s se Timeseries analysis

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?????s? ?????-se????(Time-series analysis)

• H. Isliker, 2004

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?e??e??µe??

• ??sa???? pa?ade??µata ?S, ?e????? s??p?? ?a?
  ???s?µ?t?ta, ßas???? ?????e?
• ?p??? µ???d?? pe????af?? ?????-se????
• ????d?? a????s?? ?????-se????a?t?-s?s??t?s?
  (auto-correlation)fasµat??? a????s? (spectral
  analysis)

3
??ß?????af?a

• C. Chatfield, The Analysis of Time Series,An
  Introduction, 4th ed., Chapman Hall,London,
  1989
• The Numerical Recipes(free downloadable from the
  net)

4
?? e??a? ?? ?????-se??e?
s?µa (signal) X(t)
time t
?????-se??? s?????? ap? pa?at???se?? p?? ????a?
d?ad????? st? ????? ? a????s? ?????-se????
a???e? st? stat?st???
5
?a?ade??µata ??????µ?a

• p.?. ? e?????? t?µ?? t?? pet?e?a???, ? a??a µ?a?
  µet???? (ef?µe??de? ??a t?? ??????µ?a? !)

S?f???????, µet??? µ?a? eta???a?
????
????? e?d?af???? p??ß?e?? !!!
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?a?ade??µata ?µp???? (marketing)

• p.?. p???se?? a?? µ??a
• s??p?? 1 p??ß?e??, a??? ?a? s?ed?asµ??
• s??p?? 2 s?????s?, p.?. dap??e? ??a d?af?µ?se??
  ?a? a???se?? st?? p???se?? s?????s? 2
  ?????-se????
• e????s? ? µ?a ?S e???e? t?? ????

d?af?µ?se??
p???se??
e???
e???
t
t
7
?a?ade??µata

• ??µ???af?a e?????? p????sµ??,e?d?af????
  p??ß?e??
• ??e???? d?ad??as??? pa?a?????,p???t?ta e???
  p?????t??,p.?. p???? µ?a? ß?da?

???d?a??af?, ep?d????µe?? t?µ?
??µ? µetaß??t?? e??????
t
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?a?ade??µata ?at????

• ??e?t??-???efa?????f?µa (??G, EEG)
• ??e?t??-?a?d?????f?µa (??G, ?CG)
• S??p???ata???s? ?a? e????s? t?? d??aµ???? t??
  ?a?d???, t?? e??ef??????a?????s? a??µa????,
  p??ß?e?? d?ata?a???

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?a?ade??µata F?s???

• ?p?? ?????ta? pa?at???se?? ...
• Se?sµ?????a (?ata???s? ?a? p??ß?e??)
• ?ete???????a (p??ß?e??)
• ?st??f?s???

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?a?ade??µata ?tµ?sfa????? f?s???
ta??t?ta a??µ??
?e?µ???as?a
(?a?/µ?? ??????, 2004)
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?a?ade??µata ?st??f?s???
???a?? ???aµ?? (solar flare) ?a?at???s? se
??d??-s????t?te? (300MHz)
S??p??

• pe????af? ??a s?????s? µe ?e???t??? µ??t??a,
• a??? ?a? ?ata???s? t?? d?ad??as??? p?? pa??????
  t?? ?S, ?a??d???s? st?? d?µ??????a µ??t????
• ?a ep?t??? µ??t??a ???s?µ?p?????ta? µet? ?a?
  ??ap??ß???e??

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S??p?? t?? ??S

• ?e????af? (description)
• ?????s? (explanation)
• ??????s? (prediction)
• ??e???? (control)
• ?ata???s?

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S??p?? 1 pe????af?

• ???ta ?????µe p??ta t? ??af??? pa??stas? µ?a?
  ?????-se????
• ?p?? pe????af?pe???d???t?ta (seasonal effects),
  t?s? (trend), a??a?? t?s??
• ???????µ??? pe????af? µest??ast??? µ??t??a
  (st??ast???? d?ad??as?e?), p.?. ? ?S pa??st??e?
  ?e??? ????ß?

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S??p?? 2 ?????s?

• ????µe 2 ? pe??ss?te?e? ?????-se????
• ?p??e? ? µ?a ?a e???e? t?? ?????
• ?(t) Y(t)f(X(t)),X(t) input, Y(t) output

X(t)
d?af?µ?se??
p???se??
Y(t)
e???
e???
X(t)
Y(t)
f
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S??p?? 3 ??????s?

• ?p? µ?a ?S µp????µe ?a p??ß?????µe t? µ?????, ?a?
  µe p?s? a???ße?a ?

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S??p?? 5 ?ata???s?

• ? stat?st??? d??e? f??µa??st???? pe????af?? t??
  ?S ?a?ea?t??
• St? f?s??? µp??e? ?a ????µe µ?a ?S ap? ??a ?at?
  ta ???a ????st? s?st?µa?? µp????µe ?a
  ?ata??ß??µe ap? t?? ?S ??a t? ?d?? t? s?st?µa, t?
  ?p??? ??e? pa???e? t?? ?S?
• ??. t? s?st?µa e??a? pe???d???, µe pe???d?
  ...,? t? s?st?µa e??a? e?te??? st??ast???

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?as???? ?????e?

• S??e??? (continuous) ?SX(t) pa?at??e?ta? s??e???
• d?a???t? (discrete) ?S

X(t)
t
X(ti), i1,2,3,
t
t1
t2
ti
18

• St?? ?/? ß??e???? ?? d?a???t?? ?S(p??ß??µa
  ap????e?s?? !)
• Sampling (read off, digitize) d?aß????µe ?a?
  ??at?µe ap? s??e?? ?S t?µ?? µ??? se s?µe?a µe
  sta?e?? ??????? ap?stas? ?t (sampling
  time/interval) ? µet??µe e?a???? µ??? se
  d?a???t?? ???????? st??µ??

X(t)
t
X(ti)
?p?fe????µe µ?-sta?e?? ?t, ?? pe??ss?te?e?
µ???d?? ?????ta? p?? d?s???e? !
? t
t
t1
t2
t3
19

• Se ???e? pe??pt?se??, t? X(ti) e??a? ?????sµa ?
  ????????µa ??a ??? t? ?t
• p.?. X(ti) ß???? a?? m2 a?? µ??a,?????sµa ????
  t?? ß????pt?se??
• p.?. X(ti) ??d??-a?t???ß???a ap? t?? ????
• ?p?? f(t) ? s??e??? a?t???ß???a, ?a?ti-ti-1?t

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• ??d??? st?? stat?st??? ?e???a ??a t?? a????s?
  ?????-se????s??????, ?? d?ad?????? pa?at???se??
  de? e??a? a?e???t?te?, ??a p??pe? ?a
  ??ß??µe?p???? µa? t? se??? t?? pa?at???se??
• ????ß?? a?t? ? e???t?s? ep?t??pe? t??p?????s?
  t?? µ?????t?? µe ß?s? t? pa?e????
• ???sµ???tete?µ???st??? ?S ep?t??pe? p?????s?
  µe a???ße?aSt??ast??? ?S ep?t??pe? p??ß???e??
  µ??? e? µ??e?, µe p??a??t?ta p ?a s?µße? ?, ...

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??? ßas???? p??se???se?? ??a t?? ??S

• Time-domain?? µ???d?? e??a? s??a?t?se?? t??
  ??????,p.?. a?t?-s?s??t?s? (auto-correlation)
• Frequency domain?????µe µetas??µat?sµ? Fourier
  ?a? d???e???µe st? ???? t?? s????t?t??,p.?.
  fasµat??? p????t?ta (spectral density)

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?p??? pe????af???? µ???d??

• ??af??? pa??stas? !!!!!
• µ?s?? ???? µ
• d?asp??? ?2
• t?????ta? µ?s?? ????
• stas?µ?t?ta (stationarity)
• a????s? pe???d???t?ta, t?se??, ????ß??
• f??t????sµa (filtering)

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?a??de??µa
X(t)
t

• ??p?? pe???d???
• ??e? ????ß?

?d??t?te? ?
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??s?? ???? µ
?st? ? ?S
???sµ?? t?? µ?s?? ????
d??. ? µ?s? t?µ? ???? t?? t?µ?? t?? ?S, ? ???sµ??
e??a? ?p?? s??????, ?a? ? se??? t?? X(ti) de?
pa??e? ???? !
St? pa??de??µa ? -0.77
25
?? s?µa??e? ?-0.77 ?
Plot it !
??s?? ????
S?µp?pte? µe t?? e?t?µ?s? µa?
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??asp??? ?2
?st? ?S
???sµ?? t?? d?asp????
??s?? ???? t?? ap????se?? ap? t? µ?s? t?µ? st?
tet??????
X(ti)
X(ti)-?
?
t
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St??ta?t ap????s?

• ? d?asp??? ??e? µa??µat??? p?e??e?t?µata, d??.
  st?? stat?st??? ?e???a
• ??? d?a?s??t??? e??a? ? st??ta?t ap????s? ?
  (standard deviation)µ?s? ap????s? ap? t?? µ?s?
  t?µ?

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?-0.77
?2 54.10,
? 7.36,
St? pa??de??µa
? ?
?
? - ?
?eta?? ?-? ?a? ?? ß??s???ta? ta pe??ss?te?a
s?µe?a t?? ?S, ?? d??st?µa a?t? µa? d??e? t?
d?a??µa?s? t?? t?µ?? t?? ?S
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?a??de??µa 2
? 4.24, ?2 59.50, ? 7.71
X(t)
??
?
??
t
? ?a? ? ? de? d????? ?a?? pe????af? t??
p?a?µat???? µ?s?? t?µ?? ed? ????? ?p???e? µ?a
t?s? (trend), d??. sa? ?a a????e? t? p?a?µat???
? st? ????? ? ?(t)
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?e d?a?s??s?
?(t) a tb, s???e???µ??a ?(t) t10/512
d??. ?at?s?e??saµe ??a µ??t??? ??a t?? t?s?
(trend) ??? µp????µe ?µ?? ?a t? ?????µe p??
s?st?µat??? ?
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??????ta? µ?s?? ???? (running mean, moving
average)
tiK
ti-K
ti

• ?a??µet??? K, µ???? t?? pa?a?????

32
?????? µ?s?? ????, ?40
) ? (t?????ta?) µ?s?? ???? a????e? st? ?????
33
??????ta? µ?s?? ????, ?40
) ? (t?????ta?) µ?s?? ???? a????e? st? ?????
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Stas?µ?t?ta (stationarity)

• d?a?s??t???? ???sµ??µ?a ?S e??a? st?s?µ? a? de?
  ?p???e? s?st?µat??? a??a?? t?? µ?s?? ???? ?a? t??
  d?asp???? st? ?????
• p.?. t?s? ) µ?-stas?µ?t?ta
• ? stas?µ?t?ta e??a? p???p??es? ??a ta pe??ss?te?a
  e??a?e?a t?? ??S (p.?. a?t?-s?s??t?s?, fasµat???
  a????s?)
• ) ??e?????ta? e??a?e?a µetat??p?? µ?-st?s?µ?? se
  st?s?µe? ?S !