我终于想出了正确的解决方案，感谢@Amy Teegarden 让我朝着正确的方向前进。我想我会在这里分享最终的解决方案供其他人参考！这是最终结果：
以下是真实的概率轴刻度，使用正常的 CDF 及其逆 PPF 函数，由 mu 和 sigma 参数化。
import numpy as np
from matplotlib import scale as mscale
from matplotlib import transforms as mtransforms
from matplotlib.ticker import FormatStrFormatter, FixedLocator
from scipy.stats import norm
class ProbScale(mscale.ScaleBase):
"""
Scales data in range 0 to 100 using a non-standard log transform
This scale attempts to replicate "probability paper" scaling
The scale function:
A piecewise combination of exponential, linear, and logarithmic scales
The inverse scale function:
piecewise combination of exponential, linear, and logarithmic scales
Since probabilities at 0 and 100 are not represented,
there is user-defined upper and lower limit, above and below which nothing
will be plotted. This defaults to .1 and 99 for lower and upper, respectively.
"""
# The scale class must have a member ``name`` that defines the
# string used to select the scale. For example,
# ``gca().set_yscale("mercator")`` would be used to select this
# scale.
name = 'prob_scale'
def __init__(self, axis, **kwargs):
"""
Any keyword arguments passed to ``set_xscale`` and
``set_yscale`` will be passed along to the scale's
constructor.
upper: The probability above which to crop the data.
lower: The probability below which to crop the data.
"""
mscale.ScaleBase.__init__(self)
upper = kwargs.pop("upper", 98) #Default to an upper bound of 98%
if upper <= 0 or upper >= 100:
raise ValueError("upper must be between 0 and 100.")
lower = kwargs.pop("lower", 0.2) #Default to a lower bound of .2%
if lower <= 0 or lower >= 100:
raise ValueError("lower must be between 0 and 100.")
if lower >= upper:
raise ValueError("lower must be strictly less than upper!.")
self.lower = lower
self.upper = upper
#This scale is best described by the CDF of the normal distribution
#This distribution is paramaterized by mu and sigma, these default vaules
#are provided to work generally well, but can be adjusted by the user if desired
mu = kwargs.pop("mu", 15)
sigma = kwargs.pop("sigma", 40)
self.mu = mu
self.sigma = sigma
#Need to enfore the upper and lower limits on the axes initially
axis.axes.set_xlim(lower,upper)
def get_transform(self):
"""
Override this method to return a new instance that does the
actual transformation of the data.
The ProbTransform class is defined below as a
nested class of this one.
"""
return self.ProbTransform(self.lower, self.upper, self.mu, self.sigma)
def set_default_locators_and_formatters(self, axis):
"""
Override to set up the locators and formatters to use with the
scale. This is only required if the scale requires custom
locators and formatters. Writing custom locators and
formatters: many helpful examples in ``ticker.py``.
In this case, the prob_scale uses a fixed locator from
0.1 to 99 % and a custom no formatter class
This builds both the major and minor locators, and cuts off any values
above or below the user defined thresholds: upper, lower
"""
#major_ticks = np.asarray([.2,.5,1,2,5,10,20,30,40,50,60,70,80,90,95,98])
major_ticks = np.asarray([.2,1,2,5,10,20,30,40,50,60,70,80,90,98]) #removed a couple ticks to make it look nicer
major_ticks = major_ticks[np.where( (major_ticks >= self.lower) & (major_ticks <= self.upper) )]
minor_ticks = np.concatenate( [np.arange(.2, 1, .1), np.arange(1, 2, .2), np.arange(2,20,1), np.arange(20, 80, 2), np.arange(80, 98, 1)] )
minor_ticks = minor_ticks[np.where( (minor_ticks >= self.lower) & (minor_ticks <= self.upper) )]
axis.set_major_locator(FixedLocator(major_ticks))
axis.set_minor_locator(FixedLocator(minor_ticks))
def limit_range_for_scale(self, vmin, vmax, minpos):
"""
Override to limit the bounds of the axis to the domain of the
transform. In the case of Probability, the bounds should be
limited to the user bounds that were passed in. Unlike the
autoscaling provided by the tick locators, this range limiting
will always be adhered to, whether the axis range is set
manually, determined automatically or changed through panning
and zooming.
"""
return max(self.lower, vmin), min(self.upper, vmax)
class ProbTransform(mtransforms.Transform):
# There are two value members that must be defined.
# ``input_dims`` and ``output_dims`` specify number of input
# dimensions and output dimensions to the transformation.
# These are used by the transformation framework to do some
# error checking and prevent incompatible transformations from
# being connected together. When defining transforms for a
# scale, which are, by definition, separable and have only one
# dimension, these members should always be set to 1.
input_dims = 1
output_dims = 1
is_separable = True
def __init__(self, upper, lower, mu, sigma):
mtransforms.Transform.__init__(self)
self.upper = upper
self.lower = lower
self.mu = mu
self.sigma = sigma
def transform_non_affine(self, a):
"""
This transform takes an Nx1 ``numpy`` array and returns a
transformed copy. Since the range of the Probability scale
is limited by the user-specified threshold, the input
array must be masked to contain only valid values.
``matplotlib`` will handle masked arrays and remove the
out-of-range data from the plot. Importantly, the
``transform`` method *must* return an array that is the
same shape as the input array, since these values need to
remain synchronized with values in the other dimension.
"""
masked = np.ma.masked_where( (a < self.upper) & (a > self.lower) , a)
#Get the CDF of the normal distribution located at mu and scaled by sigma
#Multiply these by 100 to put it into a percent scale
cdf = norm.cdf(masked, self.mu, self.sigma)*100
return cdf
def inverted(self):
"""
Override this method so matplotlib knows how to get the
inverse transform for this transform.
"""
return ProbScale.InvertedProbTransform(self.lower, self.upper, self.mu, self.sigma)
class InvertedProbTransform(mtransforms.Transform):
input_dims = 1
output_dims = 1
is_separable = True
def __init__(self, lower, upper, mu, sigma):
mtransforms.Transform.__init__(self)
self.lower = lower
self.upper = upper
self.mu = mu
self.sigma = sigma
def transform_non_affine(self, a):
#Need to get the PPF value for a, which is in a percent scale [0,100], so move back to probability range [0,1]
inverse = norm.ppf(a/100, self.mu, self.sigma)
return inverse
def inverted(self):
return ProbScale.ProbTransform(self.lower, self.upper)
# Now that the Scale class has been defined, it must be registered so
# that ``matplotlib`` can find it.
mscale.register_scale(ProbScale)
此外，为了获得所需的 y 轴结果，我发现对数刻度确实可以通过一些额外的调整来获得合理的图表。这是用于强制对数刻度具有适当的次要刻度的代码：
axes.set_yscale('log', basey=10, subsy=[2,3,4,5,6,7,8,9])
然后你可以用定位器和格式化器修改标签：
#Adjust the yaxis labels and format
axes.yaxis.set_minor_locator(FixedLocator([200, 500, 1500, 2500, 3500, 4500, 5000, 6000, 7000, 8000, 9000, 15000, 20000, 25000, 30000, 35000, 40000, 45000, 50000]))
axes.yaxis.set_minor_formatter(FormatStrFormatter('%d'))
axes.yaxis.set_major_formatter(FormatStrFormatter('%d'))
所以完整的坐标轴人工输入如下所示：
axes.set_ylabel('Discharge in CFS')
axes.set_xlabel('Exceedance Probability')
plt.setp(plt.xticks()[1], rotation=45)
#Adjust the scales of the x and y axis
axes.set_yscale('log', basey=10, subsy=[2,3,4,5,6,7,8,9])
axes.set_xscale('prob_scale', upper=98, lower=.2)
#Adjust the yaxis labels and format
axes.yaxis.set_minor_locator(FixedLocator([200, 500, 1500, 2500, 3500, 4500, 5000, 6000, 7000, 8000, 9000, 15000, 20000, 25000, 30000, 35000, 40000, 45000, 50000]))
axes.yaxis.set_minor_formatter(FormatStrFormatter('%d'))
axes.yaxis.set_major_formatter(FormatStrFormatter('%d'))
#Finally set the y-limit of the plot to be reasonable
axes.set_ylim((0, 2*pp['Q'].max()))
#Invert the x-axis
axes.invert_xaxis()
#Turn on major and minor grid lines
axes.grid(which='both', alpha=.9)
这提供了一个半对数比例的概率纸图！具有很好的属性，即在这些轴上绘制成直线的任何东西都表明它来自正态分布！