Time Series Analysis - 穩態時間序列簡介 Introduction to Stationary Time Series

Experience with real-world data, however, soon convinces one that both stationarity and Gaussianity are fairy tales invented for the amusement of undergraduates.
(Thomson 1994)

ㄧ、平穩過程 (Stationary process)

1. 平穩過程的定義

[用心去感覺]  這是一個限制很強的定義，因為這代表無論在哪個時間區間內，所有的degree的moment(像是期望值 expectations, 變異數 variances, third order and higher)都相同

2. 平穩過程的例子：白雜訊

• 數學期望為0：$\mu_n = \mathbb{E} \{ n(t) \} = 0$
• 其自相關函數為狄拉克δ函數 (單位脈衝函數) ：$r_{nn} = \mathbb{E} \{ n(t) n(t-\tau) \} = \delta ( \tau )$

二、廣義平穩 (弱平穩)

• 數學期望函數 $mx(t)$ 必須是常數：$\mathbb{E}\{x(t)\} = m_x(t) = m_x(t + \tau) \,\, \forall \, \tau \in \mathbb{R}$
• 相關函數 (autocorrelation function) 僅僅與 $t_1$ 和 $t_2$ 之間的差值相關：$\mathbb{E}\{x(t_1)x(t_2)\} = R_x(t_1, t_2) = R_x(t_1 + \tau, t_2 + \tau) = R_x(t_1 - t_2, 0) \,\, \forall \, \tau \in \mathbb{R}$.

[補記] 矩(moment)

Mr. Opengate：動差(矩) 與 動差生成函數 moment and moment generating function
http://mropengate.blogspot.tw/2015/04/moment-and-moment-generating-function.html

$\mu'_n=\int_{-\infty}^\infty (x - c)^n\,f(x)\,dx$

$E(x) = \int_{-\infty}^\infty x\,f(x)\,dx$

$var (x) = \int_{-\infty}^\infty \left[x - E(x)\right]^2 \,f(x)\,dx$

Reference

http://homepage.ntu.edu.tw/~sschen/Book/Slides/Ch4Arma.pdf

G. P. Nasion, chapter 11 stationary and non-stationary time series
http://www.cas.usf.edu/~cconnor/geolsoc/html/chapter11.pdf

wikipedia
http://www.wikipedia.org/

investopedia