Time Series Analysis and Its Applications Characteristics of Time Series

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

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

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

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

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

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

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

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

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 …. )

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

Time Series Statistical Models Example 1.10 Auto regressions

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

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

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

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

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

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

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

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

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

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 ]

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

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

Estimation of Correlation Sample autocovariance function Sample cross-covariance function

Estimation of Correlation -1

Estimation of Correlation