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published in: Meteorologicke Zpravy,
vol 47 (1994), No.4, p.103-112
language: English + Czech abstract; includes 7 figures and
5 tables with English+Czech captions
Martin Dubrovsky
Institute of Atmospheric Physics, Hradec Kralove‚
Czech Republic
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Forecasting thunderstorms (TS) for time projections
longer than the characteristic lifetime of TS cells often
relies on semiempirical methods relating TS activity with
predictors derived from the latest aerological data and/or
output of the numerical weather prediction model. A
prominent role among thunderstorm predictors belongs to
various stability indices and related thermodynamical
variables derived from a single station aerological
sounding. The indices may be used either autonomously or
together with information from other sources. Prognostic
equations relating the value(s) of the predictor(s) with
thunderstorm activity at given location and within given
period of the day and season of the year are typically
derived from the historical data sets. Let x = (x1,
x2,.., xr)
is the predictor vector, x IS_IN Xr,
and y IS_IN Y is a predictand. Given actual
observation of x, and the task to forecast y,
we may either estimate its value by a single number (deterministic
forecast) which is in some way optimal, e.g. minimizes
expected mean square error or maximizes likelihood, or we
may estimate the conditional probability P(y|x)
for each y IS_IN Y. The latter approach (probabilistic
forecasting) is seemingly more informative: it may be
simply converted to the deterministic forecast (postprocessing)
but it simultaneously includes information on the
uncertainty involved in the forecast. In case y is
a dichotomous variable, y IS_IN Y={0,1},
its conditional distribution may be fully expressed by
conditional probability of y taking value 1 which is
identical with a value of a regression function:
P(y=1 | x) = E(y|x) ( = r(x))................(eq.1)
where r(x) denotes a regression function. Having the
learning sample, S = {[xi,yi];
i = 1,..,n}, comprising previously observed
values of predictors and predictand, the probabilistic
prognostic function estimating the conditional
probability of event occurrence,
f(x) = (y=1 | x)
...................... (eq.2)
may be determined by regression analysis of the learning
sample. Quality of the prognostic function may be
expressed by reduction of variance (which is equivalent
with a coefficient of determination defined as a square
of a multiple correlation coefficient):
RV(f) = 1 - MSE(f)/MSEref......................(eq.3)
where MSE(f) = E(y - f(x))2 and MSEref
= E(y - y-))2. The value of RV
is maximized (MSE is minimized) if f(*) coincidates with
r(*) and RV=1 indicates 100% successful forecasts.
The present paper brings the most interesting results
found by analysis of 9y series of summer months
observations related to Prague. The main aims of the
analysis are (1) to assess the diagnostic and prognostic
potential of 38 thermodynamical predictors derived from a
TEMP-A data, (2) to select the most optimal combination
of predictors with use of three variable screening
techniques, and (3) to compare efficiency of four
regression techniques to establish relation between the
multidimensional predictors and thunderstorm activity.
The set of candidate predictors is supplemented by
persistence and information on front passages across
Prague.
The data archive available for the present study
consists of observations from warm months (May-August) of
1981-1989 period and include: TS observations in Prague,
frontal passages across Prague and aerological soundings
launched in Prague.
The dichotomous predictand, TS
The sample climatology of TS occurrence is displayed in
Fig. 1 and Fig. 2. It is seen (Fig. 1) that the mean
annual thunderstorm frequency both in Prague and its 100
km surroundings slightly increases during the learning
period. The increasing trend is statistically significant
at level ALPHA=0.018 for TS<12,18> and at level
ALPHA=0.002 for the mean (per day) number of stations
reporting thunderstorm (triangles in Fig. 1). The mean
annual frequency of days with TS occurrence in Prague (TS<0,24>=1)
displayed in Fig. 1 is higher than the average number of
stormy days for Praha-Karlov during 1946-1955 (28.6
stormy days for whole year, 24.1 stormy days for May-August;
[18]). This may be caused partly by different frequency
of thunderstorm occurrence within the two periods and
partly by different definition of thunderstorm occurrence
in the two studies ([18] vs. present study). Since the
variability of the series displayed in Fig. 1 mostly well
corresponds with the standard error of the single year
frequency of TS days we may assume that none of the years
of the learning period was anomalous from the point of
view of TS activity.
The mean annual course of TS activity (Fig. 2) displays
several maxima. The location of the best marked maxima (May
20, June 20, July 15, August 10-14) resembles the near-1-month
periodicity of TS occurrence in South Bohemia (based on
records from 1956-1986 [17] and 1921-1970 [19]) and even
in East Slovakian Lowlands ([7], 25y series). This
suggests that the peaks need not be a product of
randomness. The discussion on an origin of the peaks is
beyond the scope of present paper but may be partly found
in above referred papers.
Three groups of predictors are considered:
1) Thermodynamical predictors are derived from the
single station TEMP-A data (Tab. I) and include various
stability indices, measures of parcel energy, heights of
selected levels important from the point of view of cloud
formation and several characteristics derived from a 1D
steady state convective cloud model [16]. The predictors
are derived from both midnight and midday soundings.
2) Persistence predictors are defined and labelled
analogically with the predictand except that they relate
periods prior the target forecast period. The boundary
times of the intervals relating to the previous day are
preceded by minus sign: e.g. TS<-18,6> represents
TS occurrence in the 12h period starting at 18 GMT on the
preceding day. This group includes variable PERS giving a
number of stations within 100 km from Prague having
reported TS occurrence during 24h interval starting at 6
GMT of the previous day. When developing prognostic
equations, we consider only those persistence predictors
which cover periods ending at least 6h prior the start of
the target forecast period (i.e., e.g., PERS may be used
for target periods starting at 12 GMT but not for periods
starting at 6 GMT).
3) Frontal predictors. Since an occurrence of
fronts may reduce prognostic potential of predictors
derived from the pre-frontal soundings, and since non-negligible
portion of thunderstorms is closely associated with
frontal systems, passages of the fronts treated as binary
variables (0/1) were included into the set of predictors:
Predictor F<t1,t2> takes value 1 if at least one
front (cold or occluded) passes across Prague within time
interval <t1,t2>. The set of frontal predictors
covers 6-h, 12-h and 24-h intervals of the day.
The date-conditioned climatological probability of TS
occurrence was also tested as a predictor. However, as
the predictive power of the climatological predictors was
found to be totally insignificant, they were no longer
considered in subsequent experimentations.
Individual predictive power of the most effective
predictors is given in Tab. II and Tab. III in terms of
RV score related to the dichotomous prognostic function:
fi(xi) = P^(y=1|xi<=x)
= n((y=1) AND (xi<=x))/n(xi<=x) for
xi<=x
(eq.4) fi(xi) = P^(y=1|xi>x)
= n((y=1) AND (xi>x))/n(xi>x) for
xi>x
where xi is the variable being tested as a
predictor, x is its threshold value which maximizes the
sample RV score and n(*) is the number of elements of the
learning sample satisfying conditions given in the
parentheses. The standard error of the individual
predictive power, RVi, is about +/-0.03 for
RVi being close to 0.2 and +/-0.01 for RVi
being close to zero, as it was determined by Monte Carlo
simulations. Although the dichotomous prognostic function
is rather rough approximation of the regression function,
the obtained skill scores provide a good basis for
comparison of predictive power of the predictors:
(a) effect of the length of time projection
The effect of the length of time projection upon
predictive power of thermodynamical predictors may be
observed from two viewpoints:
(i) Having predictors derived from soundings
obtained at t0, we examine how the predictive power
depends on the delimitation of the target prognostic
period, <t1,t2>.
(ii) Having target prognostic period <t1,t2>,
we examine how the predictive power depends on the time
of acquiring the source sounding data.
Generally it holds that the predictive power decreases
with increasing length of time projection. In case of the
latter viewpoint (ii), it is seen from Tab. II and Tab.
III that elongation of the time projection leads to the
decrease of the RV score, which is best pronounced in
case of the afternoon target period: the midnight
predictors (Tab. II) gain much lower scores compared to
the midday predictors (Tab. III). In case of the former
viewpoint (i), one must consider several restraints:
Firstly, occurrence of certain weather phenomenon within
two distinct periods implies two distinct phenomena (night-time
thunderstorms are generally of different origin than the
afternoon thunderstorms). Further, comparison of the
prediction skill is complicated by different mean
climatological probability of event occurrence within the
two distinct target periods. In addition to lower
efficiency of regression techniques when applied on rarer
phenomenon, we deal with the effect of the mean frequency
of event occurrence on the RV score. To demonstrate this
effect, one may consider the following model: Let r(x) =
P(y=1|x) is given regression function which relates
binary predictand, y, and continuous predictor (scalar),
x, and px(*) is a probability distribution of
the predictor which varies in a position parameter: px(x)
= PHI(x-m). Then it may be found that RV is maximized for
such mi for which P0 = P(y=1) ( = INTEGRALr(x)PHI(x-m)dx)
= 1/2 and RV tends to 0 proportionally with (P0(1 - P0))0.5.
It means: having fixed relation between predictor and
predictand represented by r(x), attainable RV score
depends on the apriori distribution of x.
Bearing above restraints in mind we leave attempts to
compare the prediction skill related to TS occurrence
within distinct periods of the day. We just note, that
the significantly lower RV scores related to the night-time
and morning thunderstorms need not necessarily imply
lower predictability.
(b) Individual predictive power of thermodynamical
predictors.
The most powerful predictors derived from the noon
soundings are the stability indices of Faust, Showalter
and Adedokun. It may be noted that the only two indices
not considering humidity conditions (Simila and VT) are
not among fifteen best predictors derived from the noon
soundings (Tab. II), but they are among the best if
derived from the midnight soundings (Tab. III).
Assimilation of surface conditions into the definition of
the predictors may increase their predictive power, as it
follows from comparison of KMOD with K, TTMOD with TT,
ADED2 with ADED1. The former versions of the predictors
in each of the three pairs better account for the surface
conditions than the latter versions, and also gain higher
values of RV score. Predictive power of the midnight
predictors seems less dependent on the air humidity
conditions as indicated by dominance of the Simil„
index. Inclusion of the surface conditions does not
improve predictive power of the midnight predictors. On
the contrary, KMOD, TTMOD and ADED2 are even worse than
K, TT and ADED1 if derived from the midnight soundings.
The relation between two of the strongest predictors with
relatively low correlation is displayed in Fig. 3.
(c) The individual predictive power of the
persistence and frontal predictors is much lower compared
to the thermodynamical predictors. The maximal values of
the RV score were acquired by frontal predictors which
cover front passages within both target periods and 6h
interval preceding the target period. The maximum value
of RV score (0.07) is attained by F<6,18> when
applied to <12,18> and <12,24> target periods.
Of the persistence predictors (the persistence predictors
closely preceding the target period are not considered),
PERS attains the highest RV score (0.05), also for <12,18>
and <12,24> target periods. Similarly with the
thermodynamical predictors, the prediction skill of
frontal and persistence predictors is considerably lower
when applied on target periods with lower probability of
thunderstorm occurrence.
Having the binary predictand, set of candidate
predictors, learning sample of given size, and the task
to develop the probabilistic prognostic function relating
the predictand with the predictors, we stand two problems:
(1) selection of optimal set of predictors and (2)
determination of prognostic function best estimating the
conditional probability of event occurrence. The first
task aims to comprise the most of the information on the
predictand in the lowest number of predictors. The set of
selected predictors should be large enough to bear
sufficient information on the predictand, but not too
large to avoid overfitting data and instability of the
regression estimate. The second step involves choice of
the model (type of the function approximating the
regression function) and finding parameters of the model.
The choice of the model depends on the structure of the
data, types of the predictors and size of the learning
sample.
Three techniques are used in present work to select the
predictors: stepwise linear regression, stepwise logistic
regression and tree growing algorithm run in an SP mode.
The probabilistic prognostic function (2) was afterwards
estimated by REEP technique, logistic regression, k
Nearest Neighbors and Binary decision tree. The former
two techniques come from a family of the parametric
techniques which search for the regression function
within the set of functions of given type, the latter two
techniques come from the family of the robust techniques
which do not suppose any specific type of the predictor
distribution.
REEP (Regression Estimation of Event
Probabilities, [11]) approximates prognostic function by
linear regression function which is truncated to the <0,1>
range to avoid probabilities below 0 and above 1. The
REEP was originally designed by Miller [13] for multiple
category predictands. Applications of REEP to
probabilistic forecasting of various weather elements are
given in [12].
Logistic regression approximates the prognostic
function by logistic function: f(x) = {1 + exp[-(ALPHA+BETA'.x)]}-1
where ALPHA and BETA'=(BETA1,BETA2,...,BETAr)
are to be determined from the learning data (the prime
denotes the row vector here). The logistic regression
converges to the regression function, r(*), in case two
conditional distributions p(x|y=0) and p(x|y=1) have
normal distributions with the same covariance matrices.
The logistic regression was found superior to the linear
model in case the binary predictand was considered [1,4].
Method of k nearest neighbors (kNN) estimates the value
of the regression function by weighted average of k
observations selected from the learning sample which are
`closest' to the observation x, for which the prediction
is to be issued:
f(x) = SUMj=1,..,k qix(j) yix(j)
/ SUMj=1,..,k qix(j)
where ix(j) is the index of the j-th closest neighbor
selected from S such that D(x,xix(j-1)) <=
D(x,xix(j)) <= D(x,xix(j+1))
where D(*,*) is the distance of two vectors in the
predictor space; qix(j) is the weight assigned
to the ix(j)-th element of the learning sample. The
regression estimate is finetuned by choice of weight q,
distance D(*,*) and number of neighbors k. Based
on Monte Carlo simulations, an optimal value of k
was found to be about 2.(n)0.5 where n
is a number of elements in the learning sample. The tests
were made with various q-weights which smoothly
diminish to zero as the distance between the elements
increases. The best results were, however, obtained for
the q-weight being independent of the distance
between the neighbors. Two types of distance D will be
alternatively used:
(a) Weighted Mahalanobis distance:
DMAH(x,xi) =SUMk=1,..,rSUMl=1,..,r(xk-xik)ckCkl-1cl(xl-xil).................(eq.6)
where Ckl are elements of the covariance
matrix of x, c are weights proportional to
the individual predictive power (expressed by RVi) of the
respective component of x, xi*
are components of vector xi.
(b) Distance of projections of the two predictor
vectors upon the `best linear discriminant function':
Dbldf(x,xi) = |w^.(x
- xi)|...................(eq.7)
where w^ is the best linear
discriminant vector. It might be noted here that the
latter version of the D-distance is less robust. On the
contrary, greater robustness may be achieved by
transformation of components of x: x'j
= F-1[Rj(x)/(n+1)],
where x'j (j=1,..,r) are the
components of the new predictor vector, Rj(x)
is the rank of x in the learning sample
sorted according to the value of j-th component of
x and F is the distribution function
of normal distribution, N(0,1).
Construction of the binary decision tree (BDT) consists
in recursive partition of the predictor space aiming to
maximal discrimination of elements with event occurrence
from elements with event non-occurrence. The tree growing
algorithm used in the present study follows the one
described by Rez cov and Motl [16]. It was
further enhanced, tested and described in detail in [9].
Two versions of the growing algorithm are alternatively
employed for present tests:
(a) The algorithm run in an SP-mode allows only single
predictor based splits - the predictor space is
partitioned into the r-dimensional `rectangles'.
An advantage of the SP mode is a greater vividness of the
structure of the resultant tree and ability to reduce the
dimensionality of the predictor vector.
(b) The algorithm run in a BLDF-mode allows to partition
the predictor space by hyperplanes perpendicular to the
best linear discriminant functions calculated separately
in each node of the tree. In result, the predictor space
may be effectively partitioned in less number of splits.
The disadvantage of the BLDF-mode is its lower robustness.
For completeness, it should be noted here, that the
decision trees are often used to graphically visualize
human decision process and are therefore referred to as
simple expert systems. The structure of such trees is
often designed by human expert (e.g. [2,5,6,10]) which
contrasts with a present work where a computer algorithm
is employed to develop the tree from the learning data.
More general tree growing algorithm allowing for the
categorical variables may be found in [3].
Results of the variable selection analysis (Tab. IV)
show that the sets of predictors selected by both
screening techniques do not differ significantly. Number
of selected predictors generally decreases with
decreasing correlation between the predictors and
predictand. In case only thermodynamical predictors are
considered, not more then three predictors are selected.
If persistence and frontal predictors are included the
set of selected predictors is typically expanded by one
persistence and one frontal predictor. The profit due to
the persistence and frontal predictors is better
pronounced in case they supplement midnight soundings
data which are poorer correlated with the afternoon TS
occurrence. The first selected predictor in most cases
corresponds with the most powerful predictor for given
target period (cf. Tab. II and Tab. III). The skill
scores related to the standalone first-selected
predictors are higher than those given in Tab. II and Tab.
III, due to better approximation of the regression
function by linear and logistic functions compared to the
dichotomous prognostic function used in the previous
tables. Further increase of the skill score is due to
increasing dimension of the predictor vector.
The following tests were designed to compare the
performance of the four above described regression
techniques. The tests will deal with a prediction for the
afternoon period with use of a Perfect Prog approach (thermodynamic
predictors are derived from the noon soundings).
Additionally to the sets of predictors selected by the
stepwise linear regression and stepwise logistic
regression (Tab. IV), the third set of predictors was
derived from a tree grown in an SP-mode (Fig. 7; see [9]
for details):
X(lin) = (FI, POSsfc, F<12,18>, PERS)
X(log) = (FI, HEATsfc, ADED2, F<12,18>, PERS)
X(bdt) = (SICP, POSsfc, PERS, FI, F<12,18>)
Predictors in X(bdt) are arranged according to their
contribution to the total skill score. We may immediately
notice the good concordance between the three sets: the
first-selected predictor is in all cases the stability
index (correlation coefficient between FI and SICP is -0.95).
The second-selected predictors are the parcel energies
also coming from the group of the thermodynamical
predictors. All three combinations include both
persistence predictor (PERS) and frontal predictor (F<12,18>).
On each of the three predictor sets, regression
techniques described in paragraph 4 were applied.
To get a notion on the effect of errors following from
the limitness of the learning data sample, the multiple
cross-validation tests were designed: The set of 1052
elements was randomly split in two halves. The first half
served as a learning sample to develop a prognostic
function, the second half served as a testing sample to
evaluate an unbiased estimate of prediction skill, RV^.
The procedure was run ten times and the average and
sample standard deviation of RV^ were calculated. The
variability of RV^ results from two sources:
(1) Stability of an estimate of the regression function -
the prognostic function f(*) and its prediction skill, RV(f),
vary due to the variability of the learning sample. In
case of the proper choice of the parametric model or in
case we use robust methods, f(*) converges to r(*) and RV(f)
converges from below to RV(r). (2) Accuracy of sample
skill scores - RV^(f) varies about RV(f) due to the
limitness of the testing sample. Based on the author's
experience following from experimentations with randomly
generated data the second source contributes to the total
variability of RV^ about twice larger than the first
source - on the assumption of equal size of both learning
and testing samples.
The results displayed in Fig. 4 to Fig. 6 indicate that
the prediction skill in average increases with increasing
dimension of the predictor for all groups of predictors
and all tested techniques. Irregularities may be due to
relatively great dispersion of RV^ which also disables to
select reliably the most optimal statistical technique.
Anyway, the logistic regression seems to provide the most
consistent results - not only when the predictors were
selected by stepwise logistic regression which could
produce some positive bias involved in RV^. The good
performance of the logistic regression is believed to be
the consequence of the fact that the most of the
predictors is quite well behaved. Although some of the
predictors are far from being normal (e.g. POSsfc in Fig.
3), the quicker convergence of the logistic regression
seems to play greater role than the robustness of kNN and
BDT techniques. The REEP attains poorest skill. Although
the tree growing algorithm run in an SP-mode may cope
with highly dimensional predictor vector and selfishly
select the best combination of the variables it is seen
from the three figures that unnecessary high dimension of
the predictor vector may reduce quality of the tree.
Of the three sets of predictors, the one selected by
stepwise logistic regression gains the highest skill.
Prediction skill related to the predictors selected by
tree growing algorithm is about the same as for the
predictors selected by stepwise linear regression.
An explicit formulations of the logistic prognostic
function for the afternoon target period developed in a
Perfect Prog approach are given in Tab. V.
Of the predictors being tested, the stability indices
of Faust, Showalter and Adedokun are the most efficient
thunderstorm predictors. An optimal combination of
predictors selected by variable selection procedures
contains no more than five predictors: thermodynamical
predictors dominate, but the set of selected predictors
mostly include one persistence and one frontal predictor.
Of the regression techniques being tested to determine
the probabilistic prognostic function, the logistic
regression seems to provide best results. The preliminary
tests have verified, that the logistic function used as a
predictor is superior to all single indices even for 1989-1991
period (to be published). Since the differences between
the performance of the regression techniques (except of
the REEP) are only slight with respect to the variability
of the sample RV scores, one might also consider some
secondary features of the regression techniques, such as
vividness and requirements on computer resources: The
binary decision tree provides a vivid view into the
structure of the data. The kNN method is the most time
and memory consuming technique since it requires
permanently accessible data archive and may spend
relatively long time in search for the neighbors. On the
other hand kNN provides a simple mechanism of self-learning:
just enhancing the archive of the historical data
improves quality of the prognosis. Logistic regression is
the only one which extrapolates the value of the
probability prognostic function for the values of
predictors beyond the range comprised in the learning
sample. Thus it may better respond to the extreme values
of predictors, surely only in case the tails of the
logistic prognostic function sufficiently well
approximate the regression function. Further important
aspect of the regression techniques is their requirements
on the size of the learning sample: The typical property
of the robust techniques (here represented by binary
decision tree and k Nearest Neighbors) is their lower
rate of convergence to the regression function [8]. They
require more data to attain sufficient stability. On the
other hand, parametric models (logistic regression, REEP)
converge quicker and may consequently provide better
results than the robust techniques when applied on
learning sample of lower size, even if the parametric
model does not precisely suit the structure of data.
The results of the present work indicate very poor
performance of the midnight predictors for afternoon
thunderstorms. On the other hand, skill scores related to
the midday predictors are not entirely realistic -
assessment of the true prognostic potential of predictors
derived from a midday soundings would require
sufficiently long archive of the NWP model output, which
would allow to verify the Perfect Prog based prognostic
equations or possibly to design prognostic equations in a
MOS approach.
Acknowledgements. I wish to thank RNDr. Daniela
Rezacova, CSc. for providing the source computer code of
the convective cloud model. I am also indebted to her for
valuable suggestions while finalizing the manuscript of
the paper.
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87-90.
Tab I The list of predictors derived from the
aerological sounding. (references for Table I: [a] =
PEPPLER [14] or PEPPLER and LAMB [15]; [b] = REZACOVA and
MOTL [16])
Tab II The individual predictive power of 15
most powerful thermodynamical predictors derived from the
midnight TEMP-A data (t0 = 00 GMT).
Prediction skill is given in terms of the sample
reduction of variance related to the dichotomous
prognostic function (4). Size of the learning sample: N
= 993. <t1,t2> is the target
period for which the thunderstorm occurrence is forecast.
| <t1,t2> [GMT]
|
Predictor |<00,06> <00,12> <18,24> <00,24>
| <06,12> <00,18> <12,24>
-------------------------------------------------------
SIMILA | 0.04 0.02 0.04 0.09 0.04 0.09 0.11
ADED1 | 0.03 0.02 0.04 0.09 0.03 0.09 0.11
FI | 0.05 0.02 0.05 0.08 0.02 0.08 0.10
SI | 0.03 0.02 0.04 0.09 0.03 0.09 0.10
SICP | 0.03 0.02 0.04 0.08 0.03 0.09 0.10
CIIB | 0.02 0.01 0.03 0.09 0.04 0.09 0.10
EELccl | 0.04 0.01 0.03 0.07 0.04 0.08 0.10
VT | 0.03 0.02 0.04 0.09 0.02 0.08 0.09
TEI | 0.02 0.01 0.03 0.09 0.04 0.09 0.09
VZccl | 0.03 0.01 0.03 0.07 0.04 0.08 0.10
SWEAT | 0.02 0.01 0.04 0.07 0.02 0.08 0.09
SIHH | 0.02 0.01 0.03 0.07 0.02 0.08 0.09
PWBI | 0.01 0.01 0.02 0.07 0.04 0.08 0.08
Hccl | 0.02 0.02 0.03 0.06 0.03 0.07 0.08
K | 0.03 0.03 0.05 0.05 0.01 0.05 0.07
Tab III The same as in but for predictors being
derived from the midday TEMP-A data (t0 = 12
GMT; N=1052). The times preceded by plus sign relate to
the subsequent day.
| <t1,t2> GMT
|
Predictor |<12,18> <12,24> <+6,+12> <12,+12>
| <18,24> <24,+6> <24,+12>
--------------------------------------------------------
FI | 0.23 0.08 0.26 0.02 0.02 0.02 0.23
ADED2 | 0.22 0.06 0.26 0.01 0.02 0.02 0.23
SICP | 0.24 0.05 0.23 0.00 0.02 0.02 0.21
SWEAT | 0.21 0.06 0.20 0.01 0.01 0.01 0.16
Kmod | 0.19 0.07 0.20 0.01 0.01 0.02 0.16
VZlcl | 0.18 0.05 0.18 0.03 0.01 0.03 0.18
TTmod | 0.19 0.05 0.19 0.00 0.02 0.02 0.18
POSsfc | 0.19 0.06 0.19 0.01 0.01 0.02 0.16
JEFF | 0.21 0.04 0.20 0.00 0.02 0.01 0.15
SI | 0.18 0.05 0.19 0.01 0.01 0.01 0.17
ADED1 | 0.18 0.05 0.19 0.00 0.01 0.01 0.17
K | 0.16 0.07 0.19 0.01 0.01 0.01 0.16
CIIB | 0.17 0.05 0.16 0.01 0.01 0.02 0.16
TT | 0.18 0.04 0.17 0.00 0.02 0.01 0.15
VZccl | 0.16 0.04 0.16 0.02 0.01 0.03 0.15
Tab IV Predictors selected by stepwise linear regression and stepwise logistic regression. The predictors are screened out of (a) only thermodynamical predictors; (b) all predictors (thermodynamical, persistence, frontal). The numbers in the parenthesis are RV-scores related to the predictor vector consisting of all predictors ahead of the number. The value is based on the developmental sample, so that it may little bit overestimate the true quality. The stepwise logistic regression was run only for 12-h long target periods regarding the demands on computer-time resources.
================================================================================ t1,t2 |sel | candidate predictors: | candidate predictors: |(*1)| thermodynamical variables | thermodynamical variables, | | | persistence, front passages ================================================================================ Thermodynamical predictors are derived from 00 GMT soundings -------------------------------------------------------------------------------- 00,06 |lin |VZccl (0.044), FI (0.053) |VZccl (0.044), | | |F<-12,12> (0.055) -------------------------------------------------------------------------------- 06,12 |lin |SICP (0.023) |TS<-12,00> (0.022), | | |VT (0.036), F<6,12> (0.046) -------------------------------------------------------------------------------- 00,12 |lin |FI (0.050), LFCsfc (0.059), |FI (0.050), F<-12,12> (0.059), | |ELsfc (0.069) |TS<-12,-18> (0.068) | | | |log |SICP (0.053), LFCsfc (0.061), |SICP (0.053), | |Hccl (0.070) |F<-12,12> (0.058), | | |TS<-12,-18> (0.067), | | |Hccl (0.076) -------------------------------------------------------------------------------- 12,18 |lin |SIHH (0.112), VT (0.132) |SIHH (0.111), | | |F<6,18> (0.159), | | |PERS (0.175), VT (0.188) -------------------------------------------------------------------------------- 18,24 |lin |SIMILA (0.051), Tlcl (0.060) |SIMILA (0.051), | | |F<6,18> (0.075), | | |F<18,24> (0.092), Tlcl (0.102) -------------------------------------------------------------------------------- 12,24 |lin |SIMILA (0.126), SIHH |SIMILA (0.126), | |(0.146) |F<6,18> (0.169), | | |SIHH (0.189), | | |TS<-18,6> (0.201) | | | |log |SIMILA (0.134), SIHH (0.157), |SIMILA (0.134), | |LCLsfc (0.167) |F<6,18> (0.190), | | |SIHH (0.214), PERS (0.228) -------------------------------------------------------------------------------- 00,24 |lin |SIMILA (0.142), SIHH |SIMILA (0.142), | |(0.162) |F<6,18> (0.179), | | |SIHH (0.199), | | |TS<-12,-18> (0.208) -------------------------------------------------------------------------------- -------------------------------------------------------------------------------- Thermodynamical predictors are derived from 12 GMT soundings -------------------------------------------------------------------------------- 12,18 |lin |FI (0.228), POSsfc (0.284) |FI (0.230), POSsfc (0.283), | | |F<12,18> (0.301), PERS | | |(0.313) | | | |log |FI (0.279), HEATsfc (0.338), |FI(0.279), HEATsfc (0.338), | |ADED2 (0.357) |ADED2 (0.357), | | |F<12,18> (0.378), PERS(0.387) -------------------------------------------------------------------------------- 18,24 |lin |FI (0.095), POSsfc (0.107) |FI (0.095), F<0,24> (0.112), | | |POSsfc (0.121) -------------------------------------------------------------------------------- 12,24 |lin |FI (0.260), POSsfc (0.304) |FI (0.260), POSsfc (0.303), | | |F<12,18> (0.322), PERS | | |(0.333) | | | |log |FI (0.312), ADED2 (0.353), |FI(0.312), ADED2(0.353), | |CCLsfc (0.370) |F<0,24>(0.375), PERS(0.388), | | |CCLsfc(0.397) -------------------------------------------------------------------------------- 24,+6 |lin |EELccl (0.040) |EELccl (0.040), F<12,36> | | |(0.055) -------------------------------------------------------------------------------- +6,+12|lin |POSsfc (0.028) |POSsfc (0.028) | | | -------------------------------------------------------------------------------- 24,+12|lin |VZccl (0.040) |VZccl (0.040), F<12,36> | | |(0.055) | | | |log |ADED2 (0.044), CCLsfc |ADED2 (0.044), F<12,36> | |(0.054) |(0.055) -------------------------------------------------------------------------------- 12,+12|lin |FI (0.240), POSsfc (0.282) |FI (0.241), POSsfc (0.281), | | |F<0,24> (0.299) ================================================================================
(*1)variable selection technique:
Tab V Prognostic logistic functions for
prediction of afternoon thunderstorms based on predictors
derived from the noon aerological soundings.
Fig
1. The annual (May-August) frequency of thunderstorm
occurrence in Prague and its 100 km surroundings.
Triangles: daily average number of stations within 100 km
reporting TS. Upper heavy solid line: number of days with
at least one station reporting TS. The other 5 series
relate to Prague and represent number of days with TS
occurrence within given part of the day. The rectangles
on the left hand side of the graph demarcate
Fig.2
Sample climatological probability of thunderstorm
occurrence related to given day of the year and period of
the day. The curves represent 11-days running averages of
observed frequency of TS occurrence during 1981-1989. An
uppermost curve relates to thunderstorm occurrence at
least in 1 station within 100 km surroundings of Prague
and 24-h interval (starting at 06 GMT). The other curves
relate to TS occurrence in Prague within time interval
given in the brackets to the left of the respective time
series and `error' rectangle. The height of the
rectangles corresponds to
Fig
3 Scatter plot of FI vs. POSsfc derived from the
midday soundings. Thunderstorm occurrence in Prague
within 12h interval following the sounding is expressed
by the cross. Sample conditional distributions of FI and
POSsfc are represented by curves adjacent to the
respective axes: dashed lines are for TS occurrence
within the 12h interval, solid lines are for non-occurrence
of TS.
Fig
4. Comparison of performance of six regression
techniques. The vertical bars demarcate dispersion
Fig.5
The same as in but with predictors being selected by
stepwise logistic regression: X(log) = (FI,
HEATsfc, ADED2, F<12,18>, PERS).
Fig.6
The same as in but with predictors being selected by
binary decision tree: X(bdt) = (SICP, POSsfc,
PERS, FI, F<12,18>).
Fig.7
The binary decision tree developed in a Perfect Prog
approach designed for prediction of thunderstorm
occurrence in Prague in the afternoon (Predictand = TS<12,18>).
The figures in terminal nodes (ellipses) give the total
number of elements of the learning sample falling into
the terminal node (denominator = Nterminal)
and number of those with event occurrence (numerator = N(1)terminal).
The plus/minus signs below the decision nodes (rectangles)
show where to proceed if the respective decision rule is
satisfied (+) or not satisfied (-).