ArDec: Autoregressive-based time series decomposition in R Susana - - PowerPoint PPT Presentation

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ArDec: Autoregressive-based time series decomposition in R Susana - - PowerPoint PPT Presentation

Mar 04, 2024 140 likes •354 views

Introduction Method Application Summary ArDec: Autoregressive-based time series decomposition in R Susana Barbosa Universidade do Porto, Portugal : Introduction Method Application Summary Time series decomposition Approaches:

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Introduction Method Application Summary

ArDec: Autoregressive-based time series decomposition in R

Susana Barbosa

Universidade do Porto, Portugal

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Introduction Method Application Summary

Time series decomposition

Approaches:

◮ non-parametric: filtering / smoothing (eg STL, discrete

wavelet transform, ...)

◮ model-based: regression, structural models, ...

Goals:

◮ remove “known” (non-stationary) components ◮ describe components of interest (seasonal, trend, ...)

:

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Introduction Method Application Summary

Time series decomposition

Approaches:

◮ non-parametric: filtering / smoothing (eg STL, discrete

wavelet transform, ...)

◮ model-based: regression, structural models, ...

Goals:

◮ remove “known” (non-stationary) components ◮ describe components of interest (seasonal, trend, ...)

:

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Introduction Method Application Summary

Trend & seasonality

◮ “The essential idea of trend is that it is smooth.” ◮ “A trend is a consistent pattern over time. ◮ “A trend is a long-term movement in time series data after

  • ther components have been accounted for. “

◮ “A trend is a trend, is a trend, is a trend, ...”

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Introduction Method Application Summary

Trend & seasonality

◮ “the characteristics of a time series giving rise to spectral

peaks at seasonal frequencies” [Nerlove 1964]

◮ “the intra-year pattern of variation which is repeated

constantly or in an evolving fashion from year to year” [Shiskin et al. 1967]

◮ “periodic fluctuations that recur with about the same

intensity each year” [Hillmer and Tiao 1982].

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Introduction Method Application Summary

Problem

How to retrieve physically-relevant components?

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Introduction Method Application Summary

Problem

How to retrieve physically-relevant components?

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Introduction Method Application Summary

Method

  • M. West 1997

(Time series decomposition. Biometrika 84) Basic concept: Xt =

p

  • j=1

φjXt−j + εt = ⇒ Xt =

p

  • j=1

γj

t

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Introduction Method Application Summary

State-space representation of AR(p) process

Xt = F TZt Zt = GZt−1 + εt

with

F T = [1 0 ... 0] Z T

t = [Xt Xt−1 ... Xt−p+1]

G =      φ1 φ2 ... φp 1 ... . . . 1 ... . . .     

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Introduction Method Application Summary

State-space representation of AR(p) process

Xt = F TZt Zt = GZt−1 + εt G = EAE−1 − → rje±iwj a = ETF , bt = E−1Zt Xt = p

j=1 γj t , γj t = ajbj t

wj = 0 − → γj

t = rjγt−j + νt

wj = 0 − → γj

t = 2rjcos(wj)γj t−1 + r 2 j γj t−2 + ηt :

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Introduction Method Application Summary

Decomposition of sea-level records with ArDec

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Introduction Method Application Summary

Decomposition of sea-level records with ArDec

❃ ❧✐❜r❛r②✭❆r❉❡❝✮ ❃ ❝♦❡❢❂❛r❞❡❝✳❧♠✭❞❛t✮✩❝♦❡❢❢✐❝✐❡♥ts ❃ ❝♦❡❢

❳✶ ❳✷ ❳✸ ❳✹ ❳✺ ❳✻ ✵✳✸✽✻✶✽✻✹✹✻ ✵✳✵✺✵✵✵✼✺✸✻ ✵✳✵✽✽✻✹✸✹✺✾ ✵✳✵✵✷✼✸✵✵✵✹ ✵✳✵✹✺✾✶✺✷✺✵ ✲✵✳✵✵✾✺✸✾✻✹✺ ❳✼ ❳✽ ❳✾ ❳✶✵ ❳✶✶ ❳✶✷ ✵✳✵✺✽✶✸✼✺✽✽ ✵✳✵✸✷✵✶✺✽✾✼ ✲✵✳✵✼✺✸✼✽✼✵✾ ✵✳✵✻✹✽✹✼✹✹✵ ✵✳✶✶✼✼✵✺✾✺✾ ✵✳✶✻✾✸✷✷✼✽✸ ❳✶✸ ❳✶✹ ❳✶✺ ❳✶✻ ❳✶✼ ❳✶✽ ✵✳✵✻✵✷✽✽✽✽✾ ✲✵✳✵✼✼✻✷✶✻✹✵ ✲✵✳✵✼✹✽✽✵✺✾✵ ✲✵✳✵✶✷✾✶✶✷✷✸ ✵✳✵✶✵✽✻✾✵✹✸ ✲✵✳✵✵✾✼✹✷✼✸✷ ❳✶✾ ❳✷✵ ❳✷✶ ❳✷✷ ❳✷✸ ❳✷✹ ✲✵✳✵✹✽✹✾✾✻✸✻ ✲✵✳✵✵✷✻✹✸✺✵✺ ✲✵✳✵✹✹✹✷✷✼✷✷ ✵✳✵✺✹✻✾✽✸✼✷ ✵✳✵✺✷✺✷✾✶✹✼ ✵✳✶✸✾✽✹✾✼✻✾ ❳✷✺ ❳✷✻ ✵✳✵✼✷✽✵✸✽✸✻ ✲✵✳✵✽✶✻✽✼✶✵✹ :

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Introduction Method Application Summary

Decomposition of sea-level records with ArDec

❃ ❛r❞❡❝✭❞❛t✱❝♦❡❢✮ ♣❡r✐♦❞ ❞❛♠♣✐♥❣ ✶ ✧tr❡♥❞✧ ✧✵✳✾✾✻✧ ✷ ✧✶✷✳✵✽✾✧ ✧✵✳✾✽✻✧ ✸ ✧✻✳✵✵✵✧ ✧✵✳✾✽✷✧

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Introduction Method Application Summary

Decomposition of sea-level records with ArDec

❃ str✭❛r❞❡❝✳❝♦♠♣♦♥❡♥ts✭❛r❞❡❝✳♦✉t✮✮

▲✐st ♦❢ ✷ ✩ ♣❡r✐♦❞❝♦♠♣s✿▲✐st ♦❢ ✷ ✳✳✩ ♣❡r✐♦❞s✿ ♥✉♠ ❬✶✿✷❪ ✶✷✳✶ ✻✳✵ ✳✳✩ ❝♦♠♣s ✿ ♠ts ❬✶✿✾✸✻✱ ✶✿✷❪ ◆❆ ◆❆ ◆❆ ◆❆ ◆❆ ◆❆ ◆❆ ◆❆ ◆❆ ✳✳✳ ✳✳ ✳✳✲ ❛ttr✭✯✱ ✧❞✐♠♥❛♠❡s✧✮❂▲✐st ♦❢ ✷ ✳✳ ✳✳ ✳✳✩ ✿ ◆❯▲▲ ✳✳ ✳✳ ✳✳✩ ✿ ❝❤r ❬✶✿✷❪ ✧❙❡r✐❡s ✶✧ ✧❙❡r✐❡s ✷✧ ✳✳ ✳✳✲ ❛ttr✭✯✱ ✧ts♣✧✮❂ ♥✉♠ ❬✶✿✸❪ ✶✾✷✽ ✷✵✵✻ ✶✷ ✳✳ ✳✳✲ ❛ttr✭✯✱ ✧❝❧❛ss✧✮❂ ❝❤r ❬✶✿✷❪ ✧♠ts✧ ✧ts✧ ✩ tr❡♥❞❝♦♠♣ ✿ ❚✐♠❡✲❙❡r✐❡s ❬✶✿✾✸✻❪ ❢r♦♠ ✶✾✷✽ t♦ ✷✵✵✻✿ ◆❆ ◆❆ ✳✳✳

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Introduction Method Application Summary

Decomposition of sea-level records with ArDec

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Introduction Method Application Summary

Chesapeake bay

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Introduction Method Application Summary

Chesapeake bay: annual components

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Introduction Method Application Summary

Chesapeake bay: trend components

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Introduction Method Application Summary

Package ArDec

◮ implements autoregressive-based time series

decomposition

◮ model-based, additive decomposition ◮ yields periods of physically-relevant (non-damped)

components

◮ extracts flexible, time-varying estimates of such

components

◮ option of Bayesian framework for autoregressive estimation

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Introduction Method Application Summary

Thanks!

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