STATISTICAL METHODS AND WEATHER FORECASTING 
the worse the agreement is between the predicted and 
actual values if there exists a random component. In 
other words, even after the entire past of all the vari- 
ables has been exploited, there may exist an upper limit 
to the attaimable accuracy of prediction; this limit 
decreases as the forecast period increases. The complete 
problem, then, is the consideration of an ensemble of 
such sequences, all of which are known in the past up 
,to a specific present time ¢), and, regarding them as a 
eroup, the evaluation of that component which repre- 
sents the true mechanism through which the phe- 
nomenon is operating. 
In the theory of generalized harmonic analysis as it 
affects the prediction of time series, we are approach- 
ing the problem from a more general point of view than 
either that of harmonic analysis, where only certain 
frequencies or periodicities were assumed to be present, 
or that of difference equations where a differential rela- 
tionship is assumed connecting present and past values 
of the time series. In generalized harmonic analysis, 
advantage is taken of the entire spectrum of frequencies, 
and this spectrum in most geophysical phenomena does 
not consist of a series of discrete lines (exact frequencies) 
but rather of a contimuous distribution of frequencies 
over the entire range of frequency, usually from zero to 
infinity. This technique permits us to develop the best 
linear operator in a very general sense in order to predict 
the future. Actually, it is this ability to predict which 
expresses how much dynamics there exists in the series. 
The linear operator is a weighting function which is 
applied to past values of a time series in order to obtain 
estimates of future values, and this weighting function 
is determined mathematically in such a way that the 
mean square error of prediction is minimized. Actually, 
of course, this technique can be extended to more than 
one variable, and thus many time series can be con- 
sidered simultaneously. 
Unfortunately, the basic arguments and ideas are 
true only if the process or processes are stationary. 
This, of course, cannot be true in meteorological phe- 
nomena since the basic movements of the weather 
systems certainly are different, at least from season to 
season. To a certain extent this lack of stationary 
property can be overcome by dividing the data into 
what might be termed more homogeneous periods, but, 
nevertheless, this difficulty is inherent in a simple 
analysis since the seasons themselves do not arrive and 
leave at any fixed time. Another factor which markedly 
affects the analysis of time series of this type is the 
fact that most of the elements have definite trends 
which are functions of the time of year. It is always 
possible to compute what might be termed a normal 
by averaging over a large number of years, but it is 
extremely doubtful whether this normal has any par- 
ticular significance for a specific year. The dynamics 
are determined by the movement above and below a 
state of equilibrium which might be called the normal 
for a specific year, although it unquestionably is differ- 
ent from the over-all normal for all years. The shift of 
this trend line from one year to the next has a very 
marked effect upon the dynamics. Since it is necessary 
851 
to deal with relatively short periods of time in order 
to obtain compatible data, it is very hard to separate 
that part of the spectrum which deals with long-period 
phenomena from the short-period oscillations which 
tend to dominate a rather minute portion of the entire 
record. 
Actually, of course, the solution of a nonstationary 
time series has no meaning in the strict sense. The 
basic problem is to make the time series approximately 
stationary through appropriate changes in variables as 
a function of time. This approach will be increasingly 
successful as a clearer understanding of the mechanics 
of the phenomena is attained. In order to appreciate 
the immensity of the task of solving this problem, let 
us review briefly what is known about the behavior of 
some of the meteorological elements, utilizing the sim- 
ple hypothesis of a stationary time series, and consider 
what can be gleaned from this information as to possible 
future courses of fruitful action. 
One often hears the statement that abnormally warm 
winters and cold winters recur in a definite cycle. On 
this basis one might expect to find long-term effects 
in the analysis of the frequency-density function of 
average monthly temperatures taken, of course, as 
deviations from normal. If one considers either the 
monthly temperature or the daily temperature at any 
one particular locality, it may be regarded as an indi- 
vidual time series in itself. Actually there exists on the 
average a correlation of only about 0.35 between the 
predicted and observed values when one predicts one 
month in advance. This accounts for only around 10 
per cent of the variability, implying merely that there 
is shghtly more than a 50-50 chance that if the present 
month is above normal the next one will also be above 
normal. No matter how far one goes back in the past 
to obtain information (which means no matter how 
high an order of a linear differential equation one con- 
siders as representing this phenomenon) no additional 
information seems to be available, so that the past of 
the sequence has practically nothing to do with the 
future. Naturally other variables exist which might 
improve this situation, but we are dealing with a 
phenomenon which in itself shows no statistical regu- 
larity as far as long cycles are concerned. On the other 
hand one may be interested in a daily 24-hr forecast of 
temperature. If one restricts the mechanism to a given 
month during the year, a developmental sample based 
on four or five years will give a good estimate of the 
linear operator. In a 24-hr forecast, a correlation of 
about 0.75 between the predicted and actual values 
can easily be realized. Furthermore, interestingly 
enough, this same correlation will be obtained if one 
picks any other year at random for the same month 
and applies the operator to this other year. This shows 
that there exists a certain amount of high-frequency 
oscillations which are similar from year to year and 
actually represent dynamics in the time series. In other 
words there exists a true signal which can be separated 
from a disturbance which we know nothing about but 
designate as random noise. The properties of the air 
mass can be introduced by considering a network of 
