Histogram parameterization

Often one needs to parameterize the historgam in terms of some predefined function or expansion - e.g. parameterize the efficiency.
Ostap offers a wide range of embedded parameterization

• in terms of Bernstein polynomials
  • simple Bernstein sum, aka Bezier sum
  • even Bernstein sum, such as f(x)=f(2*x0-x), where x0=0.5*(xmin+xmax)
  • non-negative Bernstein sum
  • non-negative monothonic Bernstein sum
  • non-negative monothonic convex or concave Bernstein sum
  • non-negative convex or concave Bernstein sum
• in term of Legendre polynomials
• in term of Chebyshev polynomials
• in terms of Fourier series
• in terms of Fourier cosine series
• in terms of Basic splines
  • non-negative B-spline
  • non-negative monothonic B-spline
  • non-negative monothonic convex or concave B-spline
  • non-negative convex or concave B-spline

From technical side, there are three branches of methods

• methods that uses only histogram values:
  • these are safe, robust but they ignore the uncertainties
• methods that relies on ROOT.THF1.Fit
  • typically not very good CPU performance
  • sometimes fragile
• methods that relies on RooFit
  • often the best series of methods

Simple parameterization

This group of methods allows to make easy and robust histogram parameterization, ignooring histogram unncertainties

histo  = ...
b1 = histo.bernstein_sum     (  6 ) ## parameterize as degree-6 Bernstein sum
b2 = histo.bernsteineven_sum (  6 ) ## parameterize as degree-6 Bernstein "even"-sum
l  = histo.legendre_sum      (  6 ) ## parameterize as degree-6 Legendre sum
ch = histo.chebyshev_sum     (  6 ) ## parameterize as degree-6 Chebyshev sum
f  = histo.fourier_sum       ( 12 ) ## parameterize as order-12 Fourier sum
c  = histo.cosine_sum        ( 12 ) ## parameterize as order-12 Fourier Cosine sum

ROOT.TH1.Fit-based parameterizations

These methods typically have not very good CPU performance, and sometiems are fragile, but they allow more accurate treatment of parameteriztaions, in particular them takes into account the uncertainties in the historgam.

histo  = ...
b1  = histo.bernstein     (  6 ) ## parameterize as degree-6 Bernstein sum
b2  = histo.bernsteineven (  6 ) ## parameterize as degree-6 Bernstein "even"-sum
l   = histo.legendre      (  6 ) ## parameterize as degree-6 Legendre sum
ch  = histo.chebyshev     (  6 ) ## parameterize as degree-6 Chebyshev sum
f   = histo.fourier       ( 12 ) ## parameterize as order-12 Fourier sum
c   = histo.cosine        ( 12 ) ## parameterize as order-12 Fourier Cosine sum
m   = histo.polynomial    (  6 ) ## parameterize as simple degree-6 monomial sum
p1  = histo.positive      (  6 ) ## parameterize as degree-6 non-negative Bernstein sum 
p2  = histo.positiveeven  (  6 ) ## parameterize as degree-6 non-negative even Bernstein sum 
m1  = histo.monothonic    ( 6 , increasing = False ) ## parameterize as degree-6 non-negative decreasing Bernstein sum 
m2  = histo.monothonic    ( 6 , increasing = True  ) ## parameterize as degree-6 non-negative increasing Bernstein sum
c1  = histo.convex        ( 6 , increasing = False , convex = True  ) ## parameterize as degree-6 non-negative decreasing convex  Bernstein sum 
c2  = histo.convex        ( 6 , increasing = False , convex = False ) ## parameterize as degree-6 non-negative decreasing concave Bernstein sum 
c3  = histo.convex        ( 6 , increasing = True  , convex = True  ) ## parameterize as degree-6 non-negative increasing convex  Bernstein sum 
c4  = histo.convex        ( 6 , increasing = True  , convex = False ) ## parameterize as degree-6 non-negative increasing concave Bernstein sum 
cc1 = histo.convexpoly    ( 6 ) #  parameterize as degree-6 non-negative convex  Bernstein sum 
cc2 = histo.concavepoly   ( 6 ) #  parameterize as degree-6 non-negative concave Bernstein sum

Various types of splines are also provided

s1 = histo.bSpline ( degree=3 , knots = 2 ) ## parameterize as 3d order spline with 2 inner (uniform) knots 
s2 = histo.bSpline ( degree=2 , knots = [0.1,0.4,0.8,0.9] ) ## parameterize as 3d order spline with 4 inner (non-uniform) knots

and similarly for

• non-negative spline pSpline,
• non-negative monothonic spline mSpline,
• non-negative monothonic convex or concave spline cSpline,
• non-negative convex spline convexSpline,
• non-negative concave spline concaveSpline.

RooFit-based parameterizations

r1  = histo.pdf_positive           ( 5 ) ## parameterize and non-negative degree-5 Bernstein sum
r2  = histo.pdf_positiveeven       ( 5 ) ## parameterize and non-negative degree-5 even Bernstein polynomial 
r3  = histo.pdf_increasing         ( 5 ) ## parameterize and non-negative degree-5 increasing Bernstein polynomial 
r4  = histo.pdf_decreasing         ( 5 ) ## parameterize and non-negative degree-5 decreasing Bernstein polynomial 
r5  = histo.pdf_convex_increasing  ( 5 ) ## parameterize and non-negative degree-5 convex  increasing Bernstein polynomial 
r6  = histo.pdf_convex_decreasing  ( 5 ) ## parameterize and non-negative degree-5 convex  decreasing Bernstein polynomial 
r7  = histo.pdf_concave_increasing ( 5 ) ## parameterize and non-negative degree-5 concave increasing Bernstein polynomial 
r8  = histo.pdf_concave_decreasing ( 5 ) ## parameterize and non-negative degree-5 concave decreasing Bernstein polynomial 
r9  = histo.pdf_concavepoly        ( 5 ) ## parameterize and non-negative degree-5 concave Bernstein polynomial 
r10 = histo.pdf_convexpoly         ( 5 ) ## parameterize and non-negative degree-5 convex  Bernstein polynomial

Similarly there are methods that provdies the parameterization in terms of splines :

• pdf_pSpline : non-negative b-spline
• pdf_mSpline : non-negative monothonic b-spline
• pdf_cSpline : non-negative monothonic concave or convex b-spline
• pdf_convexSpline : non-negative monothonic convex b-spline
• pdf_concaveSpline : non-negative monothonic concave b-spline