Summary The interactions of environmental factors associ-ated with forest decline were analyzed by a modified multidi-mensional 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, multi-causal syndrome.
Introduction
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 ecosys-tems 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 bio-chemical, physiological and genetic control mechanisms that underlie forest decline make it difficult to obtain reliable infor-mation about the contributions of single environmental factors and about the character of their interactions.
A quantitative analysis of the effects of environmental fac-tors 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 develop-ment of a sound basis for statistically analyzing data
(Fukshan-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 physi-ological state of a plant under the measured environmental conditions.
In this paper, we have applied a general approach to non-linear 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 pre-cluded its wide application have only recently been resolved (Perekrest 1989, Khachaturova and Perekrest 1990). We as-sessed 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.
Description of method
To describe the method of functional multidimensional scal-ing, let us consider data obtained from many similar observa-tions. 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
Nonlinear regression-typological analysis of ecophysiological states of
vegetation: a pilot study with small data sets
V. T. PEREKREST,
1T. V. KHACHATUROVA,
1I. B. BERESNEVA,
1N. M. MITROFANOVA,
2E. KÜNSTLE,
3E. WAGNER
4and L. FUKSHANSKY
4,51
Economic-Mathematical Institute, Russian Academy of Science, 1 Chaikovskistreet, St. Petersburg, Russia
2
Technical University of St. Petersburg, 29 Polytechnic Street, St. Petersburg, Russia
3
Institut für Waldwachstum, University of Freiburg, D-78 Freiburg, Germany
4
Institute of Biology II, University of Freiburg, Schänzlestrasse 1, D-78 Freiburg, Germany
5
Author to whom correspondence should be sent
Received August 11, 1993
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 -dimen-sional 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 corre-lations between independent parameters (e.g., temperature, radiation) and their combinations on one hand, and the pa-rameters supposed to be affected by the independent parame-ters (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 inde-pendent 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 pa-rameters 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 neigh-boring 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 multidimen-sional 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 scal-ing 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 interconnec-tions 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 inter-relation 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 meas-ured 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 exter-nal factors and their combinations to an ecophysiological re-sponse 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).
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 ‘‘natu-ral’’ data subsets (homogeneous classes) having both parame-ter values and parameparame-ter inparame-teraction 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 (predomi-nantly linear transformations, and exponential and power func-tions are used). Thus, an investigation of the interaction of different environmental factors under natural conditions re-quires a complex approach combining elements of both ty-pological 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 experi-mental station of the Institut für Waldwachstum of the Univer-sity 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 environ-ment. Climatic variables were continuously sampled in paral-lel with measurements of CO2 gas exchange (photosynthesis and respiration) and ambient ozone concentrations. To investi-gate 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:
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 Clus-ters 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 sub-clusters differing from one another with respect to the qualita-tive 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 signifi-cant dependence of CO2 only on parameter I (Table 1). Analo-gous 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, sig-nificantly 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 homogene-ous with respect to the structure of the interrelations between the parameters. To solve this problem, a geometric repre-sentation 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 charac-teristic of the aggregate of points in the initial 4-dimensional parameter space.
The system of vectors {Vi} averaged over all the
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.
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 circum-scribed 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 gra-dients of parameters from Group B; the vertical layers, desig-nated v1 to v5, correspond to those from Group A (Groups A and B are specified in Figure 3). The values of the parameters
Figure 1. Examples of time series of the environmental and physiological parame-ters 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.
Zero term t T RH I ∆O3
Cluster 2-- 0.577 0.001 0.004 0.005 0.004 0.0004
all relationships (0.01) (0.03) (0.01) (0.0005) (0.002)
Cluster 5-- 0.432 -- -- 0.005 0.003 0.006
for single classes are presented in Table 2 (note that all parame-ters are scaled to mean zero and unit variance). The following conclusions were derived from Table 2. (1) Most of the meas-urements 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 parame-ter 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 direc-tion 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 tem-perature value over a class was accompanied by an increasing regression coefficient of I and a decreasing regression coeffi-cient of T, reflecting the nonlinear character of the relationship over the entire parameter space. (2) Pattern CO2 (I) was char-acteristic 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 dif-fered 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
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
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.
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 parame-ters 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 pa-rameters 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
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 −0.694 −0.428 −0.612 −0.482 −0.447
g4 −0.526 −0.432 −0.441 −0.510 −0.475
g3 −0.383 −0.394 −0.459 −0.427 −0.382
g2 −0.145 −0.205 −0.124 −0.160 −0.078
g1 0.447 0.348 0.291 0.281 0.237
I g5 2.677 2.693 2.712 2.596 2.609
g4 2.094 1.957 2.142 1.983 1.983
g3 1.047 0.970 0.976 0.990 0.896
g2 −0.044 −0.012 −0.135 0.005 −0.147
g1 −0.673 −0.673 −0.675 −0.671 −0.667
T g5 1.403 1.063 0.604 −0.279 −1.100
g4 1.585 0.970 0.343 −0.573 −1.025
g3 1.698 0.883 0.514 −0.450 −1.069
g2 1.604 0.837 0.355 −0.498 −1.328
g1 1.077 0.590 0.122 −0.895 −1.368
RH g5 −1.970 −0.650 0.840 −1.224 0.325
g4 −2.062 −0.703 0.359 −0.721 0.738
g3 −2.401 −0.618 0.778 −0.793 0.916
g2 −2.220 −0.777 0.449 0.078 1.299
g1 −2.056 −0.606 0.435 0.045 1.247
+O3 N% g5 100.0 48.8 60.0 57.1 57.1
Y% g5 0.0 51.2 40.0 42.9 42.9
N% g4 50.0 55.4 46.7 68.3 72.2
Y% g4 50.0 44.6 53.3 31.7 27.8
N% g3 50.0 46.0 40.0 75.8 78.4
Y% g3 50.0 54.0 60.0 24.2 21.6
N% g2 50.0 51.8 50.0 75.0 92.2
Y% g2 50.0 48.2 50.0 25.0 7.8
N% g1 48.1 50.2 49.8 66.9 90.4
Y% g1 51.9 49.8 50.2 33.1 9.6
Month M% g5 0.0 * * * *
A% g5 100.0
M% g4 *
A% g4 *
M% g3 *
A% g3 *
M% g2 *
A% g2 *
M% g1 *
points circumscribed by the closed curves contained points expressed by different symbols. This indicates that the resolu-tion of the method may be insufficient for a rigorous analysis
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
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 v2 v3 v4 v5
g5 a0 −0.7640 4.6500 −0.3710
I 0.1530 2 −2.1700 3 −0.1120 2
T −0.0477 2 2.1800 3 0.0599 2
RH 0.0636 2 −1.0700 3 0.2070 3
r0 0.399 0.907 0.955
g4 a0 −1.0660 −0.3480 −0.6670 −0.4740 −0.5450
I −0.0146 2 −0.0384 2 0.1004 2 0.0266 2 0.0009 1
T 0.1890 2 0.0050 1 −0.1150 2 0.0226 2 −0.1150 3
RH −0.2630 3 −0.0117 2 −0.0867 2 −0.1100 3 0.0631 2
r0 0.994 0.144 0.694 0.629 0.715
g3 a0 −0.3600 −0.2600 −0.0257 −0.5910
I −0.0033 1 −0.0373 2 −0.1130 3 −0.0900 2
T −0.0688 2 −0.3390 3 0.0600 2 −0.2170 3
RH −0.0379 2 0.0380 2 −0.1030 3 0.0608 2
r0 0.963 0.359 0.560 0.826
g2 a0 5.0580 0.0158 −0.3060 0.1900 −0.3650
I −0.0147 −0.2340 3 −0.5490 3 −0.4540 3 −0.3900 3
T −0.0732 2 −0.0888 2 0.0460 2 0.0401 2 −0.1200 2
RH 1.7200 3 0.1270 3 0.0307 2 0.0114 2 0.0387 2
r0 0.994 0.487 0.785 0.889 0.846
g1 a0 −1.3500 −1.0100 −0.5010 −0.3750 −0.2550
I −2.0800 2 −1.0350 3 −1.0350 3 −1.0530 3 −0.7820 3
T 0.2610 3 0.7240 3 0.0724 3 0.0902 3 0.0591 3
RH 0.0914 2 0.0215 2 0.0215 2 0.0089 2 0.0021 1
r0 0.931 0.864 0.758 0.974 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.
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
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 de-creases 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 mono-tonically 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 ecophysi-Figure 5 into 15 homo-geneous classes. The classes are specified in Figure 4.
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 −0.565 −0.559 −0.517 −0.513 −0.658
g2 −0.342 −0.478 −0.420 −0.403 −0.371
g1 0.226 0.050 0.299 0.341 0.501
I g3 2.025 2.278 1.855 1.777 1.924
g2 0.660 0.732 0.493 0.326 0.180
g1 −0.623 −0.406 −0.651 −0.602 −0.691
T g3 −1.431 −0.167 0.567 0.803 1.237
g2 −1.723 −0.347 0.357 0.492 0.881
g1 −2.163 −0.159 −0.031 0.135 0.502
RH g3 −0.473 0.401 −0.270 −0.593 −1.454
g2 0.266 1.250 −0.181 −0.010 −1.167
g1 0.803 1.105 −0.022 0.212 −1.311
∆O3 g3 2.163 0.203 0.174 −0.616 −1.510
g2 2.164 0.239 0.173 −0.915 −1.379
g1 2.159 0.316 0.179 −0.986 −1.480
+O3 N% g3 0.0 20.0 18.9 80.3 100.0
Y% g3 100.0 80.0 81.1 19.7 0.0
N% g2 0.0 0.0 19.2 100.0 93.9
Y% g2 100.0 100.0 80.8 0.0 6.1
N% g1 0.0 0.0 24.3 95.2 100.0
Y% g1 100.0 100.0 75.7 4.8 0.0
Month M% g3 0.0 * * * *
A% g3 100.0
M% g2 *
A% g2 *
M% g1 *
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 mono-tonically 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) Pat-tern 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 scal-ing 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 pro-vides a comparison of the effects of one parameter at different levels of the other parameters. Furthermore, the same compari-son 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. Sta-tistically significant information about changes in the interre-lations between parameters can be derived by comparing the
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 repre-sent the dynamics of changes in the environment and the physiological state. Moreover, explicit presentation of the dy-namics 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 regression-and classification-typological analysis of large data sets: (1) the dimension (and therefore the complexity) of the optimiza-tion 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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