import numpy as np
def _sorted_interpolate(x, y, x_eval):
"""utility function for binned_Cminus"""
# note that x should be sorted
N = len(x)
ind = x.searchsorted(x_eval)
ind[ind == N] = N - 1
y_eval = np.zeros(x_eval.shape)
# find perfect matches
match = (x[ind] == x_eval) | (x_eval > x[-1]) | (x_eval < x[0])
y_eval[match] = y[ind[match]]
ind = ind[~match]
# take care of extrapolation
ind[ind == 0] = 1
x_lo = x[ind - 1]
x_up = x[ind]
y_lo = y[ind - 1]
y_up = y[ind]
# take care of places where x_lo = x_up
y_eval[~match] = (y_lo + (x_eval[~match] - x_lo)
* (y_up - y_lo) / (x_up - x_lo))
return y_eval
[docs]def Cminus(x, y, xmax, ymax):
"""Lynden-Bell's C-minus method
Parameters
----------
x : array_like
array of x values
y : array_like
array of y values
xmax : array_like
array of maximum x values for each y value
ymax : array_like
array of maximum y values for each x value
Returns
-------
Nx, Ny, cuml_x, cuml_y: ndarrays
Nx and cuml_x are in the order of the sorted x array
Ny and cuml_y are in the order of the sorted y array
"""
# make copies of input
x, y, xmax, ymax = map(np.array, (x, y, xmax, ymax))
Nall = len(x)
cuml_x = np.zeros(x.shape)
cuml_y = np.zeros(y.shape)
Nx = np.zeros(x.shape)
Ny = np.zeros(y.shape)
# first the y direction.
i_sort = np.argsort(y)
x = x[i_sort]
y = y[i_sort]
xmax = xmax[i_sort]
ymax = ymax[i_sort]
for j in range(1, Nall):
# Making sure we don't divide with 0 later
objects = np.sum(x[:j] < xmax[j])
if objects:
Ny[j] = objects
else:
Ny[j] = np.inf
Ny[0] = np.inf
cuml_y = np.cumprod(1. + 1. / Ny)
Ny[np.isinf(Ny)] = 0
# renormalize
cuml_y *= Nall / cuml_y[-1]
# now the x direction
i_sort = np.argsort(x)
x = x[i_sort]
y = y[i_sort]
xmax = xmax[i_sort]
ymax = ymax[i_sort]
for i in range(1, Nall):
# Making sure we don't divide with 0 later
objects = np.sum(y[:i] < ymax[i])
if objects:
Nx[i] = objects
else:
Nx[i] = np.inf
Nx[0] = np.inf
cuml_x = np.cumprod(1. + 1. / Nx)
Nx[np.isinf(Nx)] = 0
# renormalize
cuml_x *= Nall / cuml_x[-1]
return Nx, Ny, cuml_x, cuml_y
[docs]def binned_Cminus(x, y, xmax, ymax, xbins, ybins, normalize=False):
"""Compute the binned distributions using the Cminus method
Parameters
----------
x : array_like
array of x values
y : array_like
array of y values
xmax : array_like
array of maximum x values for each y value
ymax : array_like
array of maximum y values for each x value
xbins : array_like
array of bin edges for the x function: size=Nbins_x + 1
ybins : array_like
array of bin edges for the y function: size=Nbins_y + 1
normalize : boolean
if true, then returned distributions are normalized. Default
is False.
Returns
-------
dist_x, dist_y : ndarrays
distributions of size Nbins_x and Nbins_y
"""
Nx, Ny, cuml_x, cuml_y = Cminus(x, y, xmax, ymax)
# simple linear interpolation using a binary search
# interpolate the cumulative distributions
x_sort = np.sort(x)
y_sort = np.sort(y)
Ix_edges = _sorted_interpolate(x_sort, cuml_x, xbins)
Iy_edges = _sorted_interpolate(y_sort, cuml_y, ybins)
if xbins[0] < x_sort[0]:
Ix_edges[0] = cuml_x[0]
if xbins[-1] > x_sort[-1]:
Ix_edges[-1] = cuml_x[-1]
if ybins[0] < y_sort[0]:
Iy_edges[0] = cuml_y[0]
if ybins[-1] > y_sort[-1]:
Iy_edges[-1] = cuml_y[-1]
x_dist = np.diff(Ix_edges) / np.diff(xbins)
y_dist = np.diff(Iy_edges) / np.diff(ybins)
if normalize:
x_dist /= len(x)
y_dist /= len(y)
return x_dist, y_dist
[docs]def bootstrap_Cminus(x, y, xmax, ymax, xbins, ybins,
Nbootstraps=10, normalize=False):
"""
Compute the binned distributions using the Cminus method, with
bootstrapped estimates of the errors
Parameters
----------
x : array_like
array of x values
y : array_like
array of y values
xmax : array_like
array of maximum x values for each y value
ymax : array_like
array of maximum y values for each x value
xbins : array_like
array of bin edges for the x function: size=Nbins_x + 1
ybins : array_like
array of bin edges for the y function: size=Nbins_y + 1
Nbootstraps : int
number of bootstrap resamplings to perform
normalize : boolean
if true, then returned distributions are normalized. Default
is False.
Returns
-------
dist_x, err_x, dist_y, err_y : ndarrays
distributions of size Nbins_x and Nbins_y
"""
x, y, xmax, ymax = map(np.asarray, (x, y, xmax, ymax))
x_dist = np.zeros((Nbootstraps, len(xbins) - 1))
y_dist = np.zeros((Nbootstraps, len(ybins) - 1))
for i in range(Nbootstraps):
ind = np.random.randint(0, len(x), len(x))
x_dist[i], y_dist[i] = binned_Cminus(x[ind], y[ind],
xmax[ind], ymax[ind],
xbins, ybins,
normalize=normalize)
return (x_dist.mean(0), x_dist.std(0, ddof=1),
y_dist.mean(0), y_dist.std(0, ddof=1))