Provided by: libpdl-stats-perl_0.83-1build3_amd64 bug

NAME

       PDL::Stats::GLM -- general and generalized linear modeling methods such as ANOVA, linear
       regression, PCA, and logistic regression.

DESCRIPTION

       The terms FUNCTIONS and METHODS are arbitrarily used to refer to methods that are
       threadable and methods that are NOT threadable, respectively. FUNCTIONS except ols_t
       support bad value. PDL::Slatec strongly recommended for most METHODS, and it is required
       for logistic.

       P-values, where appropriate, are provided if PDL::GSL::CDF is installed.

SYNOPSIS

           use PDL::LiteF;
           use PDL::NiceSlice;
           use PDL::Stats::GLM;

           # do a multiple linear regression and plot the residuals

           my $y = pdl( 8, 7, 7, 0, 2, 5, 0 );

           my $x = pdl( [ 0, 1, 2, 3, 4, 5, 6 ],        # linear component
                        [ 0, 1, 4, 9, 16, 25, 36 ] );   # quadratic component

           my %m  = $y->ols( $x, {plot=>1} );

           print "$_\t$m{$_}\n" for (sort keys %m);

FUNCTIONS

   fill_m
         Signature: (a(n); float+ [o]b(n))

       Replaces bad values with sample mean. Mean is set to 0 if all obs are bad. Can be done
       inplace.

            perldl> p $data
            [
             [  5 BAD   2 BAD]
             [  7   3   7 BAD]
            ]

            perldl> p $data->fill_m
            [
             [      5     3.5       2     3.5]
             [      7       3       7 5.66667]
            ]

       The output pdl badflag is cleared.

   fill_rand
         Signature: (a(n); [o]b(n))

       Replaces bad values with random sample (with replacement) of good observations from the
       same variable. Can be done inplace.

           perldl> p $data
           [
            [  5 BAD   2 BAD]
            [  7   3   7 BAD]
           ]

           perldl> p $data->fill_rand

           [
            [5 2 2 5]
            [7 3 7 7]
           ]

       The output pdl badflag is cleared.

   dev_m
         Signature: (a(n); float+ [o]b(n))

       Replaces values with deviations from the mean. Can be done inplace.

       dev_m processes bad values.  It will set the bad-value flag of all output ndarrays if the
       flag is set for any of the input ndarrays.

   stddz
         Signature: (a(n); float+ [o]b(n))

       Standardize ie replace values with z_scores based on sample standard deviation from the
       mean (replace with 0s if stdv==0). Can be done inplace.

       stddz processes bad values.  It will set the bad-value flag of all output ndarrays if the
       flag is set for any of the input ndarrays.

   sse
         Signature: (a(n); b(n); float+ [o]c())

       Sum of squared errors between actual and predicted values.

       sse processes bad values.  It will set the bad-value flag of all output ndarrays if the
       flag is set for any of the input ndarrays.

   mse
         Signature: (a(n); b(n); float+ [o]c())

       Mean of squared errors between actual and predicted values, ie variance around predicted
       value.

       mse processes bad values.  It will set the bad-value flag of all output ndarrays if the
       flag is set for any of the input ndarrays.

   rmse
         Signature: (a(n); b(n); float+ [o]c())

       Root mean squared error, ie stdv around predicted value.

       rmse processes bad values.  It will set the bad-value flag of all output ndarrays if the
       flag is set for any of the input ndarrays.

   pred_logistic
         Signature: (a(n,m); b(m); float+ [o]c(n))

       Calculates predicted prob value for logistic regression.

           # glue constant then apply coeff returned by the logistic method

           $pred = $x->glue(1,ones($x->dim(0)))->pred_logistic( $m{b} );

       pred_logistic processes bad values.  It will set the bad-value flag of all output ndarrays
       if the flag is set for any of the input ndarrays.

   d0
         Signature: (a(n); float+ [o]c())

           my $d0 = $y->d0();

       Null deviance for logistic regression.

       d0 processes bad values.  It will set the bad-value flag of all output ndarrays if the
       flag is set for any of the input ndarrays.

   dm
         Signature: (a(n); b(n); float+ [o]c())

           my $dm = $y->dm( $y_pred );

             # null deviance
           my $d0 = $y->dm( ones($y->nelem) * $y->avg );

       Model deviance for logistic regression.

       dm processes bad values.  It will set the bad-value flag of all output ndarrays if the
       flag is set for any of the input ndarrays.

   dvrs
         Signature: (a(); b(); float+ [o]c())

       Deviance residual for logistic regression.

       dvrs processes bad values.  It will set the bad-value flag of all output ndarrays if the
       flag is set for any of the input ndarrays.

   ols_t
       Threaded version of ordinary least squares regression (ols). The price of threading was
       losing significance tests for coefficients (but see r2_change). The fitting function was
       shamelessly copied then modified from PDL::Fit::Linfit. Uses PDL::Slatec when possible but
       otherwise uses PDL::MatrixOps. Intercept is LAST of coeff if CONST => 1.

       ols_t does not handle bad values. consider fill_m or fill_rand if there are bad values.

       Default options (case insensitive):

           CONST   => 1,

       Usage:

           # DV, 2 person's ratings for top-10 box office movies
           # ascending sorted by box office numbers

           perldl> p $y = qsort ceil( random(10, 2)*5 )
           [
            [1 1 2 4 4 4 4 5 5 5]
            [1 2 2 2 3 3 3 3 5 5]
           ]

           # model with 2 IVs, a linear and a quadratic trend component

           perldl> $x = cat sequence(10), sequence(10)**2

           # suppose our novice modeler thinks this creates 3 different models
           # for predicting movie ratings

           perldl> p $x = cat $x, $x * 2, $x * 3
           [
            [
             [ 0  1  2  3  4  5  6  7  8  9]
             [ 0  1  4  9 16 25 36 49 64 81]
            ]
            [
             [  0   2   4   6   8  10  12  14  16  18]
             [  0   2   8  18  32  50  72  98 128 162]
            ]
            [
             [  0   3   6   9  12  15  18  21  24  27]
             [  0   3  12  27  48  75 108 147 192 243]
            ]
           ]

           perldl> p $x->info
           PDL: Double D [10,2,3]

           # insert a dummy dim between IV and the dim (model) to be threaded

           perldl> %m = $y->ols_t( $x->dummy(2) )

           perldl> p "$_\t$m{$_}\n" for (sort keys %m)

           # 2 persons' ratings, eached fitted with 3 "different" models

           F
           [
            [ 38.314159  25.087209]
            [ 38.314159  25.087209]
            [ 38.314159  25.087209]
           ]

           # df is the same across dv and iv models

           F_df    [2 7]
           F_p
           [
            [0.00016967051 0.00064215074]
            [0.00016967051 0.00064215074]
            [0.00016967051 0.00064215074]
           ]

           R2
           [
            [ 0.9162963 0.87756762]
            [ 0.9162963 0.87756762]
            [ 0.9162963 0.87756762]
           ]

           b
           [  # linear      quadratic     constant
            [
             [  0.99015152 -0.056818182   0.66363636]    # person 1
             [  0.18939394  0.022727273          1.4]    # person 2
            ]
            [
             [  0.49507576 -0.028409091   0.66363636]
             [  0.09469697  0.011363636          1.4]
            ]
            [
             [  0.33005051 -0.018939394   0.66363636]
             [ 0.063131313 0.0075757576          1.4]
            ]
           ]

           # our novice modeler realizes at this point that
           # the 3 models only differ in the scaling of the IV coefficients

           ss_model
           [
            [ 20.616667  13.075758]
            [ 20.616667  13.075758]
            [ 20.616667  13.075758]
           ]

           ss_residual
           [
            [ 1.8833333  1.8242424]
            [ 1.8833333  1.8242424]
            [ 1.8833333  1.8242424]
           ]

           ss_total        [22.5 14.9]
           y_pred
           [
            [
             [0.66363636  1.5969697  2.4166667  3.1227273  ...  4.9727273]
           ...

   r2_change
       Significance test for the incremental change in R2 when new variable(s) are added to an
       ols regression model. Returns the change stats as well as stats for both models. Based on
       ols_t. (One way to make up for the lack of significance tests for coeffs in ols_t).

       Default options (case insensitive):

           CONST   => 1,

       Usage:

           # suppose these are two persons' ratings for top 10 box office movies
           # ascending sorted by box office

           perldl> p $y = qsort ceil(random(10, 2) * 5)
           [
            [1 1 2 2 2 3 4 4 4 4]
            [1 2 2 3 3 3 4 4 5 5]
           ]

           # first IV is a simple linear trend

           perldl> p $x1 = sequence 10
           [0 1 2 3 4 5 6 7 8 9]

           # the modeler wonders if adding a quadratic trend improves the fit

           perldl> p $x2 = sequence(10) ** 2
           [0 1 4 9 16 25 36 49 64 81]

           # two difference models are given in two pdls
           # each as would be pass on to ols_t
           # the 1st model includes only linear trend
           # the 2nd model includes linear and quadratic trends
           # when necessary use dummy dim so both models have the same ndims

           perldl> %c = $y->r2_change( $x1->dummy(1), cat($x1, $x2) )

           perldl> p "$_\t$c{$_}\n" for (sort keys %c)
             #              person 1   person 2
           F_change        [0.72164948 0.071283096]
             # df same for both persons
           F_df    [1 7]
           F_p     [0.42370145 0.79717232]
           R2_change       [0.0085966043 0.00048562549]
           model0  HASH(0x8c10828)
           model1  HASH(0x8c135c8)

           # the answer here is no.

METHODS

   anova
       Analysis of variance. Uses type III sum of squares for unbalanced data.

       Dependent variable should be a 1D pdl. Independent variables can be passed as 1D perl
       array ref or 1D pdl.

       Will only calculate p-value ("F_p") if there are more samples than the product of
       categories of all the IVs.

       Supports bad value (by ignoring missing or BAD values in dependent and independent
       variables list-wise).

       Default options (case insensitive):

           V      => 1,          # carps if bad value in variables
           IVNM   => [],         # auto filled as ['IV_0', 'IV_1', ... ]
           PLOT   => 0,          # plots highest order effect
                                 # can set plot_means options here
           WIN    => undef,      # for plotting

       Usage:

           # suppose this is ratings for 12 apples

           perldl> p $y = qsort ceil( random(12)*5 )
           [1 1 2 2 2 3 3 4 4 4 5 5]

           # IV for types of apple

           perldl> p $a = sequence(12) % 3 + 1
           [1 2 3 1 2 3 1 2 3 1 2 3]

           # IV for whether we baked the apple

           perldl> @b = qw( y y y y y y n n n n n n )

           perldl> %m = $y->anova( $a, \@b, { IVNM=>['apple', 'bake'] } )

           perldl> p "$_\t$m{$_}\n" for (sort keys %m)
           # apple # m
           [
            [2.5   3 3.5]
           ]

           # apple # se
           [
            [0.64549722 0.91287093 0.64549722]
           ]

           # apple ~ bake # m
           [
            [1.5 1.5 2.5]
            [3.5 4.5 4.5]
           ]

           # apple ~ bake # se
           [
            [0.5 0.5 0.5]
            [0.5 0.5 0.5]
           ]

           # bake # m
           [
            [ 1.8333333  4.1666667]
           ]

           # bake # se
           [
            [0.30731815 0.30731815]
           ]

           F       7.6
           F_df    [5 6]
           F_p     0.0141586545851857
           ms_model        3.8
           ms_residual     0.5
           ss_model        19
           ss_residual     3
           ss_total        22
           | apple | F     2
           | apple | F_df  [2 6]
           | apple | F_p   0.216
           | apple | ms    1
           | apple | ss    2
           | apple ~ bake | F      0.666666666666667
           | apple ~ bake | F_df   [2 6]
           | apple ~ bake | F_p    0.54770848985725
           | apple ~ bake | ms     0.333333333333334
           | apple ~ bake | ss     0.666666666666667
           | bake | F      32.6666666666667
           | bake | F_df   [1 6]
           | bake | F_p    0.00124263849516693
           | bake | ms     16.3333333333333
           | bake | ss     16.3333333333333

   anova_rptd
       Repeated measures and mixed model anova. Uses type III sum of squares. The standard error
       (se) for the means are based on the relevant mean squared error from the anova, ie it is
       pooled across levels of the effect.

       Will only calculate p-value ("F_p") if there are more samples than the product of
       categories of all the IVs.

       anova_rptd supports bad value in the dependent and independent variables. It automatically
       removes bad data listwise, ie remove a subject's data if there is any cell missing for the
       subject.

       Default options (case insensitive):

           V      => 1,    # carps if bad value in dv
           IVNM   => [],   # auto filled as ['IV_0', 'IV_1', ... ]
           BTWN   => [],   # indices of between-subject IVs (matches IVNM indices)
           PLOT   => 0,    # plots highest order effect
                           # see plot_means() for more options
           WIN    => undef,      # for plotting

       Usage:

           Some fictional data: recall_w_beer_and_wings.txt

           Subject Beer    Wings   Recall
           Alex    1       1       8
           Alex    1       2       9
           Alex    1       3       12
           Alex    2       1       7
           Alex    2       2       9
           Alex    2       3       12
           Brian   1       1       12
           Brian   1       2       13
           Brian   1       3       14
           Brian   2       1       9
           Brian   2       2       8
           Brian   2       3       14
           ...

             # rtable allows text only in 1st row and col
           my ($data, $idv, $subj) = rtable 'recall_w_beer_and_wings.txt';

           my ($b, $w, $dv) = $data->dog;
             # subj and IVs can be 1d pdl or @ ref
             # subj must be the first argument
           my %m = $dv->anova_rptd( $subj, $b, $w, {ivnm=>['Beer', 'Wings']} );

           print "$_\t$m{$_}\n" for (sort keys %m);

           # Beer # m
           [
            [ 10.916667  8.9166667]
           ]

           # Beer # se
           [
            [ 0.4614791  0.4614791]
           ]

           # Beer ~ Wings # m
           [
            [   10     7]
            [ 10.5  9.25]
            [12.25  10.5]
           ]

           # Beer ~ Wings # se
           [
            [0.89170561 0.89170561]
            [0.89170561 0.89170561]
            [0.89170561 0.89170561]
           ]

           # Wings # m
           [
            [   8.5  9.875 11.375]
           ]

           # Wings # se
           [
            [0.67571978 0.67571978 0.67571978]
           ]

           ss_residual 19.0833333333333
           ss_subject  24.8333333333333
           ss_total    133.833333333333
           | Beer | F  9.39130434782609
           | Beer | F_p        0.0547977008378944
           | Beer | df 1
           | Beer | ms 24
           | Beer | ss 24
           | Beer || err df    3
           | Beer || err ms    2.55555555555556
           | Beer || err ss    7.66666666666667
           | Beer ~ Wings | F  0.510917030567687
           | Beer ~ Wings | F_p        0.623881438624431
           | Beer ~ Wings | df 2
           | Beer ~ Wings | ms 1.625
           | Beer ~ Wings | ss 3.25000000000001
           | Beer ~ Wings || err df    6
           | Beer ~ Wings || err ms    3.18055555555555
           | Beer ~ Wings || err ss    19.0833333333333
           | Wings | F 4.52851711026616
           | Wings | F_p       0.0632754786153548
           | Wings | df        2
           | Wings | ms        16.5416666666667
           | Wings | ss        33.0833333333333
           | Wings || err df   6
           | Wings || err ms   3.65277777777778
           | Wings || err ss   21.9166666666667

       For mixed model anova, ie when there are between-subject IVs involved, feed the IVs as
       above, but specify in BTWN which IVs are between-subject. For example, if we had added age
       as a between-subject IV in the above example, we would do

           my %m = $dv->anova_rptd( $subj, $age, $b, $w,
                                  { ivnm=>['Age', 'Beer', 'Wings'], btwn=>[0] });

   dummy_code
       Dummy coding of nominal variable (perl @ ref or 1d pdl) for use in regression.

       Supports BAD value (missing or 'BAD' values result in the corresponding pdl elements being
       marked as BAD).

           perldl> @a = qw(a a a b b b c c c)
           perldl> p $a = dummy_code(\@a)
           [
            [1 1 1 0 0 0 0 0 0]
            [0 0 0 1 1 1 0 0 0]
           ]

   effect_code
       Unweighted effect coding of nominal variable (perl @ ref or 1d pdl) for use in regression.
       returns in @ context coded pdl and % ref to level - pdl->dim(1) index.

       Supports BAD value (missing or 'BAD' values result in the corresponding pdl elements being
       marked as BAD).

           my @var = qw( a a a b b b c c c );
           my ($var_e, $map) = effect_code( \@var );

           print $var_e . $var_e->info . "\n";

           [
            [ 1  1  1  0  0  0 -1 -1 -1]
            [ 0  0  0  1  1  1 -1 -1 -1]
           ]
           PDL: Double D [9,2]

           print "$_\t$map->{$_}\n" for (sort keys %$map)
           a       0
           b       1
           c       2

   effect_code_w
       Weighted effect code for nominal variable. returns in @ context coded pdl and % ref to
       level - pdl->dim(1) index.

       Supports BAD value (missing or 'BAD' values result in the corresponding pdl elements being
       marked as BAD).

           perldl> @a = qw( a a b b b c c )
           perldl> p $a = effect_code_w(\@a)
           [
            [   1    1    0    0    0   -1   -1]
            [   0    0    1    1    1 -1.5 -1.5]
           ]

   interaction_code
       Returns the coded interaction term for effect-coded variables.

       Supports BAD value (missing or 'BAD' values result in the corresponding pdl elements being
       marked as BAD).

           perldl> $a = sequence(6) > 2
           perldl> p $a = $a->effect_code
           [
            [ 1  1  1 -1 -1 -1]
           ]

           perldl> $b = pdl( qw( 0 1 2 0 1 2 ) )
           perldl> p $b = $b->effect_code
           [
            [ 1  0 -1  1  0 -1]
            [ 0  1 -1  0  1 -1]
           ]

           perldl> p $ab = interaction_code( $a, $b )
           [
            [ 1  0 -1 -1 -0  1]
            [ 0  1 -1 -0 -1  1]
           ]

   ols
       Ordinary least squares regression, aka linear regression. Unlike ols_t, ols is not
       threadable, but it can handle bad value (by ignoring observations with bad value in
       dependent or independent variables list-wise) and returns the full model in list context
       with various stats.

       IVs ($x) should be pdl dims $y->nelem or $y->nelem x n_iv. Do not supply the constant
       vector in $x. Intercept is automatically added and returned as LAST of the coeffs if
       CONST=>1. Returns full model in list context and coeff in scalar context.

       Default options (case insensitive):

           CONST  => 1,
           PLOT   => 0,   # see plot_residuals() for plot options
           WIN    => undef,      # for plotting

       Usage:

           # suppose this is a person's ratings for top 10 box office movies
           # ascending sorted by box office

           perldl> p $y = qsort ceil( random(10) * 5 )
           [1 1 2 2 2 2 4 4 5 5]

           # construct IV with linear and quadratic component

           perldl> p $x = cat sequence(10), sequence(10)**2
           [
            [ 0  1  2  3  4  5  6  7  8  9]
            [ 0  1  4  9 16 25 36 49 64 81]
           ]

           perldl> %m = $y->ols( $x )

           perldl> p "$_\t$m{$_}\n" for (sort keys %m)

           F       40.4225352112676
           F_df    [2 7]
           F_p     0.000142834216344756
           R2      0.920314253647587

           # coeff  linear     quadratic  constant

           b       [0.21212121 0.03030303 0.98181818]
           b_p     [0.32800118 0.20303404 0.039910509]
           b_se    [0.20174693 0.021579989 0.38987581]
           b_t     [ 1.0514223   1.404219  2.5182844]
           ss_model        19.8787878787879
           ss_residual     1.72121212121212
           ss_total        21.6
           y_pred  [0.98181818  1.2242424  1.5272727  ...  4.6181818  5.3454545]

   ols_rptd
       Repeated measures linear regression (Lorch & Myers, 1990; Van den Noortgate & Onghena,
       2006). Handles purely within-subject design for now. See t/stats_ols_rptd.t for an example
       using the Lorch and Myers data.

       Usage:

           # This is the example from Lorch and Myers (1990),
           # a study on how characteristics of sentences affected reading time
           # Three within-subject IVs:
           # SP -- serial position of sentence
           # WORDS -- number of words in sentence
           # NEW -- number of new arguments in sentence

           # $subj can be 1D pdl or @ ref and must be the first argument
           # IV can be 1D @ ref or pdl
           # 1D @ ref is effect coded internally into pdl
           # pdl is left as is

           my %r = $rt->ols_rptd( $subj, $sp, $words, $new );

           print "$_\t$r{$_}\n" for (sort keys %r);

           (ss_residual)   58.3754646504336
           (ss_subject)    51.8590337714286
           (ss_total)  405.188241771429
           #      SP        WORDS      NEW
           F   [  7.208473  61.354153  1.0243311]
           F_p [0.025006181 2.619081e-05 0.33792837]
           coeff   [0.33337285 0.45858933 0.15162986]
           df  [1 1 1]
           df_err  [9 9 9]
           ms  [ 18.450705  73.813294 0.57026483]
           ms_err  [ 2.5595857  1.2030692 0.55671923]
           ss  [ 18.450705  73.813294 0.57026483]
           ss_err  [ 23.036272  10.827623  5.0104731]

   logistic
       Logistic regression with maximum likelihood estimation using PDL::Fit::LM (requires
       PDL::Slatec. Hence loaded with "require" in the sub instead of "use" at the beginning).

       IVs ($x) should be pdl dims $y->nelem or $y->nelem x n_iv. Do not supply the constant
       vector in $x. It is included in the model and returned as LAST of coeff. Returns full
       model in list context and coeff in scalar context.

       The significance tests are likelihood ratio tests (-2LL deviance) tests. IV significance
       is tested by comparing deviances between the reduced model (ie with the IV in question
       removed) and the full model.

       ***NOTE: the results here are qualitatively similar to but not identical with results from
       R, because different algorithms are used for the nonlinear parameter fit. Use with
       discretion***

       Default options (case insensitive):

           INITP => zeroes( $x->dim(1) + 1 ),    # n_iv + 1
           MAXIT => 1000,
           EPS   => 1e-7,

       Usage:

           # suppose this is whether a person had rented 10 movies

           perldl> p $y = ushort( random(10)*2 )
           [0 0 0 1 1 0 0 1 1 1]

           # IV 1 is box office ranking

           perldl> p $x1 = sequence(10)
           [0 1 2 3 4 5 6 7 8 9]

           # IV 2 is whether the movie is action- or chick-flick

           perldl> p $x2 = sequence(10) % 2
           [0 1 0 1 0 1 0 1 0 1]

           # concatenate the IVs together

           perldl> p $x = cat $x1, $x2
           [
            [0 1 2 3 4 5 6 7 8 9]
            [0 1 0 1 0 1 0 1 0 1]
           ]

           perldl> %m = $y->logistic( $x )

           perldl> p "$_\t$m{$_}\n" for (sort keys %m)

           D0  13.8629436111989
           Dm  9.8627829791575
           Dm_chisq    4.00016063204141
           Dm_df       2
           Dm_p        0.135324414081692
             #  ranking    genre      constant
           b   [0.41127706 0.53876358 -2.1201285]
           b_chisq     [ 3.5974504 0.16835559  2.8577151]
           b_p [0.057868258  0.6815774 0.090936587]
           iter        12
           y_pred      [0.10715577 0.23683909 ... 0.76316091 0.89284423]

           # to get the covariance out, supply a true value for the COV option:
           perldl> %m = $y->logistic( $x, {COV=>1} )
           perldl> p $m{cov};

   pca
       Principal component analysis. Based on corr instead of cov (bad values are ignored pair-
       wise. OK when bad values are few but otherwise probably should fill_m etc before pca). Use
       PDL::Slatec::eigsys() if installed, otherwise use PDL::MatrixOps::eigens_sym().

       Default options (case insensitive):

           CORR  => 1,     # boolean. use correlation or covariance
           PLOT  => 0,     # calls plot_screes by default
                           # can set plot_screes options here
           WIN    => undef,      # for plotting

       Usage:

           my $d = qsort random 10, 5;      # 10 obs on 5 variables
           my %r = $d->pca( \%opt );
           print "$_\t$r{$_}\n" for (keys %r);

           eigenvalue    # variance accounted for by each component
           [4.70192 0.199604 0.0471421 0.0372981 0.0140346]

           eigenvector   # dim var x comp. weights for mapping variables to component
           [
            [ -0.451251  -0.440696  -0.457628  -0.451491  -0.434618]
            [ -0.274551   0.582455   0.131494   0.255261  -0.709168]
            [   0.43282   0.500662  -0.139209  -0.735144 -0.0467834]
            [  0.693634  -0.428171   0.125114   0.128145  -0.550879]
            [  0.229202   0.180393  -0.859217     0.4173  0.0503155]
           ]

           loadings      # dim var x comp. correlation between variable and component
           [
            [ -0.978489  -0.955601  -0.992316   -0.97901  -0.942421]
            [ -0.122661   0.260224  0.0587476   0.114043  -0.316836]
            [ 0.0939749   0.108705 -0.0302253  -0.159616 -0.0101577]
            [   0.13396 -0.0826915  0.0241629  0.0247483   -0.10639]
            [  0.027153  0.0213708  -0.101789  0.0494365 0.00596076]
           ]

           pct_var       # percent variance accounted for by each component
           [0.940384 0.0399209 0.00942842 0.00745963 0.00280691]

       Plot scores along the first two components,

           $d->plot_scores( $r{eigenvector} );

   pca_sorti
       Determine by which vars a component is best represented. Descending sort vars by size of
       association with that component. Returns sorted var and relevant component indices.

       Default options (case insensitive):

           NCOMP => 10,     # maximum number of components to consider

       Usage:

             # let's see if we replicated the Osgood et al. (1957) study
           perldl> ($data, $idv, $ido) = rtable 'osgood_exp.csv', {v=>0}

             # select a subset of var to do pca
           perldl> $ind = which_id $idv, [qw( ACTIVE BASS BRIGHT CALM FAST GOOD HAPPY HARD LARGE HEAVY )]
           perldl> $data = $data( ,$ind)->sever
           perldl> @$idv = @$idv[list $ind]

           perldl> %m = $data->pca

           perldl> ($iv, $ic) = $m{loadings}->pca_sorti()

           perldl> p "$idv->[$_]\t" . $m{loadings}->($_,$ic)->flat . "\n" for (list $iv)

                    #   COMP0     COMP1    COMP2    COMP3
           HAPPY       [0.860191 0.364911 0.174372 -0.10484]
           GOOD        [0.848694 0.303652 0.198378 -0.115177]
           CALM        [0.821177 -0.130542 0.396215 -0.125368]
           BRIGHT      [0.78303 0.232808 -0.0534081 -0.0528796]
           HEAVY       [-0.623036 0.454826 0.50447 0.073007]
           HARD        [-0.679179 0.0505568 0.384467 0.165608]
           ACTIVE      [-0.161098 0.760778 -0.44893 -0.0888592]
           FAST        [-0.196042 0.71479 -0.471355 0.00460276]
           LARGE       [-0.241994 0.594644 0.634703 -0.00618055]
           BASS        [-0.621213 -0.124918 0.0605367 -0.765184]

   plot_means
       Plots means anova style. Can handle up to 4-way interactions (ie 4D pdl).

       Default options (case insensitive):

           IVNM  => ['IV_0', 'IV_1', 'IV_2', 'IV_3'],
           DVNM  => 'DV',
           AUTO  => 1,       # auto set dims to be on x-axis, line, panel
                             # if set 0, dim 0 goes on x-axis, dim 1 as lines
                             # dim 2+ as panels
             # see PDL::Graphics::PGPLOT::Window for next options
           WIN   => undef,   # pgwin object. not closed here if passed
                             # allows comparing multiple lines in same plot
                             # set env before passing WIN
           DEV   => '/xs',         # open and close dev for plotting if no WIN
                                   # defaults to '/png' in Windows
           SIZE  => 640,           # individual square panel size in pixels
           SYMBL => [0, 4, 7, 11],

       Usage:

             # see anova for mean / se pdl structure
           $mean->plot_means( $se, {IVNM=>['apple', 'bake']} );

       Or like this:

           $m{'# apple ~ bake # m'}->plot_means;

   plot_residuals
       Plots residuals against predicted values.

       Usage:

           $y->plot_residuals( $y_pred, { dev=>'/png' } );

       Default options (case insensitive):

            # see PDL::Graphics::PGPLOT::Window for more info
           WIN   => undef,  # pgwin object. not closed here if passed
                            # allows comparing multiple lines in same plot
                            # set env before passing WIN
           DEV   => '/xs',  # open and close dev for plotting if no WIN
                            # defaults to '/png' in Windows
           SIZE  => 640,    # plot size in pixels
           COLOR => 1,

   plot_scores
       Plots standardized original and PCA transformed scores against two components. (Thank you,
       Bob MacCallum, for the documentation suggestion that led to this function.)

       Default options (case insensitive):

         CORR  => 1,      # boolean. PCA was based on correlation or covariance
         COMP  => [0,1],  # indices to components to plot
           # see PDL::Graphics::PGPLOT::Window for next options
         WIN   => undef,  # pgwin object. not closed here if passed
                          # allows comparing multiple lines in same plot
                          # set env before passing WIN
         DEV   => '/xs',  # open and close dev for plotting if no WIN
                          # defaults to '/png' in Windows
         SIZE  => 640,    # plot size in pixels
         COLOR => [1,2],  # color for original and rotated scores

       Usage:

         my %p = $data->pca();
         $data->plot_scores( $p{eigenvector}, \%opt );

   plot_screes
       Scree plot. Plots proportion of variance accounted for by PCA components.

       Default options (case insensitive):

         NCOMP => 20,     # max number of components to plot
         CUT   => 0,      # set to plot cutoff line after this many components
                          # undef to plot suggested cutoff line for NCOMP comps
          # see PDL::Graphics::PGPLOT::Window for next options
         WIN   => undef,  # pgwin object. not closed here if passed
                          # allows comparing multiple lines in same plot
                          # set env before passing WIN
         DEV   => '/xs',  # open and close dev for plotting if no WIN
                          # defaults to '/png' in Windows
         SIZE  => 640,    # plot size in pixels
         COLOR => 1,

       Usage:

         # variance should be in descending order

         $pca{var}->plot_screes( {ncomp=>16} );

       Or, because NCOMP is used so often, it is allowed a shortcut,

         $pca{var}->plot_screes( 16 );

SEE ALSO

       PDL::Fit::Linfit

       PDL::Fit::LM

REFERENCES

       Cohen, J., Cohen, P., West, S.G., & Aiken, L.S. (2003). Applied Multiple
       Regression/correlation Analysis for the Behavioral Sciences (3rd ed.). Mahwah, NJ:
       Lawrence Erlbaum Associates Publishers.

       Hosmer, D.W., & Lemeshow, S. (2000). Applied Logistic Regression (2nd ed.). New York, NY:
       Wiley-Interscience.

       Lorch, R.F., & Myers, J.L. (1990). Regression analyses of repeated measures data in
       cognitive research. Journal of Experimental Psychology: Learning, Memory, & Cognition, 16,
       149-157.

       Osgood C.E., Suci, G.J., & Tannenbaum, P.H. (1957). The Measurement of Meaning. Champaign,
       IL: University of Illinois Press.

       Rutherford, A. (2001). Introducing Anova and Ancova: A GLM Approach (1st ed.). Thousand
       Oaks, CA: Sage Publications.

       Shlens, J. (2009). A Tutorial on Principal Component Analysis. Retrieved April 10, 2011
       from http://citeseerx.ist.psu.edu/

       The GLM procedure: unbalanced ANOVA for two-way design with interaction. (2008).
       SAS/STAT(R) 9.2 User's Guide. Retrieved June 18, 2009 from http://support.sas.com/

       Van den Noortgatea, W., & Onghenaa, P. (2006). Analysing repeated measures data in
       cognitive research: A comment on regression coefficient analyses. European Journal of
       Cognitive Psychology, 18, 937-952.

AUTHOR

       Copyright (C) 2009 Maggie J. Xiong <maggiexyz users.sourceforge.net>

       All rights reserved. There is no warranty. You are allowed to redistribute this software /
       documentation as described in the file COPYING in the PDL distribution.