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<h1 class="chapter"> 64. lsquares </h1>

<table class="menu" border="0" cellspacing="0">
<tr><td align="left" valign="top"><a href="#SEC308">64.1 Introduction to lsquares</a></td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top"><a href="#SEC309">64.2 Functions and Variables for lsquares</a></td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
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<p><a name="Item_003a-Introduction-to-lsquares"></a>
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<h2 class="section"> 64.1 Introduction to lsquares </h2>

<p><code>lsquares</code> is a collection of functions to implement the method of least squares
to estimate parameters for a model from numerical data.
</p>
<div class=categorybox>
&middot;
<p>@ref{Category: Statistical estimation} &middot;
@ref{Category: Share packages} &middot;
@ref{Category: Package lsquares}
</div>
</p>
<p><a name="Item_003a-Functions-and-Variables-for-lsquares"></a>
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<h2 class="section"> 64.2 Functions and Variables for lsquares </h2>

<p><a name="Item_003a-lsquares_005festimates"></a>
</p><dl>
<dt><u>Function:</u> <b>lsquares_estimates</b><i> (<var>D</var>, <var>x</var>, <var>e</var>, <var>a</var>)</i>
<a name="IDX2557"></a>
</dt>
<dt><u>Function:</u> <b>lsquares_estimates</b><i> (<var>D</var>, <var>x</var>, <var>e</var>, <var>a</var>, initial = <var>L</var>, tol = <var>t</var>)</i>
<a name="IDX2558"></a>
</dt>
<dd><p>Estimate parameters <var>a</var> to best fit the equation <var>e</var>
in the variables <var>x</var> and <var>a</var> to the data <var>D</var>,
as determined by the method of least squares.
<code>lsquares_estimates</code> first seeks an exact solution,
and if that fails, then seeks an approximate solution.
</p>
<p>The return value is a list of lists of equations of the form <code>[a = ..., b = ..., c = ...]</code>.
Each element of the list is a distinct, equivalent minimum of the mean square error.
</p>
<p>The data <var>D</var> must be a matrix.
Each row is one datum (which may be called a `record' or `case' in some contexts),
and each column contains the values of one variable across all data.
The list of variables <var>x</var> gives a name for each column of <var>D</var>,
even the columns which do not enter the analysis.
The list of parameters <var>a</var> gives the names of the parameters for which
estimates are sought.
The equation <var>e</var> is an expression or equation in the variables <var>x</var> and <var>a</var>;
if <var>e</var> is not an equation, it is treated the same as <code><var>e</var> = 0</code>.
</p>
<p>Additional arguments to <code>lsquares_estimates</code>
are specified as equations and passed on verbatim to the function <code>lbfgs</code>
which is called to find estimates by a numerical method
when an exact result is not found.
</p>
<p>If some exact solution can be found (via <code>solve</code>),
the data <var>D</var> may contain non-numeric values.
However, if no exact solution is found,
each element of <var>D</var> must have a numeric value.
This includes numeric constants such as <code>%pi</code> and <code>%e</code> as well as literal numbers
(integers, rationals, ordinary floats, and bigfloats).
Numerical calculations are carried out with ordinary floating-point arithmetic,
so all other kinds of numbers are converted to ordinary floats for calculations.
</p>
<p><code>load(lsquares)</code> loads this function.
</p>
<p>See also
<code>lsquares_estimates_exact</code>,
<code>lsquares_estimates_approximate</code>,<br>
<code>lsquares_mse</code>,
<code>lsquares_residuals</code>,
and <code>lsquares_residual_mse</code>.
</p>
<p>Examples:
</p>
<p>A problem for which an exact solution is found.
</p>
<pre class="example">(%i1) load (lsquares)$
(%i2) M : matrix (
        [1,1,1], [3/2,1,2], [9/4,2,1], [3,2,2], [2,2,1]);
                                  [ 1  1  1 ]
                                  [         ]
                                  [ 3       ]
                                  [ -  1  2 ]
                                  [ 2       ]
                                  [         ]
(%o2)                             [ 9       ]
                                  [ -  2  1 ]
                                  [ 4       ]
                                  [         ]
                                  [ 3  2  2 ]
                                  [         ]
                                  [ 2  2  1 ]
(%i3) lsquares_estimates (
         M, [z,x,y], (z+D)^2 = A*x+B*y+C, [A,B,C,D]);
                         59        27      10921        107
(%o3)            [[A = - --, B = - --, C = -----, D = - ---]]
                         16        16      1024         32
</pre>
<p>A problem for which no exact solution is found,
so <code>lsquares_estimates</code> resorts to numerical approximation.
</p>
<pre class="example">(%i1) load (lsquares)$
(%i2) M : matrix ([1, 1], [2, 7/4], [3, 11/4], [4, 13/4]);
                                   [ 1  1  ]
                                   [       ]
                                   [    7  ]
                                   [ 2  -  ]
                                   [    4  ]
                                   [       ]
(%o2)                              [    11 ]
                                   [ 3  -- ]
                                   [    4  ]
                                   [       ]
                                   [    13 ]
                                   [ 4  -- ]
                                   [    4  ]
(%i3) lsquares_estimates (
  M, [x,y], y=a*x^b+c, [a,b,c], initial=[3,3,3], iprint=[-1,0]);
(%o3) [[a = 1.387365874920637, b = .7110956639593767, 
                                        c = - .4142705622439105]]
</pre>
<div class=categorybox>
&middot;
<p>@ref{Category: Package lsquares} &middot;
@ref{Category: Numerical methods}
</div>
</p>
</dd></dl>

<p><a name="Item_003a-lsquares_005festimates_005fexact"></a>
</p><dl>
<dt><u>Function:</u> <b>lsquares_estimates_exact</b><i> (<var>MSE</var>, <var>a</var>)</i>
<a name="IDX2559"></a>
</dt>
<dd><p>Estimate parameters <var>a</var> to minimize the mean square error <var>MSE</var>,
by constructing a system of equations and attempting to solve them symbolically via <code>solve</code>.
The mean square error is an expression in the parameters <var>a</var>,
such as that returned by <code>lsquares_mse</code>.
</p>
<p>The return value is a list of lists of equations of the form <code>[a = ..., b = ..., c = ...]</code>.
The return value may contain zero, one, or two or more elements.
If two or more elements are returned,
each represents a distinct, equivalent minimum of the mean square error.
</p>
<p>See also
<code>lsquares_estimates</code>,
<code>lsquares_estimates_approximate</code>,
<code>lsquares_mse</code>,
<code>lsquares_residuals</code>,
and <code>lsquares_residual_mse</code>.
</p>
<p>Example:
</p>
<pre class="example">(%i1) load (lsquares)$
(%i2) M : matrix (
         [1,1,1], [3/2,1,2], [9/4,2,1], [3,2,2], [2,2,1]);
                           [ 1  1  1 ]
                           [         ]
                           [ 3       ]
                           [ -  1  2 ]
                           [ 2       ]
                           [         ]
(%o2)                      [ 9       ]
                           [ -  2  1 ]
                           [ 4       ]
                           [         ]
                           [ 3  2  2 ]
                           [         ]
                           [ 2  2  1 ]
(%i3) mse : lsquares_mse (M, [z, x, y], (z + D)^2 = A*x + B*y + C);
           5
          ====
          \                 2                         2
           &gt;    ((D + M    )  - C - M     B - M     A)
          /            i, 1          i, 3      i, 2
          ====
          i = 1
(%o3)     ---------------------------------------------
                                5
(%i4) lsquares_estimates_exact (mse, [A, B, C, D]);
                  59        27      10921        107
(%o4)     [[A = - --, B = - --, C = -----, D = - ---]]
                  16        16      1024         32
</pre>
<div class=categorybox>
&middot;
<p>@ref{Category: Package lsquares}
</div>
</p>
</dd></dl>

<p><a name="Item_003a-lsquares_005festimates_005fapproximate"></a>
</p><dl>
<dt><u>Function:</u> <b>lsquares_estimates_approximate</b><i> (<var>MSE</var>, <var>a</var>, initial = <var>L</var>, tol = <var>t</var>)</i>
<a name="IDX2560"></a>
</dt>
<dd><p>Estimate parameters <var>a</var> to minimize the mean square error <var>MSE</var>,
via the numerical minimization function <code>lbfgs</code>.
The mean square error is an expression in the parameters <var>a</var>,
such as that returned by <code>lsquares_mse</code>.
</p>
<p>The solution returned by <code>lsquares_estimates_approximate</code> is a local (perhaps global) minimum
of the mean square error.
For consistency with <code>lsquares_estimates_exact</code>,
the return value is a nested list which contains one element,
namely a list of equations of the form <code>[a = ..., b = ..., c = ...]</code>.
</p>
<p>Additional arguments to <code>lsquares_estimates_approximate</code>
are specified as equations and passed on verbatim to the function <code>lbfgs</code>.
</p>
<p><var>MSE</var> must evaluate to a number when the parameters are assigned numeric values.
This requires that the data from which <var>MSE</var> was constructed
comprise only numeric constants such as <code>%pi</code> and <code>%e</code> and literal numbers
(integers, rationals, ordinary floats, and bigfloats).
Numerical calculations are carried out with ordinary floating-point arithmetic,
so all other kinds of numbers are converted to ordinary floats for calculations.
</p>
<p><code>load(lsquares)</code> loads this function.
</p>
<p>See also
<code>lsquares_estimates</code>,
<code>lsquares_estimates_exact</code>,
<code>lsquares_mse</code>,<br>
<code>lsquares_residuals</code>,
and <code>lsquares_residual_mse</code>.
</p>
<p>Example:
</p>
<pre class="example">(%i1) load (lsquares)$
(%i2) M : matrix (
         [1,1,1], [3/2,1,2], [9/4,2,1], [3,2,2], [2,2,1]);
                           [ 1  1  1 ]
                           [         ]
                           [ 3       ]
                           [ -  1  2 ]
                           [ 2       ]
                           [         ]
(%o2)                      [ 9       ]
                           [ -  2  1 ]
                           [ 4       ]
                           [         ]
                           [ 3  2  2 ]
                           [         ]
                           [ 2  2  1 ]
(%i3) mse : lsquares_mse (M, [z, x, y], (z + D)^2 = A*x + B*y + C);
           5
          ====
          \                 2                         2
           &gt;    ((D + M    )  - C - M     B - M     A)
          /            i, 1          i, 3      i, 2
          ====
          i = 1
(%o3)     ---------------------------------------------
                                5
(%i4) lsquares_estimates_approximate (
              mse, [A, B, C, D], iprint = [-1, 0]);
(%o4) [[A = - 3.67850494740174, B = - 1.683070351177813, 
                 C = 10.63469950148635, D = - 3.340357993175206]]
</pre>
<div class=categorybox>
&middot;
<p>@ref{Category: Package lsquares} &middot;
@ref{Category: Numerical methods}
</div>
</p>
</dd></dl>

<p><a name="Item_003a-lsquares_005fmse"></a>
</p><dl>
<dt><u>Function:</u> <b>lsquares_mse</b><i> (<var>D</var>, <var>x</var>, <var>e</var>)</i>
<a name="IDX2561"></a>
</dt>
<dd><p>Returns the mean square error (MSE), a summation expression, for the equation <var>e</var>
in the variables <var>x</var>, with data <var>D</var>.
</p>
<p>The MSE is defined as:
</p>
<pre class="example">                    n
                   ====
               1   \                        2
               -    &gt;    (lhs(e ) - rhs(e ))
               n   /           i         i
                   ====
                   i = 1
</pre>
<p>where <var>n</var> is the number of data and <code><var>e</var>[i]</code> is the equation <var>e</var>
evaluated with the variables in <var>x</var> assigned values from the <code>i</code>-th datum, <code><var>D</var>[i]</code>.
</p>
<p><code>load(lsquares)</code> loads this function.
</p>
<p>Example:
</p>
<pre class="example">(%i1) load (lsquares)$
(%i2) M : matrix (
         [1,1,1], [3/2,1,2], [9/4,2,1], [3,2,2], [2,2,1]);
                           [ 1  1  1 ]
                           [         ]
                           [ 3       ]
                           [ -  1  2 ]
                           [ 2       ]
                           [         ]
(%o2)                      [ 9       ]
                           [ -  2  1 ]
                           [ 4       ]
                           [         ]
                           [ 3  2  2 ]
                           [         ]
                           [ 2  2  1 ]
</pre><pre class="example">(%i3) mse : lsquares_mse (M, [z, x, y], (z + D)^2 = A*x + B*y + C);
           5
          ====
          \                 2                         2
           &gt;    ((D + M    )  - C - M     B - M     A)
          /            i, 1          i, 3      i, 2
          ====
          i = 1
(%o3)     ---------------------------------------------
                                5
</pre><pre class="example">(%i4) diff (mse, D);
         5
        ====
        \                             2
      4  &gt;    (D + M    ) ((D + M    )  - C - M     B - M     A)
        /           i, 1         i, 1          i, 3      i, 2
        ====
        i = 1
(%o4) ----------------------------------------------------------
                                  5
</pre><pre class="example">(%i5) ''mse, nouns;
               2                 2         9 2               2
(%o5) (((D + 3)  - C - 2 B - 2 A)  + ((D + -)  - C - B - 2 A)
                                           4
           2               2         3 2               2
 + ((D + 2)  - C - B - 2 A)  + ((D + -)  - C - 2 B - A)
                                     2
           2             2
 + ((D + 1)  - C - B - A) )/5
</pre>
<div class=categorybox>
&middot;
<p>@ref{Category: Package lsquares}
</div>
</p>
</dd></dl>

<p><a name="Item_003a-lsquares_005fresiduals"></a>
</p><dl>
<dt><u>Function:</u> <b>lsquares_residuals</b><i> (<var>D</var>, <var>x</var>, <var>e</var>, <var>a</var>)</i>
<a name="IDX2562"></a>
</dt>
<dd><p>Returns the residuals for the equation <var>e</var>
with specified parameters <var>a</var> and data <var>D</var>.
</p>
<p><var>D</var> is a matrix, <var>x</var> is a list of variables,
<var>e</var> is an equation or general expression;
if not an equation, <var>e</var> is treated as if it were <code><var>e</var> = 0</code>.
<var>a</var> is a list of equations which specify values for any free parameters in <var>e</var> aside from <var>x</var>.
</p>
<p>The residuals are defined as:
</p>
<pre class="example">                        lhs(e ) - rhs(e )
                             i         i
</pre>
<p>where <code><var>e</var>[i]</code> is the equation <var>e</var>
evaluated with the variables in <var>x</var> assigned values from the <code>i</code>-th datum, <code><var>D</var>[i]</code>,
and assigning any remaining free variables from <var>a</var>.
</p>
<p><code>load(lsquares)</code> loads this function.
</p>
<p>Example:
</p>
<pre class="example">(%i1) load (lsquares)$
(%i2) M : matrix (
         [1,1,1], [3/2,1,2], [9/4,2,1], [3,2,2], [2,2,1]);
                                  [ 1  1  1 ]
                                  [         ]
                                  [ 3       ]
                                  [ -  1  2 ]
                                  [ 2       ]
                                  [         ]
(%o2)                             [ 9       ]
                                  [ -  2  1 ]
                                  [ 4       ]
                                  [         ]
                                  [ 3  2  2 ]
                                  [         ]
                                  [ 2  2  1 ]
(%i3) a : lsquares_estimates (
          M, [z,x,y], (z+D)^2 = A*x+B*y+C, [A,B,C,D]);
                         59        27      10921        107
(%o3)            [[A = - --, B = - --, C = -----, D = - ---]]
                         16        16      1024         32
(%i4) lsquares_residuals (
          M, [z,x,y], (z+D)^2 = A*x+B*y+C, first(a));
                            13    13    13  13  13
(%o4)                      [--, - --, - --, --, --]
                            64    64    32  64  64
</pre>
<div class=categorybox>
&middot;
<p>@ref{Category: Package lsquares}
</div>
</p>
</dd></dl>

<p><a name="Item_003a-lsquares_005fresidual_005fmse"></a>
</p><dl>
<dt><u>Function:</u> <b>lsquares_residual_mse</b><i> (<var>D</var>, <var>x</var>, <var>e</var>, <var>a</var>)</i>
<a name="IDX2563"></a>
</dt>
<dd><p>Returns the residual mean square error (MSE) for the equation <var>e</var>
with specified parameters <var>a</var> and data <var>D</var>.
</p>
<p>The residual MSE is defined as:
</p>
<pre class="example">                    n
                   ====
               1   \                        2
               -    &gt;    (lhs(e ) - rhs(e ))
               n   /           i         i
                   ====
                   i = 1
</pre>
<p>where <code><var>e</var>[i]</code> is the equation <var>e</var>
evaluated with the variables in <var>x</var> assigned values from the <code>i</code>-th datum, <code><var>D</var>[i]</code>,
and assigning any remaining free variables from <var>a</var>.
</p>
<p><code>load(lsquares)</code> loads this function.
</p>
<p>Example:
</p>
<pre class="example">(%i1) load (lsquares)$
(%i2) M : matrix (
         [1,1,1], [3/2,1,2], [9/4,2,1], [3,2,2], [2,2,1]);
                           [ 1  1  1 ]
                           [         ]
                           [ 3       ]
                           [ -  1  2 ]
                           [ 2       ]
                           [         ]
(%o2)                      [ 9       ]
                           [ -  2  1 ]
                           [ 4       ]
                           [         ]
                           [ 3  2  2 ]
                           [         ]
                           [ 2  2  1 ]
(%i3) a : lsquares_estimates (
             M, [z,x,y], (z+D)^2 = A*x+B*y+C, [A,B,C,D]);

                  59        27      10921        107
(%o3)     [[A = - --, B = - --, C = -----, D = - ---]]
                  16        16      1024         32
(%i4) lsquares_residual_mse (
             M, [z,x,y], (z + D)^2 = A*x + B*y + C, first (a));
                              169
(%o4)                         ----
                              2560
</pre>
<div class=categorybox>
&middot;
<p>@ref{Category: Package lsquares}
</div>
</p>
</dd></dl>

<p><a name="Item_003a-plsquares"></a>
</p><dl>
<dt><u>Function:</u> <b>plsquares</b><i> (<var>Mat</var>,<var>VarList</var>,<var>depvars</var>)</i>
<a name="IDX2564"></a>
</dt>
<dt><u>Function:</u> <b>plsquares</b><i> (<var>Mat</var>,<var>VarList</var>,<var>depvars</var>,<var>maxexpon</var>)</i>
<a name="IDX2565"></a>
</dt>
<dt><u>Function:</u> <b>plsquares</b><i> (<var>Mat</var>,<var>VarList</var>,<var>depvars</var>,<var>maxexpon</var>,<var>maxdegree</var>)</i>
<a name="IDX2566"></a>
</dt>
<dd><p>Multivariable polynomial adjustment of a data table by the &quot;least squares&quot;
method. <var>Mat</var> is a matrix containing the data, <var>VarList</var> is a list of variable names (one for each Mat column, but use &quot;-&quot; instead of varnames to ignore Mat columns), <var>depvars</var> is the name of a dependent variable or a list with one or more names of dependent variables (which names should be in <var>VarList</var>), <var>maxexpon</var> is the optional maximum exponent for each independent variable (1 by default), and <var>maxdegree</var> is the optional maximum polynomial degree (<var>maxexpon</var> by default); note that the sum of exponents of each term must be equal or smaller than <var>maxdegree</var>, and if <code>maxdgree = 0</code> then no limit is applied.
</p>
<p>If <var>depvars</var> is the name of a dependent variable (not in a list), <code>plsquares</code> returns the adjusted polynomial. If <var>depvars</var> is a list of one or more dependent variables, <code>plsquares</code> returns a list with the adjusted polynomial(s). The Coefficients of Determination  are displayed in order to inform about the goodness of fit, which ranges from 0 (no correlation) to 1 (exact correlation). These values are also stored in the global variable <var>DETCOEF</var> (a list if <var>depvars</var> is a list).
</p>

<p>A simple example of multivariable linear adjustment:
</p><pre class="example">(%i1) load(&quot;plsquares&quot;)$

(%i2) plsquares(matrix([1,2,0],[3,5,4],[4,7,9],[5,8,10]),
                [x,y,z],z);
     Determination Coefficient for z = .9897039897039897
                       11 y - 9 x - 14
(%o2)              z = ---------------
                              3
</pre>
<p>The same example without degree restrictions:
</p><pre class="example">(%i3) plsquares(matrix([1,2,0],[3,5,4],[4,7,9],[5,8,10]),
                [x,y,z],z,1,0);
     Determination Coefficient for z = 1.0
                    x y + 23 y - 29 x - 19
(%o3)           z = ----------------------
                              6
</pre>
<p>How many diagonals does a N-sides polygon have? What polynomial degree should be used?
</p><pre class="example">(%i4) plsquares(matrix([3,0],[4,2],[5,5],[6,9],[7,14],[8,20]),
                [N,diagonals],diagonals,5);
     Determination Coefficient for diagonals = 1.0
                                2
                               N  - 3 N
(%o4)              diagonals = --------
                                  2
(%i5) ev(%, N=9);   /* Testing for a 9 sides polygon */
(%o5)                 diagonals = 27
</pre>
<p>How many ways do we have to put two queens without they are threatened into a n x n chessboard?
</p><pre class="example">(%i6) plsquares(matrix([0,0],[1,0],[2,0],[3,8],[4,44]),
                [n,positions],[positions],4);
     Determination Coefficient for [positions] = [1.0]
                         4       3      2
                      3 n  - 10 n  + 9 n  - 2 n
(%o6)    [positions = -------------------------]
                                  6
(%i7) ev(%[1], n=8); /* Testing for a (8 x 8) chessboard */
(%o7)                positions = 1288
</pre>
<p>An example with six dependent variables:
</p><pre class="example">(%i8) mtrx:matrix([0,0,0,0,0,1,1,1],[0,1,0,1,1,1,0,0],
                  [1,0,0,1,1,1,0,0],[1,1,1,1,0,0,0,1])$
(%i8) plsquares(mtrx,[a,b,_And,_Or,_Xor,_Nand,_Nor,_Nxor],
                     [_And,_Or,_Xor,_Nand,_Nor,_Nxor],1,0);
      Determination Coefficient for
[_And, _Or, _Xor, _Nand, _Nor, _Nxor] =
[1.0, 1.0, 1.0, 1.0, 1.0, 1.0]
(%o2) [_And = a b, _Or = - a b + b + a,
_Xor = - 2 a b + b + a, _Nand = 1 - a b,
_Nor = a b - b - a + 1, _Nxor = 2 a b - b - a + 1]
</pre>
<p>To use this function write first <code>load(&quot;lsquares&quot;)</code>.
</p>
<div class=categorybox>
&middot;
<p>@ref{Category: Package lsquares} &middot;
@ref{Category: Numerical methods}
</div>
</p>
</dd></dl>


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