def _stereo_arc(x,
y,
xy=None,
north=(1, 0, 0),
right=(0, 1, 0),
translation=-1,
**kwds):
from sage.misc.functional import n
x = vector(x)
y = vector(y)
sx = n(
_stereo_coordinates(x,
north=north,
right=right,
translation=translation))
sy = n(
_stereo_coordinates(y,
north=north,
right=right,
translation=translation))
if xy == None:
xy = x + y
sxy = n(
_stereo_coordinates(xy,
north=north,
right=right,
translation=translation))
return _arc(sx, sy, sxy, **kwds)
def _stereo_arc(x,y, xy=None, north=(1,0,0), right=(0,1,0), translation=-1, **kwds):
from sage.misc.functional import n
x=vector(x)
y=vector(y)
sx=n(_stereo_coordinates(x, north=north, right=right, translation=translation))
sy=n(_stereo_coordinates(y, north=north, right=right, translation=translation))
if xy == None:
xy=x+y
sxy=n(_stereo_coordinates(xy, north=north, right=right, translation=translation))
return _arc(sx,sy,sxy,**kwds)
def curve(nb_equipes, max_points=100, K=1, R=2, base=2, verbose=False):
r"""
INPUT:
- ``nb_equipes`` -- integer
- ``max_points`` -- integer
- ``K`` -- the value at ``p = nb_equipes``
- ``R`` -- real (default: ``2``), curve parameter
- ``base`` -- 2
- ``verbose`` - bool
EXEMPLES::
sage: from slabbe.ranking_scale import curve
sage: curve(20, 100)
-99*(p*(log(40) + 1) - p*log(p) - 20*log(40) + 20*log(20) -
20)/(19*log(40) - 20*log(20) + 19) + 1
sage: curve(64, 100)
99*(p*(7*log(2) + 1) - p*log(p) + 64*log(64) - 448*log(2) -
64)/(64*log(64) - 441*log(2) - 63) + 1
::
sage: curve(64, 100)(p=64)
1
sage: curve(64, 100)(p=1)
100
sage: curve(64, 100)(p=2)
198*(32*log(64) - 218*log(2) - 31)/(64*log(64) - 441*log(2) - 63) + 1
sage: n(curve(64, 100)(p=2)) # abs tol 1e-10
95.6871477097753
::
sage: curve(64, 100, verbose=True)
fn = -(p*(7*log(2) + 1) - p*log(p) + 64*log(64) - 448*log(2) - 64)/log(2)
aire = 147.889787576005
fn normalise = 99*(p*(7*log(2) + 1) - p*log(p) + 64*log(64) - 448*log(2) - 64)/(64*log(64) - 441*log(2) - 63) + 1
99*(p*(7*log(2) + 1) - p*log(p) + 64*log(64) - 448*log(2) - 64)/(64*log(64) - 441*log(2) - 63) + 1
The base argument seems to be useless (why?)::
sage: curve(100,100,base=3)
-99*(p*(log(200) + 1) - p*log(p) - 100*log(200) + 100*log(100) -
100)/(99*log(200) - 100*log(100) + 99) + 1
sage: curve(100,100,base=2)
-99*(p*(log(200) + 1) - p*log(p) - 100*log(200) + 100*log(100) -
100)/(99*log(200) - 100*log(100) + 99) + 1
"""
from sage.symbolic.assumptions import forget, assume
from sage.misc.functional import integrate, n
from sage.functions.log import log
from sage.calculus.var import var
x, p = var('x,p')
forget()
assume(p - 1 > 0)
assume(p - nb_equipes < 0)
fn = integrate(
log(R * nb_equipes, base=base) - log(x, base=base), x, p, nb_equipes)
if verbose: print "fn = %s" % fn
aire = fn(p=1)
if verbose: print "aire = %s" % n(aire)
fn_normalise = fn / aire * (max_points - K) + K
if verbose: print "fn normalise = %s" % fn_normalise
return fn_normalise
def curve(nb_equipes, max_points=100, K=1, R=2, base=2, verbose=False):
r"""
INPUT:
- ``nb_equipes`` -- integer
- ``max_points`` -- integer
- ``K`` -- the value at ``p = nb_equipes``
- ``R`` -- real (default: ``2``), curve parameter
- ``base`` -- 2
- ``verbose`` - bool
EXEMPLES::
sage: from slabbe.ranking_scale import curve
sage: curve(20, 100)
-99*(p*(log(40) + 1) - p*log(p) - 20*log(40) + 20*log(20) -
20)/(19*log(40) - 20*log(20) + 19) + 1
sage: curve(64, 100)
-33*(p*(7*log(2) + 1) - p*log(p) - 64*log(2) - 64)/(19*log(2) + 21) + 1
::
sage: curve(64, 100)(p=64)
1
sage: curve(64, 100)(p=1)
100
sage: curve(64, 100)(p=2)
66*(26*log(2) + 31)/(19*log(2) + 21) + 1
sage: n(curve(64, 100)(p=2)) # abs tol 1e-10
95.6871477097753
::
sage: curve(64, 100, verbose=True)
fn = -(p*(7*log(2) + 1) - p*log(p) - 64*log(2) - 64)/log(2)
aire = 147.889787576005
fn normalise = -33*(p*(7*log(2) + 1) - p*log(p) - 64*log(2) - 64)/(19*log(2) + 21) + 1
-33*(p*(7*log(2) + 1) - p*log(p) - 64*log(2) - 64)/(19*log(2) + 21) + 1
The base argument seems to be useless (why?)::
sage: curve(100,100,base=3)
-99*(p*(log(200) + 1) - p*log(p) - 100*log(200) + 200*log(10) -
100)/(99*log(200) - 200*log(10) + 99) + 1
sage: curve(100,100,base=2)
-99*(p*(log(200) + 1) - p*log(p) - 100*log(200) + 200*log(10) -
100)/(99*log(200) - 200*log(10) + 99) + 1
"""
from sage.symbolic.assumptions import forget, assume
from sage.misc.functional import integrate, n
from sage.functions.log import log
from sage.calculus.var import var
x,p = var('x,p')
forget()
assume(p - 1 > 0)
assume(p-nb_equipes < 0)
fn = integrate(log(R*nb_equipes, base=base) - log(x, base=base), x, p, nb_equipes)
if verbose: print("fn = %s" % fn)
aire = fn(p=1)
if verbose: print("aire = %s" % n(aire))
fn_normalise = fn / aire * (max_points - K) + K
if verbose: print("fn normalise = %s" % fn_normalise)
return fn_normalise