?????s? ?.2: ????d?? t?? d?af???? (differencing)

1
?????s? ?.2 ????d?? t?? d?af???? (differencing)

• a?a????a - a? f(t) a bt (??aµµ??? t?s?),
  t?te df/dt b,- a? f(t) a bt2 (µ?-??aµµ???
  t?s?), t?te d2f/dt2 2b) ? pa??????? afa??e?
  t?? t?se?? !
• ???sµ?? te?est?? d?af???s?? (difference
  operator)p??t?? t????de?te??? t????

2

• ?S t?? d?af???? (differenced time-series)
• ?ts? ?????µe ??a s?µe?? (t? te?e?ta??), ? ?S
  e??a? p?? µ????
• ? t?s? apa?e?f???e, ? pe???d???t?ta d?at??e?ta?
  (µe?????e t? p??t??)
• a??????e ? ????ί??

3

• ??at? a??????e ? ????ί??
• An EX 0, EY 0 ?a? ?,? a?e???t?te?
  VarX-Y E(X-Y)2 ??2 2
  EXEY EY2 VarX
  VarY
• a? VarX VarY ?2
  VarX-Y 2?2
• d??. ?(X-Y)
  21/2 ?

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t? ?p????p? (residual), R(ti) X(ti) - Y(ti)

• t? ?p????p? s?µp?pte? s?ed?? µe t?? a????? ?S,
  a??????e ?µ?? ? ????ί??
• R(ti) X(ti) X(ti1) - X(ti)
  2X(ti)-X(ti1)
• ? µ???d?? t?? d?af???? e??a? ?a?? ??a t??
  apa?e?f? t?? t?se??, t? ?p????p? p?? pa???e?
  de? e??a? ?µ?? ???s?µ?

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• d?af??? de?te??? t????
• ?a? ? t?s? ?a? ? pe???d???t?ta apa?e?f???a?,
  ?µe??e µ??? ? ?a?a??? ????ί??
• ? µ???d?? t?? d?af???? e??a? ?a?? ??a t??
  p??sd????sµ? (extraction) t?? ????ί??, ?a? t??
  µetat??p? t?? µ?-st?s?µ?? se st?s?µe? ?S,a????e?
  ?µ?? t? p??t?? t?? ????ί??

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?s??s? 3

• ?p?????sete t?? ?S Y(ti) t?? d?af???? p??t??
  t???? t?? ?S X(ti) t?? ?s??s?? 1
• ??af??? pa??stas?, µa?? µe t?? a????? ?S

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x,y -1,1
8
?????s? ?.3 p??sa?µ??? (fitting) µ?a? s????t?s??

• S??p?? ed? e?t?µ?s? ?a? apa?e?f? t?? t?s??
• Ge????te??? s??p?? p??sa?µ??? µ?a? s????t?s?? se
  pa?at???se??, µet??se??p.?. pe??aµa e?e??e???
  pt?s??, µet??µe t?? ap?stas? X(ti) sa? s????t?s?
  t?? ?????? ti
• µ?t??s? ??µ?? ????ί??) interpolation
  de? µa? e?d?af??e?,a??? p??sa?µ???, ? ?aµp???
  de? pe???e? ap? ta s?µe?a,p??pe? ?a pe??ae? ?µ??
  ???t? ap? a?t?

?(ti)
X
?e???a X(ti) ½ gti2
t
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?as??? ί?µata t?? p??sa?µ????

• 1. ep????? µ?a? s????t?s?? ??a p??sa?µ???
  (?e???a, ???? p??? p????f?????, ?pt??? e?t?µ?s?)
• 2. ? ?d?a ? p??sa?µ??? (fitting)
• 3. e?t?µ?s? t?? p???t?ta? t?? p??sa?µ????

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? µ???d?? t?? e?a??st?? tet?a????? (least squares)

• p.?. p??sa?µ??? µ?a? e??e?a?, f(t) a bt
• ??????µe a ?a? b ?ts? ?ste

?a ???eta? e????st? !
X(ti)
f(t)
? s??????? tet?a?????? ap?stas? e??a? e????st?
f(ti)
t
ti
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?as???? a???? t?? µe??d?? t?? e?a??st?? tet?a?????
e????st?
,
)
2 e??s?se?? ??a 2 a???st??? .. ) a, b
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• p??sa?µ??? t?? f(t) a bt st?? ?S X(ti)
• a? ? f(t) e??a? µ?-??aµµ??? t?te p????pt???
  µ?-??aµµ???? e??s?se??, ?? ?p??e? ?????ta? µ???
  a???µ?t???,p.?. f(t) ? sin(?tf)

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pa??de??µa
?p??es? ? t?s? e??a? ??aµµ??? ) p??sa?µ??? t??
s????t?s?? f(t) a bt
ί?????a? a0.65 b0.02
?p????p? R(ti) X(ti)-f(ti)
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?s??s? 4

• p??sa?µ?sete st?? ?S X(ti) t?? ?s??s?? 1 t?
  s????t?s? f(t) a bt
• ??af??? pa??stas?, µa?? µe t?? a????? ?S
• ?p?????sete t? ?p????p? ??af??? pa??stas?

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?????s? ?? pe???d???t?ta

• s??p?? d?a????sµ?? t?? pe???d???? µ????? ap? t??
  ????ί?,?e????µe ap? t? ?S ????? t?? t?s?
• µ???d??
• 1. ??aµµ??? f??t????sµa,- e?te µe t??
  t?????ta µ?s? ???, a??? µ???? pa?????? ?,-
  e?te µe e??et??? e??µ????s? (exponential
  smoothing)?a? ?p????p? (residual)
• 2. p??sa?µ??? t?? s????t?s?? f(t) A
  sin(?tf)
• (e????sta tet?????a, least squares)

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• s???p?????? 3
• ?a?a?t???st??a
• t?s?,
• pe???d???t?ta,
• ????ί??

S??p?? ap?µ???s? t?? ?a???a



pe???d???t?ta
t?s?
????ί??
?ts? µp????µe ?a ?a?a?t???s??µe p?? ?a?a?? t?
???e ?a?a?t???st???
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t?????ta? µ?s?? ???? (moving average), ? 2
a????? ?S ????? t?s?
f??t?a??sµ??? ?S
?p????p?
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e??et??? e??µ????s? (exponential smoothing), ?
0.2
a????? ?S ????? t?s?
f??t?a??sµ??? ?S
?p????p?
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p??sa?µ??? t?? f(t) A sin (?tf), p??t?
p??sp??e?a
a????? ?S, ????? t?s?
p??sa?µ???, ί?????a? ? -0.14 ? 0.994 f
3.32 pe???d?? ? 6.32
a????? ?S p??sa?µ???
) ? p??sa?µ??? ap?t??e e?te??? ...
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• ????? t?? ap?t???a? ? s????t?s? f(t) e??a?
  µ?-??aµµ???, ? ??s? ?, ?, f st? µ???d? t??
  e?a??st?? tet?a????? ί??s?eta? a???µ?t??? ?ts?
  ?ste ?a ???eta? e????st? t?p???,
  ?e?????ta? ap? µ?a a?t?µat?, a??a??et? a?????
  p??ί?e?? (initial guess) t?? Mathematica

?
?
a??a??et? a????? p??ί?e?? (initial guess)
pe??e?t??? (global) e????st?
t?p??? e????st? p?? ί?????e
??????ta? ?µ?? eµe?? ??t? (explicitly) µ?a
?a??te?? a????? p??ί?e?? (initial guess), ?
µ???d?? µp??e? ?s?? ?a pet??e? !
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p??sa?µ??? t?? f(t) A sin (?tf), de?te??
p??sp??e?a µe ??t? a????? p??ί?e?? (explicit
initial guess)
a????? ?S, ????? t?s?
a????? p??ί?e?? A0 ?, ?0 2?/T0 ?
! A0 10, ?0 2?/40
p??sa?µ??? ί?????a? ? 9.98 ? 0.16 f
-0.02 pe???d?? ? 39.49
?p????p? (pe????e? µ?a µ???? t?s?)
? µ???d?? t?? p??sa?µ???? µa? d??e? ?a? t??
pe???d? !
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?s??s? 4a

• ??a????sete t?? pe???d???t?ta ap? t?? ????ί? ??a
  t?? ?S ????? t?s? t?? ?s??s?? 1d,???s?µ?p????ta?
  t?? µ???d? t?? t?????ta µ?s?? ????.
• ??af??? pa??stas?

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?s??s? 4ί

• ??a????sete t?? pe???d???t?ta ap? t? ????ί? ??a
  t?? ?S ????? t?s? t?? ?s??s?? 1d, ???s?µ?p????ta?
  t? µ???d? t?? p??sa?µ????.
• ??af??? pa??stas?

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• p??sa?µ??? µe Mathematica
• a????? ?S s????t?s? ??a pa??µet???
• p??sa?µ???
• yfit4 FindFit yfit2r, aSinwtt ph, a,
  10, w, 2.Pi/40., ph, tt initial
  guess
• yfit4 e??a? replacement table ed?
• ? se pa???te?e? e?d???? t?? Mathematica
• ltltStatisticsNonlinearFit
• yfit4 NonlinearFit yfit2r, aSinwttph, tt,
  a,10, w,2.Pi/40., ph,0
• ?p???e?t??? d????µe a????? p??ί?e?? (initial
  guess) ??a ??e? t?? pa?aµ?t????, ?a? ? se??? t??
  pa?aµ?t??? e??a? d?af??et???
• ed? yfit4 e??a? s????t?s?

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?????s? s????? (µ???d?? t?? t?????ta µ?s?? ????)
a?x??? ?S

t?s?

pe???d???t?ta (1o ?p????p?)

????ί?? (2? ?p????p?)
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S?µp??asµa

• G?a t?? pe??pt?s? t?? ?S p?? a?a??saµe, ??
  µ???d?? t?? t?????ta µ?s?? ???? ?a? t??
  p??sa?µ???? ?ta? ep?t??e?? st?? ta?t?s? ?a? t??
  t?s?? ?a? t?? pe???d???t?ta?
• ?? µ???d?? t?? e??et???? e??µ????s?? ?a? t??
  d?af???? de? ?d?sa? t?s? ??a??p???t???
  ap?te??sµata
• ??t? t? s?µp??asµa ?µ?? de? ?e???e?eta?, a???
  e?a?t?ta? ap? t? s???e???µ??? pe??pt?s? t?? ?S
  p?? a?a????µe ...... ?a?? e??a? ?a ?????µe
  d??f??e? µe??d??? !