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1 @c Copyright (C) 1996, 1997 John W. Eaton |
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2 @c This is part of the Octave manual. |
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3 @c For copying conditions, see the file gpl.texi. |
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4 |
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5 @node Optimization, Statistics, Differential Equations, Top |
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6 @chapter Optimization |
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7 |
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8 @menu |
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9 * Quadratic Programming:: |
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10 * Nonlinear Programming:: |
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11 * Linear Least Squares:: |
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12 @end menu |
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13 |
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14 @c @cindex linear programming |
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15 @cindex quadratic programming |
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16 @cindex nonlinear programming |
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17 @cindex optimization |
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18 @cindex LP |
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19 @cindex QP |
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20 @cindex NLP |
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21 |
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22 @node Quadratic Programming, Nonlinear Programming, Optimization, Optimization |
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23 @section Quadratic Programming |
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24 |
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25 @node Nonlinear Programming, Linear Least Squares, Quadratic Programming, Optimization |
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26 @section Nonlinear Programming |
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27 |
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28 @node Linear Least Squares, , Nonlinear Programming, Optimization |
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29 @section Linear Least Squares |
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30 |
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31 @deftypefn {Function File} {[@var{beta}, @var{v}, @var{r}] =} gls (@var{y}, @var{x}, @var{o}) |
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32 Generalized least squares estimation for the multivariate model |
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33 @iftex |
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34 @tex |
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35 $y = x b + e$ |
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36 with $\bar{e} = 0$ and cov(vec($e$)) = $(s^2)o$, |
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37 @end tex |
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38 @end iftex |
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39 @ifinfo |
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40 @code{@var{y} = @var{x} * @var{b} + @var{e}} with @code{mean (@var{e}) = |
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41 0} and @code{cov (vec (@var{e})) = (@var{s}^2)*@var{o}}, |
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42 @end ifinfo |
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43 where |
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44 @iftex |
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45 @tex |
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46 $y$ is a $t \times p$ matrix, $x$ is a $t \times k$ matrix, $b$ is a $k |
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47 \times p$ matrix, $e$ is a $t \times p$ matrix, and $o$ is a $tp \times |
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48 tp$ matrix. |
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49 @end tex |
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50 @end iftex |
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51 @ifinfo |
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52 @var{Y} is a @var{T} by @var{p} matrix, @var{X} is a @var{T} by @var{k} |
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53 matrix, @var{B} is a @var{k} by @var{p} matrix, @var{E} is a @var{T} by |
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54 @var{p} matrix, and @var{O} is a @var{T}@var{p} by @var{T}@var{p} |
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55 matrix. |
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56 @end ifinfo |
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57 |
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58 @noindent |
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59 Each row of Y and X is an observation and each column a variable. |
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60 |
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61 The return values @var{beta}, @var{v}, and @var{r} are defined as |
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62 follows. |
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63 |
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64 @table @var |
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65 @item beta |
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66 The GLS estimator for @var{b}. |
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67 |
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68 @item v |
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69 The GLS estimator for @code{@var{s}^2}. |
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70 |
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71 @item r |
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72 The matrix of GLS residuals, @code{@var{r} = @var{y} - @var{x} * |
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73 @var{beta}}. |
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74 @end table |
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75 @end deftypefn |
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76 |
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77 @deftypefn {Function File} {[@var{beta}, @var{sigma}, @var{r}] =} ols (@var{y}, @var{x}) |
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78 Ordinary least squares estimation for the multivariate model |
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79 @iftex |
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80 @tex |
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81 $y = x b + e$ |
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82 with |
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83 $\bar{e} = 0$, and cov(vec($e$)) = kron ($s, I$) |
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84 @end tex |
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85 @end iftex |
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86 @ifinfo |
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87 @code{@var{y} = @var{x}*@var{b} + @var{e}} with |
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88 @code{mean (@var{e}) = 0} and @code{cov (vec (@var{e})) = kron (@var{s}, |
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89 @var{I})}. |
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90 @end ifinfo |
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91 where |
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92 @iftex |
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93 @tex |
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94 $y$ is a $t \times p$ matrix, $x$ is a $t \times k$ matrix, |
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95 $b$ is a $k \times p$ matrix, and $e$ is a $t \times p$ matrix. |
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96 @end tex |
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97 @end iftex |
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98 @ifinfo |
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99 @var{y} is a @var{t} by @var{p} matrix, @var{X} is a @var{t} by @var{k} |
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100 matrix, @var{B} is a @var{k} by @var{p} matrix, and @var{e} is a @var{t} |
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101 by @var{p} matrix. |
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102 @end ifinfo |
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103 |
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104 Each row of @var{y} and @var{x} is an observation and each column a |
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105 variable. |
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106 |
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107 The return values @var{beta}, @var{sigma}, and @var{r} are defined as |
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108 follows. |
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109 |
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110 @table @var |
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111 @item beta |
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112 The OLS estimator for @var{b}, @code{@var{beta} = pinv (@var{x}) * |
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113 @var{y}}, where @code{pinv (@var{x})} denotes the pseudoinverse of |
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114 @var{x}. |
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115 |
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116 @item sigma |
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117 The OLS estimator for the matrix @var{s}, |
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118 |
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119 @example |
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120 @group |
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121 @var{sigma} = (@var{y}-@var{x}*@var{beta})' * (@var{y}-@var{x}*@var{beta}) / (@var{t}-rank(@var{x})) |
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122 @end group |
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123 @end example |
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124 |
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125 @item r |
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126 The matrix of OLS residuals, @code{@var{r} = @var{y} - @var{x} * @var{beta}}. |
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127 @end table |
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128 @end deftypefn |
