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observe the hotel rentals in a winter resort, we find that the winter quarter index is 124. The value 124 indicates that 124 percent of the average quarterly rental occur in winter. If the hotel management records 1436 rentals for the whole of last year, then the average quarterly rental would be 359= (1436/4). As the winter-quarter index is 124, we estimate the number of winter rentals as follows:
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autocorrelation plot can help. If there is significant seasonality, the autocorrelation plot should show spikes at lags equal to the period. For example, for monthly data, if there is a seasonality effect, we would expect to see significant peaks at lag 12, 24, 36, and so on (although the intensity may decrease the further out we go).
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to the workforce upon the completion of their schooling. These regular changes are of less interest to those who study employment data than the variations that occur due to the underlying state of the economy; their focus is on how unemployment in the workforce has changed, despite the impact of the regular seasonal variations.
369:
seasonal differences (between group patterns) and also the within-group patterns. The box plot shows the seasonal difference (between group patterns) quite well, but it does not show within group patterns. However, for large data sets, the box plot is usually easier to read than the seasonal subseries plot.
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periods. This may require training, periodic maintenance, and so forth that can be organized in advance. Apart from these considerations, the organisations need to know if variation they have experienced has been more or less than the expected amount, beyond what the usual seasonal variations account for.
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An autocorrelation plot (ACF) can be used to identify seasonality, as it calculates the difference (residual amount) between a Y value and a lagged value of Y. The result gives some points where the two values are close together ( no seasonality ), but other points where there is a large discrepancy.
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Organisations facing seasonal variations, such as ice-cream vendors, are often interested in knowing their performance relative to the normal seasonal variation. Seasonal variations in the labour market can be attributed to the entrance of school leavers into the job market as they aim to contribute
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Another method of modelling periodic seasonality is the use of pairs of
Fourier terms. Similar to using the sinusoidal model, Fourier terms added into regression models utilize sine and cosine terms in order to simulate seasonality. However, the seasonality of such a regression would be represented
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Seasonal variation is measured in terms of an index, called a seasonal index. It is an average that can be used to compare an actual observation relative to what it would be if there were no seasonal variation. An index value is attached to each period of the time series within a year. This implies
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Seasonal fluctuations in a time series can be contrasted with cyclical patterns. The latter occur when the data exhibits rises and falls that are not of a fixed period. Such non-seasonal fluctuations are usually due to economic conditions and are often related to the "business cycle"; their period
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refers to the trends that occur at specific regular intervals less than a year, such as weekly, monthly, or quarterly. Seasonality may be caused by various factors, such as weather, vacation, and holidays and consists of periodic, repetitive, and generally regular and predictable patterns in the
435:
The measurement of seasonal variation by using the ratio-to-moving-average method provides an index to measure the degree of the seasonal variation in a time series. The index is based on a mean of 100, with the degree of seasonality measured by variations away from the base. For example, if we
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is the number of seasons (e.g., 4 in the case of meteorological seasons, 12 in the case of months, etc.). Each dummy variable is set to 1 if the data point is drawn from the dummy's specified season and 0 otherwise. Then the predicted value of the dependent variable for the reference season is
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into components designated with names such as "trend", "cyclic", "seasonal" and "irregular", including how these interact with each other. For example, such components might act additively or multiplicatively. Thus, if a seasonal component acts additively, the adjustment method has two stages:
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The run sequence plot is a recommended first step for analyzing any time series. Although seasonality can sometimes be indicated by this plot, seasonality is shown more clearly by the seasonal subseries plot or the box plot. The seasonal subseries plot does an excellent job of showing both the
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It is necessary for organisations to identify and measure seasonal variations within their market to help them plan for the future. This can prepare them for the temporary increases or decreases in labour requirements and inventory as demand for their product or service fluctuates over certain
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The seasonal plot, seasonal subseries plot, and the box plot all assume that the seasonal periods are known. In most cases, the analyst will in fact, know this. For example, for monthly data, the period is 12 since there are 12 months in a year. However, if the period is not known, the
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2089:. The resulting seasonally adjusted data are used, for example, when analyzing or reporting non-seasonal trends over durations rather longer than the seasonal period. An appropriate method for seasonal adjustment is chosen on the basis of a particular view taken of the
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If it is a multiplicative model, the magnitude of the seasonal fluctuations will vary with the level, which is more likely to occur with economic series. When taking seasonality into account, the seasonally adjusted multiplicative decomposition can be written as
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Now the total of seasonal averages is 398.85. Therefore, the corresponding correction factor would be 400/398.85 = 1.00288. Each seasonal average is multiplied by the correction factor 1.00288 to get the adjusted seasonal indices as shown in the above table.
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If the sum of these indices is not 1200 (or 400 for quarterly figures), multiply then by a correction factor = 1200 / (sum of monthly indices). Otherwise, the 12 monthly averages will be considered as seasonal
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that if monthly data are considered there are 12 separate seasonal indices, one for each month. The following methods use seasonal indices to measure seasonal variations of a time-series data.
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computed from the rest of the regression, while for any other season it is computed using the rest of the regression and by inserting the value 1 for the dummy variable for that season.
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as the sum of sine or cosine terms, instead of a single sine or cosine term in a sinusoidal model. Every periodic function can be approximated with the inclusion of
Fourier terms.
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A really good way to find periodicity, including seasonality, in any regular series of data is to remove any overall trend first and then to inspect time periodicity.
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whose period-lengths may be known or unknown depending on the context. A less completely regular cyclic variation might be dealt with by using a special form of an
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method. In this method, the original data values in the time-series are expressed as percentages of moving averages. The steps and the tabulations are given below.
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After establishing the seasonal pattern, methods can be implemented to eliminate it from the time-series to study the effect of other components such as
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values obtained in step(1). In other words, in a multiplicative time-series model, we get (Original data values) / (Trend values) × 100 = (
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Arrange these percentages according to months or quarter of given years. Find the averages over all months or quarters of the given years.
2057:{\displaystyle Y_{i}=a+bt+(\sum _{k=1}^{K}\alpha _{k}\cdot \sin({\tfrac {2\pi kt}{m}})+\beta _{k}\cdot \cos({\tfrac {2\pi kt}{m}}))+E_{i}}
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To use the past patterns of the seasonal variations to contribute to forecasting and the prediction of the future trends, such as in
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The description of the seasonal effect provides a better understanding of the impact this component has upon a particular series.
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Here, 359 is the average quarterly rental. 124 is the winter-quarter index. 445 the seasonalized winter-quarter rental.
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estimate the seasonal component of variation in the time series, usually in a form that has a zero mean across series;
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1420:{\displaystyle ={\frac {T\cdot S\cdot C\cdot I}{T\cdot C\cdot I}}\times 100\ ={\frac {Y}{T\cdot C\cdot I}}\times 100}
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1675:{\displaystyle {\frac {Y}{S}}\times 100={\frac {T\cdot S\cdot C\cdot I}{S}}\times 100=(T\cdot C\cdot I)\times 100}
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2. In a multiplicative time-series model, the seasonal component is expressed in terms of ratio and percentage as
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subtract the estimated seasonal component from the original time series, leaving the seasonally adjusted series:
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Now calculations for 4 quarterly moving averages and ratio-to-moving-averages are shown in the below table.
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The difference between a sinusoidal model and a regression with
Fourier terms can be simplified as below:
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The multiplicative model can be transformed into an additive model by taking the log of the time series;
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and irregular variations. This elimination of the seasonal effect is referred to as de-seasonalizing or
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model which can be structured so as to treat cyclic variations semi-explicitly. Such models represent
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Let us calculate the seasonal index by the ratio-to-moving-average method from the following data:
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Express each original data value of the time-series as a percentage of the corresponding centered
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Find the centered 12 monthly (or 4 quarterly) moving averages of the original data values in the
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This implies that the ratio-to-moving average represents the seasonal and irregular components.
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usually extends beyond a single year, and the fluctuations are usually of at least two years.
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can be used as an alternative to the seasonal subseries plot to detect seasonality
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It is important to distinguish seasonal patterns from related patterns. While a
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A completely regular cyclic variation in a time series might be dealt with in
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1. In an additive time-series model, the seasonal component is estimated as:
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However, in practice the detrending of time-series is done to arrive at
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or the time of the year, such as annual, semiannual, quarterly, etc. A
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1867:{\displaystyle Y_{i}=a+bt+\alpha \sin(2\pi \omega T_{i}+\phi )+E_{i}}
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One particular implementation of seasonal adjustment is provided by
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An ACF (autocorrelation) plot, of
Australia beer consumption data.
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Variations in data at specific regular intervals less than a year
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There are several main reasons for studying seasonal variation:
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A seasonal plot will show the data from each season overlapped
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3. The deseasonalized time-series data will have only trend (
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Taking log of the time series of the multiplicative model:
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These points indicate a level of seasonality in the data.
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Periodicity and
Stochastic Trends in Economic Time Series
361:(ACF) and a spectral plot can help identify seasonality.
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is any method for removing the seasonal component of a
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Semiregular cyclic variations might be dealt with by
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163:. Unsourced material may be challenged and removed.
2778:at NIST/SEMATECH e-Handbook of Statistical Methods
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2702:The Econometric Analysis of Seasonal Time Series
2795:NIST/SEMATECH e-Handbook of Statistical Methods
2738:Hyndman, Rob J.; Athansopoulos, George (2021).
2800:National Institute of Standards and Technology
1552:{\displaystyle {\frac {Y}{T}}=S\cdot C\cdot I}
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64:Learn how and when to remove these messages
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2497:is a more general, irregular periodicity.
445:This method is also called the percentage
335:A seasonality plot of US electricity usage
2704:. Cambridge: Cambridge University Press.
2700:Ghysels, Eric; Osborn, Denise R. (2001).
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2626:"2 Tips to Maximize Profits in Business"
1503:{\displaystyle Y=T\cdot S\cdot C\cdot I}
84:This article includes a list of general
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2305:{\displaystyle Y_{t}=S_{t}*T_{t}*E_{t}}
2233:{\displaystyle Y_{t}/S_{t}=T_{t}*E_{t}}
2160:{\displaystyle Y_{t}-S_{t}=T_{t}+E_{t}}
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658:Ratio-to-Moving-Average(%)(Y)/ (T)*100
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2685:. New York: Oxford University Press.
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2740:Forecasting: practice and principles
161:adding citations to reliable sources
2611:6.1 Time series components - OTexts
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655:2 Figures Moving Average(T)
2723:. Orlando: Academic Press.
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390:spectral density estimation
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2721:Seasonality in Regression
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262:levels of a time series.
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652:2 Figures Moving Total
646:4 Figures Moving Total
345:seasonal subseries plot
105:more precise citations.
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2426:In regression analysis
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2748:
2735:
2729:
2716:
2710:
2697:
2691:
2676:
2673:
2670:
2669:
2651:
2635:
2617:
2592:
2553:
2537:
2536:
2534:
2531:
2530:
2529:
2527:Photoperiodism
2524:
2519:
2514:
2509:
2502:
2499:
2479:, or simply a
2477:cyclic pattern
2464:
2461:
2427:
2424:
2403:
2399:
2395:
2392:
2389:
2386:
2381:
2377:
2373:
2370:
2367:
2364:
2359:
2355:
2351:
2348:
2345:
2342:
2337:
2333:
2329:
2326:
2323:
2299:
2295:
2291:
2286:
2282:
2278:
2273:
2269:
2265:
2260:
2256:
2227:
2223:
2219:
2214:
2210:
2206:
2201:
2197:
2192:
2186:
2182:
2169:
2168:
2154:
2150:
2146:
2141:
2137:
2133:
2128:
2124:
2120:
2115:
2111:
2099:
2072:Main article:
2069:
2066:
2065:
2064:
2051:
2047:
2043:
2040:
2037:
2031:
2027:
2024:
2021:
2018:
2011:
2008:
2005:
2002:
1997:
1993:
1989:
1986:
1980:
1976:
1973:
1970:
1967:
1960:
1957:
1954:
1951:
1946:
1942:
1936:
1931:
1928:
1925:
1921:
1917:
1914:
1911:
1908:
1905:
1902:
1899:
1894:
1890:
1875:
1874:
1861:
1857:
1853:
1850:
1847:
1844:
1839:
1835:
1831:
1828:
1825:
1822:
1819:
1816:
1813:
1810:
1807:
1804:
1801:
1798:
1795:
1790:
1786:
1740:
1737:
1736:
1735:
1734:
1733:
1691:Additive model
1685:
1684:
1683:
1682:
1671:
1668:
1665:
1662:
1659:
1656:
1653:
1650:
1647:
1644:
1641:
1638:
1633:
1629:
1626:
1623:
1620:
1617:
1614:
1611:
1605:
1602:
1599:
1594:
1591:
1548:
1545:
1542:
1539:
1536:
1533:
1528:
1525:
1499:
1496:
1493:
1490:
1487:
1484:
1481:
1478:
1475:
1452:
1449:
1446:
1443:
1440:
1429:
1428:
1416:
1413:
1407:
1404:
1401:
1398:
1395:
1391:
1386:
1380:
1377:
1371:
1368:
1365:
1362:
1359:
1354:
1351:
1348:
1345:
1342:
1339:
1336:
1330:
1313:
1312:
1306:
1304:
1298:
1296:
1286:
1284:
1278:
1276:
1266:
1265:
1238:
1235:
1228:
1227:
1224:
1221:
1218:
1215:
1212:
1208:
1207:
1204:
1201:
1198:
1195:
1192:
1188:
1187:
1184:
1181:
1178:
1175:
1171:
1170:
1169: —
1167:
1166: —
1164:
1161:
1158:
1154:
1153:
1150:
1147:
1144:
1141:
1137:
1136:
1133:
1130:
1127:
1124:
1120:
1119:
1117:
1114:
1111:
1110: —
1108:
1107: —
1105:
1101:
1100:
1097:
1094:
1091:
1088:
1085:
1074:
1073:
1071:
1068:
1067:
1066: —
1064:
1061:
1058:
1055:
1051:
1050:
1047:
1043:
1042:
1041: —
1039:
1036:
1033:
1030:
1026:
1025:
1022:
1018:
1017:
1014:
1011:
1008:
1005:
1001:
1000:
997:
993:
992:
989:
986:
983:
980:
977:
973:
972:
969:
965:
964:
961:
958:
955:
952:
948:
947:
944:
940:
939:
936:
933:
930:
927:
923:
922:
919:
915:
914:
911:
908:
905:
902:
898:
897:
894:
890:
889:
886:
883:
880:
877:
874:
870:
869:
866:
862:
861:
858:
855:
852:
849:
845:
844:
841:
837:
836:
833:
830:
827:
824:
820:
819:
816:
812:
811:
808:
805:
802:
799:
795:
794:
791:
787:
786:
783:
780:
777:
774:
771:
767:
766:
763:
759:
758:
755:
752:
749:
746:
742:
741:
738:
734:
733:
730:
727:
724:
721:
717:
716:
713:
709:
708:
707: —
705:
702:
699:
696:
692:
691:
688:
684:
683:
682: —
680:
677:
674:
672:
670:
667:
664:
660:
659:
656:
653:
650:
647:
644:
641:
638:
624:
623:
620:
617:
614:
611:
607:
606:
603:
600:
597:
594:
590:
589:
586:
583:
580:
577:
573:
572:
569:
566:
563:
560:
556:
555:
552:
549:
546:
543:
542:Year/Quarters
530:
527:
526:
525:
521:
520:
519:
514:
513:
512:
510:
505:) × 100.
471:moving average
467:
466:
465:
454:
451:
447:moving average
432:
429:
428:
427:
426:
425:
422:
419:moving-average
415:
408:
397:
394:
363:
362:
355:
348:
341:
338:
317:The following
314:
311:
310:
309:
308:
307:
300:
289:
279:
276:
249:
248:
231:
230:
145:
143:
136:
129:
128:
83:
81:
74:
69:
43:
42:
40:
33:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2897:
2886:
2883:
2881:
2878:
2877:
2875:
2860:
2857:
2855:
2852:
2850:
2847:
2846:
2843:
2838:
2831:
2826:
2824:
2819:
2817:
2812:
2811:
2808:
2804:
2801:
2797:
2796:
2790:
2777:
2774:
2771:
2766:
2762:
2761:
2757:
2751:
2745:
2741:
2736:
2732:
2730:0-12-363455-5
2726:
2722:
2717:
2713:
2711:0-521-56588-X
2707:
2703:
2698:
2694:
2692:0-19-877454-0
2688:
2684:
2679:
2678:
2674:
2665:
2661:
2655:
2652:
2647:
2646:
2639:
2636:
2631:
2627:
2621:
2618:
2613:
2612:
2605:
2603:
2601:
2599:
2597:
2593:
2588:
2582:
2568:on 2015-05-18
2567:
2563:
2557:
2554:
2548:
2547:"Seasonality"
2542:
2539:
2532:
2528:
2525:
2523:
2520:
2518:
2515:
2513:
2510:
2508:
2505:
2504:
2500:
2498:
2496:
2495:
2490:
2486:
2482:
2478:
2474:
2470:
2462:
2460:
2457:
2453:
2449:
2445:
2441:
2437:
2433:
2425:
2423:
2421:
2416:
2401:
2397:
2393:
2390:
2387:
2384:
2379:
2375:
2371:
2368:
2365:
2362:
2357:
2353:
2349:
2346:
2343:
2340:
2335:
2331:
2327:
2324:
2321:
2312:
2297:
2293:
2289:
2284:
2280:
2276:
2271:
2267:
2263:
2258:
2254:
2244:
2241:
2225:
2221:
2217:
2212:
2208:
2204:
2199:
2195:
2190:
2184:
2180:
2152:
2148:
2144:
2139:
2135:
2131:
2126:
2122:
2118:
2113:
2109:
2100:
2097:
2096:
2095:
2092:
2088:
2084:
2080:
2075:
2067:
2049:
2045:
2041:
2029:
2025:
2022:
2019:
2016:
2006:
2003:
2000:
1995:
1991:
1987:
1978:
1974:
1971:
1968:
1965:
1955:
1952:
1949:
1944:
1940:
1934:
1929:
1926:
1923:
1919:
1912:
1909:
1906:
1903:
1900:
1897:
1892:
1888:
1880:
1879:
1878:
1859:
1855:
1851:
1845:
1842:
1837:
1833:
1829:
1826:
1823:
1817:
1814:
1811:
1808:
1805:
1802:
1799:
1796:
1793:
1788:
1784:
1776:
1775:
1774:
1771:
1768:
1764:
1762:
1758:
1754:
1750:
1746:
1738:
1692:
1689:
1688:
1687:
1686:
1669:
1666:
1660:
1657:
1654:
1651:
1648:
1642:
1639:
1636:
1631:
1627:
1624:
1621:
1618:
1615:
1612:
1609:
1603:
1600:
1597:
1592:
1589:
1579:
1578:
1577:
1576:
1575:
1566:), cyclical (
1560:
1546:
1543:
1540:
1537:
1534:
1531:
1526:
1523:
1497:
1494:
1491:
1488:
1485:
1482:
1479:
1476:
1473:
1464:
1450:
1447:
1444:
1441:
1438:
1414:
1411:
1405:
1402:
1399:
1396:
1393:
1389:
1384:
1378:
1375:
1369:
1366:
1363:
1360:
1357:
1352:
1349:
1346:
1343:
1340:
1337:
1334:
1328:
1321:
1318:
1317:
1316:
1307:
1305:
1299:
1297:
1294:
1287:
1285:
1279:
1277:
1271:
1270:
1269:
1244:
1243:
1242:
1236:
1234:
1225:
1222:
1219:
1216:
1213:
1210:
1209:
1205:
1202:
1199:
1196:
1193:
1190:
1189:
1185:
1182:
1179:
1176:
1173:
1172:
1168:
1165:
1162:
1159:
1156:
1155:
1151:
1148:
1145:
1142:
1139:
1138:
1134:
1131:
1128:
1125:
1122:
1121:
1115:
1112:
1109:
1106:
1103:
1102:
1098:
1095:
1092:
1089:
1086:
1083:
1082:
1072:
1070:
1069:
1052:
1044:
1027:
1019:
1002:
994:
974:
966:
949:
941:
924:
916:
899:
891:
871:
863:
846:
838:
821:
813:
796:
788:
768:
760:
743:
735:
718:
710:
693:
685:
673:
671:
661:
657:
654:
651:
648:
645:
642:
639:
636:
635:
629:
621:
618:
615:
612:
609:
608:
604:
601:
598:
595:
592:
591:
587:
584:
581:
578:
575:
574:
570:
567:
564:
561:
558:
557:
553:
550:
547:
544:
541:
540:
534:
528:
522:
518:
517:
515:
511:
509:
508:
472:
468:
464:
463:
461:
457:
456:
452:
450:
448:
443:
440:
437:
430:
423:
420:
416:
413:
409:
406:
405:
404:
403:
402:
395:
393:
391:
382:
378:
374:
370:
366:
360:
356:
353:
349:
346:
342:
339:
333:
328:
324:
323:
322:
320:
312:
305:
301:
298:
294:
290:
287:
286:
285:
284:
283:
277:
275:
271:
267:
263:
260:
256:
245:
242:
227:
224:
216:
213:November 2010
205:
202:
198:
195:
191:
188:
184:
181:
177:
174: –
173:
172:"Seasonality"
169:
168:Find sources:
162:
158:
152:
151:
146:This article
144:
140:
135:
134:
125:
122:
114:
111:November 2008
104:
100:
94:
93:
87:
82:
73:
72:
67:
65:
58:
57:
52:
51:
46:
41:
32:
31:
19:
2858:
2794:
2781:
2739:
2720:
2701:
2682:
2663:
2654:
2644:
2638:
2629:
2620:
2610:
2570:. Retrieved
2566:the original
2556:
2541:
2492:
2480:
2476:
2468:
2466:
2455:
2447:
2429:
2417:
2313:
2245:
2242:
2170:
2082:
2078:
2077:
1876:
1772:
1769:
1765:
1742:
1561:
1465:
1430:
1319:
1314:
1267:
1240:
1231:
1220: 84.69
1217: 92.43
1200: 84.45
1197: 92.16
1163: 92.04
1149: 83.02
1146: 92.75
1132: 85.13
1129: 91.71
1116: 90.25
1113: 85.21
1016: 92.03
938: 83.02
913: 92.75
835: 85.13
810: 91.71
757: 90.25
732: 85.21
627:
537:Sample Data
532:
444:
441:
438:
434:
399:
387:
375:
371:
367:
364:
316:
281:
272:
268:
264:
258:
252:
237:
219:
210:
200:
193:
186:
179:
167:
155:Please help
150:verification
147:
117:
108:
89:
61:
54:
48:
47:Please help
44:
2885:Seasonality
2776:Seasonality
2770:Seasonality
2512:Oscillation
2087:time series
1747:by using a
460:time-series
396:Calculation
259:seasonality
255:time series
103:introducing
2874:Categories
2572:2015-05-13
2533:References
2420:X-12-ARIMA
278:Motivation
183:newspapers
86:references
50:improve it
2880:Inventory
2837:Inventory
2290:∗
2277:∗
2218:∗
2119:−
2020:π
2007:
2001:⋅
1992:β
1969:π
1956:
1950:⋅
1941:α
1920:∑
1846:ϕ
1830:ω
1827:π
1818:
1812:α
1753:sinusoids
1667:×
1658:⋅
1652:⋅
1637:×
1625:⋅
1619:⋅
1613:⋅
1598:×
1544:⋅
1538:⋅
1514:so that
1495:⋅
1489:⋅
1483:⋅
1448:⋅
1442:⋅
1412:×
1403:⋅
1397:⋅
1376:×
1367:⋅
1361:⋅
1350:⋅
1344:⋅
1338:⋅
1291: :
417:Ratio-to-
410:Ratio to
352:box plots
350:Multiple
313:Detection
56:talk page
2630:netsuite
2581:cite web
2501:See also
2434:such as
1739:Modeling
1063:—
1060:—
1049:—
1046:—
1038:—
1035:—
704:—
701:—
690:—
687:—
679:—
676:—
640:Quarter
524:indices.
299:of data.
293:cyclical
2485:decadal
1223:100.52
1214:122.36
1206:398.85
1203:100.23
1194:122.01
1186:300.68
1183:253.36
1180:276.49
1177:366.05
1160:120.48
1152:104.29
1143:117.45
1135:106.14
1126:128.12
1010:169.50
991:120.48
985:166.00
963:104.29
957:163.00
932:159.00
910:77.625
907:155.25
888:117.45
885:76.625
882:153.25
860:106.14
857:75.375
854:150.75
829:148.00
807:70.875
804:141.75
785:128.12
782:67.125
779:134.25
754:65.375
751:130.75
729:63.375
726:126.75
501:×
493:×
485:×
481:×
477:×
197:scholar
99:improve
2746:
2727:
2708:
2689:
2489:weekly
2473:season
1382:
1295:values
1268:where
1174:Total
1099:Total
1024:85.75
1013:84.75
999:83.75
988:83.00
971:82.25
960:81.50
946:80.75
935:79.50
921:78.25
896:77.00
868:76.25
843:74.50
832:74.00
818:73.50
793:68.25
765:66.00
740:64.75
715:62.00
421:method
414:method
257:data,
199:
192:
185:
178:
170:
88:, but
2839:types
2791:from
2481:cycle
1757:ARIMA
1293:Trend
1157:1999
1140:1998
1123:1997
1104:1996
976:1999
873:1998
770:1997
663:1996
637:Year
610:1999
593:1998
576:1997
559:1996
489:) / (
412:trend
204:JSTOR
190:books
2744:ISBN
2725:ISBN
2706:ISBN
2687:ISBN
2587:link
1717:) –
1226:400
1021:343
996:335
982:100
968:329
943:323
918:313
893:308
865:305
840:298
815:294
790:273
762:264
737:259
712:248
613:100
176:news
2450:-1
2430:In
2081:or
2004:cos
1953:sin
1815:sin
1701:= (
1670:100
1640:100
1601:100
1415:100
1379:100
1252:– (
1057:93
1032:72
1007:78
954:85
929:66
904:72
879:90
851:80
826:63
801:65
776:86
748:59
723:54
698:60
669:75
622:93
619:72
616:78
605:85
602:66
599:72
596:90
588:80
585:63
582:65
579:86
571:59
568:54
565:60
562:75
357:An
253:In
159:by
2876::
2798:.
2662:.
2628:.
2595:^
2583:}}
2579:{{
2422:.
1763:.
1729:+
1725:+
1721:=
1713:+
1709:+
1705:+
1697:–
1693::
1559:.
1463:.
1260:+
1256:+
1248:=
1096:4
1093:3
1090:2
1087:1
1054:4
1029:3
1004:2
979:1
951:4
926:3
901:2
876:1
848:4
823:3
798:2
773:1
745:4
720:3
695:2
666:1
554:4
551:3
548:2
545:1
462:.
392:.
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325:A
59:.
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2695:.
2666:.
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2632:.
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2575:.
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2456:n
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2363:+
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2341:=
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2328:g
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2264:=
2259:t
2255:Y
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2222:E
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2209:T
2205:=
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2196:S
2191:/
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2167:.
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2132:=
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2039:)
2036:)
2030:m
2026:t
2023:k
2017:2
2010:(
1996:k
1988:+
1985:)
1979:m
1975:t
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1959:(
1945:k
1935:K
1930:1
1927:=
1924:k
1916:(
1913:+
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1907:b
1904:+
1901:a
1898:=
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1860:i
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1843:+
1838:i
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1824:2
1821:(
1809:+
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1800:+
1797:a
1794:=
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1715:I
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1643:=
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1628:I
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1610:T
1604:=
1593:S
1590:Y
1572:I
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1547:I
1541:C
1535:S
1532:=
1527:T
1524:Y
1512:T
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1486:S
1480:T
1477:=
1474:Y
1451:I
1445:C
1439:S
1427:;
1406:I
1400:C
1394:T
1390:Y
1385:=
1370:I
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1329:=
1309:I
1301:C
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1273:S
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1262:I
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491:T
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306:.
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