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METHODS FOR COMPUTING SEASONALS USED IN THE ANALYSIS OF TIME SERIES 1 rhoaiori Submitted t6 t5‘tiiaulty ; or ' Michigan State College of Agriculturo and Applied Science In Pgrtial Fulfillment of tho anuironontl for tho Dogroo of Easter at Arts by Barth: Alberta Larocn 1 9 5 1 Amowmmmn 10 Hr. S. E. Grove, uh: has so faithtnlly helped me with my work 1( 21% :3 OHLPIER I Introduction With the analysis of time series in business and economic statistics comes the problem of the de- terlinstien and elimination.et the seasonal factor. As is already known. there are in all four factors which enter into the analysis of any time series. these factors are secular trend ( the long time tend- ency). cyclical variation (the wave-like movement su- perimposed on secular trend). seasonal variation ( a variation within the year due to seasonal influences) and residual variation (due to torces unforeseen). The problem here is the analysis. comparison and application of the various methods used in the measurement of seasonal variation. The important characteristics of a seasonal variation are that it is a periodic change with a period of one year and that each year it exactly repeats itself. In the analysis of such a variation. is there any way of de- termining the reliability of the different methods and. if so, which is the most reliable? In answer to this question a hypothetieal set of data has been constructed such that it the method of monthly means is used the true seasonals are known. It is the pur- pose here to show how close the other methods will come to the true results. Alas. these same methods will be applied to the analysis of motor has and truck production for a period of seven years. But before going into the study of the main problem. a review of the various methods for the de- termination of seasonal variation will be givan. CHAPEER II lhthods Used in Determining Seasonals A short description in outline form of the meth- ods used in the determination of seasonal variation will be given in this chapter. the methods are as given below: I. II. lethcd of monthly means 1. 8. 3. Compute an arithmetic mean for each month and express they; averages in terms of percent. Correct averages for secular trend. Change the corrected averages so their average will equal 100 percent. Link relative method 1. 8. 4. 5. Express each monthly figure as a percent of the figure for the previous month. These are called the link relatives. Determine the median link relative for each month. compute the chained relatives using January as a constant base. Correct the chain relatives for error. Adjust these corrected chain relatives so their average will equal 100 percent. 111. method of moving averages 1. 2. 3. 4. Compute the moving averages. Determine the moving average ratios by di- viding the actual items by the correspond- ing moving averages. Compute a suitable average ratio for each month. Adjust the average ratios so that their av- erage will equal 100 percent. IV. Ratio-to-trend 1. 2. 3. 4. Fit a suitable line of trend to the yearly averages to determine the annual increment and trend values. Express the actual item as a percent of the trend value. These are called the trend re- tics. Determine a suitable monthly average of these ratios. Adjust so the average equals 100 percent. V. Method of first differences (using trend ratios) 1. 8. Ratios of the original data to trend ordi- nates computed. First differences of the ratios next deter- mined. Cempute a suitable average of these differ- ences for each month. VI. VII. 4. Adjust so the sum of the first differ- ence averages is equal to zero. 5. With January as a base. compute the chained first differences. 6. Adjust so that the average of the chained first differences is equal to 100 percent. Method of first differences (using moving average ratios) 1. Compute ratios by dividing the original data by the corresponding moving aver- ages. 2. Compute the first differences of these ratios. 3. Get a suitable average of these first differences for each month 4. Adjust so the sum of the first differ- ence averages is equal to zero. 5. With January as base, compute the chained first differences. 6. Adjust so the average is equal to 100 percent. Thirteen-months-ratio-first-difference method 1. Compute a monthly mean for each year. 2. Obtain percent ratios by dividing ac- 3. 4. 6. 6. 7. VIII. tuals of each year by the monthly av- erage for that year. Also include the ratio of the following January . al- though the following January is not used in the determination of the monthly av- erage for the year. Leaving the January values as they are. compute the first differences of the a- bove ratios from February through the following January. Select an average for each of the thir- teen months. Cumulate the first difference averages to the January average. The annual trend increment is found by subtracting the two January values. Correct for trend making the two Janu- ary values equal. 1 Adjust the twelve monthly values so ob- tained so as to make the average equal to 100 percent. Detroit Edison method I A more detailed discussion of the develop- ment and results will be given in regard to this method and the one following (which is a modifi- cation of the Detroit Edison method). Any time series is made up of the following four factors: secular trend f (x) cyclical variation c (x) seasonal variation s (x) and residual errors 3x Let o’x . flx) s(x) s(x) e 'x where o’x repre- sents the xth term of the time series. The standard error will be equal to .1 . If the standard er- n ror is to be a minimum then the e8 must be a minimmm value. values for s(l). s(z), ........ s(12) can be found that will minimise the standard error by taking the partial derivative of e2 with respect to s(l). -s(2).--.-.-s(12) and putting this partial derivative equal to sero. The result is that 1 s(i) 8 g§§£gx . ff!) . c(x) i192)" . 02m . 1 1: Wm = rm - .(x). then an) -Zo¥x _ ‘29»: Now if !1_3. 21-2 . °""Ti+z . $1.3 represents the total production for seven years and if a sixth degree parobola is fitted in such a way that the areas under the curve for the seven equidistant points is equal to 11-5 . ’1-8 . ...... Ti ‘ 3 , we have the following result: gun" = °1:1’1 * ‘32:: T2 ‘ “an”: ‘ °4:1 (,4 O T5‘soeeee‘Tn-3) § 05:1 Tn-2 $ 06:1 tn-l I 07:1 1'11 ° This gives a formula for the determination of ;%{x so that the seasonal factor can easily be deter- mined. The values of the coefficients in the above equation have been worked out and put intable form The table of values will not be given here but can be found by referring to the list of references. IX. A second Detroit Edison method This method differs from the above in that a third degree curve is us sd’ instead of one of the sixth degree. The theory underlying this method will. therefore, be the same as the above. The two methods will differ. though, when it comes to determining a formula for the2%¢1x). In this case it is dens by us- ing a curve of the type y‘= a e bx e cxz . dx8 . £::::§% Figure 1 is an accumu- fir t lation curve represent- L ét ' ing the total production up to each successive 1'" x' Figure I .y ’ year. To : ’o-’-l 8 total production for the first year. '1 - yl-yo 8 total production for the second year. a... If the values of x are substituted in the equa- tion y a a + bx 8 ex; e dx5 we obtain when x I -1, y-1 a a - b e o - d x = O . ’0 a t x = 1 . y: : a e b . o . d I 3 2 . y: 3 C 6 2b 4 4C 4 86 These equations are solved for a. b. c. and d. The results are: a-yo o : I1.!0 ’2 - as! + r! d 3 . 5 6 v.’ As it is necessary to express the results in terms of monthly production. let Ii: January produc- tion.ll2 = February production. etc. Then: S: to N H to “J N 3 18 1;! 125 .tOe By differencing the results «m can r ._ 4 . a... 13 CHAPTER III An Application of These Methods to a Hypothetical Set of Data A hypothetical set of data has been worked out by W. L. Hart. of the University of Minnesota. such that if the method of monthly means is used the exact seasonals will be obtained. Suppose we have a series of monthly items from which the secular trend has been removed and the items are for a period of K.years. If’f(t) represents the item t months from January of the first year and if an arithmetic average of the monthly items is taken. then the results obtained by averaging is the best approx. imation to flt). Now the question arises as to how to determine a function P(t). which is periodic. that will be the best approximation to flt). The two theorems now given will be an answer. ‘ Theorem 1: If f(t) actually is a periodic func- tion whose period is one year. the monthly entries obtained by the method of monthly means are exactly the value of:f(t) at the corresponding months. If P(t) represents the periodic function with the period one year. whose value for all the Januarym. Februarys.eto.. are the corresponding monthly means it we have: Theorem 2: Let 1(t) be any function of time to known from t = O to t 8 12 X. that is. over a period of K years. Then, the sum of the squares of the re- sidualslfft)-P(t) ) for all values of t is smaller in value than it would be if any other periodic func- tion with a period of one year were used in place of P(t). Another theorem.will be stated because it gives the proper criterion of applicability of the above theory. Theorem 3: The method of monthly means gives us the actual monthly values of the seasonal variation in case f(t) is made up of the following component parts: l.- A seasonal variation. strictly periodic throughout the period of years under consideration. B -.L long term variation which consists of cer- tain independent pieces. each extending over a whole number of years. where each p1ece represents a whole number of complete oscillations of a corresponding periodic function whose period is an integral number of years (two or more). c -.A second. third. etc.. long term.variation having the characteristics specified in D. In order to obtain a hypothetical set of data so 10 £2.29. 22 12: 6d '133 (constant) 53 , 12a 122 123 If the differencing process is carried out in more detail than is given here. the second difference will remain a constant value "gig . 7 Expressing the Mfs in terms of T and using the fact that the second difference is constant the re- sults as given below are obtained: ‘1 . as: so. 754 1-1 - 145 ’2 I! 2 187 To 0814 '21 ~ 137 1'8 (I: 1 137 loo 862 Tl - 125 E: El 3 7! Tot 898 21 - 107 2 ‘ etc. where 2 p2n:= 610-1215. 622 ' Summing all the January values gives: Jan.:Q/= Eta—(see 10.1001 :1. sung. ”nun- 1) . 611 gm ~14! Tn.1). Summing all the February values gives: - l reb.j£a -';:‘;;§ (131 so. 1001 11.86elsgo....tn.1) ' 677 Tn * 137 gn*1). etc. 11 Now if the two substitutions T°=8T1 - Ti and Th‘lz 8Tn n-l be made so as to out off the first and last terms we will have a set of formulas which will readily enable one to solve for:§j1: These re- sults are: an: - .. Wmtlfilfil‘l’éll?z‘a5“’3‘ “In 2); 100nm1 {azBTn). Fail/#1575 11.577 12*854‘19-ohmghloonn 1 . . cos Tn). .izzuuz 5.731 28.85“!” «induces : margiwta n-l 9‘81 In) e Apr.§W1117 £10791 12.864(!3....o!n_z).971 2,1,1 ‘57? an). ~ ._ 1 HufW-WW97 11 0839 r24864(!3‘.. .firn-z).“7 111-1 0675 In)e m-EQB-é-y-e ”(883 1'1. 881 r,.eee(13+...rn_8)te1v ”21-1 e775 Tn)- 1 unit-mun 11. 91v re.eeetrz.,,,.,n_2).eei 13-1 40883 Tn)e 1 . Aug.i‘f=-6-.-iE-5-(-673 1-1. 947 22.eee(ez.. .iTn-z)0839 ’n-1 9997 Tn), Swt-éwifi'" 21.971 ereeetrzamrmahnl 1.1-1 .1117 an). 4 O 12 Ootfq. mama? 51. see 92.864tr34...ofn-z).vsvrn-1 .1245 an ). Noni (kg—7, 2(405 21.1001 saweuraxur n_2)+677!n_1 91376 In)e . , .. 611 Dec Exp 5'12 (325 1111007 T2e864(T3o “11-2” ’n-l .1513 Tn). By using the first three of these equations for thezi‘gflx) and differencing. formulas for D'q’ and D"? are obtained: 1 D'1W=-i§3'(-158 T1466 T2'5 Tn 1 t 73 Tn) ' c- - “'12 I D 21)) - W‘ 132 11.50 1'2 n”1. 34 an) 1 3 x . - p I)? 125 (T1 122 «and . Tn) These formulas for D'Q’and D'q/will simplify the computation work considerably because of the fact that the second difference is a constant. The only value foréiwx'x) which needs to be directly computed from the equations is the value of Jan. *Wx). 15 that the above conditions would hold. the following equations were used: f(t) : 15 . sin t (50°) . 4 sin t (10°) from t - 0 to t a 55. £(t) = 15 . sin t (50°) . 5 sin t (10°) frOm t '“t0t372e r(t) a 15 . sin t (50°) ..2 sin t (10°) from t 72 to t a 108. in the equations, 15 . sin t(50°) gives the seasonal variation and the long term variations are given by 4 sin t (10°). 6 sin t (10°). and 2 sin t (10°). From these equations the data of Table 1’3216 computed. 15 .) amm.aofl «ma.ama mmm.oo~ www.mmfi mma.mma oem.wmm mau.nea «oe.nma vom.mam oom.ona mma.ea www.ma omm.ma one.ma eom.m oea.om vom.nH one.HH oam.ma a oom.amfl ome.na oem.ma eoH.mH mmo.mH oam.oa eeo.om aaa.ma mom.aa eao.oH a ooo.oma ooo.na ooo.na ooo.oH ooo.aa ooo.aa ooo.om ooo.mH ooo.uH ooo.ma o mom.amH mvm.mH onv.na eoH.oH maa.on mmo.a~ eao.om wem.HH ooa.ma «so.ma m oom.onfi moo.mm moa.ea omn.oa «no.0 om..na oea.oa on..fla eom.nfl oem.ma s ooo.oma mom.na ooo.ma mna.aa «om.a ooo.oa eaa.ou onm.HH ooo.mm, eme.ma n oom.am~ omo.mn meo.mn ano.e~ oom.o(, «em.oa omo.om oee.aa homa.aa vam.ma a «ma.mem mam.na oma.oa mm~.afl moa.m mno.aa ema.oa oua.afl vmm.aa www.ma : ooo..ea ooo.vH ooa.>n ooo.aa ooo.oa ooo.an coo.mH coo.ma coo.ma ooo.ma 4 voa.me~ oom.ma mma.aa onm.oH 555.9 «ma.an maa.aa oum.aa www.ma emm.aa .m ocm.ana omo.nH «ne.a~ www.ma oom.a (.ooo.om «em.mm oea.aa vom.mH omH.eH -e ooo.mna(rmmm.na una.oa ooo.ma vom.a oma.om ooo.ma onm.ma eoe.ma ooo.ma n o» I). a m a e m e u a H 7 Han 17 For comparison purposes this set of data was used to test out the accuracy of the methods for com— puting seasonal variation and the results are given in Table II. OimIFGN Table II. Actual Indices as Worked Out by Method of Monthly Means Link Relative.uethod Detroit Edison.nethod Second Detroit Edison lethcd First Difference Method Thirteen-Months-Ratio-First-Difference Method method of Moving Averages (a twelve-month moving average. centered. adjusted by a two- month moving average. centered. was used) 18 )aa.oa mm.om so.am ma.mm m~.ma no.am am.em n as.vm an.em om.vm (mo.mo ae.mm(, em.mo om.eo a va.nm «o.na mm.mm Ho.vm aw.vm no.5o 95.nmtl o mm.va .ms.mo o~.ea_ 5H.uo mauve am.oa om.ea a oa.oa oo.om aa.oa mm.em mo.ea ma.ao aonmw 4 mo.mom enema oo.ooa mm.o0H ea.ooa eo.ooa oo.ooa a Hm.moa em.mofl an.moa 5H.noa ma.noa om.HoH an.moa a an.moa 55.nom, om.mofl mw.noa an.moa ee.ooa om.moa U: om.oo~ amused am.ooa 05.noa om.moa so.«om po.aoa e Hm.moa om.ooa om.moa an.eoa ma.eoa as.moa om.moa ”a as.noa ma.eofi an.nom, mo.fioa mo.smm, om.Hoa an.no~ a am.ooa em.floa oo.ooa am.aa oe.mo oo.oofi oo.ooa n a o m m, n a a maOHnaH Hdnom03 voo on.» . and hHmh OHHHHH semi and 51 £5 din Avosmvaoo. >H HummU QQO'OQGNH 29 TABLE IV Median Link Relative Method Moving Average Method Detroit Edison Method Second Detroit Edison Method Ratio-to-Trend Method First Difference “Method Using Ratio-Trend Values Thirteen-Months-Ratio-First-Differenee Method Method of First Differences Using Moving Average Ratios Method of Monthly Means 3C n#.mo 5mg. mm.nm Hm.na vv.Op o.mo on.vm 90.00 um.np vm.mb n mo.mb ao.~m mm.Hm mm.mm e.vo, oo.vm mo.mm mm.Hm n>.bm) a mm.HOH or.moH mo.moH «n.m0H «.NOH pH.mm om.mm ov.m0H Hm.QOH o no.HOH vH.nOH pv.mmm, mm.mmm 0.0m mn.mm ea.mm ma.voH Hm.HOH m mm.mm on.HOH mm.mm HH.Hm m.HOH .mo.pm H>.>m mm.c0H mH.vcH 4 v¢.m0H mm.mm mv.oaH om.omH v.uoH mm.ooH Hm.ooH mm.pm pm.mo n ma.nHH on.oOH mm.HHH Hm.HHH v.moH om.mHH pa.HHH «H.50H mm.oOH a we.mmH oH.mHH mn.¢mH mm.va >.pHH no.an om.an ma.mHH mm.mmH 0: mm.an mm.omH mm.mmH ov.mmH H.0mH mmerH po.mmH .tme.HmH on.>HH 4 mm.vHH Hm.nHH Hm.nHH mmwnHH ()meHH on.nHH 5H.nHH no.9HH an.mOH ”I lwn.mo Hm.vm mo.Hm _ mm.om 0.0m mo.mo mm.Hm mn.vm om.mo h «n.Hm em.mm oa.om an.m> m.nm em.Hm no.0m pm.mm Hm.pb a m m (L a m m, v n u H ta 4. . .. k.) -' .1 the moving average ratios. The two methods do give fairly close results. When the first difference meth- od is used with the ratio-trend values the results are quite different This result might very well be ex- pected as a straight line is not the best representa- tion of trend and would lead to quite an error. As one might expect from the problem. previously worked. using a hypothetical set of data. the link rel- ative method and the two Detroit Edison methods vary considerably from the moving average method and the first difference method using moving average ratios. The thirteen-months-ratio-firstHdifference method ap- pears to be better than any of these three methods. The method of monthly means doesn't agree with the moving average method as closely as would be ex- pected. This is probably due to the inaccuracy of trend elimination. Although there is no way of knowing what the true seasonal indices are in this practical problem, it should be noticed that if the first difference method using moving average ratios is used as a basis of com- parison our results check up as accurately as could be expected with the results from the hypothetical set of data. From this one would be led to believe that the method of moving averages or the method of first differences using moving average ratios could be fair- rm 32 1y well relied upon in the determination of seasonal variation. 33 CHAPTER V The Determination of Cyclical Variation The trend has already been removed. It is now necessary to remove the seasonal variation in order to have left a measure of the cycle. The seasonal indices obtained by the method of moving averages will be used. basing the choice upon the conclusions of Chapter IV. A detailed method as to how this is done can be found in.Mill's text book on "Statistics". The final graph. after the elimination of secu- lar trend and seasonal variation, is shown in Chart V. It represents the cyclical variation of motor bus and truck production over a period of seven years (1925-1930). It should be noticed that in only one case is the variation greater than plus or minus three standard units. o // A '/\VN£ ‘ V/\/\v/\/\ /\ ‘ H j \ xv N v \Wf \ '3. Ma n /923 Au? /923 Apr// /924 Jan. /9ZJ ’4 0y. /9Z5 Apr/7 /926 (Jan. /927 flay /9Z7 Apr/7 /9Zc5’ 4/00 /929 Aug /92 9 fl/O/x/ /930 ("HAP T I 34 BIBLIOGRAPE! Bauman. A. 0.. Thirteen-Mbnths-Retio-Pirat-Difference Method of Measuring Seasonal Variation. Journal of the American Statistical Association. Sept. 1928. Carmichael. F. I... methods of Computing Seasonal Indices. Journal of the American Statistical Association. Sept. 1987. Detroit Edison Company. A mathematical Theory of Seasonals. The Annals of Mathematical Statistics, Feb. 1950. Falkner. Helen D.. The:Measurement of Seasonal variation. Journal of the American Statistical Association. June. 1924. Hart. W. L.. The Method of Monthly means for Determination of Seasonal variation. Journal of the American Statistical Association, Sept. 1922. Mills. 1'. 0.. The Analysis of Time Series. Statistical Methods. Chapters III and VIII. "NI '3“ ’ H. 1 ‘ h-.. 3 . J . ‘ .V.“ e‘ "““'_I _.V'nw ~':‘-"3.-"~"")' .- Hes. .- .- .~ ..._ -‘.5-"" a . (‘1 y -) ’ .~; ,' Y - I "If; 4'.!$t (v? .%‘...A), e‘s‘.‘ ) '<'§. ' 1 |.' .q ‘ ..{I . ‘.r'_ (H', ‘0‘ ‘ ‘ “ \t a ' ." . >- ..'-" ' ' 'I' A" 15.. ’ ‘9,"' '1‘ .. ~ I " .r . " ‘ ‘.- 4‘ '-"“'~ i'v'i s : 3“ 1.. I J ‘T‘ ‘( ‘ It} '{I- t h A v" #3:. I. ‘~ ‘ -' ”3.3....“ _ . ‘l. J‘.. ., . e 1‘ , . ,2". . , . ., - . ‘ r ' Afill h 3“ ‘ 'l u. " "t .‘ 1 I . r I". . I ‘: ' ' ~.'. o~ iv). 'a I ' "1 ‘ I I v .‘ 4 o"\ " I ‘ - ‘\ ' e~ “’ ' Us | .1 “ . 5 H 3‘ g. ‘7“ a e . . . . 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