関数imsequence_dispersion マニュアル

(The documentation of function imsequence_dispersion)

Last Update: 2021/12/8

◆機能・用途(Purpose)

フーリエスペクトルのサンプルの値の分散を計算する。
Calculate the dispersion of the sample values of a Fourier spectral data.

◆形式(Format)

#include <sequence/statistics.h>
inline double imsequence_dispersion(struct imsequence seq)

◆引数(Arguments)

seq	計算に用いるフーリエスペクトルを表す構造体。 A structure to represent the Fourier spectral data used for the computation.

◆戻り値(Return value)

seq.valueの配列要素の分散。
The dispersion of the array components of seq.value.

◆使用例(Example)

struct imsequence seq;
double sigma2=imsequence_dispersion(seq);

◆計算式とアルゴリズム (Formula and algorithm)

a_{n} =

seq.value[n],

N =

seq.sizeとおく。この

a_{n}

(

n = 0, 1, 2, \dots, N - 1

) の分散

σ^{2}

を計算すれば良く、定義式は以下の通りである。

\begin{matrix} (1) & σ^{2} = \frac{1}{N} \sum_{n = 0}^{N - 1} | a_{n} - μ |^{2} \end{matrix}

\begin{matrix} (2) & μ = \frac{1}{N} \sum_{n = 0}^{N - 1} a_{n} \end{matrix}

(

2

)式を用いると(

1

)式は以下のように変形できる。

\begin{array}{rcl} σ^{2} & = & \frac{1}{N} \sum_{n = 0}^{N - 1} (a_{n} - μ)^{*} (a_{n} - μ) \\ = & \frac{1}{N} \sum_{n = 0}^{N - 1} (a_{n}^{*} a_{n} - μ^{*} a_{n} - μ a_{n}^{*} + μ^{*} μ) \\ = & \frac{1}{N} \sum_{n = 0}^{N - 1} a_{n}^{*} a_{n} - μ^{*} \frac{1}{N} \sum_{n = 0}^{N - 1} a_{n} - μ \frac{1}{N} \sum_{n = 0}^{N - 1} a_{n}^{*} + μ^{*} μ \frac{1}{N} \sum_{n = 0}^{N - 1} 1 \\ = & \frac{1}{N} \sum_{n = 0}^{N - 1} a_{n}^{*} a_{n} - μ^{*} μ - μ μ^{*} + μ^{*} μ \\ (3) & = & \frac{1}{N} \sum_{n = 0}^{N - 1} | a_{n} |^{2} - | μ |^{2} \end{array}

この

μ

は関数imsequence_average (sequence/statistics.h)によって計算でき、

\begin{matrix} (4) & \frac{1}{N} \sum_{n = 0}^{N - 1} | a_{n} |^{2} \end{matrix}

は関数imsequence_power2_average (sequence/statistics.h)によって計算できる。したがって、関数imsequence_averageの戻り値を

a

、 imsequence_power2_averageの戻り値を

a_{2}

とおけば、分散は

\begin{matrix} (5) & σ^{2} = a_{2} - | a |^{2} \end{matrix}

によって計算できる。この関数ではこの方法により分散を計算する。
Let us introduce notations

a_{n} =

seq.value[n] and

N =

seq.size. Then the quantity required is the dispersion

σ^{2}

for this

a_{n}

(

n = 0, 1, 2, \dots, N - 1

), which is defined by eqs. (

1

) and (

2

). Using eq. (

2

), eq. (

1

) can be arranged as (

3

). The value of

μ

and the quantity defined by eq. (

4

) can be computed by functions imsequence_average and imsequence_power2_average (both in sequence/statistics.h), respectively. Then the dispersion can be computed by eq. (

5

), where

a

and

a_{2}

are the return values of functions imsequence_average and imsequence_power2_average, respectively. This function calculates the dispersion by this approach.