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59.1 Introduction to lsquares | ||

59.2 Functions and Variables for lsquares |

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`lsquares`

is a collection of functions to implement the method of least squares
to estimate parameters for a model from numerical data.

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__Function:__**lsquares_estimates***(*`D`,`x`,`e`,`a`)__Function:__**lsquares_estimates***(*`D`,`x`,`e`,`a`, initial =`L`, tol =`t`)Estimate parameters

`a`to best fit the equation`e`in the variables`x`and`a`to the data`D`, as determined by the method of least squares.`lsquares_estimates`

first seeks an exact solution, and if that fails, then seeks an approximate solution.The return value is a list of lists of equations of the form

`[a = ..., b = ..., c = ...]`

. Each element of the list is a distinct, equivalent minimum of the mean square error.The data

`D`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`x`gives a name for each column of`D`, even the columns which do not enter the analysis. The list of parameters`a`gives the names of the parameters for which estimates are sought. The equation`e`is an expression or equation in the variables`x`and`a`; if`e`is not an equation, it is treated the same as

.`e`= 0Additional arguments to

`lsquares_estimates`

are specified as equations and passed on verbatim to the function`lbfgs`

which is called to find estimates by a numerical method when an exact result is not found.If some exact solution can be found (via

`solve`

), the data`D`may contain non-numeric values. However, if no exact solution is found, each element of`D`must have a numeric value. This includes numeric constants such as`%pi`

and`%e`

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.`load(lsquares)`

loads this function.See also

`lsquares_estimates_exact`

,`lsquares_estimates_approximate`

,`lsquares_mse`

,`lsquares_residuals`

, and`lsquares_residual_mse`

.Examples:

A problem for which an exact solution is found.

(%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

A problem for which no exact solution is found, so

`lsquares_estimates`

resorts to numerical approximation.(%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]]

__Function:__**lsquares_estimates_exact***(*`MSE`,`a`)Estimate parameters

`a`to minimize the mean square error`MSE`, by constructing a system of equations and attempting to solve them symbolically via`solve`

. The mean square error is an expression in the parameters`a`, such as that returned by`lsquares_mse`

.The return value is a list of lists of equations of the form

`[a = ..., b = ..., c = ...]`

. 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.See also

`lsquares_estimates`

,`lsquares_estimates_approximate`

,`lsquares_mse`

,`lsquares_residuals`

, and`lsquares_residual_mse`

.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 > ((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

__Function:__**lsquares_estimates_approximate***(*`MSE`,`a`, initial =`L`, tol =`t`)Estimate parameters

`a`to minimize the mean square error`MSE`, via the numerical minimization function`lbfgs`

. The mean square error is an expression in the parameters`a`, such as that returned by`lsquares_mse`

.The solution returned by

`lsquares_estimates_approximate`

is a local (perhaps global) minimum of the mean square error. For consistency with`lsquares_estimates_exact`

, the return value is a nested list which contains one element, namely a list of equations of the form`[a = ..., b = ..., c = ...]`

.Additional arguments to

`lsquares_estimates_approximate`

are specified as equations and passed on verbatim to the function`lbfgs`

.`MSE`must evaluate to a number when the parameters are assigned numeric values. This requires that the data from which`MSE`was constructed comprise only numeric constants such as`%pi`

and`%e`

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.`load(lsquares)`

loads this function.See also

`lsquares_estimates`

,`lsquares_estimates_exact`

,`lsquares_mse`

,`lsquares_residuals`

, and`lsquares_residual_mse`

.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 > ((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]]

__Function:__**lsquares_mse***(*`D`,`x`,`e`)Returns the mean square error (MSE), a summation expression, for the equation

`e`in the variables`x`, with data`D`.The MSE is defined as:

n ==== 1 \ 2 - > (lhs(e ) - rhs(e )) n / i i ==== i = 1

where

`n`is the number of data and

is the equation`e`[i]`e`evaluated with the variables in`x`assigned values from the`i`

-th datum,

.`D`[i]`load(lsquares)`

loads this function.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 > ((D + M ) - C - M B - M A) / i, 1 i, 3 i, 2 ==== i = 1 (%o3) --------------------------------------------- 5 (%i4) diff (mse, D); 5 ==== \ 2 4 > (D + M ) ((D + M ) - C - M B - M A) / i, 1 i, 1 i, 3 i, 2 ==== i = 1 (%o4) ---------------------------------------------------------- 5 (%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

__Function:__**lsquares_residuals***(*`D`,`x`,`e`,`a`)Returns the residuals for the equation

`e`with specified parameters`a`and data`D`.`D`is a matrix,`x`is a list of variables,`e`is an equation or general expression; if not an equation,`e`is treated as if it were

.`e`= 0`a`is a list of equations which specify values for any free parameters in`e`aside from`x`.The residuals are defined as:

lhs(e ) - rhs(e ) i i

where

is the equation`e`[i]`e`evaluated with the variables in`x`assigned values from the`i`

-th datum,

, and assigning any remaining free variables from`D`[i]`a`.`load(lsquares)`

loads this function.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

__Function:__**lsquares_residual_mse***(*`D`,`x`,`e`,`a`)Returns the residual mean square error (MSE) for the equation

`e`with specified parameters`a`and data`D`.The residual MSE is defined as:

n ==== 1 \ 2 - > (lhs(e ) - rhs(e )) n / i i ==== i = 1

where

is the equation`e`[i]`e`evaluated with the variables in`x`assigned values from the`i`

-th datum,

, and assigning any remaining free variables from`D`[i]`a`.`load(lsquares)`

loads this function.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

__Function:__**plsquares***(*`Mat`,`VarList`,`depvars`)__Function:__**plsquares***(*`Mat`,`VarList`,`depvars`,`maxexpon`)__Function:__**plsquares***(*`Mat`,`VarList`,`depvars`,`maxexpon`,`maxdegree`)Multivariable polynomial adjustment of a data table by the "least squares" method.

`Mat`is a matrix containing the data,`VarList`is a list of variable names (one for each Mat column, but use "-" instead of varnames to ignore Mat columns),`depvars`is the name of a dependent variable or a list with one or more names of dependent variables (which names should be in`VarList`),`maxexpon`is the optional maximum exponent for each independent variable (1 by default), and`maxdegree`is the optional maximum polynomial degree (`maxexpon`by default); note that the sum of exponents of each term must be equal or smaller than`maxdegree`, and if`maxdgree = 0`

then no limit is applied.If

`depvars`is the name of a dependent variable (not in a list),`plsquares`

returns the adjusted polynomial. If`depvars`is a list of one or more dependent variables,`plsquares`

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`DETCOEF`(a list if`depvars`is a list).A simple example of multivariable linear adjustment:

(%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

The same example without degree restrictions:

(%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

How many diagonals does a N-sides polygon have? What polynomial degree should be used?

(%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

How many ways do we have to put two queens without they are threatened into a n x n chessboard?

(%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

An example with six dependent variables:

(%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]

To use this function write first

`load("lsquares")`

.

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