[시계열] 비정상 확률과정(2)

비정상 확률과정

• 평균수준이 시간대에 따라 다름
  • -> 결정적Deterministic 추세: regression
  • -> 확률적Stochastic 추세: 차분 & ADF
• 추세Trend 지님
  • -> 결정적 계절추세: regression
  • -> 확률적 계절추세: 계절 차분
• 계절성Seasonality 지님
• 분산 변함

Type of Non-stationary Process

• $E[y_t] \neq 0$ and change with time: 평균수준이
• $Var[y_t]$ change with time -> Ranom Walk

Random Walk

• 확률과정
• $\epsilon = \text{white noise}, \alpha = intercept$
• 1) $W_t = W_{t-1} + \epsilon_t$: Pure Random Walk
• 2) ST: $W_t = \alpha + W_{t-1}+\epsilon_t$: Random Walk with Drift
• 3) $W_t = \alpha + W_{t-1} + \beta t + \epsilon_t$: Random Walk with Drift and Derministic Trend
• cf) DT: $Y_t = \alpha + \beta t + \epsilon_t$: Deterministic Trend
  

Transformation

1. A Random Walk with or without a drift

• Differencing: $\nabla W_t = W_t - W_{t-1}=\epsilon_t$ or $\alpha + \epsilon_t$
• Detrending:
  • $Y_t = \alpha + \beta t + \epsilon_t \rightarrow \nabla Y_t=$

import statsmodels.api as sm
df = sm.datasets.get_rdataset("CanPop", package="carData").data
df.plot(x='year', y='population')
plt.show()

# df = df.set_index('year')
df_diff1 = df - df.shift()
df_diff1.plot()

<matplotlib.axes._subplots.AxesSubplot at 0x7f8892339390>

reference

• 이론: http://www.irealism.org/xe/datascience/3226
• 시계열 전반: https://otexts.com/fpp2/lag-plots.html