\defmodule {gofs}

This module provides tools for computing goodness-of-fit test statistics
for testing the hypothesis $\cH_0$ that a sample of $N$ observations 
$V_1,\dots,V_N$ comes from a given univariate probability 
distribution $F$.
These test statistics generally measure, in different ways, the
distance between $F$ and the {\em empirical distribution function\/} 
(EDF) $\hat F_N$ of $V_1,\dots,V_N$.
They are also called EDF test statistics.
The observations $V_i$ are usually transformed into $U_i = F(V_i)$,
which always satisfy $0\le U_i\le 1$, and which
follow the $U(0,1)$ distribution under $\cH_0$.
These observations are also usually sorted.
Here, $U_{(1)}, \dots, U_{(N)}$ stand for $N$ observations
$U_1,\dots,U_N$ sorted by increasing order, where $0\le U_i\le 1$.

Procedures for applying certain types of transformations to the
observations $V_i$ or $U_i$ are also provided.
This includes the transformation $U_i = F(V_i)$, as well as 
the power ratio and iterated spacing transformations \cite{tSTE86a}.


\bigskip\hrule\medskip
\code\hide
/* gofs.h for ANSI C */
#ifndef GOFS_H
#define GOFS_H
\endhide
#include <testu01/bitset.h>       /* From the library mylib */
#include <testu01/fmass.h>
#include <testu01/fdist.h>
#include <testu01/wdist.h>
\endcode


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\guisec{Environment variables}

\code


extern double gofs_MinExpected;
\endcode
  \tab  Used for the chi-square tests.
  When a chi-square test statistic is computed, the expected number
  of observations in each class should be large enough if we want
  the chi-square test statistic to follow approximately the
  chi-square distribution.  Larger expected numbers are usually
  required when these numbers differ between classes \cite{tREA88a}.
  The function {\tt gofs\_MergeClasses} can be used to regroup classes
  in order to make sure that the expected number in each class is
  at least {\tt gofs\_MinExpected}.
  The default value of this variable is 10.0.
\iffalse %%%
  This is for testu01:
  For some tests, the software will merge
  classes in such a way that this is always so. For others, an error
  message will be printed if this not the case (see the {\it restrictions}
  that apply for the different tests).  
\fi %%
 \hpierre {Et si on mettait cette variable en param\`etre \`a
    {\tt MergeClasses} \`a la place?  Peser le pour et le contre.
    En fait, dans plusieurs situations, la valeur 10.0 est beaucoup
    trop \'elev\'ee. }
 \hrichard {Cette variable est vraiment d'environnement. Depuis des 
  ann\'ees, on ne l'a jamais vari\'e. Et les fonctions qui l'utilisent
  sont appel\'ees des douzaines de fois partout. Ce type de variables
  devrait demeurer environnement.}
  \endtab
\ifdetailed  %%%%
\code


extern double gofs_EpsilonAD;
\endcode
 \tab When computing the Anderson-Darling statistic $A_N^2$, 
  all observations $U_i$ are projected to the interval 
  $[\epsilon,\,1-\epsilon]$ for some $\epsilon > 0$, in order to 
  avoid numerical overflow when taking the logarithm of $U_i$ or
  $1-U_i$.  This variable gives the value of $\epsilon$;
  its default value is {\tt DBL\_EPSILON/2.0}.
  {\tt DBL\_EPSILON} (from {\tt float.h}) is usually $2^{-52}$.
 \hpierre {Autre choix possible: cacher cela dans le .c.
    Mais il ne semble pas y avoir d'avantage \`a faire cela,
    tandis que le laisser ici peut permettre aux ``experts'' de faire
    \'eventuellement des exp\'eriences avec le choix de $\epsilon$. }
 \endtab
\fi %%%% detailed

\hrichard {Il faut mettre ``{\tt extern}'' parce que sinon 
  il y aurait autant d'instances de cette variable que d'inclusions
  de ce fichier *.h dans le programme, ce qui provoquerait une erreur 
  avec plusieurs compilateurs. 
  Parmi toutes les conventions des diff\'erents compilateurs pour les 
  variables externes, celle-ci est la plus propre et est compatible
  ANSI C. Parmi les 5 conventions utilis\'ees couramment, c'est celle
  recommand\'ee par Harbison+Steele p. 94.}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\guisec{Transforming the observations}

\code

void gofs_ContUnifTransform (double V[], long N, wdist_CFUNC F,
                             double par[], double U[]);
\endcode
\tab  Applies the transformation $U_i = F(V_i)$ to the values in
  {\tt V[1..N]}, where $F$ is a {\em continuous\/} distribution function 
  given by {\tt F} and with parameters in {\tt par},
  and puts the result in {\tt U[1..N]}.
  If {\tt V} contains random variables from the distribution function
  {\tt F}, then {\tt U} will contain uniform random variables over $(0,1)$.
\endtab
\code


void gofs_DiscUnifTransform (double V[], long N, wdist_DFUNC F,
                             fmass_INFO W, double U[]);
\endcode
\tab  Applies the transformation $U_i = F(V_i)$ to the values in
  {\tt V[1..N]}, where $F$ is a {\em discrete\/} distribution function 
  specified by {\tt F} and the previously-created structure {\tt W},
  and puts the result in {\tt U[1..N]}.
  Note: If {\tt V[1..N]} are the values of random variables with
  distribution function {\tt F}, then {\tt U[1..N]} will contain 
  the values of {\em discrete\/} random variables distributed over the
  set of values taken by {\tt F}, 
  not uniform random variables over $(0,1)$.
\hrichard {Devrait probablement \^etre \'elimin\'e. Utilis\'e 1 fois dans
   smultin, mais pourrait \^etre remplac\'e.}
\hpierre {Tu as probablement raison.}
\endtab
\code


void gofs_DiffD (double U[], double D[], long N1, long N2, 
                 double a, double b);
\endcode
 \tab Assumes that the real-valued observations {\tt U[N1..N2]} 
  are already sorted in increasing order and computes the differences 
  between the successive observations.
  The difference {\tt U[i+1] - U[i]} is put in {\tt D[i]} for 
  {\tt N1 <= i < N2}, whereas {\tt U[N1] - a} is put into {\tt D[N1-1]}
  and {\tt b - U[N2]} is put into {\tt D[N2]}. 
  The sizes of the arrays {\tt U} and {\tt D} must be at least {\tt N2+1}.
\hpierre {ATTENTION:  J'ai chang\'e cette proc\'edure et la suivante
   pour les rendre plus g\'en\'erales et surtout plus {\em semblables}.
   Un appel \`a l'ancien {\tt DiffD (U, D, N)} doit se traduire par
   {\tt DiffD (U, D, 1, N, 0.0, 1.0)}, tandis qu'un appel 
   \`a l'ancien {\tt DiffL (U, D, N1, N2, L)} doit se traduire par
   {\tt DiffD (U, D, N1, N2, 0, L+U[N1])}. }
 \endtab
\code


void gofs_DiffL (long U[], long D[], long N1, long N2, long a, long b);

#ifdef USE_LONGLONG
void gofs_DiffLL (longlong U[], longlong D[], long N1, long N2,
                  longlong a, longlong b);
void gofs_DiffULL (ulonglong U[], ulonglong D[], long N1, long N2,
                   ulonglong a, ulonglong b);
#endif
\endcode
 \tab Same as {\tt gofs\_DiffD}, but for integer-valued observations.
 \endtab
\code


void gofs_IterateSpacings (double V[], double S[], long N);
\endcode
 \tab Applies one iteration of the {\em iterated spacings\/} 
   transformation \cite{rKNU98a,tSTE86a}.
   Assumes that {\tt S[0...N]} contains the {\em spacings\/} 
   between $N$ real numbers $U_1,\dots,U_N$ in the interval $[0,1]$.
   These spacings are defined by
    $$ S_i = U_{(i+1)} - U_{(i)},  \qquad  0\le i\le N, $$
   where $U_{(0)}=0$, $U_{(N+1)}=1$, and
   $U_{(1)},\dots,U_{(N)}$,  are the $U_i$ sorted in increasing order.
%  These $U_i$ do not need to be in the array {\tt V}.
   These spacings may have been obtained by calling {\tt gofs\_DiffD}.
   This procedure transforms the spacings into new
   spacings, by a variant of the  method described
   in section 11 of \cite {rMAR85a} and also by Stephens \cite{tSTE86a}:
%  See also Knuth (1998), 3th edition.
   it sorts $S_0,\dots,S_N$ to obtain
   $S_{(0)} \le S_{(1)} \le S_{(2)} \le \cdots \le S_{(N)}$,
   computes the weighted differences
  \begin {eqnarray*}
    S_{0}   &=& (N+1) S_{(0)}, \\
    S_{1}   &=& N (S_{(1)}-S_{(0)}), \\
    S_{2}   &=& (N-1) (S_{(2)}-S_{(1)}),\\
            & \vdots& \\
    S_{N}   &=& S_{(N)}-S_{(N-1)},
  \end {eqnarray*}
   and computes $V_i = S_0 + S_1 + \cdots + S_{i-1}$ for $1\le i\le N$.
   It then returns $S_0,\dots,S_N$ in {\tt S[0..N]} and
   $V_1,\dots,V_N$ in {\tt V[1..N]}.

  Under the assumption that the $U_i$ are i.i.d.\ $U(0,1)$, the new
  $S_i$ can be considered as a new set of spacings having the same
  distribution as the original spacings, and the $V_i$ are a new sample
  of i.i.d.\ $U(0,1)$ random variables, sorted by increasing order.

  This transformation is useful to detect {\em clustering\/} in a data
  set: A pair of observations that are close to each other is transformed
  into an observation close to zero.  A data set with unusually clustered
  observations is thus transformed to a data set with an 
  accumulation of observations near zero, which is easily detected by
  the Anderson-Darling GOF test.
  \hrichard {Utilis\'e dans \tt snpair}
 \endtab
\code


void gofs_PowerRatios (double U[], long N);
\endcode
 \tab  Applies the {\em power ratios\/} transformation $W$ described
   in section 8.4 of Stephens \cite{tSTE86a}.
   Assume that {\tt U[1...N]} contains $N$ real numbers 
   $U_{(1)},\dots,U_{(N)}$ from the interval $[0,1]$,
   already sorted in increasing order, and computes the transformations:
     $$ U'_i = (U_{(i)} / U_{(i+1)})^i, \qquad  i=1,\dots,N,$$
   with $U_{(N+1)} = 1$.
   These $U'_i$ are sorted in increasing order and put back in 
   {\tt U[1...N]}.
   If the $U_{(i)}$ are i.i.d.\ $U(0,1)$ sorted by increasing order, 
   then the $U'_i$ are also i.i.d.\ $U(0,1)$.

  This transformation is useful to detect clustering, as explained in
  {\tt gofs\_IterateSpacings}, except that here a pair of
  observations close to each other is transformed
  into an observation close to 1.  
  An accumulation of observations near 1 is also easily detected by
  the Anderson-Darling GOF test.
  \hrichard {Utilis\'e dans \tt snpair}
 \endtab
\code


void gofs_MergeClasses (double NbExp[], long Loc[],
                        long *smin, long *smax, long *NbClasses);
\endcode
 \tab This function is convenient for regrouping classes before 
   applying a chi-square test,
   in the case where the expected number of observations in some of
   the classes may be too small.
   It merges classes of observations so that the expected
   number of observations in each class is at least
   {\tt gofs\_MinExpected}. Initially, the expected numbers in each class
   are in {\tt NbExp[*smin...*smax]}.
   When the function returns, if ${\tt Loc}[s] = j$, this means that class
   $s$ has been merged with class $j$.
   In this case, all observations that previously belonged to class $s$ 
   are redirected to class $j$, 
   % i.e. considered as if they belong to class $j$,
   and {\tt NbExp$[s]$} has been added to {\tt NbExp$[j]$}
   and then set to zero.
   {\tt NbClasses} gives the final number of classes,
   {\tt smin} contains the new index of the lowest class, 
   and {\tt smax} the new index of the highest class.
 \endtab
\code


void gofs_WriteClasses (double NbExp[], long Loc[], 
                        long smin, long smax, long NbClasses);
\endcode
 \tab  Prints the classes before or after their regrouping by
  {\tt gofs\_MergeClasses}.
  The parameters are the same as for the latter function.
  If {\tt NbClasses > 0}, assumes that {\tt gofs\_MergeClasses} 
  has already been called to regroup classes 
% (otherwise {\tt Loc} will be undefined!)}
  and prints the classes after the regrouping.
  If {\tt NbClasses <= 0}, prints only the classes before any regrouping.
 \endtab

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\guisec{Computing EDF test statistics}

\code

double gofs_Chi2 (double NbExp[], long Count[], long smin, long smax);
\endcode
\tab  Computes and returns the chi-square statistic for the 
 observations $o_i$ in {\tt Count[smin...smax]}, for which the 
 corresponding expected values $e_i$ are in {\tt NbExp[smin...smax]}. 
 Assuming that $i$ goes from 1 to $k$, where $k =$ {\tt smax-smin+1}
 is the number of classes, the chi-square statistic is defined as
 \eq
   X^2 = \sum_{i=1}^k \frac{(o_i - e_i)^2}{e_i}.  \eqlabel{eq:chi-square}
 \endeq
 Under the hypothesis that the $e_i$ are the correct expectations and
 if these $e_i$ are large enough, $X^2$ follows approximately the
 chi-square distribution with $k-1$ degrees of freedom.
 If some of the $e_i$ are too small, one can use {\tt gofs\_MergeClasses}
 to regroup classes.
\endtab
\code


double gofs_Chi2Equal (double NbExp, long Count[], long smin, long smax);
\endcode
\tab  Similar to {\tt gofs\_Chi2}, except that the expected
  number of observations per class is assumed to be the same for 
  all classes, and equal to {\tt NbExp}.
\endtab
\code


long gofs_Scan (double U[], long N, double d);
\endcode
 \tab  Computes and returns the scan statistic $S_N(d)$,
  defined in (\ref{eq:scan}).
  The $N$ observations in the array {\tt U[1..N]} must be real numbers 
  in the interval $[0,1]$, sorted in increasing order.
  (See {\tt fbar\_Scan} for the distribution function of $S_N(d)$).
 \endtab
\code


double gofs_CramerMises (double U[], long N);
\endcode
 \tab  Computes and returns the Cram\'er-von Mises statistic $W_N^2$
   (see \cite{tDUR73a,tSTE70a,tSTE86b}), defined by
  \begin {equation}
     W_N^2 = {1\over 12N} +
            \sum_{j=1}^N \left(U_{(j)} - {(j-0.5) \over N}\right)^2,
                                                   \eqlabel {eq:CraMis}
  \end {equation}
 assuming that {\tt U[1...N]} contains $U_{(1)},\dots,U_{(N)}$ 
 sorted in increasing order. 
 \endtab
\code


double gofs_WatsonG (double U[], long N);
\endcode
 \tab  Computes and returns the Watson statistic $G_N$ 
  (see \cite{tWAT76a,tDAR83a}),  defined by
 \begin {eqnarray}
  G_N &=& \sqrt{N} \max_{\rule{0pt}{7pt} 1\le j \le N} \left\{ j/N -
         U_{(j)} + \overline U_N - 1/2 \right\}   
                                            \eqlabel {eq:WatsonG} \\[6pt]
    &=& \sqrt{N}\left (D_N^+ + \overline U_N  - 1/2\right), \nonumber
 \end {eqnarray}
  where $\overline U_N$ is the average of the observations $U_{(j)}$,
  assuming that {\tt U[1...N]} contains the sorted $U_{(1)},\dots,U_{(N)}$.
 \endtab
\code


double gofs_WatsonU (double U[], long N);
\endcode
 \tab  Computes and returns the Watson statistic  $U_N^2$ 
   (see \cite{tDUR73a,tSTE70a,tSTE86b}),  defined by
  \begin {eqnarray}
    W_N^2 &=& {1\over 12N} +
            \sum_{j=1}^N \left\{U_{(j)} - {(j- 0.5)\over N}\right\}^2, \\
    U_N^2 &=& W_N^2  - N\left (\overline U_N - 1/2\right)^2.
                                                   \eqlabel {eq:WatsonU}
  \end {eqnarray}
  where $\overline U_N$ is the average of the observations $U_{(j)}$,
  assuming that {\tt U[1...N]} contains  the sorted $U_{(1)},\dots,U_{(N)}$.
 \endtab
\code


double gofs_AndersonDarling (double U[], long N);
\endcode
 \tab Computes and returns the Anderson-Darling  statistic $A_N^2$
   (see \cite{tLEW61a,tSTE86b,tAND52a}),  defined by
  \begin {eqnarray*}
    A_N^2 &=& -N -{1\over N} \sum_{j=1}^N \left\{ (2j-1)\ln(U_{(j)})
               + (2N+1-2j) \ln(1-U_{(j)}) \right\},      \eqlabel {eq:Andar}
  \end {eqnarray*}
  assuming that {\tt U[1...N]} contains $U_{(1)},\dots,U_{(N)}$.
 \ifdetailed  %%%%
  This function uses the environment variable {\tt gofs\_EpsilonAD}.
 \fi %%%% detailed
 \endtab
\code


void gofs_KS (double U[], long N, double *DP, double *DM, double *D);
\endcode
\tab Computes the Kolmogorov-Smirnov (KS) test statistics 
 $D_N^+$, $D_N^-$, and $D_N$, defined by
 \begin {eqnarray}
  D_N^+ &=& \max_{1\le j\le N} \left(j/N - U_{(j)}\right), 
                                                    \eqlabel{eq:DNp} \\
  D_N^- &=& \max_{1\le j\le N} \left(U_{(j)} - (j-1)/N\right), 
                                                    \eqlabel{eq:DNm} \\
  D_N   &=& \max\ (D_N^+, D_N^-).                   \eqlabel{eq:DN}
 \end {eqnarray}
 and return their values in  {\tt DP, DM}, and {\tt D}, respectively.
 These statistics compare the empirical distribution of 
 $U_{(1)},\dots,U_{(N)}$, which are assumed to be in {\tt U[1...N]},
 with the uniform distribution.
\hrichard {Pourquoi avoir enlev\'e les calculs des EDF de ce fichier et
  l'avoir mis dans gofw? On calcule d\'ej\`a toutes les stats EDF 
  explicitement.}
\hpierre {Simplement pour \'eviter d'introduire {\tt TestType},
  {\tt TestArray}, etc. dans ce module, et pouvoir tout cacher 
  cela ensemble \`a la fin de {\tt gofw}.  Ces choses sont commodes
  pour Testu01, mais trop sp\'ecialis\'ees et pas trop int\'eressantes
  pour la plupart des gens. }
\endtab
\code


void gofs_KSJumpOne (double U[], long N, double a, double *DP, double *DM);
\endcode
\tab Compute the KS statistics $D_N^+(a)$ and $D_N^-(a)$ defined in the 
  description of the function 
{\tt fdist\_KSPlusJumpOne}, assuming that $F$ is the
  uniform distribution over $[0,1]$ and that
  $U_{(1)},\dots,U_{(N)}$ are in {\tt U[1...N]}.
  Returns the values in {\tt DP} and {\tt DM}.
 \endtab
\hide%%%%%

\code
#if 0
void gofs_KSJumpsMany (double X[], int N, wdist_CFUNC F, double W[],
                       double *DP, double *DM, int Detail);
#endif
\endcode
\tab We assume that $X[1...N]$ is already sorted and contains the
 $N$ empirical observations. We obtain the values of the distribution
 $U =  F(W, X[i])$ where $F$ is the theoretical
 distribution which may be discontinuous, and $W$ are the parameters
 of $F$. If {\tt Detail} $> 0$, the computed values of the
 distribution will be printed.
 Returns the values of the KS statistics in {\tt DP} and {\tt DM}.
 \endtab
\endhide%%%%%%%%%%%%%%

\code
\hide
#endif
\endhide
\endcode