Axis scales

By default Matplotlib displays data on the axis using a linear scale. Matplotlib also supports logarithmic scales, and other less common scales as well. Usually this can be done directly by using the set_xscale or set_yscale methods.

import matplotlib.pyplot as plt
import numpy as np

import matplotlib.scale as mscale
from matplotlib.ticker import FixedLocator, NullFormatter

fig, axs = plt.subplot_mosaic([['linear', 'linear-log'],
                               ['log-linear', 'log-log']], layout='constrained')

x = np.arange(0, 3*np.pi, 0.1)
y = 2 * np.sin(x) + 3

ax = axs['linear']
ax.plot(x, y)
ax.set_xlabel('linear')
ax.set_ylabel('linear')

ax = axs['linear-log']
ax.plot(x, y)
ax.set_yscale('log')
ax.set_xlabel('linear')
ax.set_ylabel('log')

ax = axs['log-linear']
ax.plot(x, y)
ax.set_xscale('log')
ax.set_xlabel('log')
ax.set_ylabel('linear')

ax = axs['log-log']
ax.plot(x, y)
ax.set_xscale('log')
ax.set_yscale('log')
ax.set_xlabel('log')
ax.set_ylabel('log')

loglog and semilogx/y

The logarithmic axis is used so often that there are a set helper functions, that do the same thing: semilogy, semilogx, and loglog.

fig, axs = plt.subplot_mosaic([['linear', 'linear-log'],
                               ['log-linear', 'log-log']], layout='constrained')

x = np.arange(0, 3*np.pi, 0.1)
y = 2 * np.sin(x) + 3

ax = axs['linear']
ax.plot(x, y)
ax.set_xlabel('linear')
ax.set_ylabel('linear')
ax.set_title('plot(x, y)')

ax = axs['linear-log']
ax.semilogy(x, y)
ax.set_xlabel('linear')
ax.set_ylabel('log')
ax.set_title('semilogy(x, y)')

ax = axs['log-linear']
ax.semilogx(x, y)
ax.set_xlabel('log')
ax.set_ylabel('linear')
ax.set_title('semilogx(x, y)')

ax = axs['log-log']
ax.loglog(x, y)
ax.set_xlabel('log')
ax.set_ylabel('log')
ax.set_title('loglog(x, y)')

Other built-in scales

There are other scales that can be used. The list of registered scales can be returned from scale.get_scale_names:

print(mscale.get_scale_names())

['asinh', 'function', 'functionlog', 'linear', 'log', 'logit', 'mercator', 'symlog']

fig, axs = plt.subplot_mosaic([['asinh', 'symlog'],
                               ['log', 'logit']], layout='constrained')

x = np.arange(0, 1000)

for name, ax in axs.items():
    if name in ['asinh', 'symlog']:
        yy = x - np.mean(x)
    elif name in ['logit']:
        yy = (x-np.min(x))
        yy = yy / np.max(np.abs(yy))
    else:
        yy = x

    ax.plot(yy, yy)
    ax.set_yscale(name)
    ax.set_title(name)

Optional arguments for scales

Some of the default scales have optional arguments. These are documented in the API reference for the respective scales at scale. One can change the base of the logarithm being plotted (eg 2 below) or the linear threshold range for 'symlog'.

fig, axs = plt.subplot_mosaic([['log', 'symlog']], layout='constrained',
                              figsize=(6.4, 3))

for name, ax in axs.items():
    if name in ['log']:
        ax.plot(x, x)
        ax.set_yscale('log', base=2)
        ax.set_title('log base=2')
    else:
        ax.plot(x - np.mean(x), x - np.mean(x))
        ax.set_yscale('symlog', linthresh=100)
        ax.set_title('symlog linthresh=100')

Arbitrary function scales

Users can define a full scale class and pass that to set_xscale and set_yscale (see Custom scale). A short cut for this is to use the 'function' scale, and pass as extra arguments a forward and an inverse function. The following performs a Mercator transform to the y-axis.

# Function Mercator transform
def forward(a):
    a = np.deg2rad(a)
    return np.rad2deg(np.log(np.abs(np.tan(a) + 1.0 / np.cos(a))))


def inverse(a):
    a = np.deg2rad(a)
    return np.rad2deg(np.arctan(np.sinh(a)))


t = np.arange(0, 170.0, 0.1)
s = t / 2.

fig, ax = plt.subplots(layout='constrained')
ax.plot(t, s, '-', lw=2)

ax.set_yscale('function', functions=(forward, inverse))
ax.set_title('function: Mercator')
ax.grid(True)
ax.set_xlim([0, 180])
ax.yaxis.set_minor_formatter(NullFormatter())
ax.yaxis.set_major_locator(FixedLocator(np.arange(0, 90, 10)))

What is a "scale"?

A scale is an object that gets attached to an axis. The class documentation is at scale. set_xscale and set_yscale set the scale on the respective Axis objects. You can determine the scale on an axis with get_scale:

fig, ax = plt.subplots(layout='constrained',
                              figsize=(3.2, 3))
ax.semilogy(x, x)

print(ax.xaxis.get_scale())
print(ax.yaxis.get_scale())

linear
log

Setting a scale does three things. First it defines a transform on the axis that maps between data values to position along the axis. This transform can be accessed via get_transform:

print(ax.yaxis.get_transform())

LogTransform(base=10, nonpositive='clip')

Transforms on the axis are a relatively low-level concept, but is one of the important roles played by set_scale.

Setting the scale also sets default tick locators (ticker) and tick formatters appropriate for the scale. An axis with a 'log' scale has a LogLocator to pick ticks at decade intervals, and a LogFormatter to use scientific notation on the decades.

print('X axis')
print(ax.xaxis.get_major_locator())
print(ax.xaxis.get_major_formatter())

print('Y axis')
print(ax.yaxis.get_major_locator())
print(ax.yaxis.get_major_formatter())

X axis
<matplotlib.ticker.AutoLocator object at 0x7d8bb858bce0>
<matplotlib.ticker.ScalarFormatter object at 0x7d8bb8589190>
Y axis
<matplotlib.ticker.LogLocator object at 0x7d8bb858ba10>
<matplotlib.ticker.LogFormatterSciNotation object at 0x7d8bd324a8d0>