TIME SERIES MODELLING OF SURFACE PRESSURE DATA

Transcription

1 INTERNATIONAL JOURNAL OF CLIMATOLOGY Int. J. Climatol. 18: (1998) TIME SERIES MODELLING OF SURFACE PRESSURE DATA SHAFEEQAH AL-AWADHI and IAN JOLLIFFE* Department of Mathematical Sciences, Uni ersity of Aberdeen, King s College, Aberdeen, AB24 3UE, UK Recei ed 22 October 1996 Accepted 10 October 1997 ABSTRACT In this paper we examine time series ling of surface pressure data, as measured by a barograph, at Herne Bay, England, during the years Autoregressive moving average (ARMA) s have been popular in many fields over the past 20 years, although applications in climatology have been rather less widespread than in some disciplines. Some recent examples are Milionis and Davies (Int. J. Climatol., 14, ) and Seleshi et al. (Int. J. Climatol., 14, ). We fit standard ARMA s to the pressure data separately for each of six 2-month natural seasons. Differences between the best fitting s for different seasons are discussed. Barograph data are recorded continuously, whereas ARMA s are fitted to discretely recorded data. The effect of different spacings between the fitted data on the s chosen is discussed briefly. Often, ARMA s can give a parsimonious and interpretable representation of a time series, but for many series the assumptions underlying such s are not fully satisfied, and more complex s may be considered. A specific feature of surface pressure data in the UK is that its behaviour is different at high and at low pressures: day-to-day changes are typically larger at low pressure levels than at higher levels. This means that standard assumptions used in fitting ARMA s are not valid, and two ways of overcoming this problem are investigated. Transformation of the data to better satisfy the usual assumptions is considered, as is the use of non-linear, specifically threshold autoregressive (TAR), s Royal Meteorological Society. KEY WORDS: autoregressive s; ARMA s; threshold autoregressive (TAR) s; surface pressure data 1. INTRODUCTION The use of time series ling has not been as prevalent in meteorology and climatology as in some other disciplines such as economics, even though climatic data often takes the form of series measured over time. Some authors have used autoregressive moving average (ARMA) s which were popularised by Box and Jenkins (1970) and are described in Section 3. Wilks (1985), Section 8.3, provides a detailed introduction to the s for atmospheric scientists, concentrating on low-order autoregressive s. Katz and Skaggs (1981) discuss some problems associated with fitting ARMA s, which will be noted later. More recently, Milionis and Davies (1994) fit simple ARMA s, and their extension to non-stationary data (ARIMA s), to a series which measures the activity of temperature inversions. Seleshi et al. (1994) use transfer function s, which extend ARMA s to look at more than one series simultaneously, to examine series of sunspots and Ethiopian rainfall. In the present paper we attempt to daily variations in atmospheric pressure, using data from South-east England. Section 2 describes the data in more detail. The first step is to fit separate ARMA s for different natural seasons. Results for different seasons are compared in Section 3. The pressure data have a number of special features, the first being that they are measured on a barograph, and hence are continuous in time. In order to fit ARMA s it is necessary to use data sampled at discrete intervals. Section 3 also examines briefly the effect of different spacings between the sampled time points on the fitted s. * Correspondence to: Department of Mathematical Sciences, University of Aberdeen, King s College, Aberdeen, AB24 3UE, UK. CCC /98/ $ Royal Meteorological Society

2 444 S. AL-AWADHI AND I. JOLLIFFE Another feature of the pressure data is that their behaviour is different at high and low pressures, so that the assumptions underlying ARMA ling are not strictly satisfied. To overcome this problem we investigate, in Section 4, the idea of transforming the data before they are led, and in Section 5 we examine whether non-linear s, specifically threshold autoregressive (TAR) s, are more appropriate than ARMA s for these data. The paper concludes, in Section 6, with some general discussion of time series ling in climatology. 2. THE DATA The data consist of continuous readings of atmospheric pressure in millibars (mb), taken from a barograph located at Herne Bay, in South-east England, for a period spanning The data are chosen mainly for illustrative purposes. They demonstrate how various features (continuous measurement, different seasonal behavior, ARMA assumptions not strictly valid) which occur more widely in climatology, can be dealt with in the context of ARMA ling and its extensions. For the purpose of our analyses, each year has been divided into six natural seasons, a classification that has been used by the UK Meteorological Office for a number of years. Each season comprises 2-monthly stretches of data, and are as follows: Winter (January February), Spring (March April), Presummer (May June), Summer (July August), Autumn (September October), and Prewinter (November December). The data have also been analysed separately for each year between 1981 and Some data, covering a period of 1 or 2 weeks in some years, were missing. Two-month periods in which these weeks occurred have been omitted from our analyses. For example in 1981 readings were missing for 11 days in January, thus leading to the exclusion of the Winter season (January February). In order to fit ARMA, and other, s readings were taken at discrete intervals. i.e. 06:00 h, 12:00 h, 18:00 h and at 00:00, respectively. For each of our analyses we used three different sampling intervals. For the first interval, all four observations per day were used, totaling between 236 and 246 observations (depending on the number of days in a month). For our second interval, two readings were taken per day, totaling between 118 and 124 observations. In this instance we obtained two series, either by taking readings at 06:00 h and 18:00 h, or by using readings at 12:00 and 00:00. Thirdly, one observation was taken per day, totaling between 59 and 61 observations. In this last case, four different series were obtained, relating to measurements taken at our four observation times. 3. APPLYING ARMA MODELS TO DIFFERENT NATURAL SEASONS AND DIFFERENT SPACINGS The surface pressure data were led using autoregressive moving average s which are defined as follows (Box and Jenkins, 1970). A series {Z t } follows an ARMA(p, q) if it can be written as: (Z t ) 1 (Z t 1 ) 2 (Z t 2 )... p (Z t p ) =a t 1 a t 1 2 a t 2... q a t q where is the mean of the series, 1, 2,..., p, 1, 2,..., q are coefficients (parameters) and {a t }is white noise (i.e. the series {a t } consists of independent identically distributed random variables). p and q are respectively, the autoregressive and moving-average orders of the, and we would ideally like both p and q to be small for simplicity. To build an ARMA, three main stages should be applied: (i) Identification: choosing a which gives the best fit, based on the autocorrelation function (acf) and partial autocorrelation function (pacf) of the series. (ii) Estimating any unknown parameters, which is done using the software package Micro TSP.

3 TIME SERIES MODELLING 445 (iii) Diagnostic checking to see if the chosen fits well. This can be examined through the autocorrelations of the residuals which should fall mainly within the interval [ (2/ n), +(2/ n)] where 1/ n is the approximate standard error of terms r k in the acf, and by the Box and Pierce (1970) test statistic =n(n+2) n 2 k=1 (r k2 )/(n k) which is compared with the n p q distribution in order to decide whether the residuals are white noise. There are certain assumptions associated with ARMA s, the most important of which is stationarity. This implies that the mean, variances and covariances are the same throughout the length of the series. In the case of variances this is not true for several seasons. To overcome this problem we can either transform the series before fitting the, or fit a more complex. Both of these approaches are illustrated later, but first an ARMA was fitted to the original data for each natural season and for each data spacing. This was done using a statistical software package called Micro Time Series (TSP) which specialises in time series ling. Choice of was determined by examining mean square error (MSE) and the significance of MA and AR coefficients. As noted above, although the data are continuous, observations were taken at discrete time intervals in order to fit ARMA s. The problem of discrete sampling has been studied by Phadke and Wu (1974) who proved that discretization of a continuous autoregressive moving average ARMA(p, q) process, at equally-spaced sampling intervals, results in a discrete autoregressive moving average process ARMA(p, p 1), regardless of q. When a continuous time series is observed discretely, the use of a frequent sampling interval would provide a more accurate approximation to the continuous time process than more widely spaced data. Therefore, the period h between observation should be sufficiently small. It has been suggested that if the process in question is well described in continuous time by a differential equation of finite order, then a mixed autoregressive moving average of finite order is implied for the sampled series (Phillips, 1973; Bergstrom, 1981). The order of the latter discrete does not depend on the sampling interval of the data, nor does the accuracy with which it can be approximated by an autoregressive process of finite order (Geweke, 1982). For example, if two lags (p=2) are adequate when taking one observation a day, it need not be the case that eight are required when taking four readings a day; only two lags might be necessary if the process is a continuous time finite order stochastic differential equation. We discuss our analyses for the winter season in some detail but summarize results for the other seasons rather briefly Winter season The Winter season comprises a 2 month stretch, January February. Years 1981 and 1987 had missing data, but we examined the remaining years between 1982 and In order to see the behaviour of each year in this season we represented all the years from 1982 to 1989 in one graph. This is shown in Figure 1. With four readings a day, all of the series can be reasonably well led by AR(2), except for 1985 and 1989 where AR(1) was adequate, as is shown in Table I. Order selection (the choice of p and q) is not a straightforward matter. Whole books have been written on the subject (Choi, 1992; Kumar and Jolliffe, 1998, see also Katz and Skaggs 1981). The choice of p and q is not always clearcut, but Figures 2 and 3 show the classic AR(2) pattern of a slowly decaying autocorrelation function, together with a partial autocorrelation with a distinct cut-off after two lags, for the 1983 data. The years with the same AR had very similar parameters, except for the constant terms, which depend on the mean of the series. This indicates that there was a fairly consistent behaviour in the atmospheric pressure over different years for the Winter season. When the same method was applied to the series with two readings a day, both AR(1) and AR(2) s fitted satisfactorily in most years. There were years in which the two winter series examined (readings of 06:00 h and 18:00h or at 12:00 and 00:00) gave different fitted s even though the differences between the two series in each year are not very great. For example, in 1983 the second series

4 446 S. AL-AWADHI AND I. JOLLIFFE Figure 1. The behaviour of atmospheric pressure in the Winter season for was best fitted with AR(4) with MSE value of mb 2 while the first series was best fitted with AR(2) with MSE value of mb 2 (for AR(4) the additional coefficients were not significant even though the MSE value was decreased). This illustrates that the choice of s can be quite sensitive to a small change in a series, and different ling does not necessarily mean a serious difference in the behaviour of the pressure data. Finally, looking at the results for the series with one reading a day, it was found that AR(1) s gave the best result for 1984, 1985, 1988 and 1989, whether the first, second, third or fourth series was chosen. The AR(1) was also best for 1983 whether we used the first reading or the fourth reading. The other readings in 1983 were led using the ARMA(1, 2). In 1982 and 1986 all the series were well fitted with ARMA(1, 1) s. A similar number of lags are needed in the three sampling intervals. As noted above, Geweke (1982) proved that if a process is described in continuous time as a differential equation of finite order, then a mixed ARMA of finite order is implied for the sampled series, and its order does not depend on the sampling interval. This explains some of our results. For example the AR(2) is adequate when taking two observations a day, but it does not follow that the AR(4) is required when taking four observations a day, if the process is a continuous time finite order stochastic differential equation. Compared to the other seasons, the atmospheric pressure in this season was more variable between years. Thus, it includes both the highest and the lowest values of atmospheric pressure from amongst all seasons. In some years low values of air pressure appeared and the variations around the mean value were Table I. The ARMA s fitted for the series with the first sampling interval (four readings per day) for in the Winter season Year Model Mean (mb) S.D. (mb) MSE (mb 2 ) Coefficients ( ˆ, 1, 2 ) 1982 AR(2) , 1.386, AR(2) , 1.398, AR(2) , 1.297, AR(1) , AR(2) , 1.359, AR(2) , 1.221, AR(1) , 0.898

5 TIME SERIES MODELLING 447 Figure 2. Acf for 1983 Winter data, four observations per day very large. For example, Figure 1 shows that the atmospheric pressure in 1988 reached a very low level, leading to a mean value of 1005 mb and an S.D. of mb, while in 1989 the atmospheric pressure was generally high, giving a mean value of 1025 mb with an S.D. of 8.57 mb Other seasons Table II summarises the results for the various seasons. In the Winter and Prewinter seasons, few different s are needed for the first two sampling intervals. However, for the third interval, although most of the s fitted were of the same type, one less lag is needed. In other seasons there is greater variation in fitted s between sampling intervals, despite what theory tells us. Note, however, that in many cases there is not a clear-cut best ; two or more fit almost equally well. In the Winter, Spring and Prewinter seasons, the which fitted best for most years was AR(2), while in the Presummer and Summer seasons the best was ARMA(3, 1). It is not possible to be precise about the physical implications of needing an ARMA(3, 1), rather than an AR(2), except to say that the more parameters we have the more wide-ranging is the set of behaviours which can be led. However, even the single AR(2) can display a substantial variety of behaviour, depending on the values of its parameters, ranging from very smooth slow variation, to rapid oscillation to quasiperiodic behaviour (see Jolliffe, 1983). In all seasons, except for Autumn, different years were fitted with almost the same type of, and also the same number of lags, but in Autumn, different years were fitted with different s. A possible explanation is that the Autumn season acts as a transitional season. Thus in some years the pressure behaves like the Summer season and in other years the Winter behaviour was more apparent, but it is generally closer to the Winter season than to the Summer season. The pattern of the atmospheric pressure in 1989 appears to have been different from the other years in most seasons because it was often fitted with different s from the other years. Extreme mean values appeared in 1989 in some seasons. Thus, in the Winter and Presummer seasons, the mean values were the highest of all years, whereas they were the lowest in the Prewinter and Spring seasons.

6 448 S. AL-AWADHI AND I. JOLLIFFE 4. TRANSFORMATIONS Earlier it was noted that the variance of the data for small values was greater than for larger ones. This is especially so in the Winter and Prewinter seasons. This means that strictly the -fitting of the previous section was inappropriate, although it was useful as a first step in the analysis and provided some useful insights into the behaviour of the series for different sampling intervals and different seasons. Transforming the data, which we do next, provides a more appropriate analysis, but gives little extra information, as the s fitted are largely the same as for the untransformed data. To make the variance more homogeneous, transformations can be applied to the atmospheric pressure data. As well as the stationarity assumptions, for some purposes it is also useful to have, approximately, a Gaussian distribution for the residuals from a, and transformations can be useful to achieve this objective too (Katz and Skaggs, 1981). We considered a number of different types of transformation in each season, including the square root, logarithm and reciprocal transformations. The one which gave the best result in all seasons, as judged by normality of residuals, was the logarithm. Such transformations also made the variances more homogeneous. Different logarithm transformations were obtained in different years. The best logarithm transformation which was chosen in each season was not log(x) but, for example in Winter, log(1055 X), which removes the negative skewness in the distribution of residuals, whereas the more usual transformation log(x) removes the more commonly encounted positive skewness. The transformation log(1055 X) seemed to be the best transformation for the Winter and Spring seasons while log(1050 X) was best for Summer and Autumn. The best transformation in the Presummer season appeared to be log(x 980) and in the Prewinter season it was log(1060 X). After transformation the mean values were very similar and so were the variances. In most seasons the transformation decreases the variances for the lower mean values, which was a problem before transformation, especially in the Winter and Prewinter seasons. However, this kind of problem did not show up in the Presummer season where the pressure was stable and rarely reached a low level, i.e. pressure values were never below 980 mb. Thus the nature of the transformation for this season was different from the Figure 3. Pacf for 1983 Winter data, four observations per day

7 TIME SERIES MODELLING 449 Table II. The ARMA s fitted for the series with different sampling intervals and for different seasons Season Sample interval 4 obs/ day Sample interval 2 obs/ day Sample interval 1 obs/ day Model NOS Model NOS Model NOS Winter AR(1) 2 AR(1) 4 AR(1) 18 AR(2) 5 AR(2) 6 ARMA(1, 1) 7 AR(3) 1 ARMA(1, 2) 2 Spring AR(1) 1 AR(2) 8 AR(1) 24 AR(2) 6 AR(3) 7 ARMA(2, 1) 9 ARMA(3, 1) 2 ARMA(1, 1) 2 ARMA(1, 1) 3 ARMA(2, 1) 1 AR(2) 1 Presummer AR(2) 1 AR(2) 7 AR(1) 15 ARMA(3, 1) 6 AR(3) 6 AR(2) 4 ARMA(4, 2) 1 ARMA(2, 1) 2 ARMA(1, 1) 10 ARMA(5, 2) 1 ARMA(3, 1) 3 ARMA(2, 1) 5 ARMA(3, 1) 3 Summer ARMA(3, 1) 5 AR(2) 5 AR(1) 8 ARMA(3, 2) 2 ARMA(2, 1) 6 AR(2) 6 ARMA(1, 1) 2 ARMA(2, 1) 8 Autumn AR(2) 4 AR(1) 1 AR(1) 12 AR(4) 2 AR(2) 4 ARMA(1, 2) 8 ARMA(2, 2) 1 AR(3) 2 AR(3) 4 ARMA(3, 1) 1 ARMA(1, 1) 5 ARMA(1, 1) 4 ARMA(3, 2) 1 ARMA(2, 1) 3 ARMA(1, 2) 1 ARMA(2, 2) 1 Prewinter AR(2) 6 AR(1) 2 AR(1) 22 AR(3) 1 AR(2) 10 ARMA(1, 1) 6 AR(3) 1 transformations for other seasons, because log(x 980) decreased the variance for higher values and removed positive skewness, whereas transformations for the other seasons decreased the variances for the lower values and removed negative skewness. At the approach of the Winter season, non-normality became more obvious in the original series while at the approach of the Summer season, normality appeared to be a better approximation. However, in all seasons the appearance of normality for the series and for the residuals was strongly improved after transformation, as was homogeneity of variances. The results of fitting s (considering four observations a day) after transformation were mostly the same as before transformation so tables corresponding to Tables I and II for transformed data are not presented. The exception was the Presummer season, where the s after transformation were different from those before transformation; the most common was AR(2) after transformation instead of ARMA(3, 1). 5. NON-LINEAR MODELS ARMA s, which are linear, have been reasonably successful as a practical tool for analysis, forecasting and control of time series. But sometimes linear s cannot be used because of some limitation. For example, linear s only deal with stationary solutions concerning a constant steady

8 450 S. AL-AWADHI AND I. JOLLIFFE state, i.e. X t will always tends to a constant called the limit point as t tends to infinity. Moreover, ARMA s are not ideally suited to data exhibiting sudden bursts of very large amplitude at irregular time intervals. In this study the problem of heterogeneity of variances (as noted above) is another reason why non-linear s may be preferred. One simple class of non-linear s is considered here. These s are called self-exciting threshold autoregressive (SETAR) s, or piecewise linear s. They been discussed thoroughly by Tong (1983), and are defined as follows: Let R 1, R 2,...,R l be non-overlapping intervals which cover the real line, so that X t R i if r i 1 X t r i, where r 1, r 2,...,r l 1 are thresholds, with r 0 =, r l =. Then X t follows a SETAR if we can write k j ( j) ( j) X t =a 0 + a i X t i + tj i=1 whenever X t d R j, j=1,2,...,l. {X t } is called a self exciting autoregressive of order (l; k 1,...,k l ) or SETAR (l; k 1,...,k l ). Here d is a delay parameter. X t is determined by an AR(k j ), but the order, the parameters of the s, and the variance of the white noise process { tj }, depend on the value of X t d. Note that a SETAR (1; k) is just a linear AR of order k. The dependence of var(x t )on X t d suggests that SETAR s may be appropriate to deal with the variance inhomogeneity in the pressure data. A similar procedure to that for building linear s can be applied to nonlinear s, and consists of the stages of identification, estimation and diagnostic checking. A may be chosen by minimising a criterion called Normalized Akaike s Information Criterion (NAIC), as well as by considering the MSE. The NAIC is related to AIC, which is discussed in the context of AR s by Wilks (1985) and Katz and Skaggs (1981). The package which was initially used to analyse the data was the STAR package, provided to us by Professor Howell Tong, which is specially designed for non-linear SETAR s. The criterion NAIC is (AIC) (Akaike, 1974) divided by the residual degrees of freedom. The following steps describe how this criterion is applied for the SETAR (l; k 1,...,k l ) when l, r 1, r 2,...,r l 1 and d are fixed and L is the maximum order for each of the l piecewise linear AR s. Let AIC(k i )= min [n i ln{ e(k i ) 2 /n i }+2(k i +1)] 0 k i L where e(k i ) 2 denotes the residual sum of squares when an AR(k i ) is fitted to the i-th set of data and n i is its sample size. Then l AIC(d, r)= AIC(k i ) i=1 and the normalized criterion is NIAC(d)=AIC(d, r)/(n N d ) where N d =max(d, L). The main purpose here is to investigate whether the data are fitted significantly better by non-linear, rather than linear, s. For illustration, only one sampling interval is considered, in which four observations are made per day, and the results discussed below again concentrate mainly on one season, Winter Winter season Various numbers and values of thresholds were tried and a choice was made from them which gave a SETAR with a small value of NAIC. This is illustrated in Table III for 1983, where a comparison between linear and non-linear s is also made.

9 TIME SERIES MODELLING 451 It is clear from Table III that as the number of thresholds increased, better s were obtained, i.e. smaller NAIC and MSE values resulted. There was evidence that the pressure had different behaviour above and below 1031 mb, since this threshold value was used in each non-linear. The behaviour of the pressure also changed at 1025 mb. Although some observations were very large (45 are 1031 mb), most of the observations (129) are smaller than 1025 mb. Table III also shows that the non-linear s were much better than the linear s, since the NAIC and MSE values for the linear were clearly larger than those for the non-linear s. It is known that criteria based on AIC tend to overestimate the complexity of the best (Hannan, 1982). We are therefore reasonably happy to take the number of thresholds (l) at a lower value than might be suggested by NAIC, in order to keep the s relatively simple. For 1983, as the number of thresholds increased, better nonlinear s were obtained. However, to simplify the comparison with the linear s, the discussion for the remaining years will concentrate on non-linear s with only one threshold. The results are given in Table IV. It is clear that the linear s were usually improved upon by the non-linear s. For all years, the two different regimes were fitted with the same type of as each other, and as in the linear, but with different coefficient values. In 1983 and 1989, the threshold values reflected their average pressure values; thus their mean values were large, (1021 mb and 1025 mb, respectively) and so were their threshold values (1031 mb and 1020 mb). This implies that high average pressure values correspond to high threshold values. However, the opposite is not true, i.e. low average pressure values do not necessarily mean low threshold values. Examining the differences between the NAIC and the MSE values of linear and non-linear s in each year, it appears that the results were more favourable to nonlinear s in 1989, 1984, 1985 and In these years there were large differences between NAIC values and also between the MSE values of linear and non-linear s. It is also the case that non-linear s were better than the linear ones in 1982, 1983 and 1986, as there were reasonable differences in both their NAIC values and their MSE values. In 1986, it may be better to choose the simplest, because the difference between the NAIC values was small, although the difference between the MSE values was reasonable Other seasons Although non-linear s appear to give improved fits compared to linear s in Winter in most years, this is not so clear in other seasons, especially Presummer and Summer. The non-linear s performed better when approaching the Winter season than when approaching the Summer season. This is probably due to better stability in the atmospheric pressure when approaching the Summer season, i.e. Table III. Linear and non-linear autoregressive s with different number of thresholds in 1983 in the Winter season No of thresholds NAIC MSE Regime (no. of observations) Model Condition (threshold) Coefficient (a 0 (j), a 1 (j), a 2 (j) ) 0 (linear (236) AR(2) , 1.399, ) (179) AR(2) X t , 1.354, (45) AR(1) X t , (129) AR(2) X t , 1.317, (44) AR(2) 1025 X t , 0.488, (45) AR(1) X t , (129) AR(2) X t , 1.317, (23) AR(1) 1025 X t , (23) AR(2) 1028 X t , 0.438, (45) AR(1) X t , 0.864

10 Table IV. Linear autoregressive s and non-linear autoregressive s with one threshold for in the Winter season Year 1982 NAIC MSE Regime (no. of observations) Model Condition (threshold) Coefficient (as in Tables I and III) Non-linear (128) AR(2) X t , 1.361, (104) AR(2) X t , 1.111, Linear AR(2) , 1.395, Non-linear (179) AR(2) X t , 1.354, (45) AR(I) X t , Linear AR(2) , 1.399, Non-linear (117) AR(2) X t , 1.015, (117) AR(2) X t , 1.446, Linear AR(2) , 1.297, Non-linear (142) AR(1) X t , (84) AR(1) X t , Linear AR(1) , Non-linear (199) AR(2) X t , 1.396, (29) AR(2) X t , 0.642, Linear AR(2) , 1.380, Non-linear (132) AR(2) X t , 1.048, (100) AR(2) X t , 1.141, Linear AR(2) , 1.231, Non-linear (66) AR(1) X t , (166) AR(1) X t , Linear AR(1) , S. AL-AWADHI AND I. JOLLIFFE

11 TIME SERIES MODELLING 453 the pressure varies less and its behaviour is similar regardless of pressure level. Approaching the Winter season the behaviour of pressure is more changeable. Differences in type of weather pattern in the Winter are larger than in the Summer, leading to different s for pressure. With the non-linear s, as the number of thresholds increased, better s were fitted. As the number of thresholds increases, so do the number of regimes and thus more s are used to represent the series. This tends to decrease the MSE values and also the NAIC values. However, in the case of the NAIC value it reaches a minimum at some stage, then increases. In most seasons, the highest threshold values for non-linear s of different years reflected the average pressure value. In addition, in most seasons, the MSE values of the higher pressure regimes were much smaller than those of the lower regimes. This was due to the pressure in the higher regimes, in all years, being much more stable. It is plausible that blocking occurred in these regimes. Conversely, at lower levels, pressure varied to a greater extent and more quickly. So threshold s capture nicely the greater variation at lower pressures. However, in most years the number of observations in the higher regime was also small compared to the lower regime. For this reason the MSE value of the linear was much closer to the MSE value of the lower regime than that of the higher regime. 6. DISCUSSION AND CONCLUSION The same ARMA s were often used for different series in the same sampling interval and their coefficient values were also similar. However, sometimes in the third sampling interval, one or two of the four series were fitted with different s, showing a surprisingly high degree of sensitivity of fitted s to small changes in the data. A comparison between the s with different sampling intervals revealed two contradictory results. In the Winter and the Prewinter seasons, the same type of s were needed for the different sampling intervals and sometimes they had similar coeffficient values. These results are consistent with the theoretical results of Geweke (1982), who showed that if mixed ARMA s of finite order are fitted to a sample series from a process described in continuous time as a stochastic differential equation of finite order, then the order of the does not depend on the sampling intervals. As a result, the first sampling interval should be sufficient to describe the behaviour of the pressure in these seasons. However, comparison between the s of different sampling intervals in other seasons shows differences between the sampling intervals. Thus, when more observations were taken, higher order s were fitted, i.e. the fitted for the series when four daily observations were taken typically had more lags than for two daily observations, which in turn was higher than that of one observation a day. This is in line with intuition, although it contradicts Geweke (1982) results. It may be due to the differences in the behaviour of the series in different sampling intervals, but also due to sampling variability. There was a consistent difference between the s for the three sampling intervals in the Presummer, Summer and Autumn seasons, with a similar difference occurring sometimes in the Spring season. For the first sampling interval, (four readings per day) AR(2) s fitted best for the Winter, Spring and Prewinter seasons. In the Presummer and Summer seasons, the best for most of the years was ARMA(3, 1). Different years in the same season were usually fitted with similar s, but this was not true in Autumn where different years were frequently fitted with different s. This was probably due to Autumn being a transient season between Summer and Winter, so that, in some years it behaved like Summer season and in other years more like Winter. The pattern of the atmospheric pressure in 1989 was different from the other years in several seasons. In particular the mean values were higher than other years in the Winter and Presummer seasons, whereas they were the lowest in the Prewinter and Spring seasons. Hence, the series in this year were often fitted with s that were different from those of other years. The smallest variations of the mean values occurred in Summer followed by Presummer. Conversely, larger variations of the mean values occurred in the Winter, followed by Prewinter. Because the variance of low pressure values was greater than that of larger values, it was suggested that a

12 454 S. AL-AWADHI AND I. JOLLIFFE transformation was required. The log(1055 X) transformation was an appropriate transformation for all years in Spring and Winter, while log(1050 X) was used in Summer and Autumn. The best transformation in Presummer was log(x 980) and in Prewinter it was log(1060 X). These transformations stabilised variances as desired. In addition, the series and the residuals appeared closer to normality after transformation. Transformations decreased the variances for the lower pressure values for all seasons except Presummer where the nature of its transformation (log(x 980) was different from that of other seasons. Pressure in this season was stable and rarely reached a low level, so that the best transformation decreased the variances at higher values and removed the positive skewness of the residuals. The different behaviour of the pressure at high and low values suggested that (non-linear) threshold autoregressive (SETAR) s might be appropriate. In general SETAR s gave better fits than linear (ARMA) ones. However, the non-linear s performed better when approaching the Winter season than when approaching the Summer season. This was probably due to greater variability in pressure levels, when approaching the Winter season. Because of the more stable pressure behaviour in Summer and Presummer, the non-linear s in some years were not much better than the linear s in these seasons. Thus NAIC values did not decrease a great deal from linear to non-linear s, and it may be preferable to choose the simplest (linear) s. In our analysis we considered at most, three thresholds in the non-linear s. As the number of thresholds increased, better s were fitted. Thresholds partition the series into subsets (regimes) and the data in each regime were led by a linear AR. A higher number of thresholds always decreases the MSE. However, the NAIC values stop decreasing at some stage and adding further thresholds is then disadvantageous. When one threshold was used in the non-linear s, the MSE values of the second regimes (high pressure) were much smaller in most seasons than those of the first regimes (low pressure). This was due to greater stability of the pressure at high pressure values, i.e. blocking may often have occurred in the second regimes. However, the MSE values of the linear s were much closer to the MSE values of the second regimes than to those of the first regimes. This is because the number of observations in the second regimes was usually much larger than in the first regimes. Our analysis using SETAR s has been exploratory and illustrative. We wished to demonstrate that the introduction of non-linearity might better explain climatological time series. The answer in our example seems to be a fairly clear Yes in some seasons. Although some insights into the behaviour of the series are gained by simple fitting ARMA s to the transformed data, extra valuable information, as well as a better fit, is obtained from SETAR ling. Even better fits to the data in all seasons might be achieved by looking at more thresholds, at a greater range of threshold values, at higher order s, or at different type of nonlinear s. It seems clear that when some of the assumptions of ARMA ling are violated for a climatological time series, there is a real potential for improved ling and explanation of data using non-linear s such as SETAR s. ACKNOWLEDGEMENTS We are grateful to a referee for comments which led to improvement in the paper. REFERENCES Akaike, H A new look at the statistical identification, IEEE Trans. Autom. Control., 19, Bergstrom, A.R Gaussian estimation of structural parameters in higher order continuous time dynamic s, Continuous Time Econ. Modelling. Recent Ad. Econ. Ser., Box, G.E.P. and Jenkins, G.M Time Series Analysis: Forecasting and Control, Revised edn, Holden Day, San Francisco. Box, G.E.P. and Pierce, D.A Distributions of residual autocorrelations in autoregressive-integrated moving average s, J. Am. Statistical Assoc., 65, Choi, B ARMA Model Identification, Springer, New York. Geweke, J The measurement of linear dependence and feedback between multiple time series, J. Am. Statistical Assoc., 77,

13 TIME SERIES MODELLING 455 Hannan, E.J Testing for autocorrelation and Akaike s criterion, in Gani, J. and Hannan, E.J. (eds.), Essays in Statistical Science, Applied Probability Trust, University of Sheffield, pp Jolliffe, I.T Quasi-periodic meteorological series and second-order autoregressive processes, J. Climatol., 3, Katz, R.W. and Skaggs, R.H On the use of autoregressive-moving average processes to meteorological time series, Mon. Wea. Re., 109, Kumar, K. and Jolliffe, I.T Specification of Time Series Models, World Scientific Publishing, Singapore, in press. Milionis, A.E. and Davies, T.D Box Jenkins univariate ling for climatological time series analysis: an application to the monthly activity of temperature inversions, Int. J. Climatol., 14, Phadke, M. and Wu, S Modelling of continuous stochastic processes from discrete observations with application to sunspots data, J. Am. Statistical Assoc., 69, Phillips, P.C.B The problem of identification in finite parameter continuous time s, J. Econ., 1, Seleshi, Y., Demarec, G.R. and Delleur, J.W Sunspot numbers as a possible indicator of annual rainfall at Addis Ababa, Ethiopia, Int. J. Climatol., 14, Tong, H Threshold Models in Non-linear Time Series Analysis, Springer-Verlag, New York. Wilks, D.S Statistical Methods in the Atmospheric Sciences, Academic Press, San Diego.