Radial tree-growth modelling with fuzzy regression

A so-called fuzzy linear regression is used in dendroecology to model empirically tree growth as a function of a bioclimatic index representing the water stress, i.e., the ratio of actual evapotranspiration to potential evapotranspiration. The response function predicts tree growth as (fuzzy) intervals, narrow in the domain where the bioclimatic index is most limiting and becoming progressively larger elsewhere. The method is tested with a population of Pinus pinea L. from the Provence region in France. It is shown that fuzzy linear regression gives results comparable with those obtained using a linear response function. The interval of credibility given by the fuzzy regression suggests that more precise expected growth is obtained for high water stress, which is typical of Mediterranean climate. Fuzzy linear regression can be also a method to test different hypotheses on several potential predictors when any further experimental approach is quite impossible as it is for trees in their natural environment. To sum up, fuzzy regression could be a first step before the construction of a kind of growth simulator adapted to different environments of a given species. In environmental sciences, the fuzzy response function thus appears to be an approach between the mechanistic and the statistical descriptive approaches. Résumé : Une régression dite linéaire floue est utilisée en dendroécologie pour modéliser empiriquement la croissance des arbres en fonction d’un indice bioclimatique représentant le stress hydrique, c’est-à-dire, le ratio de l’évapotranspiration réelle par rapport à l’évapotranspiration potentielle. La fonction prédit la croissance des arbres par intervalles (flous) qui sont étroits dans la région où l’indice bioclimatique est le plus limitatif et s’élargissent progressivement dans les autres régions. La méthode est testée avec une population de Pinus pinea L. de la région provençale de France. Il est démontré que la régression linéaire floue donne des résultats comparables à ceux obtenus avec une fonction à réponse linéaire. L’intervalle de crédibilité donné par la régression floue suggère une croissance plus precise attendue lors d’un stress hydrique élevé, typique du climat méditerranéen. La régression linéaire floue peut aussi servir à tester différentes hypothèses au sujet de prédicteurs potentiels lorsque toute approche expérimentale additionnelle est impossible, comme dans le cas des arbres dans leur environnement naturel. En conclusion, la régression floue pourrait constituer une première étape avant la construction d’une sorte de « simulateur de croissance » adapté aux différents milieux occupés par une espèce donnée. En sciences environnementales, la fonction de réponse floue semble donc être une approche qui se situe entre l’approche déterministe et l’approche des statistiques descriptives. Boreux et al. 1260 Empirical statistical models (Fritts 1976) usually describe the response of tree growth to climate. Not all climate factors are important to a tree at any given moment. Only those that limit some process can affect growth (Fritts 1982). The principle of limiting factors states that each process is governed by one factor at a time, namely, the factor that is the most stressing at that time. However, as we want to calculate response functions valid for the whole live tree, it is necessary to take into account a large number of possible limiting factors and thus to use multivariate statistics. Since the pioneer paper of Fritts et al. (1971), monthly mean temperature and monthly precipitation over a period from 12 to 16 months before the end of the growth, i.e., August or September depending on the species and (or) the climate, are used as growth predictors. As such a large number of predictors is not without statistical problems (correlation of the predictors, reduced number of degrees of freedom, etc.); the set is reduced to independent variables by using principal component analysis (Fritts et al. 1971; Fritts 1976; Guiot et al. 1982a). The effects of using an excessively large number of predictors have been well established by Monte Carlo methods (Cropper 1982) or bootstrap ones (Guiot 1989). Regrouping individual months into biological seasons according to some a priori knowledge of the tree ecology or on the basis of monthly response functions has also been tested to reduce the number of parameters in the models (Guiot et al. 1982b). More bioclimatic parameters, such as evapotranspiration, have also been introduced (Badeau et al. 1995; Bert 1992; Gadbin-Henry 1994 ; Lebourgeois 1995). Mechanistic models (Shashkin and Fritts 1995) are much more difficult to implement. In particular, the need for daily values as Can. J. For. Res. 28: 1249–1260 (1998) © 1998 NRC Canada 1249 Received December 22, 1997. Accepted May 17, 1998. J.J. Boreux. Fondation universitaire luxembourgeoise, avenue de Longwy, 185, 6700 Arlon, Belgique. e-mail: [email protected] C. Gadbin-Henry, J. Guiot,1 and L. Tessier. Institut méditerranéen d’écologie et paléoécologie, Centre national de la recherche scientifique, Faculté de St-Jérôme, case 451, 13397 Marseille Cédex 20, France. e-mail: [email protected]; [email protected], and [email protected] 1Author to whom all correspondence should be addressed. input into such bioclimatic or mechanistic models is difficult to satisfy most of the time. Mechanistic models should be the favoured approach in the future, as they are the only ones to completely take into account the biology of the problem, thus insuring a greater robustness of the predictions. Here we do not propose such a model, but we stay at a more empirical level, which is the easiest approach when large sets of data, as available in dendroecology, must be analysed. Meanwhile, by trying to incorporate maximum a priori information into the model, we think that more useful predictive models can be elaborated. Bayesian or fuzzy logic methods are able to satisfy these requirements, either by estimating the model parameters from an a priori distribution (Bayesian approach) or by chosing a reference point where the knowledge is maximum and (or) by defining the shape of the fuzzy numbers according some biological considerations. We test a so-called fuzzy linear regression on selected bioclimatic variables that take into account the uncertainties inevitable in most of the data. We also show that this approach is able to take into account the particular profile of the climatic effect on tree growth. The trees react to climate according to the limiting factor principle, which means that high values of the climatic factor do not have the same proportional effect as low values. It also means that this limiting factor can change during yearly tree growth and from one year to the next. The method is illustrated by a population of Pinus pinea L. from the Provence region in France. Field site A population of Pinus pinea L. has been retained for this study, because it grows in an healthy and large stand with an active regeneration, so it is able to reflect the average ecological conditions for this species. The sensitivity to climatic variations (called “mean sensitivity” by Douglass (1936) and denoted S) has a high level for a Mediterranean species (S = 0.26). The site is located at an altitude of 105 m, on an almost flat area near the small town of Vidauban, in the so-called Bois de Rouquan locality (43°22′N, 6°27′E). The substratum is a sandstone outcrop of the Permian depression along the crystalline massif of the Maures. In such edaphic conditions, the main limiting climatic factor is humidity; this was demonstrated previously by comparing the radial growth of Pinus pinea L. with both precipitation and bioclimatic coefficients such as real or potential evapotranspiration (Gadbin-Henry 1994). Mediterranean-type climate prevails in the studied region characterized by a cold winter and a warm and dry summer. The aridity coefficient of Emberger (1930) at the nearest meteorological station classifies the site in the subhumid zone, with a total annual precipitation of 889 mm, a mean annual temperature of 14.6°C, a mean minimum value of 1.7°C for the coldest month, and a mean maximum value of 30.9°C for the warmest month. The dominant trees were retained, because they were less affected by competition processes and thus were the best climate recorders (Schweingruber et al. 1990). Within this population, 14 dominant trees were selected to obtain the longest possible treering series, undisturbed by any accidental event. The raw data were provided by cores obtained with the Pressler borer (three samples per tree at 60° intervals around the trunk). Tree-ring width, reflecting tree growth, were directly measured on planed cores with a tree-ring measuring system including integral recording. Comparison of ring-width chronologies revealed a mean correlation of 0.95. The whole population is represented by the mean chronology, including the 42 elementary chronologies. Meteorological data for the period 1950–1985 (monthly precipitation and temperature) were provided by the nearest meteorological station having similar geographical and climatological characteristics (Fréjus; 43°26′N, 6°45′E, altitude 50 m). Bioclimatic variables We use three bioclimatic variables representing the water stress, winter frost, and thermal energy available in the growing season. These variables are derived from three monthly parameters easily available at a reasonable distance (40 km) from the tree-ring site analysed, i.e., the 12 monthly mean temperatures (°C), monthly precipitation amount (mm), and the monthly sunshine. Using the simple equations described in Harrison et al. (1993) and Prentice et al. (1993), these basic variables are transformed, for each year, into (i) the mean temperature of the coldest month (Tc in °C); (ii) the growing degree-days above 5°C (GDD5 in degree-days); and (iii) the ratio of actual evapotranspiration to potential evapotranspiration (α in %). As often, daily temperatures are not available for local meteorological stations, we have devised a method to roughly calculate them from monthly values by cubic-spline interpolation. GDD5 is obtained from these quasi-daily values by summing the part above 5°C. For the actual evapotranspiration, a simple water-balance model is used (Harrison et al. 1993). The actual evapotranspiration (AET) is taken to be the lesser of a supply function proportional to soil moisture (Federer 1982) and a demand function set equal to the potential evapotranspiration (PET). PET is empirically defined as a function of the net radiation and temperature (Jarvis and Macnaughton 1986). Net radiation is obtained as a semi-empirical function of insolation, sunshine proportion, and temperature and varies sinusoidally during the day, allowing daily actual and potential evapotranspiration to be obtained by integration (Prentice et al. 1992, 1993). Sunshine is indexed as a proportion of the maximum possible sunshine hours for the latitude and month under consideration. In addition to the three monthly climatic variables, latitude of the site and orbital parameters (eccentricity, obliquity, and phase angle indicating the timing of the perihelion relative to the equinoxes) are used as input variables. This method of calculating α is rather rough, but various tests (Gadbin-Henry 1994) have shown that this approximated parameter correlates well with the studied tree-growth series. The model requires daily precipitation values, which are obtained by dividing the monthly precipitation by the number of days in the month. The soil moisture, Ω, is obtained from January 1 to December 31 by integrating daily values calculated by adding the difference between daily precipitation and daily PET within a range [0, Ωmax]. Ωmax is the soil water-holding capacity above which the water runs off. As no data are available for the year prior to the study, we start with a value of Ω equal to Ωmax, and for the following years, we start with the last value calculated for the previous year. Response function using a bootstrap multiple regression In dendroecology, it is usual to calculate the effect of climate on tree growth by a multiple regression, called a response function. This model often includes 24 predictors and sometimes more (Fritts 1976). It can be written as follows: