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# 59. lsquares

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## 59.1 Introduction to lsquares

`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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## 59.2 Functions and Variables for lsquares

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 = 0`.

  Additional 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 `e[i]` is the equation 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 `e[i]` is the equation e evaluated with the variables in x assigned values from the `i`-th datum, `D[i]`, and assigning any remaining free variables from 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 `e[i]` is the equation e evaluated with the variables in x assigned values from the `i`-th datum, `D[i]`, and assigning any remaining free variables from 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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