How to normalize seaborn distplot?

Question

For reproducibility reasons, the dataset and for reproducibility reasons, I am sharing it [here][1].

Here is what I am doing - from column 2, I am reading the current row and compare it with the value of the previous row. If it is greater, I keep comparing. If the current value is smaller than the previous row's value, I want to divide the current value (smaller) by the previous value (larger). Accordingly, the following code:

This gives the following plots.

sns.distplot(quotient, hist=False, label=protname)

As we can see from the plots

• Data-V has a quotient of 0.8 when the quotient_times is less than 3 and the quotient remains 0.5 if the quotient_times is greater than 3.

I want to normalize the values so that we have y-axis of the second plot values between 0 and 1. How do we do that in Python?

Answer 1

Foreword

From what I understand, the seaborn distplot by default does a kde estimation. If you want a normalized distplot graph, it could be because you assume that the graph's Ys should be bounded between in [0;1]. If so, a stack overflow question has raised the question of kde estimators showing values above 1 .

Quoting one answer :

a continous pdf (pdf=probability density function) never says the value to be less than 1, with the pdf for continous random variable, f unction p(x) is not the probability . you can refer for continuous random variables and their distrubutions

Quoting first comment ofimportanceofbeingernest :

The integral over a pdf is 1 . There is no contradiction to be seen here.

From my knowledge it is the CDF (Cumulative Density Function) whose values are supposed to be in [0; 1].

Notice: All possible continuous fittable functions are on SciPy site and available in the package scipy.stats

Maybe have also a look at probability mass functions ?

---

If you really want to have the same graph normalized, then you should gather the actual data points of the plotted function (Option1), or the function definition (Option 2), and normalize them yourself and plot them again.

Option 1

import numpy as np
import matplotlib
import matplotlib.pyplot as plt
import seaborn as sns
import sys

print('System versions          : {}'.format(sys.version))
print('System versions          : {}'.format(sys.version_info))
print('Numpy versqion           : {}'.format(np.__version__))
print('matplotlib.pyplot version: {}'.format(matplotlib.__version__))
print('seaborn version          : {}'.format(sns.__version__))

protocols = {}

types = {"data_v": "data_v.csv"}

for protname, fname in types.items():
    col_time,col_window = np.loadtxt(fname,delimiter=',').T
    trailing_window = col_window[:-1] # "past" values at a given index
    leading_window  = col_window[1:]  # "current values at a given index
    decreasing_inds = np.where(leading_window < trailing_window)[0]
    quotient = leading_window[decreasing_inds]/trailing_window[decreasing_inds]
    quotient_times = col_time[decreasing_inds]

    protocols[protname] = {
        "col_time": col_time,
        "col_window": col_window,
        "quotient_times": quotient_times,
        "quotient": quotient,
    }

    fig, (ax1, ax2) = plt.subplots(1,2, sharey=False, sharex=False)
    g = sns.distplot(quotient, hist=True, label=protname, ax=ax1, rug=True)
    ax1.set_title('basic distplot (kde=True)')
    # get distplot line points
    line = g.get_lines()[0]
    xd = line.get_xdata()
    yd = line.get_ydata()
    # https://stackoverflow.com/questions/29661574/normalize-numpy-array-columns-in-python
    def normalize(x):
        return (x - x.min(0)) / x.ptp(0)
    #normalize points
    yd2 = normalize(yd)
    # plot them in another graph
    ax2.plot(xd, yd2)
    ax2.set_title('basic distplot (kde=True)\nwith normalized y plot values')

    plt.show()

Option 2

Below, I tried to perform a kde and normalize the obtained estimation. I'm not a stats expert, so the kde usage might be wrong in some way (It is different from seaborn's as one can see on the screenshot, this is because seaborn does the job way much better than me. It only tried to mimic the kde fitting with scipy. The result is not so bad i guess )

Screenshot:

Code:

import numpy as np
from scipy import stats
import matplotlib
import matplotlib.pyplot as plt
import seaborn as sns
import sys

print('System versions          : {}'.format(sys.version))
print('System versions          : {}'.format(sys.version_info))
print('Numpy versqion           : {}'.format(np.__version__))
print('matplotlib.pyplot version: {}'.format(matplotlib.__version__))
print('seaborn version          : {}'.format(sns.__version__))

protocols = {}

types = {"data_v": "data_v.csv"}

for protname, fname in types.items():
    col_time,col_window = np.loadtxt(fname,delimiter=',').T
    trailing_window = col_window[:-1] # "past" values at a given index
    leading_window  = col_window[1:]  # "current values at a given index
    decreasing_inds = np.where(leading_window < trailing_window)[0]
    quotient = leading_window[decreasing_inds]/trailing_window[decreasing_inds]
    quotient_times = col_time[decreasing_inds]

    protocols[protname] = {
        "col_time": col_time,
        "col_window": col_window,
        "quotient_times": quotient_times,
        "quotient": quotient,
    }

    fig, (ax1, ax2, ax3, ax4) = plt.subplots(1,4, sharey=False, sharex=False)
    diff=quotient_times
    ax1.plot(diff, quotient, ".", label=protname, color="blue")
    ax1.set_ylim(0, 1.0001)
    ax1.set_title(protname)
    ax1.set_xlabel("quotient_times")
    ax1.set_ylabel("quotient")
    ax1.legend()

    sns.distplot(quotient, hist=True, label=protname, ax=ax2, rug=True)
    ax2.set_title('basic distplot (kde=True)')

    # taken from seaborn's source code (utils.py and distributions.py)
    def seaborn_kde_support(data, bw, gridsize, cut, clip):
        if clip is None:
            clip = (-np.inf, np.inf)
        support_min = max(data.min() - bw * cut, clip[0])
        support_max = min(data.max() + bw * cut, clip[1])
        return np.linspace(support_min, support_max, gridsize)

    kde_estim = stats.gaussian_kde(quotient, bw_method='scott')

    # manual linearization of data
    #linearized = np.linspace(quotient.min(), quotient.max(), num=500)

    # or better: mimic seaborn's internal stuff
    bw = kde_estim.scotts_factor() * np.std(quotient)
    linearized = seaborn_kde_support(quotient, bw, 100, 3, None)

    # computes values of the estimated function on the estimated linearized inputs
    Z = kde_estim.evaluate(linearized)

    # https://stackoverflow.com/questions/29661574/normalize-numpy-array-columns-in-python
    def normalize(x):
        return (x - x.min(0)) / x.ptp(0)

    # normalize so it is between 0;1
    Z2 = normalize(Z)
    for name, func in {'min': np.min, 'max': np.max}.items():
        print('{}: source={}, normalized={}'.format(name, func(Z), func(Z2)))

    # plot is different from seaborns because not exact same method applied
    ax3.plot(linearized, Z, ".", label=protname, color="orange")
    ax3.set_title('Non linearized gaussian kde values')

    # manual kde result with Y axis avalues normalized (between 0;1)
    ax4.plot(linearized, Z2, ".", label=protname, color="green")
    ax4.set_title('Normalized gaussian kde values')

    plt.show()

Output:

System versions          : 3.7.2 (default, Feb 21 2019, 17:35:59) [MSC v.1915 64 bit (AMD64)]
System versions          : sys.version_info(major=3, minor=7, micro=2, releaselevel='final', serial=0)
Numpy versqion           : 1.16.2
matplotlib.pyplot version: 3.0.2
seaborn version          : 0.9.0
min: source=0.0021601491646143518, normalized=0.0
max: source=9.67319154426489, normalized=1.0

---

Contrary to a comment, plotting:

[(x-min(quotient))/(max(quotient)-min(quotient)) for x in quotient]

Does not change the behavior ! It only changes the source data for kernel density estimation. The curve shape would remain the same.

Quoting seaborn's distplot doc :

This function combines the matplotlib hist function (with automatic calculation of a good default bin size) with the seaborn kdeplot() and rugplot() functions. It can also fit scipy.stats distributions and plot the estimated PDF over the data.

By default:

kde : bool, optional set to True Whether to plot a gaussian kernel density estimate.

It uses kde by default. Quoting seaborn's kde doc:

Fit and plot a univariate or bivariate kernel density estimate.

Quoting SCiPy gaussian kde method doc :

Representation of a kernel-density estimate using Gaussian kernels.

Kernel density estimation is a way to estimate the probability density function (PDF) of a random variable in a non-parametric way. gaussian_kde works for both uni-variate and multi-variate data. It includes automatic bandwidth determination. The estimation works best for a unimodal distribution; bimodal or multi-modal distributions tend to be oversmoothed.

Note that I do believe that your data are bimodal, as you mentioned it yourself. They also look discrete. As far as I know, discrete distribution function may not be analyzed in the same way continuous are, and fitting may proove tricky.

Here is an example with various laws:

import numpy as np
from scipy.stats import uniform, powerlaw, logistic
import matplotlib
import matplotlib.pyplot as plt
import seaborn as sns
import sys

print('System versions          : {}'.format(sys.version))
print('System versions          : {}'.format(sys.version_info))
print('Numpy versqion           : {}'.format(np.__version__))
print('matplotlib.pyplot version: {}'.format(matplotlib.__version__))
print('seaborn version          : {}'.format(sns.__version__))

protocols = {}

types = {"data_v": "data_v.csv"}

for protname, fname in types.items():
    col_time,col_window = np.loadtxt(fname,delimiter=',').T
    trailing_window = col_window[:-1] # "past" values at a given index
    leading_window  = col_window[1:]  # "current values at a given index
    decreasing_inds = np.where(leading_window < trailing_window)[0]
    quotient = leading_window[decreasing_inds]/trailing_window[decreasing_inds]
    quotient_times = col_time[decreasing_inds]

    protocols[protname] = {
        "col_time": col_time,
        "col_window": col_window,
        "quotient_times": quotient_times,
        "quotient": quotient,
    }
    fig, [(ax1, ax2, ax3), (ax4, ax5, ax6)] = plt.subplots(2,3, sharey=False, sharex=False)
    diff=quotient_times
    ax1.plot(diff, quotient, ".", label=protname, color="blue")
    ax1.set_ylim(0, 1.0001)
    ax1.set_title(protname)
    ax1.set_xlabel("quotient_times")
    ax1.set_ylabel("quotient")
    ax1.legend()
    quotient2 = [(x-min(quotient))/(max(quotient)-min(quotient)) for x in quotient]
    print(quotient2)
    sns.distplot(quotient, hist=True, label=protname, ax=ax2, rug=True)
    ax2.set_title('basic distplot (kde=True)')
    sns.distplot(quotient2, hist=True, label=protname, ax=ax3, rug=True)
    ax3.set_title('logistic distplot')

    sns.distplot(quotient, hist=True, label=protname, ax=ax4, rug=True, kde=False, fit=uniform)
    ax4.set_title('uniform distplot')
    sns.distplot(quotient, hist=True, label=protname, ax=ax5, rug=True, kde=False, fit=powerlaw)
    ax5.set_title('powerlaw distplot')
    sns.distplot(quotient, hist=True, label=protname, ax=ax6, rug=True, kde=False, fit=logistic)
    ax6.set_title('logistic distplot')
    plt.show()

Output:

System versions          : 3.7.2 (default, Feb 21 2019, 17:35:59) [MSC v.1915 64 bit (AMD64)]
System versions          : sys.version_info(major=3, minor=7, micro=2, releaselevel='final', serial=0)
Numpy versqion           : 1.16.2
matplotlib.pyplot version: 3.0.2
seaborn version          : 0.9.0
[1.0, 0.05230125523012544, 0.0433775382360589, 0.024590765616971128, 0.05230125523012544, 0.05230125523012544, 0.05230125523012544, 0.02836946874603772, 0.05230125523012544, 0.05230125523012544, 0.05230125523012544, 0.05230125523012544, 0.03393500048652319, 0.05230125523012544, 0.05230125523012544, 0.05230125523012544, 0.0037013196009011043, 0.0, 0.05230125523012544]

Screenshot:

Answer 2

In the latest update, the sns.distplot has been deprecated and sns.displot has to be used instead. Hence, to obtain a normalized histogram/density one has to use the following syntax:

sns.displot(x, kind='hist', stat='density');

or

sns.plot(x, stat='density');

instead of

sns.distplot(x, kde=False, norm_hist=True);

PS: to get density instead of histogram the kind value has to be changed to 'kde'.

Reference:

1. https://seaborn.pydata.org/generated/seaborn.histplot.html
2. https://seaborn.pydata.org/generated/seaborn.displot.html