A. GROSJEAN

J . L. KUENY Research Engineers,

Societe ALSTHOM, Direction des Recherches,

Techniques et Produits Nouveaux, Departement d'Automatique

et d'Electrdhique, Grenoble, France

Statistical Weather Forecasting1

This paper illustrates the use of statistical methods such as multiple correlation analysis, discriminant analysis and principal component analysis for weather forecasting. Two examples are presented: the first is a qualitative local prediction of daily precip­itation for the next day, using pressure measures of the present day and of past days, by means of discriminant analysis; the second is an analysis of the 500-mbar geopoten-tial heights over Western Europe by means of principal component analysis, followed by a quantitative synoptic prediction of the evolution of these geopoteniial heights for the next week to come, by means of multiple correlation analysis. For each of these two prediction problems, good predictors are chosen among a great number of candidate ones by a special stepwise selection procedure.

Introduction Weather forecasting is not a new problem; people have been

interested in it over the ages for individual, social, economical and ecological reasons. Everbody would like to know what the weather will be tomorrow, or for the coming week or for the coming month. Though not new, ̂ weather forecasting still re­mains a challenging problem. Meteorological conditions have a great influence on human activites from many points of view, the economical one being perhaps the most important, but ecological perspectives are not to be neglected.

Before the appearance of digital computers, the approach to weather forecasting was rather empiric; it was much more sub­jective than objective. Since the availability of large-scale and high-speed computers, able to process thousands and thousands of bits of data in a short time, the approach to weather forecast­ing has become more scientific than before, with the use of com­plex forecasting models.

However, despite this important progress, it is still not pos­sible to say that meteorology is now an exact science; it is still the domain of research study on many unsolved problems. Roughly speaking, we can distinguish two quite different re­search techniques: a physical approach and a statistical ap­proach [1, 2, 3].2

1 The physical approach, which,, uses physical dynamic models, is based on the calculation of atmospheric motion and Uses mechanical and thermodynamical differential equations, which are integrated. Starting with known present initial values, the numerical integration is performed into future time and gives predictions in this manner.

This research was sponsored by the French Army Bureau (Direction des Keoherohes et Moyens d'EssaiB).

Numbers in brackets designate References at end of paper.

Contributed by the Automatic Control Division for publication in the •JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received at ASME Headquarters, June 16, 1975. Paper No. 76-Aut-A.

2 The statistical approach, which considers statistical models and is the subject of this paper, is not at all based on any physical point of view. It considers the problem as a black box, with in­puts called Xi, Xi, ..., X„ and outputs called Yi, Y%, ..., Ym, as shown in the following.

-Ai-

Inputs Xi-or

predictors Xn-

Yt Outputs or

Ym predictands

The output variables characterize what we wish to predict. The input variables are the predictors. Here, there are three problems to solve:

1 Which are the best predictors to consider? 2 What kind of relationship do we consider between the selected predictors and the output variables? 3 What is the quality of, the final prediction model?

The statistical approach tries to establish an external statistical relationship between the inputs and the outputs, using a par­ticular statistical model. The criterion of quality is also sta­tistical, such as an average success or a mean square error, for instance. The inputs may be selected among a great number of candidate predictors, in order to optimize the criterion.

Let us now review the different weather forecasting problems which are usually considered by the experimental meteorologists of the French Meteorological Office; we can distinguish several types of weather forecasts, essentially depending upon two dif­ferent criteria: a range and a spatial criterion.

The Range Criterion. Three different forecasts may be con­sidered;

1 A short term forecast, for one, two or three days ahead. For this period, prediction of atmospheric motion, physically ex­plained by mechanical differential equations, is predominant. 2 A long term forecast, for fifteen days to a month ahead. For this period, the influence of atmospheric motion vanishes, but

Journal of Dynamic Systems, Measurement, and Control MARCH 1976 / 49

Copyright © 1976 by ASMEDownloaded From: http://dynamicsystems.asmedigitalcollection.asme.org/ on 10/01/2013 Terms of Use: http://asme.org/terms

thermodynamical considerations, such as exchanges between the oceans and the atmosphere, are important. 3 A middle term forecast of between three and fifteen days ahead. Meteorologists are of the opinion that this is the most difficult range because prediction in this case involves both mechanical and thermodynamical considerations.

The Spatial Criterion. Here again, three different forecasts may be considered:

1 A short scale, in which the surface covered by the prediction is < 100 km2. Such a small surface may be affected by some very specific meteorological conditions, such as local winds or tur­bulence, which characterize a local climate or microclimate. 2 A middle scale, forecast, in which the surface covered by the predictions is > 100 km2 and < 250,000 km2; this is the scale of a big area or region; its geographical position (near seas, oceans, mountains, . . . ) and its relief contribute to its particular climate. 3 A large scale or synoptic forecast, in which the surface covered by the prediction is > 250,000 km2 and < 4,000,000 km2; this is the scale corresponding to the great atmospheric motions, cyclones, anticyclones, etc.

The physical approach, by means of numerical dynamical models, provides very good predictions for a synoptic and short range evolution of the atmosphere, i.e., the evolution of the pres­sure surface at ground level or that of geopotential heights at different pressure levels. To obtain these predictions, it is neces­sary to have a very long computing time (several hours), with the biggest up-to-date numerical computers; for a range greater than three days, the predictions rapidly become wrong. Like­wise, if the scale of forecast is too small, i.e., middle scale or local forecast, the physical approach is also unable to provide good predictions, because of the great physical complexity of the problem (the prediction has to take into account local meteoro­logical conditions, relief and microclimates, and practically, this is almost impossible). At the French Meteorological Office, some researchers are working on a middle scale, short term pre­cipitation forecasting problem by means of numerical dynamical models, taking relief into account. They have so far obtained encouraging but not definite results.

The statistical approach, forsaking the physical aspect of the problem, uses statistical information concepts for the relation­ship between predictors and predictands. Modern statistical multivariate theory provides useful tools to elaborate upon statistical forecasting models. The main tools are: multiple correlation analysis, multiple discriminant analysis, principal component analysis, canonical correlation analysis, etc. The main advantage of the statistical approach is to be able to in­clude in a simple model a great amount of physical complexity; for this reason, the approach may be useful for local forecasts, and also perhaps for middle term or long term forecasts.

Specifically, this paper presents the following problems: The first is a qualitative local prediction of daily precipitation, one day ahead, on the airport of Le Bourget (near Paris). We used pressure measures for the prediction by means of linear dis­criminant analysis. The second is a quantitative synoptic study of the 500-mbar geopotential heights over Western Europe. We proceeded by means of principal component analysis and tried then to predict the mean evolution of this surface for the next coming week, by means of multiple correlation analysis.

A Local and Short Range Precipitation Fore­casting Problem Using Multiple Discriminant Analysis

(a) The Prediction Problem. The problem was to predict, at 6 hr UT (Universal Time) on day J, the daily precipitation at the airport of Le Bourget, between 6 hr on day ,/ and 6 hr on day J + 1; let us denote P{J) as a cumulative amount of rainfall at

©4-

® 6

©?

® 9

# 1 0

n • /

0 <

<3

A

a c D E F G

Lille Rouen Le Bourqeh

Reims Hennes Orleans A,ux.erre

and po

Fig. 1 Position of the pressure measures used for predicting the occurrence of precipitation on the airport of Le Bourget

Le Bourget (or rainfall equivalent of snowfall) during this period. We desired a qualitative prediction in two classes:

Class Ci-. P(J) = 0 (no occurrence of rainfall)

Class C2: P(J) > 0 (occurrence of rainfall).

In order to predict the probability of rainfall occurrence, we used pressure values. There are two main reasons for selecting ground pressure values as possible predictors. The first is a meteorological reason; it is well known that, in most cases, good weather is associated with high ground pressure values and rainy days with low ground pressure values (especially for frontal rainfalls). The second is a statistical reason; from a probabilistic point of view, pressure values appear to be good stochastic variables, with Gaussian probability distributions.

Two groups of pressures were used. The first is composed of pressure measures at 7 French towns: Lille, Rouen, Reims, Rennes, Orleans, Le Bourget and Auxerre. These pressures are observed at 6 hr, 9 hr, 18 hr, and 21 hr UT; so for the prediction being calculated at 6 hr on day / , we could only use observations at 6 hr on day J, but also observations at 6 hr, 9 hr, 18 hr and 21 hr on days J-l, J-2, and so on. The second group is composed of 13 pressure values on particular grid points convering a part of the North Atlantic Ocean and Western Europe; these values, at sea level, are calculated at 6 hr by interpolation from several observations.

So, for our prediction problem, we could use these values at 6 hr on days J, J-l, J-2, and so on. The positions of the 7 French towns and the 13 grid points are shown in Fig. 1.

(6) The Data Set. We used a data set of N = 1298 daily samples, during winter seasons. Among these N samples, Ni = 714 samples correspond to the class Ci and Ni = 584 samples correspond to the class &.

So, the a priori frequency of no daily occurrence of rainfall is: u>i - 714/1298 no. 0.55; that of occurrence is then: to» = 1 — Wi no. 0.45. The best a priori statistical prediction, then, is to predict Ci each time; this optimal a priori prediction has a 55 percent chance of success.

(c) Selection of Good Predictors and Prediction by Linear Dis­criminant Analysis. We used the wel l known linear d iscr iminant

50 / MARCH 1976 Transactions of the ASW1E

Downloaded From: http://dynamicsystems.asmedigitalcollection.asme.org/ on 10/01/2013 Terms of Use: http://asme.org/terms

@T-l:«h.

»T-1i«h.

Fig. 2 Selected predictors for forecasting the occurrence of precipita­tion on the airport of Le Bourget during 6 hr on day J and 6 hr on day

Analysis method [5] for a two class discrimination problem; we here briefly state the object of this method: if Xh X2, . . ., Xn

are the predictors being used, the linear discriminant method provides n + 1 coefficients So, &i, • • ., &„, computed from the data set, such that an estimate of the probability that a new sample [ah, itj, . . . , x„\ corresponds to the class Ci is given by

Prob [Ci\xu xt, ..., xn] = 1/(1 + exp [do + 01X1 + . . . + anxn})

and, obviously

Prob [dlxi, x2, ..., xn] = 1 - Prob [d\xlt xh ..., x„]

These relations are based on two assumptions:

1 Xi, Xi, ..., Xn are supposed to have a multinormal prob­ability distribution for class Ci and for class C2. 2 Xi, Xi, .. ., Xn are supposed to have the same covariance matrix for class & and for class &. The accuracy of the dis­crimination is usually measured by the percent rate of success of the following deterministic prediction:

Prediction = ft if Prob [Ci\xu x2< ..., xn] > 0.5

Prediction = C2 otherwise

^ When we have to deal with a great quantity of candidate pre­dictors, an important problem which frequently occurs is the selection of the best among them. There are a number of avail­able procedures, called stepwise selection methods [8], correspond­ing to this problem; for discriminant analysis, the criterion of selection is, in most cases, the maximization of the success rate of the discrimination (for regression analysis, the criterion is the maximization of the multiple correlation coefficient). We used a special Fortran Package, called SELECT, which is described in the Appendix, based on an original and efficient selection algorithm [7]. It provides a group of good predictors, selected from among a great quantity of candidate ones.

Let us return now to our particular precipitation forecasting problem. In order to predict precipitation occurrence between 6 hr on day J and 6 hr on day J + 1, we considered as candidate predictors:

W the pressures at the 7 French towns, at 6 hr on day J (b) these pressures at 6 hr, 9 hr, 18 hr, 21 hr on days J-1 and J-2 M the pressures at the 13 grid points, at 6 hr on days J, J -1 and J-2.

Thus, we had a total of 102 candidate predictors. Then, we used SELECT, which provided a significant group of 16 "good" '

predictors. The corresponding deterministic prediction, applied to the 1298 data set samples, was successful in 79 percent of the cases.

More precisely, the class Ci was well predicted in 82 percent of the 714 samples, and the class Ci was well predicted in 75 percent of the 584 samples. Unfortunately, we did not test the discrimina­tion on an independent data set; nevertheless, we do think that our results are of good statistical significance because of the large size of the sampling data set (1298 samples) and the small num­ber (16) of selected predictors. So, the remaining degrees of freedom, 1298-16-1 = 1281, are still very important and are a factor in statistical accuracy.

For the prediction of precipitation occurrence during 6 hr on day J and 6 hr on day J + 1, the selected predictors are shown in Fig. 2. We consider these results as encouraging. We used sea level and ground level pressure measures and it is known that this information is not sufficient to explain the occurrence of coming precipitation since the latter has its origin at high altitude. These results therefore could be improved by using appropriate pre­dictors at altitude, such as geopotential heights. As we pre­viously mentioned, our results are only available for the airport of Le Bourget during winter season (December, January, February); so our forecasting model is mostly concerned with frontal winter storms and very little with convective storms which principally occur during summer season (for predicting precipitations from convective storms, it would be necessary to use vertical instability measures of the atmosphere).

Furthermore, our results are specific to Le Bourget, which is in a temperate region. The same study, but relative to Brest, which has an oceanic climate, would give other results, and, surely, better than those obtained for Le Bourget, since the a priori frequency of daily occurrence of rainfall at Brest reaches 0.70, while it was merely equal to 0.45 at Le Bourget (for winter season).

Thus, local forecasting results are to be carefully interpreted in connection with the local characteristics of the specific area concerned. Several authors have obtained interesting results in predicting precipitations with statistical models. Iver A. Lund applied stagewise and stepwise regression procedures to a prob­lem of estimating precipitation in California [6]. He used geo­potential heights (200, 500 and 850-mbar surfaces) for six classes of prediction of precipitation amounts and obtained good results. W. K. Klein, F. Lewis, F. Marshall and K. Cole developed an operational system for automated prediction of precipitation probability at 108 weather stations covering the whole of the U.S.A. [4]. As predictors, they used geopotential heights (from 1000 mbars to 500 mbars), dewpoints at various levels and pre­vious precipitations for two classes of prediction (occurrence or nonoccurrence of rainfall) and also obtained good results. In France, M. Duband also used statistical models for predicting precipitation amounts [10]. Empirical predictions of precipita­tion are also made by specialists from weather charts, but, unfortunately, we do not know their exact methods and the rate of success of their forecasts.

A Synoptic and Middle Range Geopotential Fore­casting Problem Using Principal Component Analysis and Multiple Correlation Analysis

(a) The Prediction Problem; the Data Set. This problem was to predict the evolution of the 500-mbar geopotential heights over Western Europe, for the next coming week. From a mete­orological point of view, the 500-mbar geopotential heights are very interesting because they characterize the mean state of the atmosphere; their time and spatial evolution is rather regular because they are not obscured by local disturbances coming from relief, vegetation, microclimates, etc. So, the 500-mbar geopotential surface gives a mean synoptic meteorological state which is of great importance.

Journal of Dynamic Systems, Measurement, and Control M A R C H 1976 / 51

Downloaded From: http://dynamicsystems.asmedigitalcollection.asme.org/ on 10/01/2013 Terms of Use: http://asme.org/terms

Fig. 3 Northern Hemispheric grid points and spatial delimitation of tp© prediction domain over Western Europe

For our study, we used an important data set of daily values, from 1963 to 1969, provided by the French Meteorological Office. Bach day, at 0 hr UT, we keep track of 1977 heights (500-mabr level), corresponding to 1977 grid points, as shown in Fig. 3. These grid points are regularly situated on a stereo-graphic projection of the Northern Hemisphere. On this figure is represented a square, ABCD, which delimits the spatial pre­diction domain in which we are interested, roughly speaking Western Burope. Also on this figure is represented a simplified grid composed of 63 circled points. As we shall see later, only candidate predictors corresponding to these 63 grid points will be considered. We justify this simplification by two ways:

1 Practically, it is quite impossible to deal with the totality of the 1977 daily geopotential values, from 1903 to 1969. This represents an enormous quantity of information, the statistical treatment of which would require too much computing time even with the biggest computers. 2 Statistically, the complete set of information is redundant be­cause of the high degree of spatial and time correlation between many of the grid point values.

We then decomposed our prediction problem into two suc­cessive steps: the first step was defining what we wished to predict; the second step consisted of trying to predict what we defined in the previous step.

Our data set. has been filtered in order to eliminate cyclic seasonal variations (by subtracting from original data, seasonal means calculated by interpolation between 12 monthly means) and has been decomposed into two distinct data sets: the first, called the training set, covers the period 1963-1967 and was used to elaborate the prediction model; the second, called the test set, covers the period 1968-1969 and was used to test this model.

(b) Characterization of th# Evolution of th® SOO-mbar Geopotential towel In Western Europe. Each day, at 0 hr UT, our prediction domain is characterized by 100 heights. During a week the evolution of the geopotential surface in this domain is then de-

Fig. 4 lso-(au X 100) coefficient lines for the f irst spatial componont Yi in tnts square ABCD of Fig. 3

fined by 7 X 100 = 700 heights. The statistical information contained in these 700 values is redundant because of their auto- and cross-correlations. The purpose of the characteriza­tion, then, is to reduce this large amount of data by using a minimum of variables which provides the best representation of the initial information. This reduction was obtained by means of principal component analysis; we here briefly state the object of this method: let us denote X a zero mean, stationary, stochastic vector, with n components Xi, X2, . . . , Xn.

Let us denotes Cxx 4 E(XXT)

Then, the purpose of the principal component method is to find new variables

and,

Yi, Yi, .. ., Yr, with r < n, such that

Yi & a<rX, eiT»i = 1, Yi = 1, 2, ...,r,

E(YiYi) = 0, Yi 5*je{l, 2, ...,r}

in order to maximize4 Ve £ J] Var [Yi] i~\

Solution: ai, az, . . ., a, are the r orthonormal eigenvectors of the positive definite symmetric matrix Cxx, corresponding to iM r largest (positive) eigenvalues Xi > As > As •. • > Xr. ThiUi we have

>-i

The quality of the reduction is usually measured by the propor­tion of total variance explained. This proportion is defined by

r

t - i

*E(Y) = mean of F

4Var [Y] ss variance of Y

82 / MARCH 1976 Transactions of the ASWIE

Downloaded From: http://dynamicsystems.asmedigitalcollection.asme.org/ on 10/01/2013 Terms of Use: http://asme.org/terms

Fig. S lso-(a:i X 100) coefficient lines for the second spatial compo­nent Yi in the square ABCD of Fig. 3

Fig. 6 lso-(a3i X 100) coefficient lines for the third spatial component VJ in the square ABCD of Fig. 3

If r = n, then Pe — 1. This method is very interesting when we obtain a good proportion of total variance explained Pe with a

small number r of components, In conclusion, the principal component analysis is a statistical

method which uses cross-correlations between a set of variables in order to obtain a smaller number of new variables, which provides most of the original information. For more details, see [11].

Let us return now to our particular meteorological problem. We applied the principal component method in two successive steps:

1 The first step is a spatial principal component analysis. The 100 daily heights, denoted X\, X2, . . . , Xioo, are transformed into r other variables, denoted Fi, F2, . . ., Yr. We obtained the following interesting results, with r — 3 :

• The first component \\ = ai rX, (Xy = [XiX2 . . . Xioo]), corresponds to a regular configuration of the 500-mbar surface, centered over Western Europe. The components an, i = 1,2, . . ., 100 of ai are shown in Fig. 4. Here, Xi/V, — 0.278.

' The second component Yi = a2rX amplifies the spatial variations of this 500-mbar surface between west and east. The components an, i = 1, 2, . . ., 100 of a2 are shown in Fig. 5. In this case, \,JV, = 0.199.

• The third component Y3 = a3TX amplifies the spatial varia­tions of this 500-mbar surface between south and north. The components a3„ i = 1, 2, . . ., 100 of as are shown in Fig. 6. Here, \3/Vs = 0.184.

Thus, with Yh F2, F3, we obtained P„ = 0.661; this is not a great proportion of total variance explained. However, this re-s u ' t is satisfying only because it was obtained with merely 3 components. Furthermore, the other remaining components * *> *s, etc. are of less statistical importance and will not be used; their physical significance is also less clear.

« The second step in the data reduction is a time principal component analysis. Let us first consider the component Fi M>d denote Y^J) as its value on day J. Y^J), Y^J + 1), . . .,

'(</ -f- 6) are transformed into s other variables, denoted Zn, n> • • •, Zu. We obtained the following result, with s = 3 : ' The first component £n = a n y . y , , (Yi r = [Yl(J)Yl(J + 1 ) '

T+k

- > clays

•J+k

ays

•5*-z>t

Fig. 7 Coefficients of the three first t ime components over a period of a week, of the three spatial components Vi, Yi and Yt

. . . Yi(J + 6)]) represents the time mean level of Fi during a week. To this component corresponds: Xn/Vai = 0.657.

• The second component Z12 = ai22' • Yi represents the mean first time derivative of Fi during a week. To this component corresponds: X12/F,, = 0.242.

The third component Za Yi represents the mean second time derivative of Yi during a week. To this com­ponent corresponds: \u/Vn = 0.065.

J°Urna! of Dynamic Systems, Measurement, and Control M A R C H 1976 / 53

Downloaded From: http://dynamicsystems.asmedigitalcollection.asme.org/ on 10/01/2013 Terms of Use: http://asme.org/terms

/ \ # / ^

•~^/ aoV

i i - 1

— T *

^ \ #1?

N^/* \ -

/ *\

1 •^-^.

* i

\ */

T*

Y 0 \

v *^0

/a J--*/

/*

H V

^

\

\ * V \ \ 1 ,

/* T5&-

/0T--1

n?-tfl

P JL-

/ *w X

/ • / ^\ /^

9l^Ar^ / " »

/ / * >™̂ \ . /

-"Off^

S ^ ^ T

£ / SJ^; J C 1 H 7 ^

P%A7 <J isr--l /

\ y^''

\ a J-lX/

^v"^.

Ltcl * \ ?0

A ir-'t

3> 1

s S A / ^

aVjX

5a l * t l * d pred ichors frr fi>rt*Ash'nq ZH H

- - - - - Z , .

Fig. 8 Selected predictors far forecasting the first time component of each of the three spatial components Y\, Y: and Yt, for the next coming Week

An, Xu, Xia are the eigenvalues corresponding to the 3 components Zn, Z\% and Z\%. Now,

V» - ti Var t ^ ^ + *)1 ft-0

o, Zu, Zu, and Zi3 explain 96.4 percent of the total variance of Fi(/), Yi{J + 1), . . . , Yi(J + 6); this is a very satisfying result. Analogous results were obtained for the time components of F2 and F3. For F2, we obtained 3 time components Zn = aa.r • Yz, Zn = »22T • y2 and Z ^ . y2, (y2r = [F2(J")F2(/ + 1) . . . Yi{J + 6)]), which explained 93.9 percent of the total variance of F2(/), Yt(J + 1), . . . , F»(/ + 6).

For F3 we obtained again 3 time components Z3l = asiT • Ya, Ztl = a32

r . y3 and £33 = «33r . y3, (y3r = [Y,{J)Y,(J + 1)

. . . F8( / + 6)]), which explained 95.5 percent of the total variance of Y,(J), YS(J + 1), . . . , F3( / + 6).

The components aiik, k = 0, 1, 2, . . . , 6, of the vectors ay, * = 1> 2, 3, i = 1, 2, 3, are shown in Fig. 7.

In conclusion, the 700 original variables X, (i = 1, 2, .. ., 100 for each of 7 days), which described the evolution of the 500-mbar geopotential surface over Western Europe during a week, have been reduced to 9 new Zij(i = 1, 2, 3; j = 1, 2, 3) variables which correctly describe the pattern of this evolution. This re­duction is very important and very useful for the next forecasting problem because we now do not have 700 variables to predict, but only 9.

(c) Selection of Good Predictors and Prediction by Multiple Linear Correlation Analysis. We only tried to predict the first time component of each of the three spatial components, i.e., Zn, Z21 and Zn. If these variables correspond to the week (/,

J + 1, ..., J + 6), the candidate predictors were the 500-mbar heights at the 63 grid points, shown in Fig. 4, each observed on the days / - l , J-2, ..., J-7. So, we had 63 X 7 = 441 candidal* predictors. The selection of good predictors was obtained with SELECT; the prediction is obtained by multiple linear correla­tion analysis. We here briefly state the object of this method: if Xi, X2, ..., Xn are the predictor's being used to predict F the linear correlation method provides n + 1 coefficients d, di, . . . , a„, computed from the learning data set, such that an «stimate F of F is given by

Y =• do + di-Xi + . . . + &nX„,

The accuracy of the correlation is usually measured by the multiple correlation coefficient B, defined by:

R 4 E([Y - E(Y)] • [F - #(F)])/VVar [F] • Var [F]

We have: 0 ^ R% ^ 1. Iffl2 # 0, there is a poor correlation but if B2 # 1, there is a very strong correlation.

In conclusion, the multiple correlation analysis is a statistical method which provides the best estimate of a given predictand as a linear function of given (or selected) predictors. For more details, see [11].

Prediction of Zn. SELECT provided a group of 20 predictors. R, calculated from the learning data set, was equal to 0.56. Tho test of the correlation, applied to the samples of the test data set, gave an R value equal to 0.40.

Prediction of Zn. SELECT provided a group of 10 predictors. R, from the learning set, was equal to 0.60. R, from the test set, was equal to 0.54.

Prediction of Z3i. SELECT provided a group of 15 predictors. R, from the learning set, was equal to 0.57. R, from the test set, was equal to 0.46.

So, for predicting Zu, Za, and Zn, 20 + 10 + 15 = 45 pre­dictors were selected from among 441 candidate predictors; -his is an interesting reduction. These selected predictors are shown in Fig. 8. We see that, on the average, the multiple correlation coefficients obtained took on values around 0.50; this is not very good statistically. Nevertheless, we are satisfied with these re­sults because of the great complexity of the meteorological fore­casting problem one week ahead. (This problem has not been very much studied and we cannot compare our results to those of others at this time.)

For predicting 500-mbar heights, we used only 500-mbar heights as predictors. Perhaps the predictions might be improved by using other geopotential heights as predictors, such as 700-mbar heights for instance.

Conclusion This paper illustrated the possibility of using statistical meth­

ods for weather forecasting problems. Though disconnected from physical considerations, the statistical approach is able to provide significant results in some cases, which are useful for the economist, the agriculturist, the ecologist, and finally l'ie

layman. This approach is particularly interesting for local and short range forecasts. We presented an example of the predic­tion of the occurrence of rain for tomorrow, for which the ftft" tistical prediction by means of discriminant analysis reached a rate of 80 percent success. In this same way, it is possible to predict other interesting meteorological variables, such as the occurrence of storms, or hail, or fog, for instance.

The statistical approach is also interesting for synoptic .in" middle range or long range forecasts. We presented an examP" of prediction of the mean evolution of the 500-mbar geopotential surface over Western Europe, for the next coming week. The correlation coefficient between predictions and true vaUW*

14 / MARCH 1976 Transactions of the ASME

Downloaded From: http://dynamicsystems.asmedigitalcollection.asme.org/ on 10/01/2013 Terms of Use: http://asme.org/terms

died 0.50. We hope to improve this result in the future by . kinE into account other predictors and in perfecting our pre­

diction model. Several authors have used analogous methods for long range

I recasts (a month ahead) and have obtained significant results rol In conclusion, we do think that the statistical approach is a useful tool for weather prediction.

Acknowledgments The present research was sponsored by the French Army

Huvenu D.R.M.E. (Direction des Recherches et Moyens d'Es-is). This study would not have been possible without the

meteorological data sets supplied by the French Meteorological Office. The authors are greatly indebted to Professors Foul­ard and Rabins for their kind help and encouragement.

References 1 Viaut, A., La Meteorologie, Collection que sais-je?, Presses

Universitaires de France, Paris, 1965. 2 Blanchet, J., Dettwiller, J., Fons, C , and Zeytounian,

"Etudes Critiques et Prospectives Sur la Basse Atmosphere, la Prevision A l'Echelle Locale," Societe Meteorologique de France, Convention D.R.M.E. No. 65.34.589.00.480.75.0, 1965, pp. 1-150.

3 Triplet, J. P., and Roche, G., Meteorologie GSnerale, Presses de la Meteorologie Nationale, Paris, 1971.

4 Klein, W. H., Lewis, F., Marshall, F., and Cole, H., "An Operational System for Automated Prediction of Precipitation Probability," International Symposium on Probability and Statis­tics in the Atmospheric Sciences, June 1-4, 1971, Honolulu, Hawaii, pp. 106-111.

5 Miller, R. G,, "Statistical Prediction by Discriminant Analysis," Meteoroligal Monographs, Vol. 4, No. 25, Oct. 1962, pp. 1-54.

6 Lund, I. A., "An Application of Stagewise and Stepwise Regression Procedure to a Problem of Estimating Precipitation ia California," Journal of Applied Meteorology, Vol. 10, Oct. 1974, pp. 892-902.

7 Grosjean, A., "Methode heuristique de selection de bonnes variables explicatives pour un modele de prevision par correlation multiple lineaire," C.R. Acad. Sciences Paris, t. 278, May 1974, pp. 1-3.

8 Draper, N. R., and Smith, H., Applied Regression Analy­tic, Wiley, New York, 1966.

9 Erikson, B., "Simple Methods for Statistical Prognoses," Statistical Analysis and Prognosis in Meteorology Proceedings of the W. M. O. Inter-Regional Seminar on Statistical Analysis and Prognosis in Meteorology, Paris, Oct. 8-20, 1962, W.M.O., No. 178.TP.88, Oct. 1962, pp. 107r114. t 10 Duband, D., "Reconnaissance dynamique de la forme des

situations meteorologiques; application a, la prevision quantita­tive des precipitations," Cours professe a, l'Ecole d'Hydraulique de l'Institut National Polytechnique de Grenoble, June 1970 (not published).

11 Cooley, W. W., and Lohnes, P. R., Multivariable Data Analysis, Wiley, New York, 1971.

A P P E N D I X SELECT: A Statistical Package for the Selection of Good Predictors

This package uses an original stepwise method in order to select, among a great number of candidate predictors denoted •*ii Xj, , . . ( Xn, the best ones as regards a specified criterion

which is to be optimized (for instance, a percent rate of success for a discriminant analysis or a multiple correlation coefficient for a correlation analysis).

The principle of this method is as follows:

First Step: Selection of the best individual predictor, i.e., the one which optimizes the criterion. Let us assume that it is Xu

Second Step: Selection of the best predictor, coupled with Xi. Let us assume that it is X2. Selection of the best predictor, coupled with X2. Two casses are now possible:

• If it is Xi, the second step is finished with the selection of the group [X2, Xi)

• If it is not Xh and let us assume that it is Xs, we select now the best predictor, coupled with Xs. If it is X2, the second step is finished with the selection of the group (X3, X2). Otherwise, we go on with this algorithm till we obtain a stable group (Xi, Xs) for which: firstly, Xi is the best predictor, coupled with X}: Secondly, Xj is the best pre­dictor, coupled with Xi.

Third Step: Let us assume that (X2, Xi) is the stable group found at the second step.

• Selection of the best predictor, coupled with X2 and Xi. Let us assume that it is X3.

• Then, we examine, one by one, each of the selected pre­dictors in order to see if, eventually, we can replace them by better ones. For instance, we might obtain the following sequence:

Xi, X\, A3 T

X3 means that X3 is the best predictor, coupled with X2 and Xi

r Xl, A3, A2

T

Xs, X2, Xt T

Xi, A4j A3

T

Xi, Xs, X2; the group (X4, Xs, X2) is stable and the third step T is finished.

{And so on for the following steps.) This original selection method works well and often provides

better results than those obtained with most other stepwise methods (forward and backward for instance). The correspond­ing package, called SELECT, is written in FORTRAN. I t has been, carefully optimized from both numerical and computa­tional points of view. I t provides good predictors, selected among a great number of candidate ones, with a very short computing time on digital computers. For instance, for a problem of selec­tion among 123 candidate predictors, SELECT provided the "best" groups of 1, 2, . . ., 20 predictors while requiring a total of 17 sec computing time on an IBM 360/67.

This package is particularly useful for discriminant or regres­sion problems.

Journal of Dynamic Systems, Measurement, and Control MARCH 1976 / 55

Downloaded From: http://dynamicsystems.asmedigitalcollection.asme.org/ on 10/01/2013 Terms of Use: http://asme.org/terms