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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 Statistics, Sets, Optimization, Top |
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6 @chapter Statistics |
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7 |
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8 I hope that someday Octave will include more statistics functions. If |
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9 you would like to help improve Octave in this area, please contact |
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10 @email{bug-octave@@bevo.che.wisc.edu}. |
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11 |
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12 @deftypefn {Function File} {} mean (@var{x}) |
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13 If @var{x} is a vector, compute the mean of the elements of @var{x} |
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14 @iftex |
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15 @tex |
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16 $$ {\rm mean}(x) = \bar{x} = {1\over N} \sum_{i=1}^N x_i $$ |
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17 @end tex |
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18 @end iftex |
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19 @ifinfo |
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20 |
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21 @example |
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22 mean (x) = SUM_i x(i) / N |
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23 @end example |
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24 @end ifinfo |
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25 If @var{x} is a matrix, compute the mean for each column and return them |
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26 in a row vector. |
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27 @end deftypefn |
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28 |
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29 @deftypefn {Function File} {} median (@var{x}) |
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30 If @var{x} is a vector, compute the median value of the elements of |
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31 @var{x}. |
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32 @iftex |
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33 @tex |
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34 $$ |
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35 {\rm median} (x) = |
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36 \cases{x(\lceil N/2\rceil), & $N$ odd;\cr |
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37 (x(N/2)+x(N/2+1))/2, & $N$ even.} |
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38 $$ |
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39 @end tex |
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40 @end iftex |
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41 @ifinfo |
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42 |
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43 @example |
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44 @group |
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45 x(ceil(N/2)), N odd |
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46 median(x) = |
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47 (x(N/2) + x((N/2)+1))/2, N even |
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48 @end group |
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49 @end example |
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50 @end ifinfo |
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51 If @var{x} is a matrix, compute the median value for each |
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52 column and return them in a row vector. |
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53 @end deftypefn |
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54 |
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55 @deftypefn {Function File} {} std (@var{x}) |
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56 If @var{x} is a vector, compute the standard deviation of the elements |
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57 of @var{x}. |
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58 @iftex |
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59 @tex |
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60 $$ |
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61 {\rm std} (x) = \sigma (x) = \sqrt{{\sum_{i=1}^N (x_i - \bar{x}) \over N - 1}} |
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62 $$ |
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63 @end tex |
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64 @end iftex |
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65 @ifinfo |
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66 |
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67 @example |
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68 @group |
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69 std (x) = sqrt (sumsq (x - mean (x)) / (n - 1)) |
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70 @end group |
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71 @end example |
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72 @end ifinfo |
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73 If @var{x} is a matrix, compute the standard deviation for |
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74 each column and return them in a row vector. |
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75 @end deftypefn |
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76 |
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77 @deftypefn {Function File} {} cov (@var{x}, @var{y}) |
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78 If each row of @var{x} and @var{y} is an observation and each column is |
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79 a variable, the (@var{i},@var{j})-th entry of |
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80 @code{cov (@var{x}, @var{y})} is the covariance between the @var{i}-th |
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81 variable in @var{x} and the @var{j}-th variable in @var{y}. If called |
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82 with one argument, compute @code{cov (@var{x}, @var{x})}. |
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83 @end deftypefn |
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84 |
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85 @deftypefn {Function File} {} corrcoef (@var{x}, @var{y}) |
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86 If each row of @var{x} and @var{y} is an observation and each column is |
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87 a variable, the (@var{i},@var{j})-th entry of |
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88 @code{corrcoef (@var{x}, @var{y})} is the correlation between the |
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89 @var{i}-th variable in @var{x} and the @var{j}-th variable in @var{y}. |
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90 If called with one argument, compute @code{corrcoef (@var{x}, @var{x})}. |
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91 @end deftypefn |
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92 |
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93 @deftypefn {Function File} {} kurtosis (@var{x}) |
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94 If @var{x} is a vector of length @var{N}, return the kurtosis |
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95 @iftex |
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96 @tex |
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97 $$ |
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98 {\rm kurtosis} (x) = {1\over N \sigma(x)^4} \sum_{i=1}^N (x_i-\bar{x})^4 - 3 |
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99 $$ |
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100 @end tex |
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101 @end iftex |
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102 @ifinfo |
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103 |
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104 @example |
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105 kurtosis (x) = N^(-1) std(x)^(-4) sum ((x - mean(x)).^4) - 3 |
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106 @end example |
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107 @end ifinfo |
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108 |
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109 @noindent |
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110 of @var{x}. If @var{x} is a matrix, return the row vector containing |
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111 the kurtosis of each column. |
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112 @end deftypefn |
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113 |
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114 @deftypefn {Function File} {} mahalanobis (@var{x}, @var{y}) |
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115 Return the Mahalanobis' D-square distance between the multivariate |
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116 samples @var{x} and @var{y}, which must have the same number of |
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117 components (columns), but may have a different number of observations |
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118 (rows). |
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119 @end deftypefn |
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120 |
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121 @deftypefn {Function File} {} skewness (@var{x}) |
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122 If @var{x} is a vector of length @var{N}, return the skewness |
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123 @iftex |
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124 @tex |
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125 $$ |
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126 {\rm skewness} (x) = {1\over N \sigma(x)^3} \sum_{i=1}^N (x_i-\bar{x})^3 |
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127 $$ |
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128 @end tex |
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129 @end iftex |
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130 @ifinfo |
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131 |
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132 @example |
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133 skewness (x) = N^(-1) std(x)^(-3) sum ((x - mean(x)).^3) |
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134 @end example |
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135 @end ifinfo |
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136 |
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137 @noindent |
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138 of @var{x}. If @var{x} is a matrix, return the row vector containing |
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139 the skewness of each column. |
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140 @end deftypefn |
