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Time Series Analysis and Its Applications
Characteristics of Time Series
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The Nature of Time Series Data
Johnson & Johnson quarterly earnings per share
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The Nature of Time Series Data
Yearly average global temperature deviations 자연적인 Trend? 사람의 의해 발생된 것?
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The Nature of Time Series Data
Speech recording of the syllable aaa … hhh sampled at 10,000 points per second with n = 1020 points.
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The Nature of Time Series Data
Returns of the NYSE volatility clustering ARCH, GARCH
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The Nature of Time Series Data
Monthly SOI and Recruitment (estimated new fish) 4장 period cycle and strengths 5장 lagged regression
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The Nature of Time Series Data
fMRI data (뇌신경 활동에 비례하는 신호)
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The Nature of Time Series Data
Arrival phases from an earthquake (top) and explosion (bottom) at 40points per second. Spectral analysis of variance
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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
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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 …. )
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Time Series Statistical Models
Example 1.9 Moving Averages Smoothing noise가 제거된 trend (filter)
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Time Series Statistical Models
Example 1.10 Auto regressions
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Time Series Statistical Models
Random Walk with Drift 어떤 확률변수가 서로 독립적(independent)이고 동일한 형태의 확률분포를 가 지는 경우
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Time Series Statistical Models
Example 1.12 Signal in Noise (진폭과 𝜎 𝑤 ) unknown signal white or correlated over time
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Measures of Dependence: Autocorrelation and Cross-Correlation
CDF PDF 시계열 데이터의 평균 Descriptive measure 시계열 데이터의 Autocovariance
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Measures of Dependence: Autocorrelation and Cross-Correlation
Mean Function of a Moving Average Series Mean Function of a Random Walk with Drift
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Measures of Dependence: Autocorrelation and Cross-Correlation
The autocovariance function (linear dependence) Autocovariance of White Noise
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Measures of Dependence: Autocorrelation and Cross-Correlation
Autocovariance of a Moving Average
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Measures of Dependence: Autocorrelation and Cross-Correlation
Summarize the values for all s and t 시점 차이 2 간격으로 감소 Stationarity
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Measures of Dependence: Autocorrelation and Cross-Correlation
Autocovariance of a Random Walk 편의성
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Measures of Dependence: Autocorrelation and Cross-Correlation
The cross-covariance function
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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 ]
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Stationary Time Series
Autocorrelation function (ACF) of a stationary time series = 𝛾(𝑡+ℎ,𝑡)
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Stationary Time Series
Example 1.19 Stationarity of White Noise Example 1.20 Stationarity of a Moving Average
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Estimation of Correlation
Sample autocovariance function Sample cross-covariance function
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Estimation of Correlation
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Estimation of Correlation
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