1. Time Series Analysis and Its Applications
Characteristics of Time Series

2. The Nature of Time Series Data
Johnson & Johnson quarterly earnings per share

3. The Nature of Time Series Data
Yearly average global temperature deviations
자연적인 Trend?
사람의 의해 발생된 것?

4. The Nature of Time Series Data
Speech recording of the syllable aaa … hhh sampled at 10,000 points per second with n = 1020 points.

5. The Nature of Time Series Data
Returns of the NYSE
volatility clustering
ARCH, GARCH

6. The Nature of Time Series Data
Monthly SOI and Recruitment (estimated new fish)
4장 period cycle and strengths
5장 lagged regression

7. The Nature of Time Series Data
fMRI data (뇌신경 활동에 비례하는 신호)

8. The Nature of Time Series Data
Arrival phases from an earthquake (top) and explosion (bottom) at 40points per second.
Spectral analysis of variance

9. Time Series Statistical Models
A time series is a realization of a sequence of random variables
이산형 Time Series (insufficient sampling rate)
연속형 Time Series (completely)
adjacent points in time are correlated
𝑥 𝑡 → 𝑥 𝑡+1

10. Time Series Statistical Models
White Noise (순수한 잡음)
independent and identically distributed
Time series White Noise
( 𝑥 1 , 𝑥 2 ,𝑥 3 …. )→( 𝑤 1 , 𝑤 2 ,𝑤 3 …. )

11. Time Series Statistical Models
Example 1.9 Moving Averages
Smoothing
noise가 제거된 trend (filter)

12. Time Series Statistical Models
Example 1.10 Auto regressions

13. Time Series Statistical Models
Random Walk with Drift
어떤 확률변수가 서로 독립적(independent)이고 동일한 형태의 확률분포를 가 지는 경우

14. Time Series Statistical Models
Example 1.12 Signal in Noise (진폭과 𝜎 𝑤 )
unknown signal
white or correlated over time

15. Measures of Dependence: Autocorrelation and Cross-Correlation
CDF
PDF
시계열 데이터의 평균
Descriptive measure
시계열 데이터의 Autocovariance

16. Measures of Dependence: Autocorrelation and Cross-Correlation
Mean Function of a Moving Average Series
Mean Function of a Random Walk with Drift

17. Measures of Dependence: Autocorrelation and Cross-Correlation
The autocovariance function (linear dependence)
Autocovariance of White Noise

18. Measures of Dependence: Autocorrelation and Cross-Correlation
Autocovariance of a Moving Average

19. Measures of Dependence: Autocorrelation and Cross-Correlation
Summarize the values for all s and t
시점 차이 2 간격으로 감소
Stationarity

20. Measures of Dependence: Autocorrelation and Cross-Correlation
Autocovariance of a Random Walk
편의성

21. Measures of Dependence: Autocorrelation and Cross-Correlation
The cross-covariance function

22. Stationary Time Series
A strictly stationary
A weakly stationary
E[ 𝑋 𝑡 ] 𝑖𝑠 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
Cov( 𝑋 𝑡+ℎ , 𝑋 𝑠+ℎ )= Cov( 𝑋 𝑡 , 𝑋 𝑠 ) # t,s 와 관계없이 일정함
Var[ 𝑋 𝑡+ℎ ]=Var[ 𝑋 𝑡 ]

23. Stationary Time Series
Autocorrelation function (ACF) of a stationary time series
= 𝛾(𝑡+ℎ,𝑡)

24. Stationary Time Series
Example 1.19 Stationarity of White Noise
Example 1.20 Stationarity of a Moving Average

25. Estimation of Correlation
Sample autocovariance function
Sample cross-covariance function

26. Estimation of Correlation-1

27. Estimation of Correlation