David Lu
Software Developer
Introduction to Stationary Process
Yung-Sheng Lu
Mar 7, 2017
@NCKU-CSIE
Outline
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Time Series
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Stationarity
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Stationary Process
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References
Time Series
Definition
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A time series is a variable indexed by the time :
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Examples:
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The time can be annual, monthly, daily...
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Let be the annual GDP.
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Let be the monthly temperature
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Let be the daily stocks
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X
X
t
t
X_t
Xt
X_t
Xt
Y_t
Yt
Z_t
Zt
t
t
Some Distinctions
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The time series can:
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Take discrete or continuous values.
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Be measured at discrete or continuous time.
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Be measured at regular or irregular intervals.
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Description
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To describe a time serie, we will concentrate on:
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Data Generating Process (DGP)
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The joint distribution of its elements
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Its "moments":
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Expected value
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Variance
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Autocovariance or autocorrelation of order
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\mathrm{E}[X_t] \equiv \mu _t
E[Xt]≡μt
\mathrm{var}[X_t] \equiv \sigma ^ 2 \equiv \gamma _0
var[Xt]≡σ2≡γ0
k
k
\mathrm{cov}[X_t , X_{t-k}] \equiv \gamma _k
cov[Xt,Xt−k]≡γk
X_t
Xt
Description (cont.)
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Definition of Variance
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Definition of Autocovariance
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Definition of autocorrelation of order
\mathrm{var}[X_t] \equiv \mathrm{E}[(X_t - \mathrm{E}[X_t]) ^ 2]
var[Xt]≡E[(Xt−E[Xt])2]
k
k
\mathrm{corr}[X_t , X_{t-k}] \equiv \dfrac{\mathrm{cov}[X_t , X_{t-k}]}{\sqrt{\mathrm{var}[X_t]} \sqrt{\mathrm{var}[X_{t-k}]}}
corr[Xt,Xt−k]≡var[Xt]var[Xt−k]cov[Xt,Xt−k]
\gamma _0 (t)
γ0(t)
\gamma _k (t)
γk(t)
\mathrm{cov}[X_t , X_{t-k}] \equiv \mathrm{E}[(X_t - \mathrm{E}[X_t])(X_{t-k} - \mathrm{E}[X_{t-k}])]
cov[Xt,Xt−k]≡E[(Xt−E[Xt])(Xt−k−E[Xt−k])]
Stationarity
Stationary Series
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The mean of the series should not be a function of time rather should be a constant.
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The variance of the series should not be a function of time.
Stationary Series (cont.)
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The covariance of the i-th term and the (i + m)-th term should not be a function of time.
Stationary Series (cont.)
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Example
Imagine a girl moving randomly on a giant chess board. In this case, next position of the girl is only dependent on the last position.
Random Walk
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Questions
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How to predict the position with time?
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How accurate will be?
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Random Walk (cont.)
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The randomness brings at every point in time.
X_t = X_{t-1} + Er(t)
Xt=Xt−1+Er(t)
X
X
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Recursively fit in all the
X_t = X_0 + \sum \limits_{t = 1} ^n {Er(t)}
Xt=X0+t=1∑nEr(t)
Random Walk (cont.)
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Is the mean constant?
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Expectation of any error will be 0 as it is random.
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E[x(t)] = E[x(0)] + \sum \limits_{t = 1} ^ n {E[Er(t)]}
E[x(t)]=E[x(0)]+t=1∑nE[Er(t)]
\Rightarrow E[x(t)] = E[x(0)] \in \mathbf {R}
⇒E[x(t)]=E[x(0)]∈R
Random Walk (cont.)
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Is the variance constant?
- The random walk is not a stationary process as it has a time variant variance.
- The covariance is also dependent on time.
\mathrm{var}[x(t)] = \mathrm{var}[x(0)] + \sum \limits_{t = 1} ^ n {\mathrm{var}[\mathrm{err}(t)]}
var[x(t)]=var[x(0)]+t=1∑nvar[err(t)]
\Rightarrow \mathrm{var}[x(t)] = t \times \mathrm{var}[\mathrm{err}(t)]
⇒var[x(t)]=t×var[err(t)]
Random Walk (cont.)
Stationary Process
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Coefficient
X_t = R \times X_{t-1} + Er(t)
Xt=R×Xt−1+Er(t)
- Stationary series with .
R = 0
R=0
Coefficient
R
R
R
R
X_t = R \times X_{t-1} + Er(t)
Xt=R×Xt−1+Er(t)
- Stationary series with .
R = 0.5
R=0.5
Coefficient (cont.)
R
R
X_t = R \times X_{t-1} + Er(t)
Xt=R×Xt−1+Er(t)
- Stationary series with .
R = 0.9
R=0.9
Coefficient (cont.)
R
R
X_t = R \times X_{t-1} + Er(t)
Xt=R×Xt−1+Er(t)
- Stationary series with .
\Rightarrow E[X_t] = R \cdot E[X_{t-1}]
⇒E[Xt]=R⋅E[Xt−1]
R = 1.0
R=1.0
Coefficient (cont.)
R
R
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如果有一個訊號 對於所有 都滿足以下條件，則我們稱此為一個平穩過程。
和 的聯合機率分布 (joint distribution)，只和
和 的時間差有關，與其他參數都無關。
p(X_{t_0 + k}, {t_0 + k}, X_{t + k}, {t + k}) = p(X_{t_0}, t_0, X_t, t)
p(Xt0+k,t0+k,Xt+k,t+k)=p(Xt0,t0,Xt,t)
X_t
Xt
X_{t_0}
Xt0
t_0
t0
t
t
X
X
k
k
Definition
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若為一個平穩隨機過程，則需滿足以下條件：
- 機率密度函數 (PDF) 在任意時間點 都是相同。
- 與時間無關 (time-independent) 的函式。
p(X_{t + k}, {t + k}) = p(X_t, t)
p(Xt+k,t+k)=p(Xt,t)
t
t
Definition (cont.)
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白雜訊 (white noise)
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功率密度為常數的隨機過程。
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一個時間連續隨機過程 其中 為實數是一個白雜訊若且唯若滿足以下條件：
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\mu_x = E[x(t)] = 0
μx=E[x(t)]=0
R_{xx}(t_0, t) = E[x(t_0)x(t)] = \dfrac{N_0}{2} \delta(t_0 - t)
Rxx(t0,t)=E[x(t0)x(t)]=2N0δ(t0−t)
\Rightarrow S_{xx} = \dfrac{N_0}{2}
⇒Sxx=2N0
功率密度
t
t
X_t
Xt
Example
References
References
Introduction to Stationary Process Yung-Sheng Lu Mar 7, 2017 @NCKU-CSIE
Introduction to Stationary Process
By David Lu
Introduction to Stationary Process
Introduction to Stationary Process
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