[Bug-gsl] Incorrect equation in the manual: Statistics->Autocorrelation gsl_stats_lag1_autocorrelation

[Kirill Sokolovsky]

Dear GSL developers,

I think there is an error in equation defining the lag-1 
autocorrelation in the manual (Statistics->Autocorrelation)
https://www.gnu.org/software/gsl/manual/html_node/Autocorrelation.html

Currently it is written as:

    a_1 = {\sum_{i = 1}^{n} (x_{i} - \Hat\mu) (x_{i-1} - \Hat\mu)
           \over
           \sum_{i = 1}^{n} (x_{i} - \Hat\mu) (x_{i} - \Hat\mu)}

and since the summation starts from 1, for i=1 the numerator refers to 
x_{1-1} which is undefined (if x_0 is defined it will be ignored by 
summation in the denominator, which is strange).

I think instead the equation should be written as:

    a_1 = {\sum\limits_{i = 1}^{N-1} (x_{i} - \Hat\mu) (x_{i+1} - \Hat\mu)
           \over
           \sum\limits_{i = 1}^{N} (x_{i} - \Hat\mu)^2}

The error is in the manual, not in the code.

The pice of code below shows that the suggested equation is consistent 
with what is actually implemented in the GSL code. The difference between 
values returned by my naive implementation and the library implementation 
seems to be <1e-16 which is expected for double precision computations.

With best wishes,
Kirill Sokolovsky


//################ compare_lag1.c ################//

#include <stdio.h>
#include <math.h>

#include <gsl/gsl_statistics_double.h>

double my_lag1_autocorrelation(double * data, int N){
 int i;
 double u,d,mean;
 mean=gsl_stats_mean( data, 1, N);
 u=d=0.0;
 for(i=0;i<N-1;i++)u+=(data[i]-mean)*(data[i+1]-mean);
 for(i=0;i<N;i++)d+=(data[i]-mean)*(data[i]-mean);
 return u/d;
}

int main(){
 double r_my,r_gsl;
 double data[10000];
 int i,N;
 N=0;
 while( -1<fscanf(stdin,"%lf",&data[N]) )N++;

 r_gsl=gsl_stats_lag1_autocorrelation( data, 1, N);
 r_my=my_lag1_autocorrelation( data, N);

 fprintf(stdout,"%lf - %lf = %lg\n",r_gsl,r_my,r_gsl-r_my);

 return 0;
}