EstimatedParameter( "x0",  x0Guess);
+     y0 = new EstimatedParameter( "y0",  y0Guess);
+    vx0 = new EstimatedParameter("vx0", vx0Guess);
+    vy0 = new EstimatedParameter("vy0", vy0Guess);
+
+    // inform the base class about the parameters
+    addParameter(x0);
+    addParameter(y0);
+    addParameter(vx0);
+    addParameter(vy0);
+
+  }
+
+  public double getX0() {
+    return x0.getEstimate();
+  }
+
+  public double getY0() {
+    return y0.getEstimate();
+  }
+
+  public double getVx0() {
+    return vx0.getEstimate();
+  }
+
+  public double getVy0() {
+    return vy0.getEstimate();
+  }
+
+  public void addAngularMeasurement(double wi, double ti, double ai) {
+    // let the base class handle the measurement
+    addMeasurement(new AngularMeasurement(wi, ti, ai));
+  }
+
+  public void addDistanceMeasurement(double wi, double ti, double di) {
+    // let the base class handle the measurement
+    addMeasurement(new DistanceMeasurement(wi, ti, di));
+  }
+
+  public double x(double t) {
+    return x0.getEstimate() + (t - t0) * vx0.getEstimate();
+  }
+
+  public double y(double t) {
+    return y0.getEstimate() + (t - t0) * vy0.getEstimate();
+  }
+
+  // measurements internal classes go here
+
+  private double t0;
+  private EstimatedParameter x0;
+  private EstimatedParameter y0;
+  private EstimatedParameter vx0;
+  private EstimatedParameter vy0;
+
+}
+          </source>
+        </dd>
+        <dd>The two specialized measurements class are simple internal classes that
+        implement the equation for their respective measurement type, using the
+        enclosing class to get the parameters references and the linear models x(t)
+        and y(t). The <code>serialVersionUID</code> static fields are present because
+        the <code>WeightedMeasurement</code> class implements the
+        <code>Serializable</code> interface.
+          <source>
+  private class AngularMeasurement extends WeightedMeasurement {
+
+    public AngularMeasurement(double weight, double t, double angle) {
+      super(weight, angle);
+      this.t = t;
+    }
+
+    public double getTheoreticalValue() {
+      return Math.atan2(y(t), x(t));
+    }
+
+    public double getPartial(EstimatedParameter parameter) {
+      double xt = x(t);
+      double yt = y(t);
+      double r  = Math.sqrt(xt * xt + yt * yt);
+      double u  = yt / (r + xt);
+      double c  = 2 * u / (1 + u * u);
+      if (parameter == x0) {
+        return -c;
+      } else if (parameter == vx0) {
+        return -c * t;
+      } else if (parameter == y0) {
+        return c * xt / yt;
+      } else {
+        return c * t * xt / yt;
+      }
+    }
+
+    private final double t;
+    private static final long serialVersionUID = -5990040582592763282L;
+
+  }
+          </source>
+          <source>
+  private class DistanceMeasurement extends WeightedMeasurement {
+
+    public DistanceMeasurement(double weight, double t, double angle) {
+      super(weight, angle);
+      this.t = t;
+    }
+
+    public double getTheoreticalValue() {
+      double xt = x(t);
+      double yt = y(t);
+      return Math.sqrt(xt * xt + yt * yt);
+    }
+
+    public double getPartial(EstimatedParameter parameter) {
+      double xt = x(t);
+      double yt = y(t);
+      double r  = Math.sqrt(xt * xt + yt * yt);
+      if (parameter == x0) {
+        return xt / r;
+      } else if (parameter == vx0) {
+        return xt * t / r;
+      } else if (parameter == y0) {
+        return yt / r;
+      } else {
+        return yt * t / r;
+      }
+    }
+
+    private final double t;
+    private static final long serialVersionUID = 3257286197740459503L;
+ 
+  }
+          </source>
+        </dd>
+        </subsection>
+        <subsection name="12.3 Problem solving" href="solving">
+          <p>
+            Solving the problem is simply a matter of choosing an implementation
+            of the <a href="../apidocs/org/apache/commons/math/estimation/Estimator.html">
+            Estimator</a> interface and to pass the problem instance to its <code>estimate</code>
+            method. Two implementations are already provided by the library: <a
+            href="../apidocs/org/apache/commons/math/estimation/GaussNewtonEstimator.html">
+            GaussNewtonEstimator</a> and <a
+            href="../apidocs/org/apache/commons/math/estimation/LevenbergMarquardtEstimator.html">
+            LevenbergMarquardtEstimator</a>. The first one implements a simple Gauss-Newton
+            algorithm, which is sufficient when the starting point (initial guess) is close
+            enough to the solution. The second one implements a more complex Levenberg-Marquardt
+            algorithm which is more robust when the initial guess is far from the solution.
+          </p>
+          <p>
+            The following sequence diagram explains roughly what occurs under the hood
+            in the <code>estimate</code> method.
+          </p>
+          <img src="./estimation-sequence-diagram.png"/>
+          <p>
+            Basically, the estimator first retrieves the parameters and the measurements.
+            The estimation loop is based on the gradient of the sum of the squares of the
+            residuals, hence, the estimators get the various partial derivatives of all
+            measurements with respect to all parameters. A new state hopefully globally
+            reducing the residuals is estimated, and the parameters value are updated.
+            This estimation loops stops when either the convergence conditions are met
+            or the maximal number of iterations is exceeded.
+          </p>
+        </subsection>
+        <subsection name="12.4 Fine tuning" href="tuning">
+          <p>
+            One important tuning parameter for weighted least-squares solving is the
+            weight attributed to each measurement. This weights has two purposes:
+          </p>
+          <ul>
+            <li>fixing unit problems when combining different types of measurements</li>
+            <li>adjusting the influence of good or bad measurements on the solution</li>
+          </ul>
+          <p>
+            The weight is a multiplicative factor for the <em>square</em> of the residuals.
+            A common choice is to use the inverse of the variance of the measurements error
+            as the weighting factor for all measurements for one type. On our sailing ship
+            example, we may have a range measurements accuracy of about 1 meter and an angular
+            measurements accuracy of about 0.01 degree, or 1.7 10<sup>-4</sup> radians. So we
+            would use w=1.0 for distance measurements weight and w=3 10<sup>7</sup> for
+            angular measurements weight. If we knew that the measurements quality is bad
+            at tracking start because of measurement system warm-up delay for example, then
+            we would reduce the weight for the first measurements and use for example
+            w=0.1 and w=3 10<sup>6</sup> respectively, depending on the type.
+          </p>
+          <p>
+            After a problem has been set up, it is possible to fine tune the
+            way it will be solved. For example, it may appear the measurements are not
+            sufficient to get some parameters with sufficient confidence due to observability
+            problems. It is possible to fix some parameters in order to prevent the solver
+            from changing them. This is realized by passing <code>true</code> to the
+            <code>setBound</code> method of the parameter.
+          </p>
+          <p>
+            It is also possible to ignore some measurements by passing <code>true</code> to the
+            <code>setIgnored</code> method of the measurement. A typical use is to
+            <ol>
+              <li>
+                perform a first determination with all parameters, to check each measurement
+                residual after convergence (i.e. to compute the difference between the
+                measurement and its theoretical value as computed from the estimated parameters),
+              </li>
+              <li>
+                compute standard deviation for the measurements samples (one sample for each
+                measurements type)
+              </li>
+              <li>
+                ignore measurements whose residual are above some threshold (for example three
+                time the standard deviation on the residuals) assuming they correspond to
+                bad measurements,
+              </li>
+              <li>
+                perform another determination on the reduced measurements set.
+              </li>
+            </ol>
+          </p>
+        </subsection>
      </section>
   </body>
 </document>