"""

Colormap Normalization

======================

Objects that use colormaps by default linearly map the colors in the

colormap from data values *vmin* to *vmax*. For example::

pcm = ax.pcolormesh(x, y, Z, vmin=-1., vmax=1., cmap='RdBu_r')

will map the data in *Z* linearly from -1 to +1, so *Z=0* will

give a color at the center of the colormap *RdBu_r* (white in this

case).

Matplotlib does this mapping in two steps, with a normalization from

[0,1] occurring first, and then mapping onto the indices in the

colormap. Normalizations are classes defined in the

:func:`matplotlib.colors` module. The default, linear normalization is

:func:`matplotlib.colors.Normalize`.

Artists that map data to color pass the arguments *vmin* and *vmax* to

construct a :func:`matplotlib.colors.Normalize` instance, then call it:

.. ipython::

In [1]: import matplotlib as mpl

In [2]: norm = mpl.colors.Normalize(vmin=-1.,vmax=1.)

In [3]: norm(0.)

Out[3]: 0.5

However, there are sometimes cases where it is useful to map data to

colormaps in a non-linear fashion.

Logarithmic

-----------

One of the most common transformations is to plot data by taking

its logarithm (to the base-10). This transformation is useful to

display changes across disparate scales. Using :func:`colors.LogNorm`

normalizes the data via :math:`log_{10}`. In the example below,

there are two bumps, one much smaller than the other. Using

:func:`colors.LogNorm`, the shape and location of each bump can clearly

be seen:

"""

import numpy as np

import matplotlib.pyplot as plt

import matplotlib.colors as colors

N = 100

X, Y = np.mgrid[-3:3:complex(0, N), -2:2:complex(0, N)]

# A low hump with a spike coming out of the top right. Needs to have

# z/colour axis on a log scale so we see both hump and spike. linear

# scale only shows the spike.

Z1 = np.exp(-(X)**2 - (Y)**2)

Z2 = np.exp(-(X * 10)**2 - (Y * 10)**2)

Z = Z1 + 50 * Z2

fig, ax = plt.subplots(2, 1)

pcm = ax[0].pcolor(X, Y, Z,

norm=colors.LogNorm(vmin=Z.min(), vmax=Z.max()),

cmap='PuBu_r')

fig.colorbar(pcm, ax=ax[0], extend='max')

pcm = ax[1].pcolor(X, Y, Z, cmap='PuBu_r')

fig.colorbar(pcm, ax=ax[1], extend='max')

fig.show()

###############################################################################

# Symmetric logarithmic

# ---------------------

#

# Similarly, it sometimes happens that there is data that is positive

# and negative, but we would still like a logarithmic scaling applied to

# both. In this case, the negative numbers are also scaled

# logarithmically, and mapped to smaller numbers; e.g., if `vmin=-vmax`,

# then they the negative numbers are mapped from 0 to 0.5 and the

# positive from 0.5 to 1.

#

# Since the logarithm of values close to zero tends toward infinity, a

# small range around zero needs to be mapped linearly. The parameter

# *linthresh* allows the user to specify the size of this range

# (-*linthresh*, *linthresh*). The size of this range in the colormap is

# set by *linscale*. When *linscale* == 1.0 (the default), the space used

# for the positive and negative halves of the linear range will be equal

# to one decade in the logarithmic range.

N = 100

X, Y = np.mgrid[-3:3:complex(0, N), -2:2:complex(0, N)]

Z1 = np.exp(-X**2 - Y**2)

Z2 = np.exp(-(X - 1)**2 - (Y - 1)**2)

Z = (Z1 - Z2) * 2

fig, ax = plt.subplots(2, 1)

pcm = ax[0].pcolormesh(X, Y, Z,

norm=colors.SymLogNorm(linthresh=0.03, linscale=0.03,

vmin=-1.0, vmax=1.0),

cmap='RdBu_r')

fig.colorbar(pcm, ax=ax[0], extend='both')

pcm = ax[1].pcolormesh(X, Y, Z, cmap='RdBu_r', vmin=-np.max(Z))

fig.colorbar(pcm, ax=ax[1], extend='both')

fig.show()

###############################################################################

# Power-law

# ---------

#

# Sometimes it is useful to remap the colors onto a power-law

# relationship (i.e. :math:`y=x^{\gamma}`, where :math:`\gamma` is the

# power). For this we use the :func:`colors.PowerNorm`. It takes as an

# argument *gamma* (*gamma* == 1.0 will just yield the default linear

# normalization):

#

# .. note::

#

# There should probably be a good reason for plotting the data using

# this type of transformation. Technical viewers are used to linear

# and logarithmic axes and data transformations. Power laws are less

# common, and viewers should explicitly be made aware that they have

# been used.

N = 100

X, Y = np.mgrid[0:3:complex(0, N), 0:2:complex(0, N)]

Z1 = (1 + np.sin(Y * 10.)) * X**(2.)

fig, ax = plt.subplots(2, 1)

pcm = ax[0].pcolormesh(X, Y, Z1, norm=colors.PowerNorm(gamma=0.5),

cmap='PuBu_r')

fig.colorbar(pcm, ax=ax[0], extend='max')

pcm = ax[1].pcolormesh(X, Y, Z1, cmap='PuBu_r')

fig.colorbar(pcm, ax=ax[1], extend='max')

fig.show()

###############################################################################

# Discrete bounds

# ---------------

#

# Another normaization that comes with matplolib is

# :func:`colors.BoundaryNorm`. In addition to *vmin* and *vmax*, this

# takes as arguments boundaries between which data is to be mapped. The

# colors are then linearly distributed between these "bounds". For

# instance:

#

# .. ipython::

#

# In [2]: import matplotlib.colors as colors

#

# In [3]: bounds = np.array([-0.25, -0.125, 0, 0.5, 1])

#

# In [4]: norm = colors.BoundaryNorm(boundaries=bounds, ncolors=4)

#

# In [5]: print(norm([-0.2,-0.15,-0.02, 0.3, 0.8, 0.99]))

# [0 0 1 2 3 3]

#

# Note unlike the other norms, this norm returns values from 0 to *ncolors*-1.

N = 100

X, Y = np.mgrid[-3:3:complex(0, N), -2:2:complex(0, N)]

Z1 = np.exp(-X**2 - Y**2)

Z2 = np.exp(-(X - 1)**2 - (Y - 1)**2)

Z = (Z1 - Z2) * 2

fig, ax = plt.subplots(3, 1, figsize=(8, 8))

ax = ax.flatten()

# even bounds gives a contour-like effect

bounds = np.linspace(-1, 1, 10)

norm = colors.BoundaryNorm(boundaries=bounds, ncolors=256)

pcm = ax[0].pcolormesh(X, Y, Z,

norm=norm,

cmap='RdBu_r')

fig.colorbar(pcm, ax=ax[0], extend='both', orientation='vertical')

# uneven bounds changes the colormapping:

bounds = np.array([-0.25, -0.125, 0, 0.5, 1])

norm = colors.BoundaryNorm(boundaries=bounds, ncolors=256)

pcm = ax[1].pcolormesh(X, Y, Z, norm=norm, cmap='RdBu_r')

fig.colorbar(pcm, ax=ax[1], extend='both', orientation='vertical')

pcm = ax[2].pcolormesh(X, Y, Z, cmap='RdBu_r', vmin=-np.max(Z))

fig.colorbar(pcm, ax=ax[2], extend='both', orientation='vertical')

fig.show()

###############################################################################

# Custom normalization: Two linear ranges

# ---------------------------------------

#

# It is possible to define your own normalization. In the following

# example, we modify :func:`colors:SymLogNorm` to use different linear

# maps for the negative data values and the positive. (Note that this

# example is simple, and does not validate inputs or account for complex

# cases such as masked data)

#

# .. note::

# This may appear soon as :func:`colors.OffsetNorm`.

#

# As above, non-symmetric mapping of data to color is non-standard

# practice for quantitative data, and should only be used advisedly. A

# practical example is having an ocean/land colormap where the land and

# ocean data span different ranges.

N = 100

X, Y = np.mgrid[-3:3:complex(0, N), -2:2:complex(0, N)]

Z1 = np.exp(-X**2 - Y**2)

Z2 = np.exp(-(X - 1)**2 - (Y - 1)**2)

Z = (Z1 - Z2) * 2

class MidpointNormalize(colors.Normalize):

def __init__(self, vmin=None, vmax=None, midpoint=None, clip=False):

self.midpoint = midpoint

colors.Normalize.__init__(self, vmin, vmax, clip)

def __call__(self, value, clip=None):

# I'm ignoring masked values and all kinds of edge cases to make a

# simple example...

x, y = [self.vmin, self.midpoint, self.vmax], [0, 0.5, 1]

return np.ma.masked_array(np.interp(value, x, y))

fig, ax = plt.subplots(2, 1)

pcm = ax[0].pcolormesh(X, Y, Z,

norm=MidpointNormalize(midpoint=0.),

cmap='RdBu_r')

fig.colorbar(pcm, ax=ax[0], extend='both')

pcm = ax[1].pcolormesh(X, Y, Z, cmap='RdBu_r', vmin=-np.max(Z))

fig.colorbar(pcm, ax=ax[1], extend='both')

fig.show()