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Q-Q Plots#

Preparation#

import pandas as pd
from scipy.stats import norm, skewnorm, laplace, uniform

from lets_plot import *

LetsPlot.setup_html()

df = pd.read_csv("https://raw.githubusercontent.com/JetBrains/lets-plot-docs/master/data/mpg.csv")
print(df.shape)
hwy = "hwy"
cty = "cty"
drv = "drv"
df.head()

(234, 12)
Unnamed: 0 manufacturer model displ year cyl trans drv cty hwy fl class
0 1 audi a4 1.8 1999 4 auto(l5) f 18 29 p compact
1 2 audi a4 1.8 1999 4 manual(m5) f 21 29 p compact
2 3 audi a4 2.0 2008 4 manual(m6) f 20 31 p compact
3 4 audi a4 2.0 2008 4 auto(av) f 21 30 p compact
4 5 audi a4 2.8 1999 6 auto(l5) f 16 26 p compact

Two types of Q-Q plots#

geom_qq() and geom_qq_line() functions.#

ggplot(df, aes(sample=hwy)) + \
    geom_qq(size=5, color="#3d3d3d", alpha=.3) + \
    geom_qq_line(size=1) + \
    ggtitle("Distribution of highway miles per gallon", \
            "Comparison of sample quantiles with normal distribution quantiles")

geom_qq2() and geom_qq2_line() functions.#

ggplot(df, aes(x=cty, y=hwy)) + \
    geom_qq2(size=5, color="#3d3d3d", alpha=.3) + \
    geom_qq2_line(size=1) + \
    ggtitle("City miles vs. highway miles (per gallon)", \
            "Comparison of quantiles of two sample distributions")

Quick Q-Q plot: the qq_plot() function#

In the ‘bistro’ module there is a Q-Q plot in which points and lines of both types combined to the one function with some convenient defaults.

from lets_plot.bistro.qq import qq_plot

qq_plot(df, sample=hwy) + \
    ggtitle("Distribution of highway miles per gallon", \
            "Comparison of sample quantiles with normal distribution quantiles")

Deviations investigation#

n = 1_000
norm_df = pd.DataFrame({"sample": norm.rvs(size=n, random_state=42)})
skewed_df = pd.DataFrame({"sample": skewnorm.rvs(7, size=n, random_state=42)})
neg_kurtosis_df = pd.DataFrame({"sample": uniform.rvs(size=n, random_state=42)})
pos_kurtosis_df = pd.DataFrame({"sample": laplace.rvs(size=n, random_state=42)})

p1 = ggplot(norm_df)
p11 = p1 + geom_histogram(aes(x="sample")) + ggtitle("Normal: histogram")
p12 = p1 + geom_qq(aes(sample="sample")) + geom_qq_line(aes(sample="sample")) + ggtitle("Normal: Q-Q plot")
p2 = ggplot(skewed_df)
p21 = p2 + geom_histogram(aes(x="sample")) + ggtitle("Skewed: histogram")
p22 = p2 + geom_qq(aes(sample="sample")) + geom_qq_line(aes(sample="sample")) + ggtitle("Skewed: Q-Q plot")
p3 = ggplot(neg_kurtosis_df)
p31 = p3 + geom_histogram(aes(x="sample")) + ggtitle("-Kurtosis: histogram")
p32 = p3 + geom_qq(aes(sample="sample")) + geom_qq_line(aes(sample="sample")) + ggtitle("-Kurtosis: Q-Q plot")
p4 = ggplot(pos_kurtosis_df)
p41 = p4 + geom_histogram(aes(x="sample")) + ggtitle("+Kurtosis: histogram")
p42 = p4 + geom_qq(aes(sample="sample")) + geom_qq_line(aes(sample="sample")) + ggtitle("+Kurtosis: Q-Q plot")

gggrid([p11, p12, p21, p22, p31, p32, p41, p42], ncol=2)

Choose a distribution#

The distribution parameter of the qq_plot() function.

p1 = qq_plot(df, hwy, distribution="norm", quantiles=[.1, .9]) + \
    ggtitle("Normal distribution")
p2 = qq_plot(df, hwy, distribution="uniform", quantiles=[.1, .9]) + \
    ggtitle("Uniform distribution")
p3 = qq_plot(df, hwy, distribution="t", quantiles=[.1, .9]) + \
    ggtitle("Student's t-distribution distribution")
p4 = qq_plot(df, hwy, distribution="exp", quantiles=[.1, .9]) + \
    ggtitle("Exponential distribution")

gggrid([p1, p2, p3, p4], ncol=2)

Q-Q stats with other geometries#

ggplot(df, aes(x=cty, y=hwy, color=drv)) + \
    geom_line(stat="qq2") + \
    geom_point(stat="qq2", shape=15) + \
    geom_line(stat="qq2_line", color='#636363', linetype=5) + \
    facet_grid(x=drv, scales="free") + \
    xlab("cty quantiles") + ylab("hwy quantiles")