MONTHLY WEATHER REVIEW CHdBLEB F. BEOO9e. Idltot.
CLOSED MAY 3, 1921
ISSUED JUNE 4, 1S21 MARCH, 1921. vola 49, No. 3. .
W. B. No. 736.
THEORY AND USE OF THE PERIODOCRITE.l*
By CHARLES F. MARVIN, Chief, U. S. Weather Bureau. 5s/. 5 6 /
[Washington, D. C., Mar. 1,le21.]
SPNOPBB.
Penodocrite is a word coined from Greek mota signifying a critic, a ju%, a decider of periodicities, and is a name applied to a mathe- mahcal and graphic method or device which haa been developed to aid in the conclusive separation of obscure and hidden cycles and periodi-
citiea pOareaeing a real existence from thoae whoae essential features are only such aa would result from, and can be explained by, entirely chance combinstiom of the data employed. "he periodocrite does not disclose or discover the h g t h of suspected ode or cycles. Other methods, such as the harmonic anal ais, Kwter's pendogram, or an of the many methods which have teen offered for this purpoae must bt be employed to ascertain the proper of any suspected cycle. "%&hona for clearneea in terminology are offered; the elements of the eory are briefly presented, the sigmficance of the resulta secured in applications to practical data are illustrated by examples, and methods of abridging certain mathematical computations are indicated. The paper concludes with repremxhitiona concerning the inherent charactemtics of data, the limitatiom u on the uae of the familiar least
s uare methods in problems of meteoro!ogy, and some suggestions are o&red for overcormng difiicultiea thus entailed.
INTBODUCTION AND TERMINOLOQY.
How can claims of eriodieities in the succession of values of meteorologica s and like data be proved or dis- proved P If a definite and conclusive answer to this important question were known, vast expenditures of time and labor on the part of many students might have been spared in the ast and l i e efforts conserved in the future for more frui tp ul pursuits than one so alluring and baffling as the search for obscure and hidden cycles believed by many students to lie concealed in practically any body of meteorological and like data. Science must*furnish conclusive answers to the claims and questions with which students of periodicities are confronted end this aper is an effort to su ply a few
periodic theories and claims may be successfully erected. If what is offered does not suffice full7 to segregate specious sequences from cyclical or penodic elements which have a phpical rui-son d'Zke, nevertheless it may uide the reader m forming his own conclusions and aid &e investigator to avoid wasted effort on studies which must necessarily lead only to fruitless or inconclusive resulb. Our acce tance and approval of claims of periodicities
demonstrated that the propositions involved can suc- cessfully run the gauntlet of tests and criteria such as presented herein. As if to add the last acid test to whatever may emerge as real, rather than fortuitous, from all other tests, we must invoke the quality of rrtility. The value of a scien-
basic rinciples whicg may serve as part o P a greatly neede cp solid foundation upon which any superstructure of
which may f) e advanced justified only when it has been
*Prssentsd barn Amdesp Y~rologlcal Society, Washiqeton, AprfI#),l921.
'1 Prof. C. F. Tdman su p M tbu name from mplo&or. a period + r p r e s a Judge, de- .dd.r. umpire, fmm rd.0 L seprrate, investigate, ludm.
46808-21-1
ti& truth is, of course, wholl independent of any
parallax of a remote star may be vanishingly small but of vast importance. On the other hand, a periodicity which differs from a perfectly fortuitous sequence b only a ver small margm may be an im rtant s c i e n d
casting or other urposes may be entirely inconse uential.
casting value of knowledge of this character can be measured. The ultimate goal of the student who seeks to formulate the laws of sequences of solar and terrestrial henomena and correlations thereof must be to establisf, first, the
REAWTY, second, the UTILITY of claims. For the present purposes, statistical data may be put into two classes: CZms I is illustrated by observa.tions whic.h exhibit the diurnal and annual march of temperature and pressure, the seasonal variations in rainfall, the 11-year period in sunspots, and like features in terrestrial magnetism. Sim le inspection of such data or graphs of it leaves no do& as to ita eriodic features. Either aided or not by known laws Sefining the length of the periods, the major question in these cases is to fk the amplitude and elements of the period, or, in the more complex case presented by sunspots and magnetic phenomena, to mcer- tain the red length of the changeable period and the nature of its fluctuations. Cirass II corn rises many other cases in which no real period can be gscerned b sim le inspection, and this
after steps have been taken, as is often done, to free the original material of any diurnal, annual, or other periodic features which can be evaluated with greater or less
numerical measure on which t z e fact depends. The
truth to B emonstrate, but its practica r O value for for4
I have indicate B in an unpublished note how t ll e fore-
will be true also in cases o 9 f the ata constituting Class I
115
116 MONTHLY WEATHER REVIEW. =WE, lml
Thie ie abundantly done in the textbooks, and when used in a meteorological connection those terms should carry the customarymeming as exactly as possible. If ths be done in a legitimate wa then readers will have a basis for the clear understan&yf authors. Many of the recurrent featmes of weather enomena which writers
more than i n d e h t e “sequeme+s.” The latter term seems particulllrly a propriate for
defined somewhat as follows: Sequence.-hy more or less complex succession of values of meteorological or other states, elements, etc., especially those which exhibit a tendency to cyclical or A portion of such sequences or a
Zefinite result derived therefrom which exhbib marked periodic recurrence of essential features may be properly called a qch if somewhat complex, but when the form IS
definitely periodic and pery Sim le, with essentially one
the appropriate dehitive designation, periodac element, is suggested. Harmonic elements or harmonic comppnents are terms which should be used consistently to designate
only those elements the inherent charactenstics of which are sinusoidal, whereas periodic elemenb or components of weather sequenc.es very often or nearly always are distinctly nonsmusoidal or nontrigonometric in any form, and the terminology should keep this inherent distinction clear. The problem of the meteorologist and the forecaster seeking to extend the period of hs forecasts far into the future, is to discover and formulate the bws govming mdeoroloqkal tteqwnees, if indeed those laws do not turn out to be indistinguishable from the laws of chaqce.
This statement contains a very important truth dese more recognition than it has received. Phenomena o
chance are theoretically considered to be the outcome of
fluences or causes.
call or claim are periods, cyc P eg, and the like, are little
meteorological application, and is offered P or general use,
eriodic recurrence.
maximum and one m m u m v 8Q ue in the sequence, then
7
the operation of
The othef control on dimel temperature is the inter- mittent influence of solar heating, which begins at SUB-
rise and is cut off at sunset, attaming various intensities at intermediate hours, depending on atmospheric trans- missibility, etc. As an elemental effect, this also is absolutely nonharmonic because it is intermittent. .
Of course other factors modify the hourly values of temperature, such as importation of warm or cold air from a distance, etc., but the diurnal march of tem- perature is cited as an excellent example of a element in weather sequences which is entiref?:
harmonic in its physical character. Accordingy, the
harmonic andysis applied to such data is altogether meaningless except that the sum of the harmonic ele- ments sunpl represents the original data.
monthly values of mean temperature for a station or locality form a good exam le of an annual cycle whose
,conform satisfactorily to a limited number of Fourier elements.
precipitation, the departures of elements ? rom cloudiness, average, and so-called normal conditions all quickly become very complex and nonharmonic. Resort to the use of devices and terminology of the harmonic andysis in the discus- sion of such data is more apt to mislead the student and confuse the reader than to disclose useful meteorological laws and princi les.
sion on terminology because the literature of the subject is often lacking m consistency and clearness, and the making of the distinctions and discriminations men- tioned is necessary to an understanding of what follows
On the ot % er hand, the sequence of daily, weekly, or
features are found by bot E theory and observation to
Sequences in values of pressure, humidit
It has seeme 1 necessary to offer the fore oing discus-
THE PERIODOCBTTE.
It is assumed we have at hand a lar e number, N, o
to be characterized bv one or more hidden cyces or periodic elemenb. The data may represent tempera- ture, rainfall, sunspots, intensities of radiation, or what not, and it is assumed all obvious periodicities like the diurnal, annual, and other c cles have been elimi- nated by a propriate methods, a% o that the data have otherwise ?!ea brou ht into a homogeneous body of
group or succession of p values of these data comprise a more or less complex cycle or periodicity which re eatdl iteelf over and over again. However, owing to the ar e
cycle is hidden and can not be satisfactorily discerned from a carefu1,inspection of the data.
in rows and columns in a manner designed to bring into the same columns datu oj the same suspected phase relations. The accidental variations will thereby tend to aver out, and the real features of the hidden periodicity 3 be exhibited by the sums and means of the hase columns. Emphasis is placed on the words in it a f ics because in carrying out a tabulation in rows and c o l m it is equally possible to place in each of the several colrunns the same phase values of the data when the length of the period is variuble as when it is constant. Attention to this point is necess if we are to deal proper1 with
progressively as claimed by some investigators. The
P
homogeneous values of any variant w % ich is sus ected
values of equal weig % t. Pinally, it is assumed that a
accidental variations of the values of the variant, Pg t e
The well-known method of evaluat‘ periodicity in such a cam consisb in tabu Y ating Or provhf the a h a
periodicities, the l e n z of which vary systematic a i y and
MARCH, 1991. MONTHLY WEATHER REVIEW. 11)l
only difficulty introduced b the variability in the
it entails. Let a tabulation of any portion or all of a given body of data in p phase columns and 7~ rows be indicated by letters as below, and for brevity let any group like this be designated a L'tabdatwn":
A p&dkiiy tabulation.
1ength.of the period is the a B ded labor of.computation
I .................................. .................. a 1
3 .................................. .................. r,-1
n .-.......... ..................... ...... no I n1 nr na %,-I
a. ................................. .................. $11
................................... .......................................... ............................................... ......I .............................. ..................
............................ 1 1 I ;; 1 :I 1 ;I ...... I ...... I ...... 1 sp-I
The whole body of homogeneous dnta available is sup- 'posed to be indefinite1 large, permitting of one or several relatively large " tabu.%ltions " of independent data.
, Now, only one of two results is ossible with reference
data.
(1) The sums, as likewise the means, will be constants- that is, differences will be small and negligible, and there-
i fore no eriod is indicated, or
(2) &e differences in the successive values of S or m
will be of material magnitude and may possibly signify a c cle. Jxperience with meteorological and like data tenches us that when n in a tabulation is ver small, say 3 or 4,
general, a mean of 4 sets of values will show on1 about
much range of variation as exhibited by any one of the single sets. It is also generally found that when all obvious periodicities have been eliminated, the means of a tah.lllalzon show less and less differences among them- selves as the value of .n increases. . These are relations which we know correspond entirely to the requirements of chance. Accordingly, the question a t once arises: Are the vari- ations found in the values of %, m,, q, etc., in any limited tabulation in any degree different from variations which would be found in values computed from entire1 chance combinations of the same body of data: d may. assume, for example, that all the values constitut- in the ori nal observations are completely mixed up in a %f owl an that any specified fabu.latzon is made up en- . tirely from w e chance drawings. The body of data thus secure 2 will differ from the actual observation in
only one articular, nameJy, the order of succession of the
the median, the standard deviation, etc., the whole dis- tribution of the chance drawings will be identical with like elements of the original observations which will differ only in the order of sucqession of values. We may well ask, therefore, how will values of m,, m,, etc., for actual observations diffe: from similar values derived entirely from chance drawings from the same body of numbers! The search for the ahswer to this auestion led to the
--------- sums..
Idsons.. ........................... nla ............ ......I m,-1
--
to the sums S or the means m a erived from adequate
the variations in the values of m wil f be very large. In
one-half, and of 9 sets often only about one-t i ird as
. values. I n all other features, as the mean, the mode,
amount of data will afford a very satisfachry.index nuniber which shows the realness of the periodicity. Theory.-The theory of the periodocrite is b n e 9 de- veloped as follows:
Let q,= *,/!T=the standard deviation of; the
whole body of data, in which Z T% designates the sum of the squares of the departures, and N the total number of observations. The subscripts attached here and. else- where to the s.ymbol P desi of data from. which the sum o
tures is derived,in thiscase is assumed u is computed by forming the frequency dis- tribution in the usual manner, and if the distribution is distinctly unsymmetrical, that fact should be ascertained and duly considered, together with any features of ab- normality which may affect the data and which probably can not be easily removed. Perffct fortuity.-We shall first consider the case of perfect fortuity. We must necessarily assume that. in any tabdation of a portion of the whole data, the values in the portion, n rows and p columns, for example, or np values in all, are representative of the whole body of data. Of course the failure to satisfy this requirement always
occurs in problems of chance, and it simply causes minor deviations from theory which are generally recognized nndunderstood. On the assumption made, then, in the long run we may write:
Also, if Q,, is the standard deviation of the p mean phase values of a tub&ion, then
Now if chance is the only factor which controls the results brought out in a tabulatima, then from the prin- ciples of least squares we must have
Let y = the ratio 5 which we may regard as a coefficient
of variation. .
Hence from (1)
QO
1 Also let' - -2.
y-2 -- -- -- -- (2),
of coordinates at an angle of 45' to the axes. From T t e
which is the equation of a straight line through the
derivation of its e uation subh a line represents the results of perfectly P ortuitous 'combinations of the data employed, (See fig. 1 .)
Since n .is any integer from 1 to + infinity, the values of z lie between z = 0 and z = $- 1. Likewise, the d u e s
of y range between y = 0, and y = + 1, because for perfectly fortuitous control when n = +infinity u,, = 0, hence y -0
and when n =l , u,, on the averqe=u,, . - . y-1.
The application of the foregoing to actual observational data is very simple. Having computed the standard deviation uo for all the data, find mean phase values a. m - - - - - -for one or several tabulations. Then deduce
Jn -
development of the eriodocrite. It is a method and a aphic device whicf serves not only to segregate real K m accidental periodicities, but with an adequate
vduei of z and y as explained. If these are equal, or nearly so, then the variations in the values of w, m,, m, etc., are no greater than those due entirely to chance, and
118 MONTHLY WEATHER REVIEW. -E, 1@1
at least in so f a r as the amplitude of variations is con- cerned no claims of periodicity are justifted.
FIO. l.-Rainfallperiodocrite. Star: Annualcycfe five stations in Iowa 38gesrrecord. Heavy dots: Annual ycle Washington rainfall, 50 ear record dmsw Annual cycle Boston Yaps. myear 1 4 , very feebly &med. Ckclea: A &month sequence, I o L rsinhl. Other sequences 15 months, 16 months. oneninth the varlable Sunspot period, llke the mrcles, all All Ln the class of perfect fortuity.
P4ec.t perkdicity.-Let us consider data which exhibits perfectly cyclical succession of values, aa for example consecutive values of the average hourly tem eratures repeating the sequence over and over ?aim 8, course
the periodic feature would be perfectly o vious in such a
case and there would be no need to resort to analytical demonstrations to establish a periodicity, however the case aids in developing the theory of the periodocrite. It is known that uo is entirely inde endent of any uestion of theorder of succession of the I f ata. However,
?I p values in sequence constitute a complete cycle, then because the cycle is deet a i2zbdu4h of n p observations will give the same p \ ase values of m,, m,, q, etc., what- ever the value of n . * . u,=constant=u, for perfect
periodicity and . - . y =:= constant = 1 - - - - - - - - (3) ,
which is the equation of a line parallel to the axis of X a t distance 1 and is a l i e of perfect periodicity. ing we see in general that to test a
periodicity it s 2 ces to form one or more tub&tiona as
explained and corn ute values of z and y for one or more group of data. &en y is substantially and consistently eater than x a real periodicity is indicated of greater or amplitude. If z and y are nearly equal, especially y smaller than 2, the amplitude of the eriodic variations is less or no greater than that due who 5 J to chance. In
the face of such a result, the probability of the c cle being real is very small or nil. An entirel new bo B y of data e of values of p E mes, but the order of succession may Y e quite different. If, however, the
en d! ent data, then the conclusion must be that a perio-
From the fo
may give a like r
order and features of succession of the phase values should prove to remain sensibly invariable even while z and remain nearly-equal for different tah&tim of inde-
%city exists of amplitude no greater than chance alone will produce, and the period tends to vanish as the length of record increases.
It will be noticed that theoretically y can not 8xc8Bd unity; however, when n ia large y may become several times larger than x. This shows strong1 marked
Faes the same fact, and the calculation of x and y
simply serve to express in numerical terms the propor- tionate effects of eriodic control aa compared with
eriodicity, but in a3l such cases inspection tJ one estab-
chance represented E y z.
short stud of rainfal P in annual and appromate 15-
month cyces, s
SUMMABY OF APPIJCATIONS.
The principle of the eriodocrite haa been applied to a
both fked and variable m len th. The resulta are shown partly in figure 2. Only 8 e annual cycles show any real existence. Even this is very feeble a t Boston, Mass., for a long record of 103 years. All other cases examined show a variation over the cycle just such as the laws of chance would lead us to expect, that is no period except the annual one has been found.
ho. 2.-(1) Annual cycle, 36 years. (a) Effect of distributin the annual cycle a 15months sequence. (3) Actual lslqonth cycle after elfmmatmg annual c y v y ratios. (4) Same cycle, secured by fortuitous dramngs from 36 years of monthly ratios.
ABRIDGED METHODS OF COMPUTATION.
same accuracy?
In ftict, this index of diversity, V ,,=q of any group of
vdues, as of S , S,, S,, etc., or of mol m, m,, etc in a
periodicity tub&tion, is quite as dependabie for present
purposes as U, = f ,/c or a value of the probab levaria-
tion, E,= f .6745 $Zl because in any one of these c-
n -
&RGE, 1921. MONTHLY WEATRER REVIE'W. 119
.......................
.........................
............
a,.:
ar
ar
4
an
Sums Za
we are compelled to measure the variation of relatively oups of numbers. As a random sample of the whole small r ody of data, such groups are not repreaentative.
As a eneral rule, a small ortion of the data will tend to show%w zmriation than t % e whole body of data, because
thevery large de arturea which are due to occur infre- quentl are not li II ely to be found in a small sample taken at ran B om. On the other hand, when such extremes do occur the variation for the small group then ap ears to be too great. All these considerationsfully just 4 y using
simply V, = *: as a measure of variation. f i e c ~c u -
lation of is much easier than the quantity II and ZV Z P
.......................................................
.......................................................
...................................................... --
b ,-M T i .
a-arrf.
-
28--1 M T:!.
there is indicated below a quick method of com uting Z V
V. In this connection it is shown in the textbooks on least squarea a that there is a direct relation between ZV and the standard deviation u. Thus it is shown that the probable error of a single observation is
without even forming the individual values of a) epartures
.a453 sv T =--
d m
Prom this it is easy to show that
==.7979 n u, (4)
This relation is of much practical used in the present connection in the
To enter upon any rational study body of data a h t im ortant step is to to as homo eneous an a as elemental a state of values of equal weig 5l t as possible. Then form a frequency classification and compute the standard deviation.
designate this value
We
From (4) we easily obtain
Vo=-- VN -.7979 uo
It is seen, therefore, to be ood practice to compute the
and from this to evaluate indirectly, because this
value is subsequently needed for comparative purpose ZV with other values of 7 derived by abridged methods of
computation, as follows :
Let a,, a,, a,, a,. ... a, be any oup of values for
mean is desired. Tabulating the usual operation of effecting such a
calculation we get a table as follows in which the I positive and u n ative departures are set out in different columns in whic8,, b,, etc., indicates values of a which are reater than the mean and c,, e,, etc., values of a equf to or leas than the mean.
standard deviation directly f rom the whole body of data ZV
which the arithmetical sum of the (g. epartures from the
~~ ~
. a M. Yerrimn: Method of Least Squares. John Why & Sons. 1915. P. 92.
Tabulation of dspmhuU,/ionr man.
DqmrtlUea.
Positive. Negativa
Za T
n n Mean = - = M-C - in which T is a remainder in division
which may be equal to but is generally less than
is rejected.
the departures =d,
and
Now, from the lilw of the mean the algebraic sum of
(4)
T T
. ' . ' -~-~M F ~~+S C -U M F -U =O n
Let Sb-td%=B and Ze--uM= -C.
(5)
T
. * . B- OFi (t+u> =o
Since t + u =n, B- C = f T, an equation which checks the accuracy of calculations when B and C are calculated separately. Chan the signs of the quantities composing the negativsepartures in equation (4) and substituting B and C, we get for the arithmetical sum of departures,
ZV=B+C+(u-t) T
which gives rigorously the sum of departures. Since
71 is a maximum possible value of T and since (u-t) 2 tends to be zero or only small integral numbers, the
term - (u-t) is a small corrective term which can in
ZV=B+ c
The practical meaning of all this may be stated in a
sim le rule for computing any values of ZV, thus: If not already known, compute the mean M of all the values, noting the arnount.of any discarded remainder r. Form the sum of all the values of the variant which rqre eater than the mean. Let the number be t. Subtract gom the sum, t times the value of the mean. The differ- ence is the sum of the positive de artures, B. The sum
If desired, the work can be checked by computing C independently from the sum of the values of a equal to or less than the mean. It will be noted for check purposes we also have the relation
T .
n
general be wholly neglected. ...
of the negative departures is C= B ~r :.Z v =2 B T r .
C= Ea- Zb - u M
120 MONTHLY WEATHER REVIEW. .MARCH, 1921
n=/2 4
Formulae similar to the foregoing could be developed for accomplishing the same results by the use of some number R instead of the mean M. I t is
mean would be lost in the mofe complex calculations re uired if an arbitrary number is used.
'x single example illustrates the comparative simplicity of this method of computing the mean departure of the results of a tabulation. The data are the monthly sums of a 16-year record of rainfall. We get the mean monthly departure without computing either the departures or the monthly means and the result is almost ri orously
the end and are a minimum. The tabulation in figure 3
shows the work as carried out on a listing machine which greatly facilitates the computation.
AM/%ED CALCULA77ON OF MEAN DEWRWIPE
believe arbitrardr the abridgment of the work secured by use of the
accurate-that is, fractional excesses are rejecte i only a t
' 27.32
22.57
33.56
44.51
70.22
53-25
52.8 I
51.8 I
47.62
33.94
25.28
SUR7 = S f f = 486.45
sums
than Meon
9PiZQAZl-
44.511 70.22
243.24 = t . = UM
+76 98 = 0
- -77.01 = t Diff = -3=r
/53.99=s I/ '
FIQ. 8.
The mean departure, however, computed, is probably the best and most easily evaluated measure or index of variation we can em loy when such a measure is needed.
ondary maxima and minima in a small grou of values. Of course, when periodicities are very e P emental and
closely harmonic in character, the amplittde of the various harmonic elements of a complex cycle is an all- sufEcient index. On the other hand, when we are dealing with a com lex cycle with many aexions, little signifi-
mum and the minimum values as an index of variation.
This is especially t % e case when there are several sec-
cance attac i es to the extreme range between the maxi-
THE INHERENT CHARACTERISTICS OF DATA.
All detailed records of meteorological conditions and many like phenomena of solar and terrestrial activities
are of an exceed.ing1 corn lex character. Red progress in t e stu y and analysis of such data and
interrelations thereof is greatly promoted by a recognition
of its inherenk characteristics and attention to quegtions of comparability, homogeneity, uniform weight of values, period of time covered, and other such factors. It is impossible to discuss these questions exhaustive1 in this note, and attention will be directed to on1 the f o l
the resent connection:
(17 Variation or diversity of similar values.
(2) Order of succession and obvious periodicities.
(3) Frequemy distribution :
(a) Elemental. (b ) Composite. (c ) S mmetrical, Gaussian and non-Gaussian. (d) d e w .
lowing charaoteristics which are of special signi z came in
(1) The varhtion or the dhmity among a considerable number of assumed similar values of any meteoroloqcal element is of great im ortance when various combina-
the numerical results secured. Ever student of the
are good or poor; depending upon how much variation there is in the individual values. Long records, that is, a
large number of individual values are necessary to fix normals of tempertaure and rainfall which in show great variations, whereas shorter records su ce to fix normals of elements like pressure, which exhibits smaller variations. The same elements show systemat- ically much eater va.riation in one section of the country
the vrtriat.ions are greater than in others. These consider- ations which guide us in fixing our degee of confidence in the values of important normals are just as applicable to the averages of a given number of observtltions for what- ever pur ose they may be combined as for normds, and
the diversity of the individual vahes is an znherent charac- teristic o f mii.ch. importance. which mu t be. propedy evalu- ated and reckoned with. The metho ! s of doing this by proper allowance for strong and weak observational values, or by weights and otherwise, are so fully covered in the testbooks and generally practiced by students in the more esact sciences, that it is needless to go ikto such details here. Few students of physical meteorology appear to realize the s lendid opportunity the enormous
of statistical anal is and discussion than is requently practiced. The ocect of the present effort is to secure attention to these important details of investigation and research.
(2) Order of succession and obvious perio&ities.-The fundamental feature which identifies a periodicity k the orderly recurrent succession over and over aqain of iden- tical phase values. Elastic vibrations and many like physical phenomena exhibit very perfect periodicity even when the amplitude is very, ver small. In meteorolo
diurnal and annual changes in values of temperature, pressure, rainfall, etc., may also be very definite. In these cases, the Zenglh of the period is invariable and %xed
by astronomical causes and relations. All such periodici- ties in general are well defined and perfectly obvious by mere inspection and in the majorit7 of cases the am litude of the periodic features is relatively large. cases like the succession of HIGHS and LOWS moving eastward in the extratropical latitudes, the interval between events is, very roughly, three t.0 five days, but is extremely variable and at times there seems to be com- plete interruption or suspension of the orderly succes-
sion of events which, however, tw resqmed agam after a,
tions of values are ma 1 e and conclusions deduced from
subject knows that so-called normals .9 rom short records
%-Y"
than in anot Y er. Also, during certain months or seasons
thus it P ollows that in all careful investigations of data
f
body of meteorologica P statistics offers for a hi her order
fY and the inexact sciences, examp T es of periodicity like t e
&GH, 1921. MONTHLY WEATHER REVIEW. iai
short time. These secondary features of the major phenomena of the general circulation of the atmosphere are attended b a whole train of characteristic changes
new, precipitation, winds, etc., all of which recur in se uences of highly irregular le we think m units of ,on%, years, etc., obscure and indefinite sequences of the same irregular t p e and character as just described, all- tending to semcyclical recurrence, are marked features of any long record which ma be invest' ated.
4 is f 3 y representative of practically an body of data which may be presented. A very simple c ange of scale and interval of time between successive values suffices to adapt the di ams to become representative of a great many records x c h may be subjects of investigation. One of the greatest roblems in physical meteoroloq is to formulate bona 2 de laws of se uence of the semi- cyclical succession of values such aslave just been dis- cussed. Indeed, the problem is broad1 general in many branches of the inexact sciences. d a n r fragmentary solutions or discoveries of alleged cyces have been offered, as a cycle of seven or eight years in temperature or the Bruckner cycle of 35 years. An 11-yearc clical correlation of small percentage between tropicaf tem- peratures and suns ots is hesitatingly conceded to be of possible redity. &arching criticisms of such claims weakens rather than strenghtens their foundations of proof. In a last analysis the element of pure chance is such a large factor of domination of the events claimed, or, stating the matter in other words, the m reality over purely fortuitous recurrence is so smzthx', such claims have no practical forecasting value. No useful margin of successful verification is possible. In the face of suc.h facts one is puzzled to know how much may be acce ted as a matter of physical reality and how much shoul a be rejected as only the operations of c.hance. Pro ess in the study of obscure eriodicities requires:
liihed as f a r as possible by the use of more or less rigorous and anal tical methods rather than by resort to arbitrary graphicY methods often practiced and which too often
in the values o s pressure, temperature, sunshine, cloudi-
th and amplitude.
i
(I) q h a t the reality of the res J ts claimed be estab-
tend to nurse into realism the creatures of the imqgha- tion.
Bp
for any body of data whatever. Such a rocedure brings the investi ator face to face with the re problem of periodicity. &clical se uences derived from the real data that are indistinguishhe in their principal features from like sequences deduced from the fortuitous drawings can not be claimed to have ph sical reality. Demon- strations of reality must be base B on results drawn from the real data which can not in any way be duplicated from the fortuitous drawings. It is wonderfully mstruc- tive to any investigator to tr out and compare for him-
exclusively with the one question of the order of suo cession of any body of data.
(3) Frequency dlstribtition-General remarks.-For ur- poses of physical investigation of meteorolo ical and s ike data, fre uency distribution must be reduce d to the most elementA form possible. The almost universal practice of classifying data on the basis of departure from a mean value is satisfactor only for data like tem erature, pres- sure, etc., where t % ere is no theoretical f unit upon the magnitude of either plus or minus departures. Certain classes of data of which rainfall and wind velocity fur- nish good illustrations have definite zero classes. In such cases values less than zero are hypothetical and impossi- ble. If this kind of data is classified on actual values from zero to the hi hest, then the resulting ditrihution for any long recorf will he hi hly composite and will
Departure* rom a mean valiie.-Whether departures
data are emplo ed a classification by departures, of ata
in either case the zero values of data fall indiscnminahly in different places in the frequency distribution. Ratios to the mean.-A very satisfactory remedy for this difficulty is found by taking the ratio of t.he indi- vidual values of any variant to the mean either of B
month, a year, or any other goup, or the mean of the whole body of data may be the basis of the ratio. This method brings all zero values into coincidence a t zerb class and all mean values are coincident a t unity or class 1.000. "here is no limit to the values which may occur in excess of the mean. The method is essential in classi-
self the results procurable 9 rom drawings, which deals
most likely tend to be multimo 5 al as shown in figure 5.
from a mont d y mean or a mean for the whole grou of
having a zero c 9 ass like rainfall, is unsatisfactory because 1
zero class, but it is equally any other data. The compute seem to require much additional shows not only that suhsequent
erwise widely diverse, com arable the use of ratios but that such
on a reasonable basis o€ equality, Regions of lig t and R
1 22 MONTHLY WEATHER REVIEW. MARCH, 1921
heavy winds, or of light and heavy precipitation, may be thus readily compared. Moreover, taking the ratios to the mont-bly mean ef- fectively eliminates features like the annual cycle. This result is also secured by taking departures from the same means. The latter method, however, introduces a hetero- neous mixture of plus and minus signs which are a
K t f u l source of errors of computation and are otherwise ob 'ectionable. bigwe 6 illustrates a classification of rainfall by ratios, attainin thereb what appears to be a closer approach
haps otherwise possible. The curves and fortuitous drawings in figure 4 are derived from the same data.
to the e P 9 ementa charactenstics of the data than IS per-
U
4
I I - I
I 3
n
0 C 4 6 8 b fC El I6 M 2 0 2 2 24
no. 5.4lassification and frequency polygon of Iowa rainfall; actual monthly means.
Normal and skew disti-ibution.~, elemental and compo8-
ite.-It is a question if any meteorological data exhibit
a strictly symmetrical and normal dist.ribution as defined b the Gaussian equations. Variations of temperature
weight ani free from durnal and annual systematic chapges appear to be very near1 symmetrical in distri-
even these values obey Gauss's law of distribution at all
As a general proposition, all meteorological data form skew distributions well illustrated by dia rams like fig-
apply to these only in the crudest possible wa . Even
slight, conformity to the Gaussian law is not satisfactory, because of some inherent complexity of the {ata. 'fake for example a very simple case in which the
nwiations are due to only two separate causes, each of
w T l en brou ht into a homogeneous state of values of equal
bution, but I do not think it has TJ een demonstrated that
ures 5 and 6. The Gaussian equations o 9 probabilities
in many cases where the skewness of a distri 3: ution is
closely.
robabl
which is Gaussian in its action, we may write from a well-known principle in least squares-
in which a and b are the standard deviations of the sep- arate causes of variation, respectively. Now if the two systems of variations have approximately the same mean value, then the com os$e distribution will be symmetri-
distribution can not be fitted b a single Gaussian curve
for the two systems differ more or less the resulting dis- tribution will tend to be unsymmetrical and may even have two definite modes.
u = * &a+ bZ
cttl but not elementa P , and if a and b differ materially the
with any satisfaction. If, furt z ermore, the mean values
FIO. 6.4lasslBcation and Ire uency lygon of Iowa rainfall made homogemem and the annual cycle elidnated 6 ratios; also Oausslan curve best flt.
Pearson' has invest' ated analytically the case of a composite consisting o P two normal distributions and finds its solution both uncertain and very difficult. He finds the result depends upon evaluating the roots of an equation of the ninth degree and selecting those which seem most plausible. Studies and considerations of this character place the mathematical treatment of composite and unsymmetri- cal frequency distributions entirely beyond practical possibilities and the grasp of the ordinary investigator. Even the well-known formulae and methods of Pearson for treatment of certain elemental types of skew distri- bution are prohibitive because of the great labor of com- utation entailed, es ecially when solutions are required
For perhaps hundre a s of cases. In addition, the result
4 Pearson, Karl: Contributions to mathematical theory of evolution, gtil, w,q Roy. Soc., vol. 185, pt. 1, A, 1894, p. 71.
lKAsm, 1921. MONTHLY WEATHER REVIEW. 123
in the end may be disappointing because of the com-
posite character of matenal handled and the imperfect fit of the theoretical curve to the actual data. Notwithstanding all such serious obstacles the search-
ing analysis of the frequency distribution of any body of data under discussion is a most essential as well as fruitful source of information, and the case is by no means so
hopeless as it seems, because important and very useful phases of the whole roblem can be solved most satis-
lously simple and certain c0mpare.d with the laborious and possibly disappointing mathematical methods.
factorily by empiric a f graphic methods which are ridicu-
FIG. l.-Didribution of Iowa rainfall ratios repwenting graphic integration of skew curve of best At.
The skew distribution of Iowl rainfall ratios redrawn as fi re 7 furnishes a good example, illustrating how the grapcc method may be employed. The more important quantities we wish to determine are :
(1) The modal or most frequent monthly amount of rainfall.
(2) The probability of the modal amount. (3) The mid or median amount, than which a monthly amount is just as likely to be greater as to be less.
(4) The probability that the mont,hly rainfall will be the normal for month.
(5) The relations of the mode and median values to the mean. (6) The probabilities that a departure from the mean will be (a) greater than the mean, (b) less than the mean. (7) The most probable positive departure, the one havi a 50-50 chance. (8%e most probable negative departure. Method-Draw the frequency distribution carefully to scale on section paper. A size about 10 by 12 inches is ample. Squares about 8 or 10 to the inch are preferable aa easy to read. Draw a smooth skew curve avoiding secondary inflections such that the curve incloses an area practically equal to the combined area of the polygon of
The answers to all of the above questions could be found (generally, however, with great labor of mathe- matical computation) if the equation of the curve were known and qf we could integrate the expression ydx. In
general, however, no satisfactory e uation of the curve
or im ossible. It is perfectly easy, however, to perform a mec r;la nieal integration of the expression y& for any free-hand curve, as in figure 7, by simply adding together the coordinates or y values at each intersection of the
curve with successive vertical lines of the section ruling.
rectangles.
can be found, and the integration o Y ydx is also difficult
One of the elemenh y& is shown in dotted lines in the figure. In this case &z = 0.05. This operation of summs
tion is very greatly facilitated by the use of a listing ma-
and permits introducing subtotals a t certain desirabe 8"
chine which preserves a record of the successive readin
points, such as (a) a t an ordinate which divides the area at the left of the mean into two equal halves. The place of this ordinate is estimated approximately by eye and the subtotal introduced as the listing proceeds; (6) at or near the modal value; (c) at or near the mean value of z; finally, (d) a subtotal should be taken at or near an ordinate which divides the area to the ri ht of the mean in two
. The grand total o f the whole summation gives t tarts e total area of the distribution in units of squares of the coordinate rulings. If the readings of the 0rdmat.m are estimated to the tenth of a unit, the size and scale of diagram recommended will give a total area of about 1,000 squares, witsh the nearest tenth added, which is abundant.ly accurate. Let A = total area of the distribution. (If this area is not quite equal to the total area (expressed in squares) of the polygons themselves, the curve ma need to be ad- justed in places to include better the B esired area.) eas to in-
areas :
Aided by the subtotals a, b, c, d, it is ve
terpolate ordinates which accurately fix P T t e folowing
area from zero to mean
2 a'=
The abscissa for the ordinate defining this area is the most probable negative departure (a 50-50 chance). Question (8).
The subtotal 6'=5 defines the mid or medianordinate
which divides the distribution into two equal Darts.
A
Question (3).
*
The subtotal c' defines the area from 0 to the mean or normal axis, viz, abscissa= 1.00
Subtotal d' locates
area from mean to positive limit of ratios
2
This defines the most probable value of rainfall ratio greater than the normal, the on? having a 50-50 chance of occurring. Question (7).
From the data t.hus secured numerical answers are eaaily found for the Iowa rainfall data for 36 years represented by the distribution.
(1) The most frequent (the modal) rainfall we see by inspection is very approximately at x=.700; that 1, 70 per cent of the monthly normal is the most probable value of monthly rainfall.
(2) The probability that the rainfall in any month will be the modal amount within a narrow lmit, say between ratios 0.675 and 0.725 will be 858.6+39=about once in 22 times-that is, although the modal rainfall is most frequent of any it will occur as a monthly amount within these limits only about once in 22 months. (3) The mid or median value of rainfall ratio by inter- polat,ion from subtotal c' is 0.917-that is, it is an even chance that, any month1 rainfall for this section of Iowa
normal. (4) The frequency for a mean monthly rainfall (ratio 1.00) from the curve is 30.3. The probability that the amount of a monthly rainfall will ocour between a ratio say 0.995 and 1.035 will be 858.6+30.3=28-that is, the monthly mean rainfall within f 0.025 will occur only once in about 28 months.
will be greater or less t % an 98 per cent of the monthly
124 MONTHLY WEATHER REVIEW. , &f.ARGH, 1921
(5) We have seen the mode falls a t x-0.700, the median a t 0.917, and the mean or normal a t 1.00.
(6) The probability that a monthly value of rainfall
will be greater than the normal is measured by the ratio
(8) If a monthly rainfall is less than the normal it d be an even chance that the amount will be greater or lese than 66 per cent of the normal, and it WM s h o e under
(1) that the most frequent of all monthly amountg was 70
er cent of the normal. Thus it appears that the most frequent monthly amounts and the probable amounts below the average are both about two-thirds the monthly normal. Such are answers that are easily deduced by inteq+
such scale drawings of frequency distributions as shown in figure 7.
------ area greater than mean 855.6 - 481.9 -0.44
Hence the monthly rainfall will equal or be reater than
equal or be less than the normal 56 months.
whole area ( = A) - 858.6
the m m d about 44 months in 100 and 0 P Course Will lations from the mechanical or approximate integration of
percentage 1.36 is the probab 5 e amount. (7) E'or monthly amounts reater than the normal the
A STATISTICAL COMPARISON OF METEOROLOQICAL DATA WITH DATA OF RANDOM OCCURRENCE.
&-5/.5&&. : 5s1.501 By H. W. CLOUGH.
[Weather Bureau, Washington, D. C.. Apr. 18,1921.]
SYNOPSIS.
Daily, monthlv and annual means of meteorological data show fluc- tuations of varying orders of magnitude, which may be regarded aa either of a fortuitous character or as resenting more or less systematic characteristics. Certain precise regtions which are distinct,ive of purely fortuitous data are derived by both theoretical and empirical methods. Theae relations constitute crit,eria for determining the extent to which meteorological dah differ from such fortuitous data.. Monthly and annual means of temperature are nearly Gaussian in their distribution, their deviations being of the nature of accident,al errors. but the order of succession of their occurrence is not fortuitous. Rainfall data are more forti.iitous in their characteristics than tempera- ture. In a plot of unrelated numbers the two-interval is predominant, while in the case of moet meteorological annual means the three-year interval ia the most frequent. The variations of mean annual tem- eratuea show systematic characteristics to a greater extent in the iouthern Hemiaphere and the low latitudes of the Northern Hemis- phere than in the higher latitudes of the Er'orthern Hemi Statistical criteria a plied to the variat,ions of the perixyfhe solar
spotg disclose mathdry svst,emdtic characteristics. . A period of recurrence of extremes of pressure at Toronto, averaging 32 to 34 days seems to he dimdosed by a purely statistical method of treatment of the dates of hichest and lowest preseure in each month for a long aeries of years.
Variability is a dominant characteristic of weather, In the Tr0pic.s t.he gay-to-day fluctuations are negligible and the seasonal changes occur with clock-like regularity. The inter- diurnal variability of temperature increases with latitude to about the Arctic Circle, then decreases somewhat. A lot of daily mean temperatures exhibits characteristic [uctuations with crests separated by intervals varying irregularly from 3 to 7 days or more. If these daily values be combined into weekly means and plotted there are again shown similar fluctuations but with longer intervals varying from 2 to 5 or 6 weeks. The same data combined into monthly means show, when the residuals are plotted, fluctuations apparently analogous to those of the daily data but with intervals between the succes-
sive crests varying from 2 to 6 months or more. Yearly mean temperatures at an locality when plotted show fluctuations which are inJstinguishable from a plot of monthly residuals, the intervals being measured in years instead of months. Thus daily, weekly, monthly, and annual means of meteorological data present fluctuations of varying orders of magnitude. The smaller day-today fluctuations are superposed upon the larger weekly fluctuations, the weekly upon the larger monthly, and so on until we arrive a t the long secular variations measured by decades or even centuries. The question arises as to the character of these appar- ently irregular fluctuations. Are they to be regarded as purely accidental and fortuitous or do they present char-
actenstics which show them to be deviations partaking
articularly in temperate latitudes.
of a systematic nature and if so, susceptible of prediction 4 Obviously, if they are of a purely fortuitous nature long- ran e forecastin is out of the question. Ansiderable fiversit of opinion regarding this particu-
ture of the subject. A conservative element re ards the monthly, seasonal, and annual variations as % ue to a complex set of many varying influences whose resultant effect is a series of nearly fortuitous deviations about the normal which can be represented by the well-known Gaussian law of errors. Another element regards the variations as .controlled by more or less systematic laws and as being essentially sequences of a quasi-periodic nature. Po ular weather lore has for its basis an almost universal beyief in the tendency of weather changes to in other words for one extreme to be
lar aspect of weather c x anges is gleaned from the litera-
within a short period. . by the employment of statis- the estent to which a given succession of meteorological data conforms to a purely fortuitous selection of similar data, and it will be the ur-
which re resent purely accidental deviations about a mean an1 to illustrate by exam les of meteorolo ical
of random occurrence.
pose of this paper to set forth the characteristics of B ata
data how and to what extent the P atter differ from i ata
CHARACTERISTIC FEATURES OF FORTUITOUS DATA.
There are two classes of data whose deviations present purely fortuitous characteristics: (1) A series of unrelated numbei-s, illustrated b a random selection of numbers
e ually probable. (2) A sam le of the component data
sums of ten digits of random selection. The data of this class are also unrelated, but the various possible values of the variant are of unequal robability.
classes of data rigorously conform and which constitute criteria for testin the conformity of any series of .obser-
suc.h a conformity indicates some systematic influence operating which results in a frequency curve of either a skew type or a symmetrical but composite type. In either case the curve of best fit by least square methods exhibits escesses in one part of the curve and deficiencies in another part. Relatiom between indices of dispcrsion.-There are various measures of the dispersion or scatter, which,
in the case of data showing Q distribution ..of a
between 0 and 100. 9 n this class of data all values are
o?' a normal frequency distri g ution, illustrated by .the
There are certain precise re P ations to which these two
vational data to t E ese requirements. Any deviation from