Visualising the spectral analysis of time series
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1 Visualising he specral analysis of ime series Adam M. Sykulski Marie Curie Research Fellow NorhWes Research Associaes (NWRA), Seale, USA & Universiy College London (UCL), UK Slides available online a:
2 Noaion x() : coninuous real-valued saionary process, 2 R x : discree real-valued saionary process, 2 Z : angular frequency, =2 f (f is measured in herz) : ime-lag (posiive or negaive) i = p 1
3 Jus a few equaions Fourier Transform: f x () = R 11 x()e i d, 2 R Inverse Fourier Transform: x() = 1 2 R 1 1 f x()e i d, 2 R Power Specral Densiy: S x () =lim T 1 E 1 2T R T T x()e i d 2 Relaionship wih auocovariance sequence s x ( ) =E[x()x( )]: S x () = R 1 1 s x( )e i d () s x ( ) = 1 2 R 1 1 S x()e i d Percival and Walden, page 65
4 x() =sin(.1) 2 ime series 6 power specral densiy x() S x () (db) db = 1 log 1
5 x() = sin(.1)+.6 sin(.13) 2 ime series 6 power specral densiy x() S x () (db)
6 x() = sin(.1)+.6 sin(.13)+.2 sin(.18) 2 ime series 6 power specral densiy x() S x () (db)
7 x = ", " N (, 1) x ime series S x () (db) power specral densiy
8 x =.9x 1 + ", " N (, 1) x ime series S x () (db) power specral densiy
9 x = ".7" 1, " N (, 1) x ime series S x () (db) power specral densiy
10 Longiude Oceanographic roary specra db " Laiude Complex-valued ime series: z = x + iy 33 32
11 Longiude Oceanographic roary specra db " Laiude Complex-valued ime series: z = x + iy 33 32
12 Solomon Islands 1991 Earhquake X # Y # Y # X # ime (UTC) Complex-valued ime series: z = x + iy
13 ime series ime series 1 1 Y # 1 3 Y # X # X # 1 3 db roary specra : -:/2 :/2 : db roary specra : -:/2 :/2 :
14 Specral Densiy Esimaion Observe: X 1,...,X N from he process x() or x Discree Fourier Transform: J X () = 1 p N P N =1 X e i, 2 R Periodogram: Ŝ X () = J x () 2, 2 R Fourier Frequencies: 2 2 N, 1,...,b N 2 c Percival and Walden, page 196
15 x =.9x 1 + ", " N (, 1) S x () = 2 " cos()+ 2 1 x ime series S x () (db) power specral densiy Periodogram Specral Densiy
16 More lives have been los looking a he raw periodogram han by any oher acion involving ime series John W. Tukey
17 Why? Warning 1: Beware of he Decibels E[1 log 1 {ŜX()}] 1 log 1 S x () = 1 log 1 (e) = (Euler-Mascheroni consan) 2.5 Percival and Walden, page 28
18 x =.9x 1 + ", " N (, 1) S x () = 2 " cos() ime series 3 2 power specral densiy Periodogram Specral Densiy SD (db bias adjused) x 2-2 S x () (db)
19 x =2.767x x x x 4 + ", " N (, 1) S x () = 1 2 " P p k=1 ke i 2 1 ime series 5 4 power specral densiy Periodogram Specral Densiy 5 3 x S x () (db)
20 Wha else? Warning 2: Sampling isn forever Theory: S x () =lim T 1 E 1 2T R T T x()e i d 2 Pracice: Ŝ X () = P N =1 X e i 2 o Convoluion: E nŝx () = R F( )S x ( )d where F( ) ishefejérkernel:f() = 1 2 N sin 2 (N/2) sin 2 (/2) Percival and Walden, page 198
21 x =2.767x x x x 4 + ", " N (, 1) S x () = 1 2 " P p k=1 ke i 2 F() (db) Fejer kernel (N=16) S x () (db) power specral densiy Periodogram Specral Densiy
22 x =2.767x x x x 4 + ", " N (, 1) S x () = 1 2 " P p k=1 ke i 2 F() (db) Fejer kernel (N=1) S x () (db) power specral densiy Periodogram Specral Densiy
23 Poenial Soluion: Tapering o Reminder: E nŝx () = R F( )S x ( )d Propose a daa aper h, such ha we work wih h X New specral esimae: Ŝ (d) X () = P N =1 h X e i 2 Define: H() = P N =1 h e i 2 Then: E nŝ(d) o X () = R H( )S x ( )d Require: R H( )d = P N =1 h2 =1 Percival and Walden, page 26
24 .3 no/recangular aper (N=64) 2 power specral densiy.25 h H() (db)
25 .3 dpss(1) aper (N=64) 2 power specral densiy.25 h H() (db)
26 .3 dpss(2) aper (N=64) 2 power specral densiy.25 h H() (db)
27 .3 dpss(4) aper (N=64) 2 power specral densiy.25 h H() (db)
28 .3 dpss(8) aper (N=64) 2 power specral densiy.25 h H() (db)
29 x =2.767x x x x 4 + ", " N (, 1) 2 1 Fejer kernel (N=1) 5 4 power specral densiy Periodogram Specral Densiy F() (db) S x () (db)
30 x =2.767x x x x 4 + ", " N (, 1) 2 1 dpss(1) kernel (N=1) 5 4 power specral densiy dpss(1) esimae Specral Densiy H() (db) S x () (db)
31 x =2.767x x x x 4 + ", " N (, 1) 2 1 dpss(2) kernel (N=1) 5 4 power specral densiy dpss(2) esimae Specral Densiy H() (db) S x () (db)
32 5 4 3 power specral densiy (N=128) Periodogram Specral Densiy S x () (db)
33 5 4 3 power specral densiy (N=128) dpss(1) esimae Specral Densiy S x () (db)
34 5 4 3 power specral densiy (N=128) dpss(2) esimae Specral Densiy S x () (db)
35 5 4 3 power specral densiy (N=128) dpss(4) esimae Specral Densiy S x () (db)
36 5 4 3 power specral densiy (N=128) dpss(8) esimae Specral Densiy S x () (db)
37 X() Maérn(A =1, =.6,h=.1) S x () = A 2 ( 2 + h 2 ) ime series 3 2 power specral densiy periodogram Specral Densiy x 2-2 S x () (db)
38 X() Maérn(A =1, =.6,h=.1) S x () = A 2 ( 2 + h 2 ) ime series 3 2 power specral densiy dpss(2) aper Specral Densiy x 2-2 S x () (db)
39 X() Maérn(A =1, =.6,h=.1) S x () = A 2 ( 2 + h 2 ) ime series 3 2 power specral densiy dpss(8) aper Specral Densiy x 2-2 S x () (db)
40 Wha s wrong his ime? Warning 3: Sampling isn coninuous aliasing ime series sample x.5 x Percival and Walden, page 97
41 Wha s wrong his ime? Warning 3: Sampling isn coninuous aliasing ime series sample x coninuous process x()?.5 x
42 Wha s wrong his ime? Warning 3: Sampling isn coninuous aliasing ime series sample x coninuous process x()?.5 x
43 Wha s wrong his ime? Warning 3: Sampling isn coninuous aliasing ime series sample x coninuous process x()?.5 x
44 Wha s wrong his ime? Warning 3: Sampling isn coninuous aliasing ime series sample x coninuous process x()?.5 x
45 Wha s wrong his ime? Warning 3: Sampling isn coninuous aliasing ime series sample x coninuous process x()?.5 x
46 Wha s wrong his ime? Warning 3: Sampling isn coninuous aliasing ime series sample x coninuous process x()?.5 x
47 Wha s wrong his ime? Warning 3: Sampling isn coninuous aliasing ime series sample x coninuous process x()?.5 x
48 In 2-D he aliasing problem is also known as he wagon-wheel effec Link o Video
49 X() Maérn(A =1, =.6,h=.1) 1X A 2 S x () = (( +2 k) 2 + h 2 ) k= ime series 3 2 power specral densiy periodogram Specral Densiy Aliased Specral Densiy x 2-2 S x () (db)
50 Summary Specral analysis of ime series can reveal hidden srucure in a ime series Care mus be aken when using periodograms log-scale: adjus for Euler-Mascheroni Consan shor N or high dynamic range: aper coninuous or subsampled: accoun for aliasing
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