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<h1 class="chapter"> 57. lsquares </h1>
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<h2 class="section"> 57.1 Definitions for lsquares </h2>
<dl>
<dt><u>Global variable:</u> <b>DETCOEF</b>
<a name="IDX1780"></a>
</dt>
<dd><p>This variable is used by functions <code>lsquares</code> and <code>plsquares</code> to store the Coefficient of Determination which measures the goodness of fit. It ranges from 0 (no correlation) to 1 (exact correlation).
</p>
<p>When <code>plsquares</code> is called with a list of dependent variables, <var>DETCOEF</var> is set to a list of Coefficients of Determination. See <code>plsquares</code> for details.
</p>
<p>See also <code>lsquares</code>.
</p></dd></dl>
<dl>
<dt><u>Function:</u> <b>lsquares</b><i> (<var>Mat</var>,<var>VarList</var>,<var>equation</var>,<var>ParamList</var>)</i>
<a name="IDX1781"></a>
</dt>
<dt><u>Function:</u> <b>lsquares</b><i> (<var>Mat</var>,<var>VarList</var>,<var>equation</var>,<var>ParamList</var>,<var>GuessList</var>)</i>
<a name="IDX1782"></a>
</dt>
<dd><p>Multiple nonlinear equation adjustment of a data table by the
"least squares" method. <var>Mat</var> is a matrix containing the data,
<var>VarList</var> is a list of variable names (one for each <var>Mat</var> column),
<var>equation</var> is the equation to adjust (it must be in the form:
<code>depvar=f(indepvari,..., paramj,...)</code>, <code>g(depvar)=f(indepvari,..., paramj,...)</code>
or <code>g(depvar, paramk,...)=f(indepvari,..., paramj,...)</code>), <var>ParamList</var> is the
list of the parameters to obtain, and <var>GuessList</var> is an optional list of initial
approximations to the parameters; when this last argument is present, <code>mnewton</code> is used
instead of <code>solve</code> in order to get the parameters.
</p>
<p>The equation may be fully nonlinear with respect to the independent
variables and to the dependent variable.
In order to use <code>solve()</code>, the equations must be linear or polynomial with
respect to the parameters. Equations like <code>y=a*b^x+c</code> may be adjusted for
<code>[a,b,c]</code> with <code>solve</code> if the <code>x</code> values are little positive integers and
there are few data (see the example in lsquares.dem).
<code>mnewton</code> allows to adjust a nonlinear equation with respect to the
parameters, but a good set of initial approximations must be provided.
</p>
<p>If possible, the adjusted equation is returned. If there exists more
than one solution, a list of equations is returned.
The Coefficient of Determination is displayed in order to inform about
the goodness of fit, from 0 (no correlation) to 1 (exact correlation).
This value is also stored in the global variable <var>DETCOEF</var>.
</p>
<p>Examples using <code>solve</code>:
</p><table><tr><td> </td><td><pre class="example">(%i1) load("lsquares")$
(%i2) lsquares(matrix([1,2,0],[3,5,4],[4,7,9],[5,8,10]),
[x,y,z], z=a*x*y+b*x+c*y+d, [a,b,c,d]);
Determination Coefficient = 1.0
x y + 23 y - 29 x - 19
(%o2) z = ----------------------
6
(%i3) lsquares(matrix([0,0],[1,0],[2,0],[3,8],[4,44]),
[n,p], p=a4*n^4+a3*n^3+a2*n^2+a1*n+a0,
[a0,a1,a2,a3,a4]);
Determination Coefficient = 1.0
4 3 2
3 n - 10 n + 9 n - 2 n
(%o3) p = -------------------------
6
(%i4) lsquares(matrix([1,7],[2,13],[3,25]),
[x,y], (y+c)^2=a*x+b, [a,b,c]);
Determination Coefficient = 1.0
(%o4) [y = 28 - sqrt(657 - 216 x),
y = sqrt(657 - 216 x) + 28]
(%i5) lsquares(matrix([1,7],[2,13],[3,25],[4,49]),
[x,y], y=a*b^x+c, [a,b,c]);
Determination Coefficient = 1.0
x
(%o5) y = 3 2 + 1
</pre></td></tr></table>
<p>Examples using <code>mnewton</code>:
</p><table><tr><td> </td><td><pre class="example">(%i6) load("lsquares")$
(%i7) lsquares(matrix([1.1,7.1],[2.1,13.1],[3.1,25.1],[4.1,49.1]),
[x,y], y=a*b^x+c, [a,b,c], [5,5,5]);
x
(%o7) y = 2.799098974610482 1.999999999999991
+ 1.099999999999874
(%i8) lsquares(matrix([1.1,4.1],[4.1,7.1],[9.1,10.1],[16.1,13.1]),
[x,y], y=a*x^b+c, [a,b,c], [4,1,2]);
.4878659755898127
(%o8) y = 3.177315891123101 x
+ .7723843491402264
(%i9) lsquares(matrix([0,2,4],[3,3,5],[8,6,6]),
[m,n,y], y=(A*m+B*n)^(1/3)+C, [A,B,C], [3,3,3]);
1/3
(%o9) y = (3.999999999999862 n + 4.999999999999359 m)
+ 2.00000000000012
</pre></td></tr></table>
<p>To use this function write first <code>load("lsquares")</code>. See also <code>DETCOEF</code> and <code>mnewton</code>.
</p></dd></dl>
<dl>
<dt><u>Function:</u> <b>plsquares</b><i> (<var>Mat</var>,<var>VarList</var>,<var>depvars</var>)</i>
<a name="IDX1783"></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="IDX1784"></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="IDX1785"></a>
</dt>
<dd><p>Multivariable polynomial adjustment of a data table by the "least squares"
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 "-" 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><table><tr><td> </td><td><pre class="example">(%i1) load("plsquares")$
(%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></td></tr></table>
<p>The same example without degree restrictions:
</p><table><tr><td> </td><td><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></td></tr></table>
<p>How many diagonals does a N-sides polygon have? What polynomial degree should be used?
</p><table><tr><td> </td><td><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></td></tr></table>
<p>How many ways do we have to put two queens without they are threatened into a n x n chessboard?
</p><table><tr><td> </td><td><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></td></tr></table>
<p>An example with six dependent variables:
</p><table><tr><td> </td><td><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></td></tr></table>
<p>To use this function write first <code>load("lsquares")</code>.
</p></dd></dl>
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