Smoothing

[HSMOOTH]

Histograming is the least expensive and most popular density estimator, but has several statistical drawbacks. To name only two, it fails to identify structures that are much narrower than the bin size, and exhibits sharp discontinuities (statistical fluctuations) among adjacent low population bins.

The first problem is usually solved by adapting the bin width to the experimental resolution, or by re-binning after looking at the histogram. To filter out the statistical fluctuations, smoothing algorithms can be applied.

Two such techniques are implemented in HBOOK, the so called 353QH ( HSMOOF) and the method of B-splines ( HSPLI1, HSPLI2, HSPFUN). Before trying them out references

[bib-DATA], [bib-SPLINE] and [bib-LISS]

should be consulted, and results taken with care.

                     +------------------------------+
                     | CALL  HSMOOF (ID,ICASE,CHI2*) |
                     +------------------------------+
                                  

Action: This rouitne smoothes a 1-dimensional histogram according to the algorithm 353QH, TWICE (see [bib-DATA]).

Input parameters:

ID

Histogram identifier

ICASE

0 and 1 replace original histogram by smoothed 2 superimpose as a function when editing

Output Parameter:

2

CHI2

chisquare chi between original and smoothed histogram. 

Remark:

1. The mean value and standard deviation are recalculated if ICASE=1
2. The routine can be called several times for the same histogram identifier ID, for ICASE=1 or 2.

                        +------------------------------+
                        |CALL  HSPLI1 (ID,IC,N,K,CHI2*) |
                        +------------------------------+
                                     

   Action: B-splines smoothing of a 1-dimensional histogram.

   Input parameters:

   ID

   Identifier of an existing 1-dimensional histogram

   IC

   Superimposition flag (IC=0 is identical to IC=1) 1 Replace original contents by the value of the spline 2 Superimpose the spline function when editing

   N

   Number of knots (when N<=0 then N=13).

   K

   Degree of the splines (when K>=3 then K=3).

   Output Parameter:

   2

   CHI2

   chisquare chi between original and smoothed histogram.

   Remark:

   
   1. HSPLI1 can be called several times for the same histogram identifier ID, for any value of the parameters
   2. If the distribution to be smoothed exibits NP statistically relevant peaks then a rule of thumb to define the number of knots is, N = 4*NP+6 for a spline of degree 3.

                           +------------------------------+
                           | CALL  HSPLI2 (ID,NX,NY,KX,KY) |
                           +------------------------------+
                                        

      Action: B-splines smoothing of a 2-dimensional histogram.

      Input parameters:

      ID

      Identifier of an existing 2-dimensional

      NX

      Number of knots in the X interval (when NX<=0 then NX=13).

      NY

      Number of knots in the Y interval (when NY<=0 then NY=13).

      KX

      Degree of the spline in X (when KX>=3 then KX=3).

      KY

      Degree of the spline in Y (when KY>=3 then KY=3).

      Remark:

      1. The original contents of the histogram are replaced by the value of the spline approximation.
      2. See the remark about the number of knots for routine HSPLI1.

         
                                  +----------------------+
                                  |S =  HSPFUN (ID,X,N,K) |
                                  +----------------------+
                                           

         Action: Performs a B-spline smoothing of a 1-dimensional histogram and returns the value at a given abscissa point.

         Input parameters:

         ID

         Identifier of an existing 1-dimensional histogram

         X

         Abscissa

         N

         Number of knots (when N<=0 then N=13).

         K

         Degree of the splines (when K>=3 then K=3).