初めに
Just a moment, but what is actually a moment? ．．．なんてね.
Methods
期待値（Expectation），分散（Variance）
連続型だけ考える．
期待値，分散を
\begin{align}
E \ [ X ]
&= \int_{-\infty}^{\infty} x p \left( x \right) \, dx
\\
V \ [ X ]
&= \int_{-\infty}^{\infty} \left( x - \mu \right)^2 p \left( x \right) \, dx
\end{align}
と定めると，
\begin{align}
V \ [ X ]
&= \int_{-\infty}^{\infty} \left( x - \mu \right)^2 p \left( x \right) \, dx
\\
&= \int_{-\infty}^{\infty} \left( x^2 - 2 \mu x + \mu^2 \right) p \left( x \right) \, dx
\\
&= \int_{-\infty}^{\infty} x^2 p \left( x \right) \, dx
- 2 \mu \int_{-\infty}^{\infty} x p \left( x \right) \, dx
+ \mu^2 \int_{-\infty}^{\infty} p \left( x \right) \, dx
\\
&= E \ [ X^2 ]
- 2 \mu E \ [ X ]
+ \mu^2
\\
&= E \ [ X^2 ] - \left( E \ [ X ] \right)^2
\quad \left( \because \mu = E \ [ X ] \right)
\end{align}
が成り立つ．
Moment (raw/central)
ところで，
\begin{align}
E \ [ X ]
&= \int_{-\infty}^{\infty} x p \left( x \right) \, dx
\\
V \ [ X ]
&= \int_{-\infty}^{\infty} \left( x - \mu \right)^2 p \left( x \right) \, dx
\end{align}
を見つめると，
\begin{align}
E \ [ X ]
&= E \ [ \left( X - 0 \right)^1 ]
\\
V \ [ X ]
&= E \ [ \left( X - \mu \right)^2 ]
\end{align}
と書いても差し支えないことに気付く．ここで，
\begin{align}
m_{0}^{(n)}
&= E \ [ \left( X - 0 \right)^{n} ]
\\
m_{\mu}^{(n)}
&= E \ [ \left( X - \mu \right)^{n} ]
\end{align}
を導入し，それぞれ，
- $m_{0}^{(n)}$: 原点周りの$n$次モーメント（$n$-th raw moment）
- $m_{\mu}^{(n)}$: 期待値周りの$n$次モーメント（$n$-th central moment）
と呼ぶ．これに従えば，
- 期待値: 原点周りの1次モーメント
- 分散 : 期待値周りの2次モーメント
である．
歪み度（Skewness），尖り度（Kurtosis）
特に，
\begin{align}
S \ [ X ]
&= \frac{E \ [ \left( X - \mu \right)^3 ]}{\sigma^3}
= \frac{m_{\mu}^{(3)}}{\sigma^3}
\\
K \ [ X ]
&= \frac{E \ [ \left( X - \mu \right)^4 ]}{\sigma^4}
= \frac{m_{\mu}^{(4)}}{\sigma^4}
\end{align}
を，歪み度，尖り度と呼ぶ（本当は，歪度（わいど），尖度（せんど）と呼ぶのだけれど．．．音からイメージするのが難しいので，暫くの間はこう呼んでおく）．
Moment-generating function
$t \in \mathbb{R}$に対して，
\begin{align}
M_{X} \left( t \right)
&= E \ [ e^{t X} ]
\\
&= \int_{-\infty}^{\infty} e^{t x} p \left( x \right) \, dx
\end{align}
を，$X$のMoment母関数（Moment-generating function）と呼ぶ．原点周りで多項式展開すると，
\begin{align}
M_{X} \left( t \right)
&= \int_{-\infty}^{\infty} e^{t x} p \left( x \right) \, dx
\\
&= \int_{-\infty}^{\infty}
\left(
1 + t x + \frac{(t x)^2}{2!} + \frac{(t x)^3}{3!} + \cdots
\right)
p \left( x \right)
\, dx
\\
&= \int_{-\infty}^{\infty} p \left( x \right) \, dx
+ t \int_{-\infty}^{\infty} x p \left( x \right) \, dx
+ \frac{t^2}{2!} \int_{-\infty}^{\infty} x^2 p \left( x \right) \, dx
+ \cdots
\\
&= 1
+ t E \ [ X ]
+ \frac{t^2}{2!} E \ [ X^2 ]
+ \cdots
\\
&= 1
+ t m_{0}^{(1)}
+ \frac{t^2}{2!} m_{0}^{(2)}
+ \cdots
\end{align}
より，
\begin{align}
M_{X} \left( t=0 \right)
&= 1
\\
\frac{d}{d t} M_{X} \left( t=0 \right)
&= m_{0}^{(1)}
\\
\frac{d^2}{d t^2} M_{X} \left( t=0 \right)
&= m_{0}^{(2)}
\\
\vdots
\\
\frac{d^n}{d t^n} M_{X} \left( t=0 \right)
&= m_{0}^{(n)}
= E \ [ X^{n} ]
\end{align}
として，確かにモーメントたちが生まれている．
Results
色々な分布から
$X \sim \mathcal{N} \ (\mu=0, \sigma^2=1), \ \sim \mathcal{t} \ (\nu=5), \ \Gamma \ (k=2, \theta=1), \ \mathcal{L} \ (\mu=0, b=1)$
歪み，尖りが確認できる．
終わりに
あれ，重み付き残差法でも，冪乗でテストする方針を，method of momentsと呼んでいたっけ．
そう言えば，構造力学にも，断面1次・2次モーメントという概念がある．
うーん，みんな，冪を積分する形をmomentと呼んでいる．何か背景があるのかな．
Appendix
import numpy as np
import scipy as sp
import matplotlib.pyplot as plt
import seaborn as sns
################################################################################
plt.style.use("default")
plt.style.use("classic")
plt.rcParams["font.family"] = "STIXGeneral"
plt.rcParams["mathtext.fontset"] = "cm"
plt.rcParams["figure.figsize"] = (7, 5)
plt.rcParams["figure.autolayout"] = True
plt.rcParams["axes.grid"] = True
plt.rcParams["axes.axisbelow"] = True
plt.rcParams["grid.alpha"] = .3
plt.rcParams["legend.framealpha"] = .8
plt.rcParams["legend.facecolor"] = "w"
plt.rcParams["savefig.dpi"] = 300
################################################################################
def compute_moments(data, name, about=None):
if about==0. or about=="zero" or about=="origin" or about=="raw":
# print("\nraw moments")
m1 = sp.stats.moment(data, order=1, center=0.)
m2 = sp.stats.moment(data, order=2, center=0.)
m3 = sp.stats.moment(data, order=3, center=0.)
m4 = sp.stats.moment(data, order=4, center=0.)
elif about==None or about=="mean" or about=="center" or about=="central":
# print("\ncentral moments")
m1 = sp.stats.moment(data, order=1, center=None)
m2 = sp.stats.moment(data, order=2, center=None)
m3 = sp.stats.moment(data, order=3, center=None)
m4 = sp.stats.moment(data, order=4, center=None)
else:
raise ValueError(f"about {about} is not a valid argument")
print(f"\n--- {name} ---")
print(f"1st moment: {m1:.6f}")
print(f"2nd moment: {m2:.6f}")
print(f"3rd moment: {m3:.6f}")
print(f"4th moment: {m4:.6f}")
return m1, m2, m3, m4
################################################################################
size = 1000
data_gauss = np.random.normal(loc=0., scale=1., size=size)
data_student = np.random.standard_t(df=5., size=size)
data_gamma = np.random.gamma(shape=2., scale=1., size=size)
data_laplace = np.random.laplace(loc=0., scale=1., size=size)
data = [data_gauss, data_student, data_gamma, data_laplace]
names = ["Gaussian", "Student's t", "Gamma", "Laplace"]
colors = ["C0", "C1", "C2", "C3"]
bins = 32
density = True
alpha = .8
nrows, ncols = 1, 4
fig, ax = plt.subplots(nrows, ncols, figsize=(20, 5))
for i, (data, name, color) in enumerate(zip(data, names, colors)):
ax[i].hist(data, bins=bins, density=density, alpha=alpha, color=color, label=name)
sns.kdeplot(data, ax=ax[i], color=color, label=f"{name}: KDE", fill=False, linestyle="--", linewidth=5)
m1, m2, m3, m4 = compute_moments(data, name, about="raw")
ax[i].legend()
ax[i].set(
xlabel="Value",
ylabel="Density",
xlim=(-10., 10.),
ylim=(0., .5),
title=rf"$m_{{0}}^{{(1)}}: {m1:.2f}, m_{{0}}^{{(2)}}: {m2:.2f}, m_{{0}}^{{(3)}}: {m3:.2f}, m_{{0}}^{{(4)}}: {m4:.2f}$",
# title=rf"$m_{{\mu}}^{{(1)}}: {m1:.2f}, m_{{\mu}}^{{(2)}}: {m2:.2f}, m_{{\mu}}^{{(3)}}: {m3:.2f}, m_{{\mu}}^{{(4)}}: {m4:.2f}$",
)
fig.tight_layout()
fig.savefig("./fig.png")
plt.show()