A friend of mine, an expert in statistical genomics, told me the following story: At a dinner party, an attractive lady asked him, "What do you do for a living?" He replied, "I model." As my friend is a handsome man, the lady did not question his statement and continued, "What do you model?" "Genes." She then looked at him up and down and said, "Mh, you must be very much in demand." "Yes, very much so, especially after I helped discover a new culprit gene for a common childhood disease." The lady looked puzzled. In this snapshot, I will give you an insight into Statistics, the field that fascinated my friend (and myself) so much. I will concentrate on phenomena that change over time, in other words, dynamical events

Tong, Howell

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Snapshots of modern mathematicsfrom Oberwolfach№ 6/2014Stat ist ics and dynamicalphenomenaHowel l TongA friend of mine, an expert in statistical genomics,told me the following story: At a dinner party, anattractive lady asked him, “What do you do for a liv-ing?” He replied, “I model.” As my friend is a hand-some man, the lady did not question his statementand continued, “What do you model?” “Genes.” Shethen looked at him up and down and said, “Mh, youmust be very much in demand.” “Yes, very much so,especially after I helped discover a new culprit genefor a common childhood disease.” The lady lookedpuzzled.In this snapshot, I will give you an insight into Sta-tistics, the field that fascinated my friend (and my-self) so much. I will concentrate on phenomena thatchange over time, in other words, dynamical events.11 Chaos and autoregressive modelsConsider the following doubling map, where n denotes an integer: We choosey0 > 0, and define a series whose nth element is given byyn = 2yn−1 mod 1. (1)In words this means that we double the previous y-value, throw away the integerpart and keep only the decimal part 1 . The map is also called a saw-tooth mapbecause it is equivalent to the iteration defined by the saw-tooth functionf(y) ={2y, 0 ≤ y < 0.52y − 1, 0.5 ≤ y < 1.(2)Plotting the graph of the function shows that it contains two straight lines witha break (threshold) at 0.5. Such a function is nonlinear, although the singlepieces are linear.10.5 1 yf(y)Figure 1: A plot of the saw-tooth function with threshold at 0.5Now, select any positive number as a starting point and iterate forward togety1, y2 = f(y1), y3 = f(y2), . . . .We will quickly realize that the output shows no regularity no matter howlong we run it, that is, the output looks quite chaotic (hence we need ChaosTheory), almost indistinguishable from a random series. This seemingly puzzlingfeature of generating randomness from a wholly deterministic mechanism can beunderstood as soon as we realize that the function f is highly sensitive to initialvalues: Two initial values differing only in, say, the 8th decimal place will quicklydiverge (that is, spread apart) upon repeated application of f . The famous1 For example if y0 = 0.7, we get y1 = 2y0 − 1 = 0.4.2French mathematician Henry Poincaré (1854-1912) listed such sensitivity asone of the sources of randomness. Another curious thing about the doubling(or saw tooth) map is that if we think of n as a parameter for time and run themap “backwards” in time, we will discover that we need to introduce “externalrandomness” denoted by εn. We setXn = 0.5 ·Xn−1 + εn, (3)where n is an integer, εn equals 0 and 0.5 with equal probability, and theconnection to the series of the yn given above is Xn = y−n. The εn termappears because the inverse of f maps one point to two possible points equallylikely. For example we get f−1(0.5) = 0.25 or 0.75.This leads us to yet another curious feature: We started with a deterministic 2nonlinear map (2) and end up with a random (a more fancy word is stochastic)linear equation (3).An equation like (3) defines what is called a time series model in Statistics,and it describes the dynamics of a system over discrete time (n). In general,the εn can have a more general probability distribution than the uniformdistribution from our example, such as the Gaussian distribution (after thefamous German mathematical genius Carl Friedrich Gauß (1977-1855)) overthe set of real numbers. An equation of the formXn = a1 ·Xn−1 + . . .+ ap ·Xn−p + εn (4)is called a (linear) autoregressive model (AR model for short), which wasinvented by the British statistician Udny Yule (1871-1951) in 1927 when hestudied the annual number of sunspots. Here, the coefficients aj are the definingparameters to be estimated from observations.The AR model finds appliciations in a broad variety of fields both in itsoriginal form and its multidimensional or piecewise linear generalizations. Wewill illustrate this by considering some real-life exmaples.2 Some real examples• US hog dataProfessors George Box and George Tiao from the USA analyzed the dataseen in Figure 2 by using a 5-dimesional AR model. They used the variablesHp, Hs, Rp, Rs,W for the hog price, hog supply, corn price, corn supply and2 Deterministic means that each time we plug in the same value into our function, it givesus the same result. The opposite of deterministic is stochastic, such as seen in equation (3):The input 0.5 gives 0.25 (for ε = 0) or 0.75 (for ε = 0.5) with equal probability.3Figure 2: Hog data from the US (Box and Tiao, 1977)the farmer’s wage respectively. They concluded thatHpHs(RpRs)0.75W 0.50is approximately independently distributed about a fixed mean. In otherwords, using real-life observations, they have justified thatprofit of the farmerexpense of the farmerfollows a stable economic law! It is a remarkable historical fact that such asimple empirical relationship had never been discovered for the above classicdata set, until the two statisticians invented a new time series method basedon another statistical technique called canonical correlations in 1977.• Economics and FinanceThe piecewise linear deterministic model (2) has its stochastic equivalentby adding εn. The result is commonly called the threshold AR (or TAR)model first introduced by the author in 1978. In an even more general form,it looks likeXn ={a0 + a1Xn−1 + . . .+ apXn−p + εn, Xn−d < rb0 + b1Xn−1 + . . .+ bpXn−q + cεn, Xn−d ≥ r.(5)The variable r is called the threshold parameter and has to be estimated fromthe observed data, along with the aj , bj , p, q, d, c, and the variance of εn.4The model has made an enormous impact in economics and finance.In 2011, the US econometrician Professor Bruce Hansen published a com-prehensive review of 75 papers related to TAR models. A typical examplefor the application of the TAR model is the modelling of aggregate outputas measured by GNP growth rates. It has for instance been shown that theGNP of the United States is subject to floor and ceiling effects. The use ofthe TAR model to study the relationship between long and short interestrates has helped to reveal the strong asymmetric response of interest ratechanges to the spread between the long and short rates.• Plagues in KazakhstanEarly this century, the statistician Kung-Sik Chan and his then doctoralstudent, Noelle Samia, worked with a team of biologists/epidemiologistsled by Professor Nils Chr. Stenseth, formerly President of the NorwegianAcademy of Science and Letters, on an extensive scale study of plagueepidemics in Central Asia. Concerning the bacterium Yersinia pestis thatcauses bubonic plague, they concluded in their report in the Proceedings ofthe National Academy of Sciences (USA) in 2006 that“Y. pestis prevalence in gerbils increases with warmersprings and wetter summers... Climatic conditions favour-ing plague apparently existed in Central Asia at the onsetof the Black Death as well as when the most recent plaguepandemic arose in the same region, and they are expectedto continue or become more favorable as a result of climatechange.”This conclusion is based on a modified TAR model which the participatingstatisticians developed and fitted from observed data. In the model tobe given below, Nt,` is the number of great gerbils examined at time t in“large square” 3 `. The number of great gerbils testing positive under abacteriological test does not follow a Gaussian distribution. Instead, it ismore likely to follow a binomial distribution and hence we modify the TARmodel slightly while retaining the piecewise linear structure.We distiguish between data collected in the spring and in the fall and marktheir parameters with s and f respectively. The binomial distribution modelhas parameters (Nt,`, Pt,`), where if t is a spring, the true prevalence rate isPt,` = 0 when the lag-ds occupancy (i.e. how many burrows are occupied bythe great gerbils ds units of time ago, namely Xt−ds,`), is below the spring3 The large squares are the result of dividing Kazakhstan into non-overlapping 40 × 40 km2squares.5threshold rs` . Otherwise, it follows a logistic regression model shown below.A similar specification applies to the fall data. We should not worry abouttoo many details as long as we get the message that the dynamics underlyingbacterium Yersinia pestis operates in 4 regimes depending on the season(spring or fall) and the lag-ds occupancy rate (above or below the seasonalthreshold).Pt,` =0, if Xt−ds,` < rs` and t is a springlogit−1{(βs0 + bs0,`) + (βs1 + bs1,`)Tsp,t + bs2,`Rsp,t + εt,`},if Xt−ds,` ≥ rs` and t is a spring;0, if Xt−df ,` < rf` and t is a falllogit−1{(βf0 + bf0,`) + βf1Rsu,t + βf2Xt−1/2,` + εt,`},if Xt−df ,` ≥ rf` and t is a fall;Here, the superscript f signifies fall, X denotes the great gerbil occupancy,Tsp,t is the spring temperature, Rsp,t is the log spring rainfall, and Rsu,tis the log summer rainfall. The βs are tuning parameters to be estimatedfrom the observed data. The logistic function logit−1(x) = 11+exp(−x) is justa mathematical trick to transform the binomial model setting into a TARmodel format.3 Looking to the futureThe present information age poses many exciting challenges to Statistics. Datacollection is so fast and plentiful that suddenly we find ourselves flooded withdata.Let us start with a simple illustration: We saw the example of the hog data,taken at one particular site. Suppose we have similar data at say 100 sites.Does a similar empirical law hold for all of them? Will there be differences?Similar panel time series can and do occur in many situations, e.g. a panel ofstock price time series across different stock markets, a panel of death ratesdue to a particular infectious disease at different locations in the world, and soon. Many interesting questions then need to be answered, e.g. are the differentstock markets equally volatile or do they cluster in some way? Is the infectiousdisease spreading?It is clear that new statistical methodologies will be necessary in order to copewith the new challenges. This is where fresh thinking is needed. To do so,6we may have to re-examine existing statistical methodologies and even ourphilosophy. Statistics as a scientific discipline has a rather recent history, notmuch longer than 100 years. Over these 100 years, the discipline has beendominated by a concept called likelihood. It is based on the assumption thatthe real underlying model is known and the only thing unknown are its tuningparameters. The observed data then enables us to estimate them. Two schoolsof thought, the frequentist school and the Bayesian school, have been the pillarsof Statistics. Despite their sometimes heated and colourful polemics, they sharethe common ground of likelihood.But what if the true model does not exist? What if we know that the model givento us by our scientist friend is wrong but it is the best available? I think theseare legitimate questions that we need to address. Although some opening shotshave been fired by people like Professor Laurie Davies in Germany, ProfessorYingcun Xia in Singapore, the author and others, the main contributions willhave to come from younger brains like yours!7Image credi tsFig. 2: U.S. hog data, original series. Cited from [1]References[1] G. E. P. Box and G. C. Tiao, A canonical analysis of multiple time series,Biometrika 64 (1977), no. 2, 355–365, http://biomet.oxfordjournals.org/content/64/2/355.abstract.[2] K.S. Chan and H. Tong, Chaos: A statistical perspective, Springer Seriesin Statistics, Springer, 2001, ISBN 9780387952802, http://books.google.de/books?id=DurlS0WgVZEC.[3] P.L. Davies, Approximating data, Journal of the Korean Statistical Society37 (2008), no. 3, 191 – 211, ISSN 1226-3192, http://www.sciencedirect.com/science/article/pii/S1226319208000380.[4] B. Hansen, Threshold autoregression in economics, Statistics and Its Interface4 (2011), 123 – 128.[5] N.C. Stenseth, N.I. Samia, H. Viljugrein, K.L. Kausrud, M. Begon, S. Davis,H. Leirs, V.M. Dubyanskiy, J. Esper, V.S. Ageyev, N.L. Klassovskiy, S.B.Pole, and K.S. Chan, Plague dynamics are driven by climate variation.,Proceedings of the National Academy of Sciences of the United States ofAmerica 103 (2006-08-29 00:00:00.0), no. 5, 13110–5, ISSN 0027-8424.[6] H. Tong, On a threshold model, Pattern recognition and signal processing(C.H.Chen, ed.), NATO ASI Series E: Applied Sc., no. 29, Springer, 1978,pp. 577 – 586.[7] Yingcun Xia and Howell Tong, Feature matching in time series model-ing, Statistical Science 26 (2011), no. 1, 21–46, http://dx.doi.org/10.1214/10-STS345.8Howel l Tong is professor emer i tus forStat ist ics at the London School ofEconomics.Mathematical subjectsProbabi l i ty Theory and Stat ist icsConnect ions to other f ie ldsPhysicsLicenseCreat ive Commons BY-NC-SA 3.0DOI10.14760/SNAP-2014-006-ENSnapshots of modern mathematics from Oberwolfach are written by participants inthe scientific program of the Mathematisches Forschungsinstitut Oberwolfach (MFO).The snapshot project is designed to promote the understanding and appreciationof modern mathematics and mathematical research in the general public worldwide.It is part of the mathematics communication project “Oberwolfach meetsIMAGINARY” funded by the Klaus Tschira Foundation and the OberwolfachFoundation. All snapshots can be found on www.imaginary.org and onwww.mfo.de/snapshots.Junior Edi torLea Rennerjunior- edi tors@mfo.deSenior Edi torCar la Cederbaumsenior- edi tor@mfo.deMathematisches Forschungsinst i tutOberwolfach gGmbHSchwarzwaldstr. 9 –1177709 OberwolfachGermanyDirectorGerhard Huisken

Statistics and dynamical phenomena

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Statistics and dynamical phenomena

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