MODELLING AND ANALYZING INTERVAL DATA

Transcription

1 MODEING AND ANAYZING INTERVA DATA Paula Brio FEP / NIAAD-IACC - NIV. PORTO PORTGA

2 Ouline Inerval daa Dynamical clusering Sandardizaion A regression model Disersion, associaion and linear combinaions of inerval variables Discriminan analysis Inerval ime-series Conclusions 2

3 From classical o symbolic daa Classical daa analysis : Daa is reresened in a n x marix each of n individuals (in row) ake one single value for each of variables (in column) Too simle model o reresen variabiliy uncerainy 3

4 From classical o symbolic daa Symbolic daa new variable yes: Se-valued variables : variable values are subses of an underlying se Inerval variables Caegorical muli-valued variables Modal variables : variable values are disribuions on an underlying se Hisogram variables 4

5 From classical o symbolic daa : Inerval Variables Ω {ω,..., ω n } Y wih underlying domain O R I se of inervals of O Y : Ω I ω i [l i, u i ] 5

6 Inerval Daa Y Y j... Y ω [l, u ]... [l j, u j ]... [l, u ] ω i [l i, u i ]... [l ij, u ij ] [l i, u i ] ω n [l n, u n ] [l nj, u nj ] [l n, u n ] 6

7 Inerval Daa Inerval daa may occur in many differen siuaions: Naive' inerval daa, describing ranges of variable values Ex : daily sock rices, monhly emeraure range Imrecise daa, coming from reeaed measures or confidence inerval esimaion Aggregaion of huge daa bases, when real values are generalized by inervals Symbolic daa Ex : descriions of biological secies or echnical secificaions 7

8 Inerval arihmeic Given wo inervals X and Y X o Y { x o y,x X,y Y} The resul is an inerval comrehending all ossible oucome of he oeraion beween values of X and Y. Moore(966), Case(999) 8

9 Dynamical clusering of inerval daa De Carvalho, Brio & Bock : Pariioning clusering mehod 2 disance beween inervals Dynamic clusering aroach Assignmen funcion Reresenaion funcion nil convergence 9

10 Assignmen Funcion : Given reresenaives (l,, l k ) he ariion P {P,, P k } is defined by: P h {ω Ω : D({ω}, l h ) D ({ω},l m ), m k} Reresenaion Funcion : Given a ariion {P,, P k }, he reresenaives (l,, l k ) are defined by : l h : D(P h, l h ) minimizes D(P h, ) l ( l,u ], K,[l,u ] ) h [ h h h h

11 Dynamical clusering of inerval daa Alying ieraively he assignemen funcion followed by he reresenaion funcion in urn decreases seadily he values of W (,P) k D(P, l h h h ) unil a local miminum is aained. The mehod hence minimizes W(,(P)) k ([l h i P j h ij l hj ] 2 + [u ij u hj ] 2 ) wih resec o ariion P.

12 Sandardizaion Dissimilariy values and clusering resuls are srongly affeced when modifying he scales of variables. Some sandardizaion mus be erformed rior o he clusering rocess in order o aain an 'objecive' or 'scale-invarian' resul. The same ransformaion mus be alied o boh he lower and he uer limis of each inerval. 2

13 Sandardizaion sing he disersion of he inerval ceners The firs mehod considers he mean and he disersion of he inerval ceners and sandardizes such ha he resuling ransformed midoins have zero mean and disersion in each dimension. I ij [l ij, u ij ] I ij [l ij, u ij ] 3

14 Sandardizaion 2 sing he disersion of he inerval boundaries Evaluae he disersion of an inerval variable by he disersion of inerval boundaries: I ij [l ij, u ij ] I ij [l ij, u ij ] 4

15 Sandardizaion 3 sing he global range The hird sandardizaion mehod ransforms, for a given variable, he inervals I ij [l ij, u ij ] (i,...,n) such ha he range of he n rescaled inervals is he uni inerval [0,]. Min j Min { l ij, u ij, i,,n } Min { l ij, i,,n } Max j Max { l ij, u ij, i,,n } Max { u ij, i,,n } I ij [l ij, u ij ] I ij [l ij, u ij ] 5

16 Exerimenal resuls Simulaion sudies showed ha sandardizaion grealy imroves he qualiy of he cluser resuls (recover of an imosed srucure). Sandardizaion 2 slighly beer for illsearaed clusers where inervals have large ranges. 6

17 Sandardizaion 2 : Consequences Covariance Measure Correlaion 7

18 ooking for he regression model Y deenden variable X indeenden variable 8

19 Regression Model i,..., n α and β minimize () Model obained indeendenly by Neo and De Carvalho 9 by direc minimizaion of he crierion ().

20 Exerimenal resuls Mone-Carlo exeriences Simulaing inerval daa wih differen degrees of lineariy Differen degrees of variabiliy (inerval ranges) and qualiy of adjusmen Performance similar o ha of mehod based on midoins (Billard and Diday) (MSE on lower and uer bounds, R 2 of lower and uer bounds) 20

21 Disersion, associaion and linear combinaions (Duare Silva & Brio) I[I ij ij ] i,...,n, j,..., I ij ij [l ij, ij, u ij ] β[β ij ij ] i,...,, j,...,r Z[Z ij ij ] i,...,n, j,...,r Z [ z, zij] ij ij Z I β 2

22 Disersion, associaion and linear combinaions β β β S S (P2) ij j j I i β I β l l (P) 22 β β j ij j i j ij j i u z l z l l l l (C) β + β β + β < β > β < β > β 0 ij j 0 ij j i 0 ij j 0 ij j i j j j j l u z u l z l l l l l l l l l l (C2) β β β I S I S (P2)

23 Disersion, associaion and linear combinaions C is aroriae when lower (resec. uer) bounds of differen variables end o occur simulaneously: Posiive Inner Correlaion C does no saisfy P C2 saisfies P and is aroriae in he absence of inner correlaion. 23

24 Disersion, associaion and linear combinaions Disersion s j2 and associaion s jj measures deend on l ij and u ij symmerically C and C2 saisfy P2 Variances of linear combinaions are given by quadraic forms : raios are maximized by a radiional 24 eigenanalysis.

25 Discriminan funcions. Disribuional Aroach Equidisribuional hyohesis (Berrand, Gouil, 2000) : An uniform disribuion is assumed for each observed inerval Emirical Dis. of each inerval variable is a mixure of n uniform laws K grous : mixure of K mixures 25

26 Discriminan funcions. Disribuional Aroach m j n n i l ij + 2 u ij s n j ij ij ij ij 2 ij uij 3n i 4n i ( n l + l u + u ) ( l + u ) s jj' 4n 2 4n i n n i ( l + u )( l + u ) ( ) ( ) l + u l + u ij ij ij ij ij' n i ij' ij' ij' 26

27 Discriminan funcions. Disribuional Aroach From hese measures a decomosiion in w jj w jj ' b jj' wihin-grous and (j j ) and beween-grous comonens is obained. Discriminan linear funcions are hen given by he eigenvecors of W - B. 27

28 2. Verices aroach Each individual is reresened by he corresonding hyercube verices : he original marix is exanded ino a (n 2 ) marix A classical analysis of he verices marix is erformed. imis for he l-h discriminan funcion on individual ω i : z il Min{z ql,q Q i } z il Max{z ql,q Q i } 28

29 3. Midoins and ranges aroach Each inerval is reresened by is midoin c ij and range r ij Two classical analysis are hen erformed : o Searaely for C[c ij ] and R[r ij ] o Joinly for he marix [C R] 29

30 Classificaion rules Classificaion rules are derived from discriminan sace reresenaions Poinwise Euclidean Dis. Mahalanobis Dis. Inervalwise Hausdorff Dis. δ ( z, z ) Max{ z z, zil z jl il jl Oher inerval disances il jl 30 }

31 Exerimenal resuls Searaion by midoins only: Mehods which ake ranges exlicily ino accoun have wors erformance Searaion by midoins and ranges: Mehods which ake ranges exlicily ino accoun have bes erformance Mehods based on inerval aroaches caure range informaion in a limied exen 3

32 Modelling Inerval Time Series Daa (Teles, Brio) Inerval-valued daa colleced as an ordered sequence hrough ime (or any oher dimension) Inerval ime series X lower limis X uer limis of he observed inervals wihx X, K,N Inerval ime series : [ X,X ], [ X,X ], K, [ X,X ]. 2 2 N N Aroriae mehodology o sudy and model i: ime series analysis ARMA models 32

33 Main model Assumion : boh limis are realizaions of second-order saionary sochasic rocesses. Assumion 2: he limis of he inerval ime series follow ARMA models wih he same orders and ARMA arameers. Main assumion, basis of our aroach 33

34 X X Proosed model for he inerval ime series where C C +φ X +φ X [ X,X ],, K, N + K+φ + K+φ X X + a + a θa θ a K θ K θ q q a a q q () C µ ( φ K φ ) C µ ( φ K φ ) and accoun for he differen means of X and X wih µ > µ C C ; > φ, K φare he auoregressive arameers; θ, K θ are he moving average arameers;,, q 34

35 Proosed model for he inerval ime series a a and are indeenden whie noise sequences wih 2 zero mean and variances σ and. 2 a σ a I is also assumed ha a and a are indeenden of each oher. Thus, i is assumed ha X and have differen means and variances and follow ARMA(,q) rocesses wih he same ARMA arameers. X 35

36 Proosed model for he inerval ime series Model () may also be wrien as ( ) ( q ) φb K φb X C + θ θ B K qb ( φ φ ) + ( q B K B X C θ B K θ B ) j where B is he backshif oeraor, BX X j. In a more comac form, φ φ ( B) X C + θ( ) ( B) X C + θ( ) B a B a q a a (2) (3) where φ( B) φ K φ and ( ) θ B θ K θq are he auoregressive and moving average olynomials, resecively. 36

37 Proosed model for he inerval ime series We assume ha o he olynomials have no roos in common, ( B ) o he roos of θ are ouside he uni circle so ha hese rocesses are inverible, φ ( B) o he roos of are ouside he uni circle so ha hese rocesses are saionary. The rocedures we roose o model inerval ime series will be based on model (). 37

38 Model for boh inerval limis e X & X µ and & wih E ( X & ) EX (& ) 0. X X µ From (), we have X& X& φ X & + K +φ X & + a θ a K θ a φ X& + K+φ X& + a θ a K θ q q a q q (4) and esimaion of he model arameers is required. 38

39 Since boh inerval limis follow ARMA(,q) rocesses wih he same ARMA coefficiens, i is ossible o base he esimaion mehod on a ime series ha joins all he uer and lower observaions, i.e., This doubles he samle size used in esimaion..,x,,x,x, X N N K K This doubles he samle size used in esimaion. Rewriing (4) as and assuming he iniial condiions (5) φ φ + + θ + θ φ φ + + θ + θ q q q q X X X a a a X X X a a a & K & & K & K & & K q q a a a 0 a a a + + K K

40 The condiional sum of squares funcion is S N [ ] ( ) 2 ( ) 2 + ( ) ( 2 2 φ, θ a φ, θ a φ, θ a a )( φ θ) + N +, + N + (6) φ and θ ( θ K, θ ). where ( φ, K φ ),, q Therefore, condiional leas squares esimaors of are obained from minimizing S. φ and θ

41 Afer obaining he arameer esimaes and θ ˆ ( θˆ,, ˆ K θ q ), 2 a noise variances σ and. ( φˆ,, φˆ ) φ ˆ K i is necessary o esimae he whie 2 σ a This is accomlished wih he uer and he lower ime series searaely: σ ˆ σˆ 2 a 2 a S N S N N ( ) a ( ˆ, ˆ ) 2 ˆ, ˆ φ θ φ θ + ( 2 + q) N ( 2 + q) ( φˆ, θˆ ) ( 2 + q) N + N a 2 ( φˆ, θˆ ) ( 2 + q) where S and S are he uer and lower condiional sums of squares resecively and he denominaors are he degrees of freedom, i.e., N (+q) N (2 + q).

42 Finally, he esimaors of he means μ and μ are he samle means of and resecively: X X X N X X ˆ N N µ µ The esimaion rocess is now comlee. N X ˆ µ and ( ) ( ). ˆ ˆ X Ĉ ˆ ˆ X Ĉ φ φ φ φ K K

43 Forecasing fuure values of he inerval ime series is based on he esimaed model (), i.e., he uer and he lower limis are forecas searaely from he resecive models, which is a sandard rocedure in ime series analysis.

44 Model for he differences of he inerval limis A simler aroach is based on he differences of X & andx & in model (): D X& X& This rocedure leads o a roblem of modeling a single ime series D follows an ARMA(,q) rocess wih he same ARMA arameers as model (): D + ( ) ( ) X& X& φ X& X& + K+φ X& X& ( ) ( ) ( ) a a θ a a K θ a a φd + K+φ D +ε θ ε q K θ q q ε q q sandard ime series analysis 44

45 This simle rocedure reduces he roblem of modeling an inerval ime series o he sandard analysis of a single ime series sraighforward and very easy o imlemen. 45

46 Exerimenal resuls The roosed rocedures erform very well in erms of esimaion accuracy in erms of forecasing erformance. Esimaion accuracy: The mehod for boh inerval limis doubles he number of observaions used in esimaion I shows higher accuracy and clearly ouerforms he oher mehod, being more efficien. 46

47 Exerimenal resuls Forecasing erformance: The wo mehods show he same forecasing erformance as single ime series models. In sie of he beer esimaion accuracy of he mehod for boh inerval limis, he forecasing erformance is he same. 47

48 Conclusions The exension of classical mehodologies o he analysis of inerval daa raises new roblems: How o evaluae disersion? How o define linear combinaions? Which roeries remain valid?... 48

49 Conclusions Reresenaions in lower dimensional saces may assume differen forms : Inervals, emhasize variabiliy inheren o each observaion Poins, can dislay disinc conribuions o he searaion beween grous In general : need of models 49

50 Mehods for inerval daa inear regression : Billard,. ; Diday, E `From he Saisics of Daa o he Saisics of Knowledge:Symbolic Daa Analysis', JASA Neo, E. A.., De Carvalho, F.,Tenório, C nivariae and Mulivariae inear Regression Mehods o Predic Inerval-Valued Feaures 50

51 Mehods for inerval daa Princial Comonen and Discriminan Analysis Chouakria, A., Cazes, P. Diday, E Symbolic Princial Comonen Analysis auro, C. Palumbo, F Princial Comonen Analysis for Non-Precise Daa auro, C., Verde, R. Palumbo, F Facorial Discriminan Analysis on Symbolic Objecs 5

52 Mehods for inerval daa Oher Mehods Rossi, F., Conan Guez, B Mulilayer Perceron on Inerval Daa Simoff, S. J. 996 Handling ncerainy in Neural Neworks: an Inerval Aroach Do T.-N., Poule, F Kernel Mehods and Visualisaion for Inerval Daa Mining 52