C.H.1983/D:2 Statistics Committee Time Series Models of Mean Monthly Temperature on the AtlanticContinental Shelf Trina A. 1I0smer University Computing Center University of Massachusetts Amherst, MA 01003 ABSTRACT Box-Jenkins Time Series techniques autoregressive, integrated, moving average water temperature data collected for the Continental Shelf, (Cape Hatteras, NC, to 1950 through 1980 by lunar month. are used to fit seasonal (ARIMA) time series models to Northeastern United States Cape Sable, NS) for the years , Ce For geographical as well as oceanographical reasons the area is subdivided into six major areas: Nova Scotia, Gulf of Maine East, Gulf of MaineWest, Georges Bank, Middle Atlantic, and Gulf Stream. Each major area is further split into a shallow area (water depth less the 150 meters) and a deep area (water depth greater than 150 meters). For each of the twelve regions, ARIMA models are developed. The process used to develop the ARIMA models as well as the parameter estimates, residual mean-squares and Q-statistics are given. Tho performance of the fitted models is displayed by plotting the observed and forecastcd temperature over time. ARIMA models are developed primarily for forecasting purposes. In this paper we are more interested in comparing the form and content of the various time series models in order to better und erstand the ocean dynamics. . For example, we find one basic ARIMA model that fi ts all the series but with variations on the necessary lag time parameters. ThiiJ variation supports existing hypotheses about theunderstanding of water movement in the particular areas.
Transcript
C.H.1983/D:2Statistics Committee
Time Series Models of Mean Monthly Temperatureon the AtlanticContinental Shelf
Trina A. 1I0smerUniversity Computing CenterUniversity of Massachusetts
Amherst, MA 01003
ABSTRACT
Box-Jenkins Time Series techniquesautoregressive, integrated, moving averagewater temperature data collected for theContinental Shelf, (Cape Hatteras, NC, to1950 through 1980 by lunar month.
are used to fit seasonal(ARIMA) time series models toNortheastern United States
Cape Sable, NS) for the years
,Ce
For geographical as well as oceanographical reasons the area issubdivided into six major areas: Nova Scotia, Gulf of Maine East, Gulfof MaineWest, Georges Bank, Middle Atlantic, and Gulf Stream. Eachmajor area is further split into a shallow area (water depth less the150 meters) and a deep area (water depth greater than 150 meters). Foreach of the twelve regions, ARIMA models are developed. The processused to develop the ARIMA models as well as the parameter estimates,residual mean-squares and Q-statistics are given. Tho performance ofthe fitted models is displayed by plotting the observed and forecastcdtemperature over time.
ARIMA models are developed primarily for forecasting purposes. Inthis paper we are more interested in comparing the form and content ofthe various time series models in order to better und erstand the oceandynamics. . For example, we find one basic ARIMA model that fi ts all theseries but with variations on the necessary lag time parameters. ThiiJvariation supports existing hypotheses about theunderstanding of watermovement in the particular areas.
funk-haas
Neuer Stempel
C.M.1983/D:2Statistics Committee
This paper not to be cited without prior reference to the author
Time Series Models of Mean Monthly Temperatureon the Atlantic Continental Shelf
by
Trina HosmerUniversity Computing CenterUniversity of Massachusetts
Amherst, MA 01003
C.M.1983/D:2Statistics 'Committee
Des Mod~les type Serie Chronologique de la' TemperatureMensuelle Moyenne sur la eSte Continentale Atlantique.
RESUME
Les techniques de s~rie chronologique Box-Jenkins sont utilisees pourajuster un modele ARI~~ aux temperatures d'eau enregistrees sur la cStecontinentale du nord-est des Etats-Unis (Cape Hatteras, NC, to Cape Sable,NS) par mois lunaire durant les annees de 1950 ~ 1980.
!'()lll- t!l'H COIll-d t!~ rati um, allHH.i Id eil l'.~ograp" i <tUCH qll' oecmli qlleH, LI
rcgion üst suht!ivisee en six: La Nouvelle Ecosse, liest du golfe du Maine,l'ouest dll Rolfe du Maine, Ceorge Rank, 1e Moyen Atlnntique ct 1e Culf St~cam.
Cllaque sub region Cl:>t U l:>on tour t!ivil:>ce en zonc a bas-font! (pro[ont!eurd'eau de moins de 150 metres) et en zone profonde (profondeur d'eau de plusde 150 metres). Pour chacune des douze subregions des mod~les ARIMA ont etedeveloppes. Le procede utilise pour developper les modelcs ARIMA de m~meque les estimations des paramitres, la moyenne des carres des residuels etles Q statistiques sont donnes. La performance de l'ajustement des modelesest presentee en tra~ant les graphiques des temperatures observees et projeteessur une echelle de temps.
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Des modeles ARlMA sont developpes generalement pour permettre des previsions. Dans cet article nous sommes plus interesses a comparer 1a formeet 1e contenu des differents modeles de serie chronologique afin de mieuxcomprendre la dynamique des oceans. Nous avons trouve par exemple UI1 modlUeA~lMAde base qui s'ajuste a toutes les series etudiees mais avec des variations quant aUK param~tres de decalage dans le temps. Cette variation supportedes hypotheses existantes sur 1a connaissance du mouvement d'eau dans ces ~sub regions particulieres.
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1. Introduction
Since the early 1900's, data items such as temperature, salinity,
phytoplankton and fi sh population have been collected from the waters of the
Northeastern Continental Shelf by the scientific cruises of the National
Marine Fisheries Service and its predessors. Bigelow(1927,1933) and Bigelow
nnd Scars(1935) were the first to fully discuss the oceanographic features
of temperature and salinity by month and season for this region. More
recently. atlases of long ter'm temperature means were drawn by Wal ford and
Wickland(1968), Golton and Stoddard(l972). Erneryand Uchupi (1972) have
drawn similar charts for surface salinity. Pawlowski and Wright(1978) have
prepared charts for Spring and Fall sea-surface temperature and salinity for
the years 1972-1977. From a statistical point of view these studies have
done little more than present the observed data in an organized fashion. No
attempt h3S been made to fit statistical models to the data. This paper
presents an approach to the time series statistical models to analyze the
water ternperature data. The advantage of this approach is that it allows an
extensive and complex set' of data to be characterized by a few parameters
without losing the essential oceanographic information in the data. These
models can be used to better understand the ocean dynamies. They can also
be used to compare and contrast various areas, determine short and long tenn
trends wi thin and between areas and forecast future observations.
The data have been collected over time and the Box and Jenkins time
series methods are used to develop statistical models. Section Ir describes
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how the 11 time series analyzed in this paper were formed. In Section !II,
we give a techl'1ical description of the Box and Jenkins teehnique. This
seetion eontains the teehnical details of model formulation and readers not
farniliar with the Box and Jenkins methods may want to skip to Seetion IV.
Seetion IV contains a discussion of the fitted models and the conclusions
are given in Section V.
H. Data Description
The ri<lta analyzed in this paper are the largest collection of water
temperature recordings from the Northeastern U.S. Continental She1f (west
of lJt1gitllde 60 degrees Wand north of 30 degrees 20 minutes N), to be
amassed and ana1yzed in one study. It eonsists of over one million water
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ternperature recordings colleoted by the National Marine Fisheries
Service(NMFS) cruise ships and other sources beginning in 1908. Using a
bathythermograph, water temperatures were reeorded for 13 standard dp.pths:
0, 10,20,30,40,50,75,100,125,150,200,250,300 and 400 meters for over 200,000
~tations or loeations. Since the rnajority of data was colleeted post Wor1d
W:'lr II and time series anal ysi s requires comp1ete data throughout the year,
we chose to use only the data collected from 1950 through 1980.
From these data we created 11 series which consists of 13 month1y ~ean
water temperatures for each of the 31 years (1950-1980). For geographie as
weU as oceanographie reasons, th~ series were determined by dividing the
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Northeastern U.S. Continental Shelf into six major areas: Nova
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Seotia(NSOT), Georges Bank(GBNK), Gulf of Maine East(GME), Gulf of Maine
Hest (GMW) , Middle Atlantie(MATL), and Gulf' Stream(GSTR). Refer to Figure 1
for exaet lati tude and longitude loeation of eaeh of the six major areas.
Eaeh major area was further divided into a shallow area (water depth less
than 150 meters) and a deep area (water depth greater than 150 meters); thus
rcsultinU in 12 series. fuwever, due to the faet that in the Gulf Stream
area there is so little water wi th depth less than 150 meters, the Gulf
Stream Shallow series was dropped from the analyses •
The 13 lunar months were determined by dividing the ealendar year into
28 day periods, with the exeeption of the last Month which eontains 29 days
and on leap ycar 30 day~. The monthly means for each year were then
computed by averaging a11 the temperature reeordings within that month for
that year for depths 0,10,20,30,40,50, 15,100,125 and 150 meters for a given
area. \~e have 31 years each with 13 months per year yielding a total of 403
data points per series.
One requirement for the time series techniques used in this paper is
• that the series be complete with no rnissing data. However, none of tha 11
series met this criteria for numerous reasons: bad weather, mechanical
failures,reducert nurn')er of cruises during the Christmas Holidays, or lirnit8d
budget. Each series has at most 10% missing with the exception of Gulf
Maine East, Gulf Maine West Shallow and Deep which have approximately 15%
missing. Before the full analysis W3S performed a forecasting technique was
used to replace each missing value with apredicted value to cornplete each
series. The longest stretch of contiguous years wi th no missing data was
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used from each series to fit a model. Values were then forecasted using the
fitted model for the missing data points. The effect of using a time series
model to forecast missing val ues will be to bias the analysi s in favor of
the model. However, for most series less than 10% of the data were missing
so this effect should be negligible.
ur. Time Series Methods
The time series methods of Box and Jenkins were used to fit seasonal
autoregressive, integrated, moving average(ARIMA) models to the 11 water
temperature series. The general form for a p-th order autoregressive , d-th
order iritegrated, q-th order moving average and seasonal P-th order
autoregressive, D-th order integrated, Q-th order moving average model with
s observations per season, also referred to as a (p,d,q)x(P,D,Q) ARIMA
model is
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where
and
r (BsP ) S 2s sPP l-rlB -r2B - ••• -rpB
8 (B) 2 PP
1-8I B-82B - .•• -8pB
A (BsQ) s 2s sQQ I-AlB -A2B - ... -AQB
ep (B) 2 qq l-epIB-ep2B - •.• -epqB
d DV Yt Yt-Yt - d Vs Yt Yt-Ytt- SD
BYt Yt - l Bm= Yt - mYt
monthly water temperature
2zero mean white noise with variance 0 a
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° The identification and selection of the appropriate time series models
for each of the 11 series was accomplished following the methods of Box and
Jenkins (1970, Chapters 6,9). The generation of the autocorrelations,
parOtial-autocorrelations and parameter estimates was performed by the
BMDP2T(1981) computer program.
Box and Jenkins technique for determing the appropriate time series
•model is a three step procedure: (1) identification- use of the data and any
information on how the series was formed to suggest an appropriate model,
(2) estimation of the parameters using a non-linear least squares algorithm.
(3) diagnostic checking- examination of the fitted model in its relation to
The first step in identification is to determine if the data is
the data to determine model inadequacies and how to improve the fit.
autocorrelations Which have been computed from the data for lags of
we ex amine thestationari ty,forcheckTo
the data displays an affinity to a mean value).
If the data is stationary the computed autocorrelations should
(i .e.
technique.
1.2••••• K•
stationary
Stationarity is a necessary property in order to perform the Box and Jenkins
modeling
• die out gradually as the lags increase. If the data is not stationary, by
differencing the data an dappropriate number of times (V =y - Y )d we cant t-
achieve stationarity. If the data is seasonal, like the water temperature
data. then we will have to consider a seasonal di fferencing (VD= y -y ) ast t-D
weIl. Table I gives the typical sampie autocorrelations for the water
temperature series (y ) for (1) no differencing (y =y ) (2)differenced witht t t
respect to lunar month only (Vd=~ ;y =y ) (3)differenced with respect tot t t-l
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year only (VD=~ ;y =y ) (4)differenced with respect to lunar month andt t t-l3
year . (vd=lvD=ly ;y =y +y -y ). From Table I, we can see thet t t-l t-l3 t-l4
autocorrelations for Yt are large and fail to die out. With the simple
difference (Vd=l), the autocorrelations die out butthere remains a high
autocorrelation at seasonal lags. With the simple differencing with respect
~l .to seasonality (V), the autocorrelations are first persistently positive
and then persistently negative. By contrast the differencing with respect
d=l D=lto month and year (V V ) reduces the autocorrelations throughout. Since
this pattern was typical with all the series, they were all differenced
mcnthly as weIl as yearly to obtain stationarity.
Once the appropriate differencing has been decided, then we examine the
autocerrelations and partial-autocorrelations to indicate the possible
chocies of order for p,P,q,Q for autoregressive, seasonal autoregressive and
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mcving average, seasonal moving average operators. In general, for a
nonseasonal model, if the autocorrelations tail off to zero after lag p and
the partial-autocorrelations. truncate after lag p, then we have an
truncate after lag q and the partial-autocorrelations tail off after lag q,
autoregressive model of order p. Conversely, if' the autocorrelations
•then we have a moving average model of order q. If both autocorrelations
and partial-autocorrelations tail off, then we have a mixed process. The
autocorrelation and partial-autocorrelation structure for seasonal models is
ccmpletely analagous to that of ncnseasonal models except we only examine
lags S,25 and so forth. That is, if the autocorrelations appear to truncate
after lag s or 2s and the partial- autocorrelations at seasonal lags s,2s,
and 3s appear to tail off towards zero, then a seasonal moving average
operator of order 1 or 2 should be included in the model.
TABtE I
Autccorrelations for No Di fferencing. Monthly. Seasonaland both Monthly and Seasonal
Bige1ow, H. B. 1927. Physica1 Oceanography of the Gu1f of Maine. Bu11.U. S. Bur. Fish 40(2):511-1027.
Bige1ow, H. B. 1933. Studies of the Waters on the Continental Shelf,Cape Cod to Chesapeake Bay. I. The cyc1e of temperature. Pap. Phys.Oceanogr. and Met. 4(1). 135 pp.
Bige1ow, H. B., ·and M. Sears. 1935. Studies of the Waters on the Continental Shelf, Cape Cod to Chesapeake Bay. 11. Pap. Phys.Oceanogr. and Met. 94 pp.
Box, G. and Jenkins, G. 1976. Time Series Analysis: Forecasting andContro1, Ho1den-Day, San Francisco.
Colton, J. B., and R. R. Stoddard. 1972. Average Monthly Seawater Temperatures, Nova Scotia to Lang Is1and, 1940-1959. Amer. Geogr.Soc. Sero Atlas Mar. Envir. Folio 21. 2pp.
McCleary, R. and Hay, R. 1980. Applied Time Series Analysis for theSocial Science, Sage Publications, Beverly Hi1ls.
Nelson, C., 1973. Applied Time Series Analysis for Managerial Forecasting, Holden-Day, San Francisco.
O'Donovan, T. M. 1983. Short Term Forecasting, An Introduction to theBox-Jenkins Approach, John Wiley & Sons, New York.
Pawlowski, R. J. and Wright, W. R. 1978. Spring and Fall Sea-SurfaceTemperature and Salinity on the Northeastern Continental Shelf;Cape Hatteras to Cape·Sab1e, 1972-1978.
Walford, L. A. , and R. I. Wick1and. 1968. Monthly Sea TemperatureStructure from the Florida Keys to Cape Cod.Amer. Geogr. Soc.Sero Atlas Mar. Envir. Folio 15. 2pp.
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