Modelling Record Times in Sport with Extreme Value Methods

Transcription

1 utput 2017/5/9 5:13 page 1 #1 Malaysian Jurnal f Mathematical Sciences *(*): ** ** (201*) MALAYSIAN JOURNAL OF MATHEMATICAL SCIENCES Jurnal hmepage: Mdelling Recrd Times in Sprt with Extreme Value Methds authr 1 1 and authr affil 1 2 affil 2 Crrespnding authr ABSTRACT We explit cnnectins between extreme value thery and recrd prcesses t develp inference methds fr recrd prcesses. Recrd prcesses have trends in them even when the underlying prcess is statinary. We study the prblem f estimating the underlying trend in times achieved in sprts frm recrd data. We develp new methds fr inference, simulating recrd series in nn-statinary cntexts, and assessing fit which accunt fr the censred characteristic f recrd data and we apply these methds t athletics and swimming data. Keywrds: Athletics, extreme value thery, Pissn prcess, recrds, swimming.

2 utput 2017/5/9 5:13 page 2 #2 authr 1. Intrductin In many fields recrds are f particular interest such as in weather (highest fld, httest temperature, strngest wind speed) and in finance (largest insurance claim, biggest stck market crash). In sprting events, such as athletics and swimming, recrds are the vital events in characterising the develpment f the sprt, they are the target fr cmpetitrs, and they receive mst publicity. Recrds are hwever simply the mst extreme values in a series at the time f their ccurrence and s there are natural cnnectins between the thery fr recrds and the thery fr extreme values, this has been explited in prbabilistic results fr recrds by Ahsanullah (2004), Arnld et al. (1998), Ahsanullah and Bhj (1996), Bairamv (1996), David and Nagaraja (2003), Glick (1978), Sibuya and Nishimura (1997), Smith (1988) and Benested (2004). In this paper we apply and adapt ur knwledge f extreme value methds t the analysis f recrd data fr sprting events, fcusing n athletics track events and swimming cmpetitins. T illustrate the issues in this paper first cnsider the minimum recrd prcess {Y t } which arises frm a statinary sequence {X t }, i.e. Y t = min(x 1,..., X t ). We assume that {X t } are cntinuus randm variables. Let R t = I(X t < Y t 1 ) be an indicatr variable fr the ccurrence f a recrd at time t and let N n be the number f recrds until time n, with N 1 = 1 by definitin. Then R t Bernulli(t 1 ) and E(N n ) = n t=1 E(R t) = n t=1 1 t, s fr large n, E(N n ) lg n γ, with γ is Euler s cnstant Arnld et al. (1998) pint ut that abut 7 recrds are expected t ccur in a sequence f 100,000 bservatins f a statinary sequence, s when studying recrds f statinary prcesses we need t recgnise that recrds are brken rarely. Furthermre, there is a decreasing trend in the {Y t } prcess as Y t1 Y t fr all t 1. If an analysis is t be undertaken frm a series f recrd data {Y t } alne then at first sight it may appear that the infrmatin in the series is limited as many f the values in the series will be identical. The few values in the series where the recrd changes, i.e. Y t < Y t 1 tell us directly that X t = Y t. Hwever, the recrd remaining unchanged des als prvide infrmatin abut the underlying {X t } prcess as Y t = Y t 1 tells us that X t > Y t 1, s this gives censred infrmatin abut the underlying {X t } prcess. Despite the infrmatin frm censred data, fr statinary series the infrmatin abut the recrd prcess is gathered very slwly because when the prbability f nt breaking the recrd is high there is very limited infrmatin in the censred data. As sprts have increasingly becme prfessinal, their training techniques 2 Malaysian Jurnal f Mathematical Sciences

3 utput 2017/5/9 5:13 page 3 #3 Mdelling Recrd Times in Sprt with Extreme Value Methds Figure 1: Plt A - wmen s athletics 3000 m with annual minima (circle) and annual recrd data (dt) frm 1972 t 1992; Plt B - wmen s athletics 1500 m with annual minima (circle) and annual recrd data (dt) frm 1972 t have been refined, dietary knwledge has been imprved, mre peple are training t high standard in every sprt, and it is clear that the underlying marginal distributin f elite perfrmance in sprt has changed. This is seen in the athletics and swimming data we study. Figure 1 shws the annual best times achieved in recgnised internatinal cmpetitins ver the perid f fr the wmen s athletics 1500 m and 3000 m track events, see Rbinsn and Tawn (1995). The recrd prcess, derived frm these annual data, is als n the figure. It shws that the number f recrds being brken is greater than wuld be expected if the annual best perfrmance was statinary ver time. Similarly, Figure 2 shws the FINA wrld recrd series f the men s 400 m freestyle swimming event pltted against the actual times in the year when the recrds are achieved. Unlike the athletics data, this type f recrd data shws when the recrd has been brken mre than nce in a year. Again the number f recrds is greater than wuld be expected if the series was statinary, indicating a trend in elite perfrmances. We call the tw types f data annual recrd data and actual recrd data respectively. As indicated earlier, the recrd prcess exhibits a trend even when the Malaysian Jurnal f Mathematical Sciences 3

4 utput 2017/5/9 5:13 page 4 #4 authr times(s) year Figure 2: The men s Olympic 400 m freestyle gld medallist s times (empty dt) and the men s actual wrld recrd data fr 400 m freestyle (dark dt) frm 1912 t underlying prcess is statinary. We can tell frm the number f recrds in the actual recrd data that there must be nn-statinarity in the underlying prcess but it is clear that separating the cmpnent f the trend due t recrds frm that due t nn-statinarity is nt bvius frm infrmatin n the recrd prcess alne. Fr the annual recrds data in athletics we als have annual best data which allws us t identify the separatin as the recrds decrease ver time faster than the mean f the annual best data. Mst ften data n recrds takes the frm f actual recrd data as annual recrd data exclude recrds if the recrd is brken mre than nce in a year. Furthermre, annual best data (frm which annual recrd data are typically derived) are difficult t find fr early perids in the histry f an event whereas all actual recrds are well dcumented. As annual best data, and their assciated annual recrd data, prvide infrmatin n bth the underlying prcess and the recrd prcess we first study these data, this helps in identifying the level f lss f infrmatin in analysing annual recrds in cmparisn t annual best data and identifies the need fr sme independent infrmatin n the trend in the underlying series. 4 Malaysian Jurnal f Mathematical Sciences

5 utput 2017/5/9 5:13 page 5 #5 Mdelling Recrd Times in Sprt with Extreme Value Methds As the data are minima times the apprpriate distributin fr the annual best is the generalised extreme value distributin fr minima, GEVM(µ,, ξ), which has a distributin functin { ( )] } 1/ξ x µ H(x) = 1 exp 1 ξ (1) with a lcatin parameter µ R, a scale parameter > 0, a shape parameter ξ R and y] = max(y, 0), see Cles (2001). The negative Gumbel is given by the GEVM distributin with ξ 0, the negative Fréchet by ξ > 0 and the Weibull by ξ < 0. The justificatin fr this distributin is that it is the nly pssible nn-degenerate distributin fr the limiting distributin f the minima f statinary sequences after linear nrmalisatin, see Leadbetter et al. (1983). Statinarity ver a year may be a reasnable apprximatin t justify the chice f the GEVM distributin, but the bvius nn-statinarity in the annual best times suggest that at least sme f the parameters f the GEVM need t change ver time, as in the mdelling framewrk f Davisn and Smith (1990) and Cles (2001). Here we assume that the GEVM parameter µ(t) varies smthly ver time t, with and ξ being a cnstant, i.e. the nn-statinarity is purely a lcatin shift in the elite perfrmance in a year. This is justified by findings in Adam and Tawn (2011) and Adam and Tawn (2012) mre generally fr data f this type but als given the limited data in ur applicatins it is als unlikely that we can find evidence that the scale and shape parameter changes ver time. Fr the athletics example we take µ(t) as a simple expnential decay but fr the swimming data n simple parametric mdel seems apprpriate s we use nn-parametric methds fr estimating µ(t). We study annual recrds using the infrmatin that they are simply a partially censred series f annual best data. Fllwing the study f annual recrds we then develp a general Pissn prcess limiting characterisatin fr mdelling the actual recrd data. Specifically, pint prcess results fr extremes, Pickands (1971) and Smith (1989), are adapted t recrd values recgnising that recrds are a frm f censred extreme value data. Again mdelling the nn-statinarity f the underlying data needs t be addressed. Evidence frm the analysis f annual recrds shws the need fr infrmatin n the trend in elite perfrmance (in the frm f annual best data r smething equivalent) t supplement infrmatin frm the annual recrd t get any frm f useful inference. In rder t make inferences frm the actual recrd data we need additinal data that prvides infrmatin abut the trend f the underlying series. Annual best data are nt available generally fr ther recrd series s sme frm f prxy data are required. Fr the actual recrd Malaysian Jurnal f Mathematical Sciences 5

6 utput 2017/5/9 5:13 page 6 #6 authr data fr swimming we use the Olympic gld medallists times as prxy data fr inference n the trend in elite perfrmances. The structure f this paper is as fllws. In Sectins 2 and 3 we derive mdels and inference methds fr annual recrd data and actual recrd data respectively. As a large amunt f data are censred this infrmatin needs t be accunted fr nt just in the fitting but als in methds f gdness-f-fit fr bth methds. Then we apply the thery t athletics data in Sectin 5 and swimmming data in Sectin Mdels and Inference fr Annual Recrds If Z 1,..., Z n is the sequence f annual minima frm {X t } with Z t the annual minima in year t then we define Y t = min(z 1, Z 2,..., Z t ) fr t 1 s that {Y t } is the annual recrd prcess. We assume that {Z t } are independent with Z t GEVM(µ(t),, ξ) and s µ(t), t = 1,..., n, and ξ determine the distributin f the annual recrd prcess. Suppse that we have data fr {Y t } but nt {Z t }. When Y t < Y t 1, i.e. a recrd ccurs, then Y t = Z t is an bservatin frm a GEVM(µ(t),, ξ) variable. When Y t = Y t 1, i.e. nt a recrd, then Z t > Y t s Z t is a GEVM(µ(t),, ξ) variable censred belw at Y t. Cnsequently, the lg-likelihd functin fr the recrd sequence y = (y 1,..., y n ) is where Ψ t = l(y; µ,, ξ) = 1 ξ n I t lg t=1 n t=1 I t Ψ 1/ξ t n t=1 ( I t 1 1 ) lg Ψ t ξ n (1 I t )Ψ 1/ξ t t=1 (2) ( )] yt µ(t), with I t = I(Y t < Y t 1 ) and µ = (µ(1),..., µ(n)). If µ(t) can be parametrically specified then maximum likelihd estimates are btained by maximizing (2) directly. Hwever if µ(t) is specified nly t be a smth functin we use a penalized lg-likelihd functin fr the GEVM distributin f the frm l(y; µ,, ξ) λ 2 µt Kµ, (3) where K is a symmetric n n matrix f rank n 2, defined in Green and Silverman (1994) which depends nly n (1, 2,..., n) and λ is a smthing parameter we select using the AICc criteria. The secnd term in the penalized 6 Malaysian Jurnal f Mathematical Sciences

7 utput 2017/5/9 5:13 page 7 #7 Mdelling Recrd Times in Sprt with Extreme Value Methds lg-likelihd (3) impses a penalty fr the smth µ values. We maximize Equatin (3) t get the best estimate values f ˆµ, ˆ and ˆξ using the Fisher s scring methd explained in Adam (2007). As the estimated parameters determine the distributin f {Y t } thrugh {Z t } we need t derive the distributin f {Y t }. When {Z t } are independent and identically distributed (IID) then Pr(Y t > y) = Pr(Z 1 > y) Pr(Z 2 > y)... Pr(Z t > y) { ( )] } y µ 1/ξ = exp 1 ξ t, (4) where µ t = µ ξ (tξ 1) and t = t ξ, s Y t GEVM(µ t, t, ξ), s the expected value fr Y t is E(Y t ) = µ t t ξ 1 Γ(1 ξ)]. Fr the nn-statinary case as Pr(Y t > y) is cmplex we resrt t deriving the distributin f {Y t } by simulatin frm {Z t }. We can assess the fit f the mdel thrugh cmparisn f Y m and its estimated expected value Ê(Y m). This cmparisn is cmplicated by the strng dependence in {Y t }. Instead we prefer t explit the prperty that if Z t GEVM(µ(t),, ξ) then Z t = µ(t) E t, where µ(t) is the trend in Z t and E t GEVM(0,, ξ). Then the estimated sequence f {E t } = {Z t ˆµ(t)} fr t = 1,..., n are IID. We use estimated values f {E t } t cnstruct the P-P and Q-Q plts fr assessing the gdness f fit f the recrd prgressin data. The initial step is t get the E t value. If Y t is a recrd then Z t = Y t, we get E t = Y t µ(t). Otherwise, if Y t is nt a recrd then Z t > Y t 1 s E t > Y t µ(t). Data n {E t } are therefre censred when n recrd is brken. T estimate the distributin f E we need t accunt fr censring, with the standard apprach being the Kaplan-Meier estimatr. Let e (1) <... < e (m) dente the residuals f the bserved recrds and d i be the number f {E t } > e (i). The Kaplan-Meier estimate fr a survival functin is given by ĤKM (z) = ) i:e (i) <z (1 1di. The mdel-based estimate f the distributin functin fr E is { Ĥ(z) = 1 exp 1 ˆξ ( ) ] } 1/ˆξ z ˆ. By cmparing the mdel-based survival prbability and empirical Kaplan Meier the P-P and Q-Q plts can be cnstructed. Let e (j) be the jth largest f e 1,..., e n (i.e. bserved and censred values tgether) then the P-P plt is cnstructed by pltting Ĥ(e (i) ) against Ĥ KM (e (i)) fr i = 1,..., n. t Malaysian Jurnal f Mathematical Sciences 7

8 utput 2017/5/9 5:13 page 8 #8 authr and the Q-Q plt, using the expnential quantile plt, fr the {e i } is ] ] lg 1 Ĥ(e (i) ) versus lg 1 ĤKM (e (i) ) fr i = 1,..., n. The transfrmatin expnential quantile scale changes the lwer tail int the upper tail. 3. Mdels and Inference fr Actual Recrds 3.1 Pint prcess mdel First cnsider a statinary sequence {X t }. We define the pint prcess P n = {i/(n 1), (X i b n )/a n ] : i = 1,..., n}, where the sequence f cnstants {a n > 0} and {b n } are such that ] min(x1,..., X n ) b n n Pr > x 1 H(x) a n with H(x) a nn-degenerate limit distributin. Then H(x) is GEVM(µ,, ξ) given by (1.1), see Leadbetter et al. (1983). Let P be the limit as n f P n. Let x F = min{x : F X (x) > 0} where F X is the distributin functin f {X t }, and b l = lim n (x F b n )/a n. Fr A 0, 1] (, b l ) let N n (A) be the number f pints f P n that fall in A and N(A) be the number f pints f P in A. Fllwing Pickands (1971) and Smith (1989) it fllws that P is nn-hmgeneus Pissn prcesses with intensity λ(t, r) = 1 1 ξ ( r µ )] 1 1/ξ. (5) Nw, if we let A = t 0, t 1 ] (, r), with 0 < t 0 < t 1 < 1 and r < b l, the limiting distributin f N n (A) is Pissn ( Λ(A) ) with Λ(A) = (t 1 )] 1/ξ. Nw, we cnsider the nn-statinary case with lcatin t 0 ) 1 ξ ( r µ parameter µ(t). The intensity measure fr this nn-hmgeneus Pissn prcesses is then λ(t, r) = 1 0, 1] each fr t in µ(t). 1 ξ ( r µ(t) )] 1 1/ξ where here time is scaled nt Suppse that between times t 0 and t m1 the times and values f a sequence f m cnsecutive actual recrds are (t 1, r 1 ),..., (t m, r m ), with r i the recrd value btained at time t i with r 1 > > r m 1 > r m. It is assumed that the current recrd at time t 0 is r 0. We derive the likelihd fr this actual 8 Malaysian Jurnal f Mathematical Sciences

9 utput 2017/5/9 5:13 page 9 #9 Mdelling Recrd Times in Sprt with Extreme Value Methds recrd sequence in stages. Using the Pissn prcesses the prbability f n new recrd in the time interval (t 0, t 1 ) is t1 r0 ] { t1 ( )] } 1/ξ r0 µ(t) exp λ(t, r)drdt = exp 1 ξ dt. t 0 t 0 The prbability f ne recrd in the interval (r i δr, r i ) in the time interval (t i, t i δt), { tiδt ri ( )] } 1 1/ξ 1 r µ(t) λ(t i, r i )δrδt exp 1 ξ drdt fr small δr > 0 and δt > 0. t i r i δr The prbability f n recrd in the time interval (t i δt, t i1 ) is { ti1 ri ( )] } 1 1/ξ 1 r µ(t) exp 1 ξ drdt. t iδt S the jint prbability f a recrd r i at time t i and then n recrd until t i1 is prprtinal t { ti1 ri ( )] } 1 1/ξ 1 r µ(t) λ(t i, r i ) exp 1 ξ drdt. t i The likelihd fr the recrd prgressin frm t 0 t t m1 is prprtinal t m ] { m ti1 ( )] } 1/ξ ri µ(t) λ(t i, r i ) exp 1 ξ dt. (6) i=1 i=0 t i As µ(t) changes slwly we expect the integral in Equatin (6) t als change smthly with t, and s fr numerical simplificatin we apprximate the sum f integrals in the lg-likelihd functin. In rder t get an accurate value f the apprximatin f the integral, we break the time frm t j t t j1 with t j1 = t j k j, where k j is the number f partitins frm t j t t j1 with equal distance f and s j,i = t j (t j1 t j )(i 1)/k j, i = 1,..., k j 1 t give m k i ( ri µ(s j,i ) 1 ξ i=0 j=1 )] 1/ξ. (7) Malaysian Jurnal f Mathematical Sciences 9

10 utput 2017/5/9 5:13 page 10 #10 authr As in Sectin 2 nn-parametric inference fr µ(t) can be btained by penalized likelihd with a penalty term identical t that in Equatin (2). 3.2 Cnstructing tlerance bands In rder t develp new methds fr diagnsing whether the mdel is acceptable r nt fr the actual recrd data, we need t deal with tw aspects which are the time when the recrd ccurred, t, and the value f the new recrd, r, given that a recrd ccurs at time t. Our first apprach is t generate tlerance intervals fr the recrd prgressin under the assumptin that the fitted mdel is crrect. In cnstructing such tlerance intervals, the fitted mdel is used t simulate replicates f the recrd prgressin data. The resulting pintwise tlerance intervals prduced prvide a regin f values f recrd prgressins which can be used as a criteria fr judging whether ur fitted mdel is acceptable when cmpared t the bserved recrd prgressin data. The bserved recrd prgressin data shuld be within the regin t indicate that the mdel is gd. Critical t this apprach is the need t be able t simulate a recrd prgressin fr the fitted mdel. Fr efficiency reasns we nly want t simulate recrd times and values. T generate the recrd prgressin there are tw stages given a current recrd time t j and a current recrd value r j. The first stage is t generate the next recrd time t j1 (t j1 > t j ) and the secnd stage is t generate the next recrd value r j1 (r j1 < r j ). Furthermre, we need an initialisatin stage given (t 0, r 0 ) and a way fr terminating the simulatin. We give details f all these stages belw. Fundamental t ur apprach in each case is the use f the limiting nn-hmgeneus Pissn prcess P. Refer t Equatin (5). First let us derive the waiting time distributin fr the next recrd given that a recrd value f r j ccurred at time t j. Let this waiting time randm variable be W j1. Then Pr(W j1 > w) = Pr( n pints f P in A j ) where A j = t j, t j w] (, r j ). Pr( n pints f P in A j ) = exp{ Λ(A j )} = exp tjw rj t j λ(s, r)drds. ] T simulate T j1, the time f the (j 1)th recrd value, we use the prbability integral transfrmatin fr the survivr functin f W j1. Specifically if u is a realisatin f U U(0, 1), then slving u = Pr(W j1 > w) gives a realisatin w frm the waiting time distributin. The next recrd time is then generated as t j1 = t j w, this is a realisatin f T j1. Hence we slve fr 10 Malaysian Jurnal f Mathematical Sciences

11 utput 2017/5/9 5:13 page 11 #11 Mdelling Recrd Times in Sprt with Extreme Value Methds t j1 in u = exp t j1 ] rj t j λ(s, r)drds, we get lg u = tj1 rj t j λ(s, r)drds k j1 i=1 { 1 ξ } 1/ξ r j µ(s i )]. (8) The secnd stage is t derive the distributin f the decrease V j in the recrd value given that the (j 1)th recrd value ccurs at time t j1. The previus recrd value r j is nw a threshld fr any further data t be the next new recrd value r j1 = r j v fr v > 0. The prbability f getting a decrease in recrd level greater than v at time t j1 is Pr(V j1 < v T j1 = t j1 ) = Pr(R j1 < r j v R j1 < r j, T j1 = t j1 ) where R j1 is the new recrd prgressin at t j1. Hwever, Pr(R j1 < r j1 R j1 < r j, T j1 = t j1 ) = { v 1 ξ ξ(r j µ(t j1 )) ]} 1/ξ is the Generalised Paret type distributin, GPD(, ξ) with parameter = ξ(r j µ(t i1 )). We simulate a new recrd value frm the decrease randm variable be V i. T simulate r j1 we use the prbability integral transfrm fr the distributin functin f V i. Using the realisatin u f U U(0, 1). Then slving u = Pr(V j < v) gives a realisatin frm getting new recrd value distributin given time f new recrd ccur. The next recrd value is then generated as r j1 = r j v. Hence slve fr v frm u = we get, { 1 ξ v ξ(r j µ(t j1)) ]} 1/ξ ] r j1 = r j ξ (r j µ(t j1 )) (1 u ξ ). (9),, The algrithm fr cnstructing the tlerance bands is as fllws: 1. Initiate = ˆ rec, ξ = ˆξ rec and µ = ˆµ = {ˆµ 1,..., ˆµ n }. 2. Selectin f r 0 ( r 1 ) and t 0 ( t 1 ). 3. First stage: Simulate t j1 frm Equatin (8) using t j and r j. 4. Secnd stage: Simulate r j1 frm Equatin (9) using r j and t j1. 5. If t j1 < T repeat steps 3 and 4 with T is the maximum time f interest. Stp if t j1 T. Malaysian Jurnal f Mathematical Sciences 11

12 utput 2017/5/9 5:13 page 12 #12 authr We replicate the algrithm abve k times t get k realisatins f the recrd prgressin prcess f fitted mdel. We used the pintwise quantiles as the upper and lwer bund f pssible recrd value fr the bserved recrd prgressin. If the real recrd prgressin falls reasnably within the pintwise acceptance bunds, it indicates than the estimated value f the parameters are apprpriate t the selected mdel and the mdel is reasnable descriptin f the data. As already we knew that Z i GEVM(µ i,, ξ) then Z i = µ i E i, where µ i is the trend in Z i and E i GEVM(0,, ξ). Then the sequence f {E i } fr i = 1,..., n are independent and identically distributed by separating the trend frm the data. The E i is used t cnstruct the P-P and Q-Q plts fr assessing the gdness f fit f the recrd prgressin data. The initial step is t get the E i value. If Y i is a recrd then y i < y i 1 where Z i = Y i, we get E i = Y i µ i. Otherwise, if Y j is nt a recrd then y j = y j 1 as Z j > Y j 1. We get E j > y j 1 µ j 1. Data n E j are therefre censred when n recrd is brken. T estimate the distributin f E we need t accunt fr censring, with the standard apprach being the Kaplan-Meier estimatr. Initially we need t plt the Kaplan-Meier estimate, KM j = dj j (1 t j ), t estimate the empirical survivr functin f E, where d j is the number f recrds ccur until time j and t j is the number f events left frm time j. The mdel-based } estimate f the survivr functin fr E i is { Ŝ(z) = exp 1 ˆξ ( z ˆ ) ] 1/ˆξ = 1 Ĥ(z). By cmparing the mdelbased survival prbability and empirical Kaplan Meier estimate frm graph, the P-P and Q-Q plts can be cnstructed. When we pltting KM j against S(e j ), where e j is the jth largest f e 1,..., e n value, r transfrmatins f these quantities give P-P and Q-Q plts. The P-P plt is cnstructed by pltting Ĥ(e i ) = 1 Ŝ(e i ) versus 1 KM i fr i = 1,..., n. When studying the minima data, the fit f the upper tails is needed t relcate t the right hand f the graph. The Q-Q plt using the expnential quantile plt fr the {e i } is ] lg Ŝ(e i ) versus lg KM i ] fr i = 1,..., n. In summary, t vercme the nn-iid difficulties, we separate the trend frm the data t get the iid residual data. Then cnstruct the P-P and Q-Q plts frm the residual data. 12 Malaysian Jurnal f Mathematical Sciences

13 utput 2017/5/9 5:13 page 13 #13 Mdelling Recrd Times in Sprt with Extreme Value Methds 4. Estimating Trend frm Additinal Data As the recrd data d nt prvided enugh infrmatin t fully separate the recrd prgressin frm a trend we need anther different set f data frm similar type f prcess which is able t give additinal infrmatin t enable better separatin. This additinal surce f data shuld be able t represent whle data including the recrd data. Adam (2007) shwn the difficulties f analysing extreme value data using parametric methds cmpared t nn-parametric methds. The nn-parametric apprach allws the data t "speak fr themselves" with less assumptins being made. Cmplex trends can be handled better using nn-parametric apprach cmpared t parametric apprach. We assume that Z 1, Z 2,... are the extreme additinal data which can prvide an infrmatin f the mdel trend in lcatin fr the recrd prgressin data. In this case we will using the annual minima type f data. 4.1 Nn-parametric apprach fr GEVM T allw fr nn-statinarity we replace µ with the smth functin g as we are estimating the trend in lcatin using nn-parametric methd in this case f study. Then the density functin f the GEVM at time t with smth functin g(t) is defined as h t (z) = 1 1 ξ ( z g(t) )] 1 1/ξ exp { 1 ξ ( z g(t) The penalized lg-likelihd functin fr the GEVM distributin is )] 1/ξ l λg (z; g,, ξ) λ g 2 gt Kg (10) with g = ( g(t 1 ),..., g(t n ) ) = (g 1,..., g n ) and l λg (z i ; g i,, ξ) is the lg-likelihd cntributin fr z = {z 1,..., z n } where n ( )] zi g i l λg (z; g,, ξ) = n lg (1 1/ξ) lg 1 ξ i=1 n ( )] 1/ξ zi g i 1 ξ (11) i=1 and λ g, the smthing parameter value, is selected using the AICc criteria. We smth g thrugh the penalty functin f (10) and maximize (11), t get the }. Malaysian Jurnal f Mathematical Sciences 13

14 utput 2017/5/9 5:13 page 14 #14 authr best estimate values f ĝg, ˆ and ˆξ using the Fisher s scring methd explained in Adam (2007). As ur pririty is t mdel and t analyse the recrd prgressin data, the nly infrmatin relevant frm this inference is the estimated ĝg. 4.2 Nn-parametric apprach t estimate µ t in a Pissn prcess mdel f recrd prgressin T mdel µ t Adam (2007) shwn that using a nn-parametric apprach is preferable. When a nn-parametric is implemented in the analysis, the µ t values in Equatin (7) are replaced by the smth functin g t at time t with t = 1,..., m. The lg-likelihd functin with g t is m m k i ( ri g j l(r; g t,, ξ) = lg λ(t i, r i ) 1 ξ i=1 i=0 j=1 )] 1/ξ. (12) In rder t get an accurate value f apprximatin f the integratin, we break the time frm t i t t i1 t a small equal distance f frm partitins s 0, s 1 ] t s ki 1, s ki ] we get the mdified lg-likelihd as m m k i ( ri g sj l(r; g t,, ξ) = lg λ(t i, r i ) 1 ξ i=1 i=0 j=1 )] 1/ξ (13) where the distance frm s 0 t s ki is the distance frm t i t t i1 fr i = 1,..., n with the distances between s 0 t s ki has been partitins t s 0, s 1 ],..., s ki 1, s ki ]. 5. Applicatin t Athletics Annual Recrds The five fastest annual rder statistics fr wmen s athletics 1500 m and 3000 m track events frm recgnized internatinal events ver the perid f are used in this study. See Rbinsn and Tawn (1995). Fr this analysis we assume times run by every athlete in all events are independent. There is a trend fr bth events, it is based n the expnential decay f the annual minima ver time. We use the trend, µ t = α β1 exp( γt)] where β > 0, γ > 0 and t is the year, taking 1971 as the base year. This mdel was used by Rbinsn and Tawn (1995). We redefined ur recrd as Y i = min(z 1,..., Z i ), i = 1972,..., 1992 as in Figure 1. Let Z i GEV M(ˆµ i, ˆ, ˆξ) fr i = 72,..., 92, where this is independent but nt identically distributed randm variables. The randm variable fr 14 Malaysian Jurnal f Mathematical Sciences

15 utput 2017/5/9 5:13 page 15 #15 Mdelling Recrd Times in Sprt with Extreme Value Methds the number f recrd brken ver the 21 years is N 21 = 92 i=72 I i, where I i = 1 if Z i < min(z 72,..., Z i 1 ) and is zer therwise. In Figure 3 the distributin f the number f recrds in annual minimum is shwn fr three situatins: firstly if the annual minima are iid; secnd fr the annual minima with trend and year t year variatin as estimated fr the wmen s athletics 1500 m data; finally the trend and year t year variatin as estimated fr the wmen s athletics 3000 m data. Fr iid case, it is pssible fr the recrd nt t be brken fr lng time intervals. The distributin f the number f recrds is skewed t the right. Fr nn-iid case, at least six recrds brken is predicted fr the wmen s athletics 3000 m (5 times fr the wmen s athletics 1500 m). This is because fr the athletics, which is nt identically distributed as ˆµ i gives sme bvius trend early in the perid, we predict that the recrd tends t be brken frequently. The bx-plt in Figure 3 summarise the discussin f a numbers f breaking recrd fr the wmen s athletics 1500 m and 3000 m. In 1500 m track event the recrds have been brken 3 times Bxplt f IID, nn IID 3000m and nn IID 1500m m1500 m3000 IID Numbers f recrd breaking Figure 3: The bx-plt f numbers f recrd breaking fr annual minima fr iid case and nniid case fr the wmen s athletics 3000 m and 1500 m, evaluated by Mnte Carl with repetitins. Malaysian Jurnal f Mathematical Sciences 15

16 utput 2017/5/9 5:13 page 16 #16 authr since 1972 and sme breaking recrds is predicted t happen. The wmen s athletics 3000 m at 1992, current number f breaking recrd is 6 times and mre recrd als will be predicted t happen. We nw cnsider the wmen s athletics 1500 m and 3000 m events f Figure 1, fcusing n the recrd data nly. This led t the maximum likelihd estimate frm Equatin (2), we get the parameter estimates in Table 1. The 95 % CI fr ξ fr 1500 m and 3000 m events are (-5.585,5.018) and (-2.196,2.156). This is are unacceptably large cnfidence intervals. When maximizing the lg likelihd with fixed α, β and γ values at their estimated values (using all annual minima data), the standard errr fr the shape are lwered, this is because separating the trend frm the recrd prcess is hard if nly the recrd are bserved. Table 1: The parameter estimatin fr recrd fr the wmen s athletics 1500 m and 3000 m track events frm 1972 t Parameter α β γ ξ 1500 m 248(29) 8.9(31.1) 0.251(0.609) 4.79(14.03) (2.546) 3000 m 548(9) 37.7(13.9) 0.212(0.212) 4.12(7.23) (1.110) 1500 m 3000 m KMeier estimate KMeier estimate e e Figure 4: Kaplan-Meier estimates fr the wmen s athletics 1500 m and 3000 m f e i with the dtted line is survival functin, Ŝ(e ). 16 Malaysian Jurnal f Mathematical Sciences

17 utput 2017/5/9 5:13 page 17 #17 Mdelling Recrd Times in Sprt with Extreme Value Methds 1500 m 1500 m 1 Survival prbability lg(survival prbability) Kaplan Meier estimate lg(kaplan Meier Estimate) 3000 m 3000 m 1 Survival prbability lg(survival prbability) Kaplan Meier estimate lg(kaplan Meier Estimate) Figure 5: The P-P and Q-Q plts fr the wmen s athletics 1500 m and 3000 m, with recrd pints marked with black circle. In rder t fit the mdel, as the recrd data invlving the censr data, we replace the empirical density functin with a Kaplan-Meier estimate in P-P and Q-Q plts, see Figure 4. The P-P and Q-Q plts in Figure 5 shws that the mdel s fit is acceptable fr the wmen s athletics 1500 m and 3000 m events as there are signs f linear pattern fr in plts fr censred data and the recrd (slid circle). We nted an imprtant pint, i.e. a number f the recrd depend whlly n the frm f the trend f the mean. 6. Applicatin t Actual Recrds Swimming We applied the methds f analysis develped in this chapter using data n the recrds prgressin f the men s 400 m Freestyle swimming event with data which are recgnised all arund the wrld by FINA. Additinal infrmatin is given by the men s Olympic 400 m Freestyle data. We implement the thery and fitting the mdel prpse in Sectins 2-3. We used the men s annual recrd prgressin fr 400 m Freestyle swimming event within 97 years time frm 1908 t In the early stage f the cmpetitin, the Freestyle event is als included the Breaststrke and the Butterfly until Malaysian Jurnal f Mathematical Sciences 17

18 utput 2017/5/9 5:13 page 18 #18 authr 1953, when these tw styles where separated frm the Freestyle event. Dealing with recrd data always led us t lse infrmatin regarding the trend in mean f true perfrmance f the men s 400 m Freestyle. In rder t vercme this missing infrmatin frm recrd prgressin data, we will used the gld medallist f the men s Olympic 400 m Freestyle swimming events frm 1908 t 2004 t btain the trend n the mean. We initially minimize the Equatin (10) fr the Olympic event t btain the nn-parametric trend in mean ĝ t (nt shwn here). This Fisher scring methd is capable in dealing with cmplexes trend in mean cmpared t parametric methd, see Adam (2007). secnd year Figure 6: The maximum (upper line) and the minimum (lwer line) value f the simulatin recrd prgressin with m = 33 where m is the number f recrd have been simulated using ĝ t, ˆ rec = 5.32 and ˆξ rec = Figure 6 shws that using the parameter estimates fr the recrd prgressin using infrmatin frm the men s Olympic events gives a fairly gd fit cmpared t the bserved recrds. Mre than 80% f the bserved recrds are within the acceptance area (between the maximum and the minimum f simulated recrds). In Figure 7, the prpsed P-P and Q-Q plts shwed a gd fit fr the mdel. We treated the event in 1908 as r 0, which means that it is nt a cnsidered as a recrd. Then there are 45 recrd have been recrded until 2004 fr the 18 Malaysian Jurnal f Mathematical Sciences

19 utput 2017/5/9 5:13 page 19 #19 Mdelling Recrd Times in Sprt with Extreme Value Methds 400 m Freestyle 400 m Freestyle 1 Survival prbability lg(survival prbability) Kaplan Meier estimate lg(kaplan Meier Estimate) Figure 7: The P-P (left) and Q-Q (right) plts using Kaplan Meier estimatin fr recrd prgressin and censred data using methd II in Sectin 3. recgnised men s 400 m Freestyle swimming events. Frm simulatin f m = 33 times f recrd prgressin, the recrd have been brken between 34 t 50 times tabulated in Table 2, inclusively with the average recrd breaking is 43 times (further indicate that the simulated mdel is apprximately similar with the real mdel). Nted that we nly cnsidered the annual prgressin f the best perfrmance by men s 400 m Freestyle swimmer each years. Table 2: Summary result f the number f recrds brken frm m = 33 f simulatin data using ĝ t, ˆ rec and ˆξ rec. m true minima median mean maximum Frm the minimizing the lg-likelihd f Equatin (13), the parameter estimates fr ˆ rec is 5.32(0.79) and fr ˆξ rec is (0.089). The estimate f rec values is quite similar with Olympic (5.625) indicate that the mdel prpsed frm Equatin (6) is gd. Malaysian Jurnal f Mathematical Sciences 19

20 utput 2017/5/9 5:13 page 20 #20 authr 7. Cncluding Remarks We have intrduced a new methd t analysis the prgressin f recrd data. We use the Pissn prcesses in rder t build the jint density functin fr the recrd data. We have als intrduced a new test fr gdness--fit fr the recrds. T accunt fr trends in the underlying data the recrds are drawn frm we used infrmatin frm anther existing events e.g. fr the swimming recrd analysis we used the perfrmance f the swimmers frm Olympic event t estimate the nn-parametric trend in mean. We limited the study t annually data which mean that if in a year few recrd brken events ccurred we will stick t the latter recrd in ur study. We have als intrduced tw gdness-f-fit methds hw t test the fitting mdel fr the recrd prgressin data i.e. the tlerance regins f the mdel using sme simulatin data frm estimated parameters and if we treated the recrd data as a censred type f extreme data, we used the P-P and Q-Q plts as alternative mdel diagnstics. T prceed the tlerance regins and plts we still need an infrmatin frm anther existing events. The nly drawback f the methds prpsed here is that they require a higher capability f cmputer pwer. Tw main aspects cnsume mst cmputer ability: Fisher scring methd in estimating ĝ t and when cnstructing the tlerance regin fr the new prpsed gdness f fit prcedure. References Adam, M. B. (2007). Extreme Value Mdelling fr Sprts Data. PhD thesis, Lancaster University, Lancaster, UK. Adam, M. B. and Tawn, J. A. (2011). Mdificatin f Pickand s dependent fr rdered bivariate extreme distributin. Cmmunicatins in Statistics: Thery and Methds, 40(9): Adam, M. B. and Tawn, J. A. (2012). Bivariate extreme analysis f Olympic swimming data. Jurnal f Statistical Thery and Practice, 6(3): Ahsanullah, M. (2004). Recrd Values -Thery and Applicatin. University Press f America, Oxfrd. Ahsanullah, M. and Bhj, D. S. (1996). Recrd values f extreme value distributins and a test fr dmain f attractin f type i extreme value distributin. Sankhyâ: The Indian Jurnal f Statistics, 58: Series B. 20 Malaysian Jurnal f Mathematical Sciences

21 utput 2017/5/9 5:13 page 21 #21 Mdelling Recrd Times in Sprt with Extreme Value Methds Arnld, B. C., Balakrishnan, N., and Nagaraja, H. N. (1998). Recrds. Wiley, Canada. Bairamv, I. G. (1996). Sme distributin free prperties f statistics based n recrd values and characterizatins f the distributins thrugh a recrd. Jurnal f Applied Statistical Science. Benested, R. E. (2004). Recrd-values, nn-statinary tests and extreme value distributins. Glbal and Planetary Change, pages Cles, S. G. (2001). An Intrductin t Statistical Mdeling f Extreme Values. Springer-Verlag, Lndn. David, H. A. and Nagaraja, H. N. (2003). Order Statistics. Wiley, New Jersey, third editin. Davisn, A. C. and Smith, R. L. (1990). Mdels fr exceedances ver hight threshlds. Jurnal f the Ryal Statistical Sciety Series B (Methdlgical), 52: Glick, N. (1978). Breaking recrds and breaking bards. The American Mathematical Mnthly, 85(1):2 26. Green, P. J. and Silverman, B. W. (1994). Nnparametric Regressin and Generalized Linear Mdels. Chapman & Hall, Lndn. Leadbetter, M. R., Lindgren, G., and Rtzen, H. (1983). Extremes and Related Prperties f Randm Sequences and Prcesses. Springer, Berlin. Pickands, J. (1971). The tw-dimensinal pissn prcess and extremal prcesses. Jurnal f Applied Prbability, 8: Rbinsn, M. E. and Tawn, J. A. (1995). Statistics fr exceptinal athletics recrds. Applied Statistics, 44(4): Sibuya, M. and Nishimura, K. (1997). Predictin f recrd breaking. Statistica Sinica, 7: Smith, R. L. (1988). Frecasting recrds by maximum likelihd. Jurnal f the American Statistical Assciatin, 83: Smith, R. L. (1989). Extreme value analysis f envirnmental time series: An example based n zne data (with discussin). Statistical Sciences, 4: Malaysian Jurnal f Mathematical Sciences 21