Time Series Analysis Chapter 01 (4up).pdf - 2 Outline 0 1 2...

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TIME SERIES ANALYSIS Chapter 1, Introduction Outline 2 0. Brief review on moments 1. Examples of time series 2. Objectives of time series analysis 3. Simple time series models 4. Stationary models and autocorrelation function (ACF) 5. Trend and seasonality removal 6. Testing the estimated noise sequence 0. Brief review on moments Expectation : ° ± = ² ³ See Appendix A.1.3 Variance : ´ ± µ = ¶·¸ ³ = ² ³ − ° ± µ Covariance and correlation : ¹ ³, º = »¼½ ³, º = ² ³ − ° ± º − ° ¾ ¿ ³, º = »¼¸ ³, º = ÀÁ ±,¾ Ã Ä Ã Å if 0 < ´ ± , ´ ¾ < ∞ 3 Properties of expectation ² ·³ + Æ = ·² ³ + Æ If Ç ³ ≥ · = 1 , then ² ³ ≥ · ; if Ç ³ ≤ Æ = 1 , then ² ³ ≤ Æ . ² ³ # + ⋯ + ³ % = ² ³ # + ⋯ + ² ³ % If ³ and º are independent, then ² ³º = ² ³ ² º . 4
Properties of variance 5 ¶·¸ ³ = ² ³ µ ² ³ µ ¶·¸ ·³ + Æ = · µ ¶·¸ ³ ¶·¸ ³ = 0 if and only if there exists a constant such that Ç ³ = ’ = 1 If ³ and º are independent, then ¶·¸ ³ + º = ¶·¸ ³ + ¶·¸ º Note, independence of ³ and º does not imply ¶·¸ ³º = ¶·¸ ³ ¶·¸ º unless ² ³ = ² º = 0 Properties of covariance and correlation 6 »¼½ ³, º ≤ ´ ± ´ ¾ and hence |¿ ³, º | ≤ 1 »¼½ ³, º = ² ³º − ² ³ ² º If ³ and º are independent, then »¼½ ³, º = 0 The converse is not true (for example, ³~* 0,1 & º = ³ µ ) ¶·¸ ³ + º = ¶·¸ ³ + ¶·¸ º + 2»¼½ ³, º »¼½ ·³ + ƺ, ’, + -¶ = ·’»¼½ ³, , + ·-»¼½ ³, ¶ + Æ’»¼½ º, , + Æ-»¼½ º, ¶ Moments ² ³ . is called the / 01 moment of ³ . ² ³ − ° ± . is called the / 01 central moment of ³ . If ³~* 0,1 , then ² ³ µ.2# = 0 and ² ³ µ. = 2/ − 1 2/ − 3 ⋯ 1 for / = 1,2, ⋯ . 7 Conditional expectation & variance Conditional expectation : ° ¾|± = ² º|³ Conditional variance : ´ ¾|± µ = ¶·¸ º|³ ≝ ²[ º − ² º|³ µ |³] Properties: ² º = ² ² º|³ (iterated expectation) ¶·¸ º = ² ¶·¸ º|³ + ¶·¸ ² º|³ 8
Example #1 Suppose ³~789: 0,1 and º|³~789: 0, ³ . Find ² º|³ ² º ¶·¸ º|³ ¶·¸ º 9 Example #2 (Problem 1.1) Suppose ³ & º are 2 random variables with ² º = ° & ² º µ < ∞ . Show that ’ = ° minimizes ² º − ’ µ . Show that : ³ = ² º|³ minimizes ² ;<º − : ³ = µ |³> . Show that : ³ = ² º|³ also minimizes ² º − : ³ µ . 10 1. Examples of time series A time series is a set of observations ? 0 , each being recorded at a specific time @ ? 0 could be discrete or continuous for a given ? Time series plots We examine a time series plot for trend over time seasonal/cyclical/periodic component 11 . @ @ 12
Examples 13 Australian red wine sales; wine.txt ? 0 = monthly sales of red wine (in kiloliters) by Australian winemakers @ = (Jan, 1980), (Feb, 1980), …, (Oct, 1991) Features? Examples (cont’d) 14 Accidental deaths, USA, 1973–1978; deaths.txt Features?

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