Introduction to Statistical Analysis of Time Series Richard A. Davis Department of Statistics

Transcription

1 Iroduio o Saisial Aalysis of Time Series Rihard A. Davis Deparme of Saisis Oulie Modelig obeives i ime series Geeral feaures of eologial/eviromeal ime series Compoes of a ime series Frequey domai aalysis-he sperum Esimaig ad removig seasoal ompoes Oher ylial ompoes Puig i all ogeher

2 Time Series: A olleio of observaios x, eah oe beig reorded a ime. (Time ould be disree,,,3,, or oiuous > 0.) Obeive of Time Series Aalaysis Daa ompressio -provide ompa desripio of he daa. Explaaory -seasoal faors -relaioships wih oher variables (emperaure, humidiy, polluio, e) Sigal proessig -exraig a sigal i he presee of oise Prediio -use he model o predi fuure values of he ime series

3 Geeral feaures of eologial/eviromeal ime series Examples.. Maua Loa (CO,, O `58-Sep `90) CO Feaures ireasig red (liear, quadrai?) seasoal (mohly) effe. 3

4 . Ave-max mohly emp (vegeaioudra,, ) emp Feaures seasoal (mohly effe) more variabiliy i Ja ha i July Go o ITSM Demo 4

5 emp July: mea.95, var.6305 Ja : mea -.486, var emp Sep : mea 7.5, var.466 Lie:

6 Compoes of a ime series Classial deomposiio X m + s + Y m red ompoe (slowly hagig i ime) s seasoal ompoe (kow period d4(hourly), d(mohly)) Y radom oise ompoe (migh oai irregular ylial ompoes of ukow frequey + oher suff). Go o ITSM Demo 6

7 Esimaio of he ompoes. X m + s + Y Tred m filerig. E.g., for mohly daa use mˆ (.5x 6 + x x x + 6) / polyomial fiig m ˆ a + a a k k 7

8 Esimaio of he ompoes (o). Seasoal s X m + s + Y Use seasoal (mohly) averages afer deredig. (sadardize so ha s sums o 0 aross he year. sˆ ( x + x + + x+ 4 ) / N, N umber of years harmoi ompoes fi o he ime series usig leas squares. ˆ s π Aos( ) + π Bsi( ) Irregular ompoe Y Yˆ X mˆ sˆ 8

9 x s s 0 3 π0 π0 (os( ),, os( 6) )'/sqr(6) 6 6 π π (os( ),, os( 6) )' /sqr(3) 6 6 π π (si( ),, si ( 6) )' /sqr(3) 6 6 π π (os( ),,os( 6))'/sqr(3) 6 6 π π (si( ),, si( 6) )'/sqr(3) 6 6 π π (os( ),,os( 6))'/sqr(6 The sperum ad frequey domai aalysis Toy example. (6) 0 s s 3 X(4.4, 3.6, -3.4, -3.4, 0.739, 3.04) 0 +5( +s )-.5( +s )

10 0 Fa: Ay veor of 6 umbers, x (x,..., x 6 ) a be wrie as a liear ombiaio of he veors 0,,, s, s, 3. More geerally, ay ime series x (x,..., x ) of legh (assume is odd) a be wrie as a liear ombiaio of he basis (orhoormal) veors 0,,,, [/], s, s,, s [/]. Tha is, ] / [, 0 0 m b a b a a m m m m s s x ω ω ω ω ω ω ) si( ) si( ) si(, ) os( ) os( ) os(, / / / 0 s

11 Properies:. The se of oeffiies {a 0, a, b, } is alled he disree Fourier rasform ] / [, 0 0 m b a b a a m m m m s s x ω ω x b x a x a / / / / / 0 0 ) si( ), ( ) os( ), ( ), ( s x x x

12 . Sum of squares. x a 0 + m ( a + ) b 3. ANOVA (aalysis of variae able) Soure DF Sum of Squares Periodgram ω 0 a 0 I(ω 0 ) ω π/ a + b I(ω ) ω m πm/ a m + b m I(ω m ) x

13 Applied o oy example Soure DF Sum of Squares ω 0 0 (period 0) a ω π/6 (period 6) a + b 50.0 ω π/6 (period 3) a + b 4.5 ω 3 π3/6 (period ) a Tes ha period 6 is sigifia H 0 : X µ + ε, {ε } ~ idepede oise H : X µ + A os (π/6) + B si (π/6) + ε Tes Saisi: (-3)I(ω )/(Σ x -I(0)-I(ω )) ~ F(,-3) (6-3)(50/)/( )5.79 p-value.003 x 3

14 The sperum ad frequey domai aalysis Ex. Siusoid wih period. x π π 5 os( ) + 3si( ),,,,0. Ex. Siusoid wih periods 4 ad. Ex. Maua Loa ITSM DEMO 4

15 Differeig a lag Someimes, a seasoal ompoe wih period i he ime series a be removed by differeig a lag. Tha is he differeed series is x y x x Now suppose x is he siusoid wih period + oise. The π π 5 os( ) + 3si( ) + ε,,,,0. y x x ε ε whih has orrelaio a lag. 5

16 Oher ylial ompoes; searhig for hidde yles Ex. Suspos. period ~ π/ years Fisher s es sigifiae Wha model should we use? ITSM DEMO 6

17 Noise. The ime series {X } is whie or idepede oise if he sequee of radom variables is idepede ad ideially disribued. x_ ime Baery of ess for hekig whieess. I ITSM, hoose saisis > residual aalysis > Tess of Radomess 7

18 Residuals from Maua Loa daa. x_{+} x_{+} x_{+} Cor(X, X +5 ).074 r r Cor(X Cor(X, Cor(X, X +3 ) + ) , X + ) x_ x_ x_ 8

19 Auoorrelaio fuio (ACF): ACF Maua Loa residuals Cof Bds: ±.96/sqr() ACF lag whie oise lag 9

20 irregular par seasoal -3-3 red CO Puig i all ogeher Example: Maua Loa

21 Sraegies for modelig he irregular par {Y }. Fi a auoregressive proess Fi a movig average proess Fi a ARMA (auoregressive-movig average) proess I ITSM, hoose he bes fiig AR or ARMA usig he meu opio Model > Esimaio > Prelimiary > AR esimaio or Model > Esimaio > Auofi

22 How well does he model fi he daa?. Ispeio of residuals. Are hey ompaible wih whie (idepede) oise? o diserible red o seasoal ompoe variabiliy does o hage i ime. o orrelaio i residuals or squares of residuals Are hey ormally disribued?. How well does he model predi. values wihi he series (i-sample foreasig) fuure values 3. How well do he simulaed values from he model apure he haraerisis i he observed daa? ITSM DEMO wih Maua Loa

23 Model refieme ad Simulaio Residual aalysis a ofe lead o model refieme Do simulaed realizaios refle he key feaures prese i he origial daa Two examples Suspos NEE (Ne eosysem exhage). Limiaios of exisig models Seasoal ompoes are fixed from year o year. Saioary hrough he seasos Add ierveio ompoes (fores fires, volai erupios, e.) 3

24 Oher direios Sruural model formulaio for red ad seasoal ompoes Loal level model m m - + oise Seasoal ompoe wih oise s s - s -... s - +oise X m + s + Y + ε Easy o add ierveio erms i he above formulaio. Periodi models (allows more flexibiliy i modelig rasiios from oe seaso o he ex). 4