Directory UMM :Data Elmu:jurnal:T:Tree Physiology:Vol15.1995:
Tree Physiology 15, 765--774
© 1995 Heron Publishing----Victoria, Canada
Nonlinear regression-typological analysis of ecophysiological states of
vegetation: a pilot study with small data sets
V. T. PEREKREST,1 T. V. KHACHATUROVA,1 I. B. BERESNEVA,1 N. M. MITROFANOVA,2
E. KÜNSTLE,3 E. WAGNER4 and L. FUKSHANSKY4,5
1
2
3
4
5
Economic-Mathematical Institute, Russian Academy of Science, 1 Chaikovskistreet, St. Petersburg, Russia
Technical University of St. Petersburg, 29 Polytechnic Street, St. Petersburg, Russia
Institut für Waldwachstum, University of Freiburg, D-78 Freiburg, Germany
Institute of Biology II, University of Freiburg, Schänzlestrasse 1, D-78 Freiburg, Germany
Author to whom correspondence should be sent
Received August 11, 1993
Summary The interactions of environmental factors associated with forest decline were analyzed by a modified multidimensional scaling method. The method subdivides the entire
data set into homogeneous classes; linear regression is then
applied within each single class. A nonlinear picture of the
interdependence of the effects of different factors is developed
as a composite of the contributions from each single class. The
analysis was performed on a restricted data set, and the results
compared with some expected effects and with results obtained
by standard linear regression. Even with the limited data set,
multidimensional scaling not only explained expected effects
but also revealed new information. We conclude that the
method will be useful for analyzing complex time series data
because it is able to detect complex interactions between
environmental variables that affect physiological parameters.
Keywords: forest decline, functional multidimensional scaling,
interdependence of the effects of environmental factors, multicausal syndrome.
sky 1992). To develop a comprehensive statistical analysis of
the correlations between environmental parameters and the
physiological characteristics of the system, we need tools for
the quantitative description and classification of the physiological state of a plant under the measured environmental
conditions.
In this paper, we have applied a general approach to nonlinear functional multidimensional scaling. This approach has
been used in a restricted way in ecology, medicine, economics,
production control and other fields (Bechtel 1978, Sibson
1979, Takane 1981, Shepard 1989). Some problems that precluded its wide application have only recently been resolved
(Perekrest 1989, Khachaturova and Perekrest 1990). We assessed the advantages and limitations of the method based on
an analysis of real sets of environmental data. Because of the
small number of parameters in these data sets, this study was
limited to a comparison of our results with some expected
effects and also with those of conventional linear regression
analysis.
Introduction
Description of method
It seems increasingly likely that air pollution, drought and high
irradiances are the major factors contributing to the ‘‘new forest
decline’’ syndrome. The syndrome can occur in forest ecosystems of different structure as a result of complex interactions
between several stress-causing factors that are dependent on
climatic and soil conditions. The complex, finely tuned biochemical, physiological and genetic control mechanisms that
underlie forest decline make it difficult to obtain reliable information about the contributions of single environmental factors
and about the character of their interactions.
A quantitative analysis of the effects of environmental factors is necessary before a strategy can be developed for the
prevention and control of forest decline. A rational system for
monitoring the environment is also needed as is the development of a sound basis for statistically analyzing data (Fukshan-
To describe the method of functional multidimensional scaling, let us consider data obtained from many similar observations. Each observation provides measured values of N
different parameters and represents the state of a single object
from a population of similar objects, or a current state from a
time series of states of a single object. A typical example is a
series of measurements of N environmental and physiological
parameters (e.g., radiation, air temperature, soil temperature,
humidity, photosynthesis, transpiration) of a plant community,
repeated at different times. Parameters of qualitative origin,
assuming discrete values (e.g., degree of oxidative damage) as
well as all-or-none variables (e.g., irrigated versus nonirrigated
area), are also permitted.
The data are used to build an aggregate of points in the
N-dimensional space of the studied variables. The space is
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PEREKREST ET AL.
characterized by a metric, i.e., the distance between any two
points can be defined. The distance between two points can be
calculated, for example, as the sum of squares of differences
between the values of corresponding parameters (which for
this purpose are scaled to zero mean and unit variance). In
addition, a stochastic sample measure arises in the N-dimensional space, as a result of the spatial pattern of the aggregate
of points. The sample measure reflects the density of points in
different areas of the N-dimensional space. The larger the data
sample, the better the sample measure approximates the actual
distribution of the values of N parameters.
In principle, the distribution of points in the N-dimensional
parameter space, with known distances between any two
points, contains all the information that can be extracted from
the measurements. Our goal was to derive quantitative correlations between independent parameters (e.g., temperature,
radiation) and their combinations on one hand, and the parameters supposed to be affected by the independent parameters (e.g., photosynthesis) on the other. We also determined
how the effect of one parameter was influenced by other
parameters.
The simplest procedure is to perform a linear regression over
the entire data set; that is, to regress dependent against independent parameters, and thereby obtain ‘‘average’’ statements
about the degree and direction of interconnections between the
parameters. However, such information is usually of limited
value because it does not reflect the true interconnections,
which are nonlinear and have a heterogeneous structure over
the entire space.
Nonlinear functional scaling overcomes this difficulty by
subdividing the whole data set into classes with homogeneous
structure. These classes are composed of neighboring states
that have approximately the same parameter values and, in
addition, similar relationships among state parameters. Within
each class, linear regression reveals these relationships. The
global picture of the nonlinear relationships between the parameters over the entire space is assembled from the linear
contributions of the single classes.
Thus, the core of the multidimensional scaling method is the
subdivision of the data set into homogeneous classes. The
scaling function f: X → R2, a nonlinear mapping of the initial
N-dimensional parameter space X = {xi} into a plane (or
generally, into an Euclidean space of small dimension), is
constructed so that the binary distances between the points are
preserved as much as possible. Formally written, this means
finding a scaling vector-function f that minimizes the function:
V( f ) = (( f(x) − f(y))2 − r (x,y))2 dµ(x) dµ(y),
where r(x,y) is the distance between objects x and y in the
N-dimensional space X, and µ is the joint distribution of N
parameters (approximated by the sample measure). The neighboring points in the plane can be easily found because they are
also neighbors in the initial N-dimensional space. This is the
first step in constructing homogeneous classes. The second
step is to identify the states with similar interconnections
between parameters, performed as follows.
The recently developed theory of functional multidimensional scaling (Perekrest 1989, Khachaturova and Perekrest
1990) provides the best possible f mapping within the broad
class of all measurable and integrable functions to the fourth
degree. The theory also permits estimation of the distortion of
the distances. If the distortion is too large, the dimension of the
image space should be increased (i.e., the 3-dimensional rather
than the 2-dimensional space should be considered). The scaling function can be differentiated with respect to the measured
parameters. Comparison of its partial derivatives in different
areas of the initial space reveals the character of interconnections between the measured parameters in these areas and
provides the basis for the second step in constructing the
homogeneous classes. For this purpose, we consider a system
of vectors {Vi(x)}, where each vector Vi(x) corresponds to
parameter i of the entire N-dimensional space. Coordinates of
Vi(x) are determined as (Vi(x))j = (∂fj / ∂ xi), where fj(x) is the jth
coordinate of the vector-function f (when the range of values
of f is a plane j = 1or 2). Vector Vi(x) is interpreted as the vector
of the movements of the image of point x on the model plane
when the ith coordinate of this point in the initial parameter
space is increased by a unit, while all other coordinates remain
fixed. The system of vectors {Vi(x)} may be different for
different measurements of x, reflecting different types of interrelation between the parameters. A homogeneous class by
definition can only contain points with similar interrelations,
i.e., points with similar vectors {Vi(x)}.
Linear regression analysis is then performed within each
class. The combined picture of linear models of all classes
provides the nonlinear model for the entire space of the measured parameters. This global model yields information about
the interconnections of the parameters and the modifications
of these interconnections over the entire parameter space.
Although determination of the contributions of single external factors and their combinations to an ecophysiological response can be approached by different statistical methods, the
complexity of both the noncontrolled experimental conditions
and the nonlinear objects with memory (i.e., objects with
responses that also depend on previous treatments) should not
be underestimated. Because the approach used must be able to
treat a dynamic process that is affected by external factors in a
nonlinear way, and must also be able to accommodate the
effects of different factors that are both interdependent and
dependent on the process history, we conclude that elementary
statistical approaches, for example, correlation coefficients or
linear regressions between two or several quantities, are not
appropriate (cf. Burgeois et al. 1992).
More complex treatments, especially those dealing with
grouping of data with respect to some expected similarity
within a group are required. An example of such a treatment is
the clustering analysis of multivariate data sets in comparative
plant ecology used by Grime et al. (1988). The rationale for
this treatment is that because certain attributes of a group of
species evolve through simultaneous selection (a composite
response to a set of external environmental pressures), one can
expect clustering with respect to these attributes. However, the
analysis of grouping solves only one part of the problem.
NONLINEAR REGRESSION-TYPOLOGICAL ANALYSIS
Additionally, weighting of the contributions of single external
factors within groups (and therefore over the entire parameter
space) is required.
Recently, several studies have shown that direct nonlinear
regressions of entire data sets can yield important information
about the interaction of external factors in many situations
(e.g., Hinckley et al. 1975, Hall 1982, Penning de Vries 1983,
Chen and Kreeb 1989, Kreeb and Chen 1991). Although this
approach is restricted to a few parameters, we conclude that it
is a useful analysis that can supplement more complex analysis
and can be performed with standard software. However, there
are some limitations when applying straightforward nonlinear
regression analysis to data observed in a natural environment.
In contrast to multidimensional scaling, which reveals ‘‘natural’’ data subsets (homogeneous classes) having both parameter values and parameter interaction rules nearly constant,
straightforward regression analysis requires a superimposed
subdivision of the parameter values; however, there is no
guarantee that, within an interval between two subsequent
values of a parameter, the interaction rule remains constant.
Another limitation to the straightforward regression treatment
is the use of a fixed set of nonlinear model functions (predominantly linear transformations, and exponential and power functions are used). Thus, an investigation of the interaction of
different environmental factors under natural conditions requires a complex approach combining elements of both typological and regression analyses. The only method that does
not need to be supplemented by other methods is multivariate
time series analysis (see, for example, Anderson 1971). This is
a family of procedures that requires modification for each class
of problems. The time series procedures applied to stochastic
processes with memory (so-called procedures with distributed
lags) are especially difficult to construct and analyze.
Experimental data sets
Linear regression
We analyzed time series measurements made at the experimental station of the Institut für Waldwachstum of the University of Freiburg at Schauinsland at an elevation of 1230 m.
Growth and metabolic processes of individual trees were
monitored under the influence of ozone in the natural environment. Climatic variables were continuously sampled in parallel with measurements of CO2 gas exchange (photosynthesis
and respiration) and ambient ozone concentrations. To investigate the impact of ozone, various ozone concentrations were
examined.
Two data samples were measured, the first one during 15
days in May 1991, the second one during 15 days in August
1991. Individual measurements were performed every 30 min.
The set of measured parameters included photosynthesis,
measured as CO2 uptake (CO2, mg), irradiance (I, µmol m −2
s −1), air temperature (T, °C), relative humidity (RH, %), ozone
concentration outside the gas exchange cuvette (O3A, µg m −3),
ozone concentration inside the gas exchange cuvette (O3K, µg
m −3), and current time (t). In addition, each measurement was
also characterized by the following qualitative parameters:
767
ozone into the gas exchange cuvette (+O3, yes/no), season
(Month, May or August), and day of the month (Day). As
illustrative examples, some time series of CO2, I, T and RH are
shown in Figure 1. Before analysis, each parameter was scaled
to zero mean and unit variance.
The entire data set was subdivided into clusters of data
measured under homogeneous conditions as follows. First, we
considered two overall clusters, hereafter referred to as Clusters 1 and 2. Cluster 1 contained 1863 points with no gaps for
the values of the parameters CO2, t, T, RH, I, +O3, Month and
Day. Cluster 2 contained 1234 points with no gaps for the
values of parameters CO2, t, T, RH, I, O3K, O3A, +O3, Month
and Day.
More detailed linear analysis was performed on four subclusters differing from one another with respect to the qualitative parameters Month and +O3: Subcluster 3 (516 points)
comprised measurements in August without ozone addition;
Subcluster 4 (554 points) comprised measurements in May
without ozone addition; Subcluster 5 (589 points) comprised
measurements in August with ozone addition; and Subcluster
6 (552 points) comprised measurements in May with ozone
addition.
Linear analysis of the data from Cluster 2 revealed significant dependence of CO2 only on parameter I (Table 1). Analogous analysis of Cluster 1 gave similar results to the analysis
of Cluster 2 performed excluding the parameter ∆O3 = O3K −
O3A (∆O3 was excluded to obtain comparable results for the
two clusters). Analyses of Subclusters 3, 4 and 6 yielded results
analogous with those obtained with Cluster 2. For Subcluster
5, however, the independent parameters, I, RH and ∆O3, significantly affected CO2 (see Table 1).
Because the data appeared heterogeneous and contradictory
when subjected to linear regression analysis, we subjected the
data to nonlinear analysis using multidimensional functional
scaling based on two models. The first model, constructed
from the data from Cluster 1 (1863 points), was based on a
mapping of the initial 4-dimensional parameter space (CO2, I,
T, RH) into a plane. The second model, constructed from the
data from Cluster 2 (1234 points), was based on a mapping of
the initial 5-dimensional parameter space (CO2, I, T, RH, ∆O3)
into a plane.
Nonlinear analysis: visualization and description of the
homogeneous classes from the 1st cluster of data
The entire sample of data from Cluster 1 was subdivided into
groups that had similar parameter values and were homogeneous with respect to the structure of the interrelations between
the parameters. To solve this problem, a geometric representation of the measurements in the form of an aggregate of
points on the Euclidean plane was constructed (Figure 2). This
aggregate of points preserved the proximity structure characteristic of the aggregate of points in the initial 4-dimensional
parameter space.
The system of vectors {Vi} averaged over all the measurements from Cluster 1 is shown in Figure 3. The set of four
measured parameters was subdivided into two almost orthogonal groups: Group A = CO2 and I, and Group B = T and RH.
768
PEREKREST ET AL.
Figure 1. Examples of time series of the
environmental and physiological parameters subjected to statistical analysis: CO2
uptake, light intensity (I), temperature
(T), and air humidity (RH).
Table 1. Regression coefficients for linear regression of CO2 against independent parameters on Clusters 2 and 5. Values in parenthesis denote 95%
confidence intervals.
Cluster 2-all relationships
Cluster 5-only significant relationships
Zero term
t
T
RH
I
∆O3
0.577
(0.01)
0.432
0.001
(0.03)
--
0.004
(0.01)
--
0.005
(0.0005)
0.005
(0.003)
0.004
(0.002)
0.003
(0.0001)
0.0004
The upper positions in Figure 2 are occupied by the states with
maximal values of I and minimal values of CO2. At the extreme
left are the states with the maximal T values and minimal RH
values.
The system of vectors shown in Figure 3 is the average over
all such vector systems for each point x on the image plane.
These vector systems may differ at different points as a result
of the different types of interactions between the parameters.
To visualize this aspect, we specified points with similar vector
systems by the same symbol on our computer monitor. Thus,
a class was built up from all the points that are neighbors on
the plane and additionally have the same symbol.
0.006
(0.005)
The entire sample of data was subdivided into 25 classes in
this way. This subdivision is shown in Figure 4, where the
neighboring points having similar vector systems are circumscribed by closed curves. The classes are not overlapping, i.e.,
the chosen size of the classes and the 2-dimensional image
space are satisfactory for proper subdivision and analysis. The
classification can be seen as the intersection of two sets of
layers that spread horizontally and vertically in Figure 4. The
horizontal layers, designated g1 to g5, correspond to the gradients of parameters from Group B; the vertical layers, designated v1 to v5, correspond to those from Group A (Groups A
and B are specified in Figure 3). The values of the parameters
NONLINEAR REGRESSION-TYPOLOGICAL ANALYSIS
Figure 2. The aggregate of points arising from the mapping of the
4-dimensional parameter space into a Euclidean plane for the 1st
cluster of data.
Figure 3. The system of the averaged vectors {Vi} representing the
directions of action of different parameters on the image plane from
Figure 2.
769
for single classes are presented in Table 2 (note that all parameters are scaled to mean zero and unit variance). The following
conclusions were derived from Table 2. (1) Most of the measurements belong to the horizontal layer g1; the higher the
horizontal layer, the less populated it is. (2) Each horizontal
layer has its own value of irradiance (constant over the layer).
Layer g1 has the lowest irradiance, and layer g5 has the highest
irradiance. (3) The value of CO2 decreases monotonically from
g1 to g5 and is approximately constant within each horizontal
layer. (4) Parameter T decreases monotonically across the
vertical layers from v1 to v5. There is a large difference in T
between layers v3 and v4; the difference between the mean
values of these layers exceeds 0.9. (5) The qualitative parameter Month shows that all the points from August are contained
in the vertical layers v1--v3, whereas the measurements from
May belong to layers v4 and v5. (6) The distribution of the
qualitative parameter +O3 over the classes is interesting.
Within the vertical layers v1--v3, each class contains almost
equal numbers of states achieved with and without introducing
O3 in the chamber. In contrast, for the spring measurements
(layers v4 and v5), classes with lower temperature contain only
a small percentage of cases with O3 (below 10% in classes
g1v5 and g2v5). It seems that in the presence of an enriched
O3 concentration the same ecophysiological state is reached at
a higher temperature; however, this is true only for the spring
season. (7) The distribution of humidity over the classes is
nonlinear. In the horizontal direction, humidity increases
within each season (spring and summer) in the opposite direction to temperature. Along the vertical layers, humidity is
approximately constant in summer, whereas during the spring,
it decreases monotonically with increasing irradiance.
Regressions of CO2 against the environmental parameters (I,
T, RH) were performed for each class and revealed several
patterns of influence (Table 3). (1) Pattern CO2 (I, T) was valid
for the entire horizontal layer g1. The decrease in mean temperature value over a class was accompanied by an increasing
regression coefficient of I and a decreasing regression coefficient of T, reflecting the nonlinear character of the relationship
over the entire parameter space. (2) Pattern CO2 (I) was characteristic for the three classes of the layer g2 with lower
temperature: g2v3, g2v4 and g2v5. The regression coefficient
of I increased with the decreasing mean temperature value of
a class. (3) Pattern CO2 (I, RH) was represented by classes
g2v2, g3v4 and g4v2; however, the type of relationship differed among classes. Within the class g2v2 (summer, moderate
I, high T), an increase in RH (with fixed I) was accompanied
by an increase in CO2 uptake, whereas within the classes g3v4
(spring, high I, rather low T) and g4v2 (summer, both I and T
high), an increase in RH (with fixed I) led to a decrease in CO2
uptake.
Nonlinear analysis: visualization and description of the
homogeneous classes from the 2nd cluster of data
Figure 4. Subdivision of the data presented in Figure 2 into 25 homogeneous classes. The classes are specified by the number of vertical
and horizontal layers to which they belong. For example, the class
g1v2 belongs to the horizontal layer g1 and the vertical layer v2.
Analysis of Cluster 2 was performed in the same way as for
Cluster 1. Cluster 2 contained fewer points than Cluster 1, but
because the quantitative parameter ∆O3 was measured over the
entire cluster, it was possible to analyze the 5-dimensional
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PEREKREST ET AL.
Table 2. Parameter values for the single homogeneous classes presented in Figure 4 (average values for a class). All the values are scaled to zero
mean and unit variance over the 1st cluster of data. A homogeneous class of data is determined by specifying the horizontal and vertical layers
containing this class; for example, the class v1g5 is contained in vertical layer v1 and horizontal layer g5. The parameter Month has the value of
0% for May (M%) and 100% for August (A%) (indicated by asterisks).
Parameter
Layer
v1
v2
v3
v4
v5
CO2
g5
g4
g3
g2
g1
−0.694
−0.526
−0.383
−0.145
0.447
−0.428
−0.432
−0.394
−0.205
0.348
−0.612
−0.441
−0.459
−0.124
0.291
−0.482
−0.510
−0.427
−0.160
0.281
−0.447
−0.475
−0.382
−0.078
0.237
I
g5
g4
g3
g2
g1
2.677
2.094
1.047
−0.044
−0.673
2.693
1.957
0.970
−0.012
−0.673
2.712
2.142
0.976
−0.135
−0.675
2.596
1.983
0.990
0.005
−0.671
2.609
1.983
0.896
−0.147
−0.667
T
g5
g4
g3
g2
g1
1.403
1.585
1.698
1.604
1.077
1.063
0.970
0.883
0.837
0.590
0.604
0.343
0.514
0.355
0.122
−0.279
−0.573
−0.450
−0.498
−0.895
−1.100
−1.025
−1.069
−1.328
−1.368
RH
g5
g4
g3
g2
g1
−1.970
−2.062
−2.401
−2.220
−2.056
−0.650
−0.703
−0.618
−0.777
−0.606
0.840
0.359
0.778
0.449
0.435
−1.224
−0.721
−0.793
0.078
0.045
0.325
0.738
0.916
1.299
1.247
+O3
N% g5
Y% g5
N% g4
Y% g4
N% g3
Y% g3
N% g2
Y% g2
N% g1
Y% g1
100.0
0.0
50.0
50.0
50.0
50.0
50.0
50.0
48.1
51.9
48.8
51.2
55.4
44.6
46.0
54.0
51.8
48.2
50.2
49.8
60.0
40.0
46.7
53.3
40.0
60.0
50.0
50.0
49.8
50.2
57.1
42.9
68.3
31.7
75.8
24.2
75.0
25.0
66.9
33.1
57.1
42.9
72.2
27.8
78.4
21.6
92.2
7.8
90.4
9.6
Month
M% g5
A% g5
M% g4
A% g4
M% g3
A% g3
M% g2
A% g2
M% g1
A% g1
0.0
100.0
*
*
*
*
*
*
*
*
*
*
*
*
parameter space with the parameter ∆O3 added to the four
parameters from Cluster 1.
The aggregate of points emerging on the Euclidean image
plane and the system of the averaged vectors {vi} are shown in
Figures 5 and 6, respectively. The set of five measured parameters was subdivided into three groups, of which Groups A (CO2
and I) and B (T and ∆O3) were almost orthogonal and Group
C (RH) was interrelated with the parameters of the other two
groups. The upper positions in Figure 5 are occupied by states
with maximal values of I and minimal values of CO2. At the
extreme left are the states with minimal T values and maximal
∆O3 values. The states with maximal values of RH are located
on the lower left, and those with minimal RH values are located
on the upper right.
The whole data sample was subdivided into 15 classes
(Figure 7) which can be seen as intersections of the three
horizontal layers (g1--g3) with the five vertical layers (v1--v5).
The horizontal layers correspond to the gradients of the parameters from Group B; the vertical layers correspond to those
from Group A.
The classes in Figure 7, which were built in the same way as
those in Figure 4, overlap, i.e., the cross sections of the sets of
NONLINEAR REGRESSION-TYPOLOGICAL ANALYSIS
771
Table 3. Coefficients of the regression equations for the single classes presented in Figure 4. I, T and RH are the environmental parameters, g1 to
g5 are horizontal layers, and v1 to v5 are vertical layers. A class is determined by specifying the horizontal and vertical layers to which it belongs;
for example, the regression equation for the class g1v1 reads: CO2 = −1.35 − 2.08 I + 0.26 T + 0.09 RH. The term a0 is the zero term of a regression
equation, and r0 is the multiple correlation coefficient, which exceeds 0.8 for most classes, demonstrating the suitability of the linear model within
a class. To the right-hand side of some regression parameters, the strength of its effect on CO2 uptake is specified: 3 = strong, 2 = moderate, and
1 = weak influence.
v1
g5
a0
I
T
RH
r0
g4
a0
I
T
RH
r0
g3
a0
I
T
RH
r0
g2
a0
I
T
RH
r0
g1
a0
I
T
RH
r0
v2
v3
−0.7640
0.1530 2
−0.0477 2
0.0636 2
0.399
−1.0660
−0.0146 2
0.1890 2
−0.2630 3
0.994
v4
v5
4.6500
−2.1700 3
2.1800 3
−1.0700 3
0.907
−0.3710
−0.1120 2
0.0599 2
0.2070 3
0.955
−0.3480
−0.0384 2
0.0050 1
−0.0117 2
0.144
−0.6670
0.1004 2
−0.1150 2
−0.0867 2
0.694
−0.4740
0.0266 2
0.0226 2
−0.1100 3
0.629
−0.5450
0.0009 1
−0.1150 3
0.0631 2
0.715
−0.3600
−0.0033 1
−0.0688 2
−0.0379 2
0.963
−0.2600
−0.0373 2
−0.3390 3
0.0380 2
0.359
−0.0257
−0.1130 3
0.0600 2
−0.1030 3
0.560
−0.5910
−0.0900 2
−0.2170 3
0.0608 2
0.826
5.0580
−0.0147
−0.0732 2
1.7200 3
0.994
0.0158
−0.2340 3
−0.0888 2
0.1270 3
0.487
−0.3060
−0.5490 3
0.0460 2
0.0307 2
0.785
0.1900
−0.4540 3
0.0401 2
0.0114 2
0.889
−0.3650
−0.3900 3
−0.1200 2
0.0387 2
0.846
−1.3500
−2.0800 2
0.2610 3
0.0914 2
0.931
−1.0100
−1.0350 3
0.7240 3
0.0215 2
0.864
−0.5010
−1.0350 3
0.0724 3
0.0215 2
0.758
−0.3750
−1.0530 3
0.0902 3
0.0089 2
0.974
−0.2550
−0.7820 3
0.0591 3
0.0021 1
0.923
Figure 5. The aggregate of points arising from the mapping of the
5-dimensional parameter space into a Euclidean plane for the 2nd
cluster of data.
points circumscribed by the closed curves contained points
expressed by different symbols. This indicates that the resolution of the method may be insufficient for a rigorous analysis
Figure 6. The system of the averaged vectors {Vi} representing the
directions of action of different parameters on the image plane from
Figure 5.
and that the class size should be reduced. If reducing the class
size does not decrease the overlapping, changing from the
2-dimensional to the 3-dimensional image space may be the
only way to resolve the classes. However, because the main
772
PEREKREST ET AL.
higher the horizontal layer, the less populated it is. (2) Each
horizontal layer has its corresponding (constant over the layer)
value of irradiance. Layer g1 has the lowest irradiance, and
layer g3 has the highest irradiance. (3) The value of CO2
decreases monotonically from g1 to g3 and is approximately
constant within each horizontal layer. (4) Parameter T decreases monotonically across the vertical layers from v5 to v1.
There is a large difference in T between layers v1 and v2; the
difference between the mean values of these layers exceeds
1.0. (5) The qualitative parameter Month shows that all the
points from August are contained in the vertical layers v2--v5,
whereas the measurements from May are contained in layer
v1. (6) To analyze the quantitative parameter ∆O3, we had to
exclude all the points without measured ∆O3 from Cluster 2
(that is why Cluster 2 contains fewer points than Cluster 1). As
a result, all the spring measurements with ozone had to be
discarded, and so it was not possible to estimate the real
percentage of points with ozone contained within layer v1.
Within the summer classes, parameter ∆O3 increased monotonically from v5 to v2. The values of the qualitative parameter
+O3 show that the composition of the classes changed from v2
to v5 as the percentage of points with ozone decreased. Thus,
for the summer points in Cluster 2, we found that in the
presence of an enriched O3 concentration, the same ecophysi-
Figure 7. Subdivision of the data presented in Figure 5 into 15 homogeneous classes. The classes are specified in Figure 4.
purpose of this paper was to introduce the method, we did not
attempt to eliminate the overlap among the classes presented
in Figure 7. The values of the parameters for single classes are
presented in Table 4.
The following conclusions were derived from Table 4. (1)
Most measurements belong to the horizontal layer g1; the
Table 4. Parameter values for the single classes presented in Figure 7. The designations are the same as in Table 2.
Parameter
Layer
v1
v2
v3
v4
v5
CO2
g3
g2
g1
−0.565
−0.342
0.226
−0.559
−0.478
0.050
−0.517
−0.420
0.299
−0.513
−0.403
0.341
−0.658
−0.371
0.501
I
g3
g2
g1
2.025
0.660
−0.623
2.278
0.732
−0.406
1.855
0.493
−0.651
1.777
0.326
−0.602
1.924
0.180
−0.691
T
g3
g2
g1
−1.431
−1.723
−2.163
−0.167
−0.347
−0.159
0.567
0.357
−0.031
0.803
0.492
0.135
1.237
0.881
0.502
RH
g3
g2
g1
−0.473
0.266
0.803
0.401
1.250
1.105
−0.270
−0.181
−0.022
−0.593
−0.010
0.212
−1.454
−1.167
−1.311
∆O3
g3
g2
g1
2.163
2.164
2.159
0.203
0.239
0.316
0.174
0.173
0.179
−0.616
−0.915
−0.986
−1.510
−1.379
−1.480
+O3
N% g3
Y% g3
N% g2
Y% g2
N% g1
Y% g1
0.0
100.0
0.0
100.0
0.0
100.0
20.0
80.0
0.0
100.0
0.0
100.0
18.9
81.1
19.2
80.8
24.3
75.7
80.3
19.7
100.0
0.0
95.2
4.8
100.0
0.0
93.9
6.1
100.0
0.0
Month
M% g3
A% g3
M% g2
A% g2
M% g1
A% g1
0.0
100.0
*
*
*
*
*
*
*
*
NONLINEAR REGRESSION-TYPOLOGICAL ANALYSIS
ological state was reached at a lower temperatures, which is the
opposite of the interrelation between ozone and temperature
found for Cluster 1. (7) The parameter RH decreased monotonically along all the horizontal layers when moving from v1
to v5 and increased monotonically almost everywhere along
the vertical layers when moving from g3 to g1.
On the basis of the quantitative description of the classes,
regression of CO2 against the environmental parameters (I, T,
RH, ∆O3) was performed for each class (Table 5). Regression
analysis revealed the following patterns of influence. (1) Pattern CO2 (I, ∆O3) is valid for class g2v3 (moderate values of I
and T). (2) Pattern CO2 (∆O3) is valid for classes g3v2, g3v3
and g3v4. (3) Pattern CO2 (I) is valid for the four neighboring
classes of layer g1: g1v1, g1v2, g1v3 and g1v4. This pattern is
characteristic of the high temperatures and lower irradiances
during the summer season.
Conclusions
We have developed a comprehensive statistical treatment of
the complex and still unknown processes leading to forest
decline. We conclude that the nonlinear multidimensional scaling method provides analytical tools to account for (1) the
large number of measurements and parameters, and (2) the
absence of any obvious assumption concerning the nature of
the mechanism underlying the phenomenon. The method provides a comparison of the effects of one parameter at different
levels of the other parameters. Furthermore, the same comparison can be carried through for combinations of parameters.
Classes of measured points having homogeneous structure
with respect to the ecophysiological state and the interrelations
between the environmental parameters can be visualized. Statistically significant information about changes in the interrelations between parameters can be derived by comparing the
773
linear regression models constructed for the different classes.
The experimental information used for the analysis appears
in the form of a time series of events superimposed on random
fluctuations (dynamic noise). This can be considered as a
stochastic process with memory plus a complicated dynamics
of the mutual influences between different parameters. Though
there was no explicit time dependence in our treatment, the
three time parameters (month, day, time) can be used to represent the dynamics of changes in the environment and the
physiological state. Moreover, explicit presentation of the dynamics is possible by using the regression coefficients as
functions of the time parameters.
Multidimensional functional scaling has two features that
suggest that it can be used as a universal tool for regressionand classification-typological analysis of large data sets: (1)
the dimension (and therefore the complexity) of the optimization procedure is determined not by the number of objects
under study but by the complexity of the metrics (proximity
relation); and (2) the number of computations required within
one iteration increases linearly with the dimension of the
calculations, whereas for the most comparable methods, this
relationship is quadratic.
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0.0098 2
−0.2020 2
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water relations. In Encyclopedia of Plant Physiology. Vol. 12B. Eds.
O.L. Lange, P.S. Nobel, C.B. Osmond and H. Ziegler. Springer
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1975. Effect of several environmental variables and xylem pressure
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Penning de Vries, F.W.T. 1983. Modeling of growth and production.
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P.S. Nobel, C.B. Osmond and H. Ziegler. Springer Verlag, Berlin,
Germany, pp 118--150.
Perekrest, V.T. 1989. Biquadratic functional models of the parameterization of the experimental data. Habilitation Thesis (Dr. of Science). Academy of Science, Moscow, Russia, 29 p.
Shepard, R.N. 1989. Multidimensional scaling, tree-fitting and clustering. Science 210:390--398.
Sibson, R. 1979. Studies in the robustness of MDS: pertubational
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© 1995 Heron Publishing----Victoria, Canada
Nonlinear regression-typological analysis of ecophysiological states of
vegetation: a pilot study with small data sets
V. T. PEREKREST,1 T. V. KHACHATUROVA,1 I. B. BERESNEVA,1 N. M. MITROFANOVA,2
E. KÜNSTLE,3 E. WAGNER4 and L. FUKSHANSKY4,5
1
2
3
4
5
Economic-Mathematical Institute, Russian Academy of Science, 1 Chaikovskistreet, St. Petersburg, Russia
Technical University of St. Petersburg, 29 Polytechnic Street, St. Petersburg, Russia
Institut für Waldwachstum, University of Freiburg, D-78 Freiburg, Germany
Institute of Biology II, University of Freiburg, Schänzlestrasse 1, D-78 Freiburg, Germany
Author to whom correspondence should be sent
Received August 11, 1993
Summary The interactions of environmental factors associated with forest decline were analyzed by a modified multidimensional scaling method. The method subdivides the entire
data set into homogeneous classes; linear regression is then
applied within each single class. A nonlinear picture of the
interdependence of the effects of different factors is developed
as a composite of the contributions from each single class. The
analysis was performed on a restricted data set, and the results
compared with some expected effects and with results obtained
by standard linear regression. Even with the limited data set,
multidimensional scaling not only explained expected effects
but also revealed new information. We conclude that the
method will be useful for analyzing complex time series data
because it is able to detect complex interactions between
environmental variables that affect physiological parameters.
Keywords: forest decline, functional multidimensional scaling,
interdependence of the effects of environmental factors, multicausal syndrome.
sky 1992). To develop a comprehensive statistical analysis of
the correlations between environmental parameters and the
physiological characteristics of the system, we need tools for
the quantitative description and classification of the physiological state of a plant under the measured environmental
conditions.
In this paper, we have applied a general approach to nonlinear functional multidimensional scaling. This approach has
been used in a restricted way in ecology, medicine, economics,
production control and other fields (Bechtel 1978, Sibson
1979, Takane 1981, Shepard 1989). Some problems that precluded its wide application have only recently been resolved
(Perekrest 1989, Khachaturova and Perekrest 1990). We assessed the advantages and limitations of the method based on
an analysis of real sets of environmental data. Because of the
small number of parameters in these data sets, this study was
limited to a comparison of our results with some expected
effects and also with those of conventional linear regression
analysis.
Introduction
Description of method
It seems increasingly likely that air pollution, drought and high
irradiances are the major factors contributing to the ‘‘new forest
decline’’ syndrome. The syndrome can occur in forest ecosystems of different structure as a result of complex interactions
between several stress-causing factors that are dependent on
climatic and soil conditions. The complex, finely tuned biochemical, physiological and genetic control mechanisms that
underlie forest decline make it difficult to obtain reliable information about the contributions of single environmental factors
and about the character of their interactions.
A quantitative analysis of the effects of environmental factors is necessary before a strategy can be developed for the
prevention and control of forest decline. A rational system for
monitoring the environment is also needed as is the development of a sound basis for statistically analyzing data (Fukshan-
To describe the method of functional multidimensional scaling, let us consider data obtained from many similar observations. Each observation provides measured values of N
different parameters and represents the state of a single object
from a population of similar objects, or a current state from a
time series of states of a single object. A typical example is a
series of measurements of N environmental and physiological
parameters (e.g., radiation, air temperature, soil temperature,
humidity, photosynthesis, transpiration) of a plant community,
repeated at different times. Parameters of qualitative origin,
assuming discrete values (e.g., degree of oxidative damage) as
well as all-or-none variables (e.g., irrigated versus nonirrigated
area), are also permitted.
The data are used to build an aggregate of points in the
N-dimensional space of the studied variables. The space is
766
PEREKREST ET AL.
characterized by a metric, i.e., the distance between any two
points can be defined. The distance between two points can be
calculated, for example, as the sum of squares of differences
between the values of corresponding parameters (which for
this purpose are scaled to zero mean and unit variance). In
addition, a stochastic sample measure arises in the N-dimensional space, as a result of the spatial pattern of the aggregate
of points. The sample measure reflects the density of points in
different areas of the N-dimensional space. The larger the data
sample, the better the sample measure approximates the actual
distribution of the values of N parameters.
In principle, the distribution of points in the N-dimensional
parameter space, with known distances between any two
points, contains all the information that can be extracted from
the measurements. Our goal was to derive quantitative correlations between independent parameters (e.g., temperature,
radiation) and their combinations on one hand, and the parameters supposed to be affected by the independent parameters (e.g., photosynthesis) on the other. We also determined
how the effect of one parameter was influenced by other
parameters.
The simplest procedure is to perform a linear regression over
the entire data set; that is, to regress dependent against independent parameters, and thereby obtain ‘‘average’’ statements
about the degree and direction of interconnections between the
parameters. However, such information is usually of limited
value because it does not reflect the true interconnections,
which are nonlinear and have a heterogeneous structure over
the entire space.
Nonlinear functional scaling overcomes this difficulty by
subdividing the whole data set into classes with homogeneous
structure. These classes are composed of neighboring states
that have approximately the same parameter values and, in
addition, similar relationships among state parameters. Within
each class, linear regression reveals these relationships. The
global picture of the nonlinear relationships between the parameters over the entire space is assembled from the linear
contributions of the single classes.
Thus, the core of the multidimensional scaling method is the
subdivision of the data set into homogeneous classes. The
scaling function f: X → R2, a nonlinear mapping of the initial
N-dimensional parameter space X = {xi} into a plane (or
generally, into an Euclidean space of small dimension), is
constructed so that the binary distances between the points are
preserved as much as possible. Formally written, this means
finding a scaling vector-function f that minimizes the function:
V( f ) = (( f(x) − f(y))2 − r (x,y))2 dµ(x) dµ(y),
where r(x,y) is the distance between objects x and y in the
N-dimensional space X, and µ is the joint distribution of N
parameters (approximated by the sample measure). The neighboring points in the plane can be easily found because they are
also neighbors in the initial N-dimensional space. This is the
first step in constructing homogeneous classes. The second
step is to identify the states with similar interconnections
between parameters, performed as follows.
The recently developed theory of functional multidimensional scaling (Perekrest 1989, Khachaturova and Perekrest
1990) provides the best possible f mapping within the broad
class of all measurable and integrable functions to the fourth
degree. The theory also permits estimation of the distortion of
the distances. If the distortion is too large, the dimension of the
image space should be increased (i.e., the 3-dimensional rather
than the 2-dimensional space should be considered). The scaling function can be differentiated with respect to the measured
parameters. Comparison of its partial derivatives in different
areas of the initial space reveals the character of interconnections between the measured parameters in these areas and
provides the basis for the second step in constructing the
homogeneous classes. For this purpose, we consider a system
of vectors {Vi(x)}, where each vector Vi(x) corresponds to
parameter i of the entire N-dimensional space. Coordinates of
Vi(x) are determined as (Vi(x))j = (∂fj / ∂ xi), where fj(x) is the jth
coordinate of the vector-function f (when the range of values
of f is a plane j = 1or 2). Vector Vi(x) is interpreted as the vector
of the movements of the image of point x on the model plane
when the ith coordinate of this point in the initial parameter
space is increased by a unit, while all other coordinates remain
fixed. The system of vectors {Vi(x)} may be different for
different measurements of x, reflecting different types of interrelation between the parameters. A homogeneous class by
definition can only contain points with similar interrelations,
i.e., points with similar vectors {Vi(x)}.
Linear regression analysis is then performed within each
class. The combined picture of linear models of all classes
provides the nonlinear model for the entire space of the measured parameters. This global model yields information about
the interconnections of the parameters and the modifications
of these interconnections over the entire parameter space.
Although determination of the contributions of single external factors and their combinations to an ecophysiological response can be approached by different statistical methods, the
complexity of both the noncontrolled experimental conditions
and the nonlinear objects with memory (i.e., objects with
responses that also depend on previous treatments) should not
be underestimated. Because the approach used must be able to
treat a dynamic process that is affected by external factors in a
nonlinear way, and must also be able to accommodate the
effects of different factors that are both interdependent and
dependent on the process history, we conclude that elementary
statistical approaches, for example, correlation coefficients or
linear regressions between two or several quantities, are not
appropriate (cf. Burgeois et al. 1992).
More complex treatments, especially those dealing with
grouping of data with respect to some expected similarity
within a group are required. An example of such a treatment is
the clustering analysis of multivariate data sets in comparative
plant ecology used by Grime et al. (1988). The rationale for
this treatment is that because certain attributes of a group of
species evolve through simultaneous selection (a composite
response to a set of external environmental pressures), one can
expect clustering with respect to these attributes. However, the
analysis of grouping solves only one part of the problem.
NONLINEAR REGRESSION-TYPOLOGICAL ANALYSIS
Additionally, weighting of the contributions of single external
factors within groups (and therefore over the entire parameter
space) is required.
Recently, several studies have shown that direct nonlinear
regressions of entire data sets can yield important information
about the interaction of external factors in many situations
(e.g., Hinckley et al. 1975, Hall 1982, Penning de Vries 1983,
Chen and Kreeb 1989, Kreeb and Chen 1991). Although this
approach is restricted to a few parameters, we conclude that it
is a useful analysis that can supplement more complex analysis
and can be performed with standard software. However, there
are some limitations when applying straightforward nonlinear
regression analysis to data observed in a natural environment.
In contrast to multidimensional scaling, which reveals ‘‘natural’’ data subsets (homogeneous classes) having both parameter values and parameter interaction rules nearly constant,
straightforward regression analysis requires a superimposed
subdivision of the parameter values; however, there is no
guarantee that, within an interval between two subsequent
values of a parameter, the interaction rule remains constant.
Another limitation to the straightforward regression treatment
is the use of a fixed set of nonlinear model functions (predominantly linear transformations, and exponential and power functions are used). Thus, an investigation of the interaction of
different environmental factors under natural conditions requires a complex approach combining elements of both typological and regression analyses. The only method that does
not need to be supplemented by other methods is multivariate
time series analysis (see, for example, Anderson 1971). This is
a family of procedures that requires modification for each class
of problems. The time series procedures applied to stochastic
processes with memory (so-called procedures with distributed
lags) are especially difficult to construct and analyze.
Experimental data sets
Linear regression
We analyzed time series measurements made at the experimental station of the Institut für Waldwachstum of the University of Freiburg at Schauinsland at an elevation of 1230 m.
Growth and metabolic processes of individual trees were
monitored under the influence of ozone in the natural environment. Climatic variables were continuously sampled in parallel with measurements of CO2 gas exchange (photosynthesis
and respiration) and ambient ozone concentrations. To investigate the impact of ozone, various ozone concentrations were
examined.
Two data samples were measured, the first one during 15
days in May 1991, the second one during 15 days in August
1991. Individual measurements were performed every 30 min.
The set of measured parameters included photosynthesis,
measured as CO2 uptake (CO2, mg), irradiance (I, µmol m −2
s −1), air temperature (T, °C), relative humidity (RH, %), ozone
concentration outside the gas exchange cuvette (O3A, µg m −3),
ozone concentration inside the gas exchange cuvette (O3K, µg
m −3), and current time (t). In addition, each measurement was
also characterized by the following qualitative parameters:
767
ozone into the gas exchange cuvette (+O3, yes/no), season
(Month, May or August), and day of the month (Day). As
illustrative examples, some time series of CO2, I, T and RH are
shown in Figure 1. Before analysis, each parameter was scaled
to zero mean and unit variance.
The entire data set was subdivided into clusters of data
measured under homogeneous conditions as follows. First, we
considered two overall clusters, hereafter referred to as Clusters 1 and 2. Cluster 1 contained 1863 points with no gaps for
the values of the parameters CO2, t, T, RH, I, +O3, Month and
Day. Cluster 2 contained 1234 points with no gaps for the
values of parameters CO2, t, T, RH, I, O3K, O3A, +O3, Month
and Day.
More detailed linear analysis was performed on four subclusters differing from one another with respect to the qualitative parameters Month and +O3: Subcluster 3 (516 points)
comprised measurements in August without ozone addition;
Subcluster 4 (554 points) comprised measurements in May
without ozone addition; Subcluster 5 (589 points) comprised
measurements in August with ozone addition; and Subcluster
6 (552 points) comprised measurements in May with ozone
addition.
Linear analysis of the data from Cluster 2 revealed significant dependence of CO2 only on parameter I (Table 1). Analogous analysis of Cluster 1 gave similar results to the analysis
of Cluster 2 performed excluding the parameter ∆O3 = O3K −
O3A (∆O3 was excluded to obtain comparable results for the
two clusters). Analyses of Subclusters 3, 4 and 6 yielded results
analogous with those obtained with Cluster 2. For Subcluster
5, however, the independent parameters, I, RH and ∆O3, significantly affected CO2 (see Table 1).
Because the data appeared heterogeneous and contradictory
when subjected to linear regression analysis, we subjected the
data to nonlinear analysis using multidimensional functional
scaling based on two models. The first model, constructed
from the data from Cluster 1 (1863 points), was based on a
mapping of the initial 4-dimensional parameter space (CO2, I,
T, RH) into a plane. The second model, constructed from the
data from Cluster 2 (1234 points), was based on a mapping of
the initial 5-dimensional parameter space (CO2, I, T, RH, ∆O3)
into a plane.
Nonlinear analysis: visualization and description of the
homogeneous classes from the 1st cluster of data
The entire sample of data from Cluster 1 was subdivided into
groups that had similar parameter values and were homogeneous with respect to the structure of the interrelations between
the parameters. To solve this problem, a geometric representation of the measurements in the form of an aggregate of
points on the Euclidean plane was constructed (Figure 2). This
aggregate of points preserved the proximity structure characteristic of the aggregate of points in the initial 4-dimensional
parameter space.
The system of vectors {Vi} averaged over all the measurements from Cluster 1 is shown in Figure 3. The set of four
measured parameters was subdivided into two almost orthogonal groups: Group A = CO2 and I, and Group B = T and RH.
768
PEREKREST ET AL.
Figure 1. Examples of time series of the
environmental and physiological parameters subjected to statistical analysis: CO2
uptake, light intensity (I), temperature
(T), and air humidity (RH).
Table 1. Regression coefficients for linear regression of CO2 against independent parameters on Clusters 2 and 5. Values in parenthesis denote 95%
confidence intervals.
Cluster 2-all relationships
Cluster 5-only significant relationships
Zero term
t
T
RH
I
∆O3
0.577
(0.01)
0.432
0.001
(0.03)
--
0.004
(0.01)
--
0.005
(0.0005)
0.005
(0.003)
0.004
(0.002)
0.003
(0.0001)
0.0004
The upper positions in Figure 2 are occupied by the states with
maximal values of I and minimal values of CO2. At the extreme
left are the states with the maximal T values and minimal RH
values.
The system of vectors shown in Figure 3 is the average over
all such vector systems for each point x on the image plane.
These vector systems may differ at different points as a result
of the different types of interactions between the parameters.
To visualize this aspect, we specified points with similar vector
systems by the same symbol on our computer monitor. Thus,
a class was built up from all the points that are neighbors on
the plane and additionally have the same symbol.
0.006
(0.005)
The entire sample of data was subdivided into 25 classes in
this way. This subdivision is shown in Figure 4, where the
neighboring points having similar vector systems are circumscribed by closed curves. The classes are not overlapping, i.e.,
the chosen size of the classes and the 2-dimensional image
space are satisfactory for proper subdivision and analysis. The
classification can be seen as the intersection of two sets of
layers that spread horizontally and vertically in Figure 4. The
horizontal layers, designated g1 to g5, correspond to the gradients of parameters from Group B; the vertical layers, designated v1 to v5, correspond to those from Group A (Groups A
and B are specified in Figure 3). The values of the parameters
NONLINEAR REGRESSION-TYPOLOGICAL ANALYSIS
Figure 2. The aggregate of points arising from the mapping of the
4-dimensional parameter space into a Euclidean plane for the 1st
cluster of data.
Figure 3. The system of the averaged vectors {Vi} representing the
directions of action of different parameters on the image plane from
Figure 2.
769
for single classes are presented in Table 2 (note that all parameters are scaled to mean zero and unit variance). The following
conclusions were derived from Table 2. (1) Most of the measurements belong to the horizontal layer g1; the higher the
horizontal layer, the less populated it is. (2) Each horizontal
layer has its own value of irradiance (constant over the layer).
Layer g1 has the lowest irradiance, and layer g5 has the highest
irradiance. (3) The value of CO2 decreases monotonically from
g1 to g5 and is approximately constant within each horizontal
layer. (4) Parameter T decreases monotonically across the
vertical layers from v1 to v5. There is a large difference in T
between layers v3 and v4; the difference between the mean
values of these layers exceeds 0.9. (5) The qualitative parameter Month shows that all the points from August are contained
in the vertical layers v1--v3, whereas the measurements from
May belong to layers v4 and v5. (6) The distribution of the
qualitative parameter +O3 over the classes is interesting.
Within the vertical layers v1--v3, each class contains almost
equal numbers of states achieved with and without introducing
O3 in the chamber. In contrast, for the spring measurements
(layers v4 and v5), classes with lower temperature contain only
a small percentage of cases with O3 (below 10% in classes
g1v5 and g2v5). It seems that in the presence of an enriched
O3 concentration the same ecophysiological state is reached at
a higher temperature; however, this is true only for the spring
season. (7) The distribution of humidity over the classes is
nonlinear. In the horizontal direction, humidity increases
within each season (spring and summer) in the opposite direction to temperature. Along the vertical layers, humidity is
approximately constant in summer, whereas during the spring,
it decreases monotonically with increasing irradiance.
Regressions of CO2 against the environmental parameters (I,
T, RH) were performed for each class and revealed several
patterns of influence (Table 3). (1) Pattern CO2 (I, T) was valid
for the entire horizontal layer g1. The decrease in mean temperature value over a class was accompanied by an increasing
regression coefficient of I and a decreasing regression coefficient of T, reflecting the nonlinear character of the relationship
over the entire parameter space. (2) Pattern CO2 (I) was characteristic for the three classes of the layer g2 with lower
temperature: g2v3, g2v4 and g2v5. The regression coefficient
of I increased with the decreasing mean temperature value of
a class. (3) Pattern CO2 (I, RH) was represented by classes
g2v2, g3v4 and g4v2; however, the type of relationship differed among classes. Within the class g2v2 (summer, moderate
I, high T), an increase in RH (with fixed I) was accompanied
by an increase in CO2 uptake, whereas within the classes g3v4
(spring, high I, rather low T) and g4v2 (summer, both I and T
high), an increase in RH (with fixed I) led to a decrease in CO2
uptake.
Nonlinear analysis: visualization and description of the
homogeneous classes from the 2nd cluster of data
Figure 4. Subdivision of the data presented in Figure 2 into 25 homogeneous classes. The classes are specified by the number of vertical
and horizontal layers to which they belong. For example, the class
g1v2 belongs to the horizontal layer g1 and the vertical layer v2.
Analysis of Cluster 2 was performed in the same way as for
Cluster 1. Cluster 2 contained fewer points than Cluster 1, but
because the quantitative parameter ∆O3 was measured over the
entire cluster, it was possible to analyze the 5-dimensional
770
PEREKREST ET AL.
Table 2. Parameter values for the single homogeneous classes presented in Figure 4 (average values for a class). All the values are scaled to zero
mean and unit variance over the 1st cluster of data. A homogeneous class of data is determined by specifying the horizontal and vertical layers
containing this class; for example, the class v1g5 is contained in vertical layer v1 and horizontal layer g5. The parameter Month has the value of
0% for May (M%) and 100% for August (A%) (indicated by asterisks).
Parameter
Layer
v1
v2
v3
v4
v5
CO2
g5
g4
g3
g2
g1
−0.694
−0.526
−0.383
−0.145
0.447
−0.428
−0.432
−0.394
−0.205
0.348
−0.612
−0.441
−0.459
−0.124
0.291
−0.482
−0.510
−0.427
−0.160
0.281
−0.447
−0.475
−0.382
−0.078
0.237
I
g5
g4
g3
g2
g1
2.677
2.094
1.047
−0.044
−0.673
2.693
1.957
0.970
−0.012
−0.673
2.712
2.142
0.976
−0.135
−0.675
2.596
1.983
0.990
0.005
−0.671
2.609
1.983
0.896
−0.147
−0.667
T
g5
g4
g3
g2
g1
1.403
1.585
1.698
1.604
1.077
1.063
0.970
0.883
0.837
0.590
0.604
0.343
0.514
0.355
0.122
−0.279
−0.573
−0.450
−0.498
−0.895
−1.100
−1.025
−1.069
−1.328
−1.368
RH
g5
g4
g3
g2
g1
−1.970
−2.062
−2.401
−2.220
−2.056
−0.650
−0.703
−0.618
−0.777
−0.606
0.840
0.359
0.778
0.449
0.435
−1.224
−0.721
−0.793
0.078
0.045
0.325
0.738
0.916
1.299
1.247
+O3
N% g5
Y% g5
N% g4
Y% g4
N% g3
Y% g3
N% g2
Y% g2
N% g1
Y% g1
100.0
0.0
50.0
50.0
50.0
50.0
50.0
50.0
48.1
51.9
48.8
51.2
55.4
44.6
46.0
54.0
51.8
48.2
50.2
49.8
60.0
40.0
46.7
53.3
40.0
60.0
50.0
50.0
49.8
50.2
57.1
42.9
68.3
31.7
75.8
24.2
75.0
25.0
66.9
33.1
57.1
42.9
72.2
27.8
78.4
21.6
92.2
7.8
90.4
9.6
Month
M% g5
A% g5
M% g4
A% g4
M% g3
A% g3
M% g2
A% g2
M% g1
A% g1
0.0
100.0
*
*
*
*
*
*
*
*
*
*
*
*
parameter space with the parameter ∆O3 added to the four
parameters from Cluster 1.
The aggregate of points emerging on the Euclidean image
plane and the system of the averaged vectors {vi} are shown in
Figures 5 and 6, respectively. The set of five measured parameters was subdivided into three groups, of which Groups A (CO2
and I) and B (T and ∆O3) were almost orthogonal and Group
C (RH) was interrelated with the parameters of the other two
groups. The upper positions in Figure 5 are occupied by states
with maximal values of I and minimal values of CO2. At the
extreme left are the states with minimal T values and maximal
∆O3 values. The states with maximal values of RH are located
on the lower left, and those with minimal RH values are located
on the upper right.
The whole data sample was subdivided into 15 classes
(Figure 7) which can be seen as intersections of the three
horizontal layers (g1--g3) with the five vertical layers (v1--v5).
The horizontal layers correspond to the gradients of the parameters from Group B; the vertical layers correspond to those
from Group A.
The classes in Figure 7, which were built in the same way as
those in Figure 4, overlap, i.e., the cross sections of the sets of
NONLINEAR REGRESSION-TYPOLOGICAL ANALYSIS
771
Table 3. Coefficients of the regression equations for the single classes presented in Figure 4. I, T and RH are the environmental parameters, g1 to
g5 are horizontal layers, and v1 to v5 are vertical layers. A class is determined by specifying the horizontal and vertical layers to which it belongs;
for example, the regression equation for the class g1v1 reads: CO2 = −1.35 − 2.08 I + 0.26 T + 0.09 RH. The term a0 is the zero term of a regression
equation, and r0 is the multiple correlation coefficient, which exceeds 0.8 for most classes, demonstrating the suitability of the linear model within
a class. To the right-hand side of some regression parameters, the strength of its effect on CO2 uptake is specified: 3 = strong, 2 = moderate, and
1 = weak influence.
v1
g5
a0
I
T
RH
r0
g4
a0
I
T
RH
r0
g3
a0
I
T
RH
r0
g2
a0
I
T
RH
r0
g1
a0
I
T
RH
r0
v2
v3
−0.7640
0.1530 2
−0.0477 2
0.0636 2
0.399
−1.0660
−0.0146 2
0.1890 2
−0.2630 3
0.994
v4
v5
4.6500
−2.1700 3
2.1800 3
−1.0700 3
0.907
−0.3710
−0.1120 2
0.0599 2
0.2070 3
0.955
−0.3480
−0.0384 2
0.0050 1
−0.0117 2
0.144
−0.6670
0.1004 2
−0.1150 2
−0.0867 2
0.694
−0.4740
0.0266 2
0.0226 2
−0.1100 3
0.629
−0.5450
0.0009 1
−0.1150 3
0.0631 2
0.715
−0.3600
−0.0033 1
−0.0688 2
−0.0379 2
0.963
−0.2600
−0.0373 2
−0.3390 3
0.0380 2
0.359
−0.0257
−0.1130 3
0.0600 2
−0.1030 3
0.560
−0.5910
−0.0900 2
−0.2170 3
0.0608 2
0.826
5.0580
−0.0147
−0.0732 2
1.7200 3
0.994
0.0158
−0.2340 3
−0.0888 2
0.1270 3
0.487
−0.3060
−0.5490 3
0.0460 2
0.0307 2
0.785
0.1900
−0.4540 3
0.0401 2
0.0114 2
0.889
−0.3650
−0.3900 3
−0.1200 2
0.0387 2
0.846
−1.3500
−2.0800 2
0.2610 3
0.0914 2
0.931
−1.0100
−1.0350 3
0.7240 3
0.0215 2
0.864
−0.5010
−1.0350 3
0.0724 3
0.0215 2
0.758
−0.3750
−1.0530 3
0.0902 3
0.0089 2
0.974
−0.2550
−0.7820 3
0.0591 3
0.0021 1
0.923
Figure 5. The aggregate of points arising from the mapping of the
5-dimensional parameter space into a Euclidean plane for the 2nd
cluster of data.
points circumscribed by the closed curves contained points
expressed by different symbols. This indicates that the resolution of the method may be insufficient for a rigorous analysis
Figure 6. The system of the averaged vectors {Vi} representing the
directions of action of different parameters on the image plane from
Figure 5.
and that the class size should be reduced. If reducing the class
size does not decrease the overlapping, changing from the
2-dimensional to the 3-dimensional image space may be the
only way to resolve the classes. However, because the main
772
PEREKREST ET AL.
higher the horizontal layer, the less populated it is. (2) Each
horizontal layer has its corresponding (constant over the layer)
value of irradiance. Layer g1 has the lowest irradiance, and
layer g3 has the highest irradiance. (3) The value of CO2
decreases monotonically from g1 to g3 and is approximately
constant within each horizontal layer. (4) Parameter T decreases monotonically across the vertical layers from v5 to v1.
There is a large difference in T between layers v1 and v2; the
difference between the mean values of these layers exceeds
1.0. (5) The qualitative parameter Month shows that all the
points from August are contained in the vertical layers v2--v5,
whereas the measurements from May are contained in layer
v1. (6) To analyze the quantitative parameter ∆O3, we had to
exclude all the points without measured ∆O3 from Cluster 2
(that is why Cluster 2 contains fewer points than Cluster 1). As
a result, all the spring measurements with ozone had to be
discarded, and so it was not possible to estimate the real
percentage of points with ozone contained within layer v1.
Within the summer classes, parameter ∆O3 increased monotonically from v5 to v2. The values of the qualitative parameter
+O3 show that the composition of the classes changed from v2
to v5 as the percentage of points with ozone decreased. Thus,
for the summer points in Cluster 2, we found that in the
presence of an enriched O3 concentration, the same ecophysi-
Figure 7. Subdivision of the data presented in Figure 5 into 15 homogeneous classes. The classes are specified in Figure 4.
purpose of this paper was to introduce the method, we did not
attempt to eliminate the overlap among the classes presented
in Figure 7. The values of the parameters for single classes are
presented in Table 4.
The following conclusions were derived from Table 4. (1)
Most measurements belong to the horizontal layer g1; the
Table 4. Parameter values for the single classes presented in Figure 7. The designations are the same as in Table 2.
Parameter
Layer
v1
v2
v3
v4
v5
CO2
g3
g2
g1
−0.565
−0.342
0.226
−0.559
−0.478
0.050
−0.517
−0.420
0.299
−0.513
−0.403
0.341
−0.658
−0.371
0.501
I
g3
g2
g1
2.025
0.660
−0.623
2.278
0.732
−0.406
1.855
0.493
−0.651
1.777
0.326
−0.602
1.924
0.180
−0.691
T
g3
g2
g1
−1.431
−1.723
−2.163
−0.167
−0.347
−0.159
0.567
0.357
−0.031
0.803
0.492
0.135
1.237
0.881
0.502
RH
g3
g2
g1
−0.473
0.266
0.803
0.401
1.250
1.105
−0.270
−0.181
−0.022
−0.593
−0.010
0.212
−1.454
−1.167
−1.311
∆O3
g3
g2
g1
2.163
2.164
2.159
0.203
0.239
0.316
0.174
0.173
0.179
−0.616
−0.915
−0.986
−1.510
−1.379
−1.480
+O3
N% g3
Y% g3
N% g2
Y% g2
N% g1
Y% g1
0.0
100.0
0.0
100.0
0.0
100.0
20.0
80.0
0.0
100.0
0.0
100.0
18.9
81.1
19.2
80.8
24.3
75.7
80.3
19.7
100.0
0.0
95.2
4.8
100.0
0.0
93.9
6.1
100.0
0.0
Month
M% g3
A% g3
M% g2
A% g2
M% g1
A% g1
0.0
100.0
*
*
*
*
*
*
*
*
NONLINEAR REGRESSION-TYPOLOGICAL ANALYSIS
ological state was reached at a lower temperatures, which is the
opposite of the interrelation between ozone and temperature
found for Cluster 1. (7) The parameter RH decreased monotonically along all the horizontal layers when moving from v1
to v5 and increased monotonically almost everywhere along
the vertical layers when moving from g3 to g1.
On the basis of the quantitative description of the classes,
regression of CO2 against the environmental parameters (I, T,
RH, ∆O3) was performed for each class (Table 5). Regression
analysis revealed the following patterns of influence. (1) Pattern CO2 (I, ∆O3) is valid for class g2v3 (moderate values of I
and T). (2) Pattern CO2 (∆O3) is valid for classes g3v2, g3v3
and g3v4. (3) Pattern CO2 (I) is valid for the four neighboring
classes of layer g1: g1v1, g1v2, g1v3 and g1v4. This pattern is
characteristic of the high temperatures and lower irradiances
during the summer season.
Conclusions
We have developed a comprehensive statistical treatment of
the complex and still unknown processes leading to forest
decline. We conclude that the nonlinear multidimensional scaling method provides analytical tools to account for (1) the
large number of measurements and parameters, and (2) the
absence of any obvious assumption concerning the nature of
the mechanism underlying the phenomenon. The method provides a comparison of the effects of one parameter at different
levels of the other parameters. Furthermore, the same comparison can be carried through for combinations of parameters.
Classes of measured points having homogeneous structure
with respect to the ecophysiological state and the interrelations
between the environmental parameters can be visualized. Statistically significant information about changes in the interrelations between parameters can be derived by comparing the
773
linear regression models constructed for the different classes.
The experimental information used for the analysis appears
in the form of a time series of events superimposed on random
fluctuations (dynamic noise). This can be considered as a
stochastic process with memory plus a complicated dynamics
of the mutual influences between different parameters. Though
there was no explicit time dependence in our treatment, the
three time parameters (month, day, time) can be used to represent the dynamics of changes in the environment and the
physiological state. Moreover, explicit presentation of the dynamics is possible by using the regression coefficients as
functions of the time parameters.
Multidimensional functional scaling has two features that
suggest that it can be used as a universal tool for regressionand classification-typological analysis of large data sets: (1)
the dimension (and therefore the complexity) of the optimization procedure is determined not by the number of objects
under study but by the complexity of the metrics (proximity
relation); and (2) the number of computations required within
one iteration increases linearly with the dimension of the
calculations, whereas for the most comparable methods, this
relationship is quadratic.
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