Modeling spatio-temporal crop allocation patterns by a stochastic decision tree method, considering agronomic driving factors

Agricultural Systems 103 (2010) 647–655

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Modeling spatio-temporal crop allocation patterns by a stochastic decision tree method, considering agronomic driving factors Luc Sorel, Valérie Viaud ⇑, Patrick Durand, Christian Walter INRA, UMR1069, Sol Agro et hydrosystème Spatialisation, F-35000 Rennes, France Agrocampus Ouest, UMR1069, Sol Agro et hydrosystème Spatialisation, F-35000 Rennes, France

a r t i c l e

i n f o

Article history: Received 19 June 2009 Received in revised form 13 July 2010 Accepted 17 August 2010 Available online 18 September 2010 Keywords: Crop rotation Decision tree Land cover change modeling Transition probability matrices Stochastic modeling

a b s t r a c t Evaluating the environmental impacts of agricultural practices increasingly involves the use of spatially distributed simulation models that account for crop allocations across fields as an input factor. Our objective was to develop a model for spatio-temporal allocation of crops to a field pattern that was able to account for agronomic and spatial driving factors including crop production objectives, spatial distribution of the crops around farmsteads, and preferential allocation of crops on soil waterlogging classes. We developed a model based on stochastic decision trees (SDTs) to integrate farm type and field characteristics (area, distance to farmstead, waterlogging, and current crop) in the spatio-temporal allocation process without prior expert knowledge, and we compared the model to a reference model based on firstorder Markov chains or transition matrices. A case study comparing both models was performed in the Naizin catchment (Western France), where crop allocation to fields was known for the period 1993– 2006. The SDTs built had a general structure similar to transition matrices. SDTs and transition matrices exhibited similar performances in predicting crop transitions in time and in allocating crops to the proper soil waterlogging class. However, SDTs proved to better reproduce the spatial distribution of crops around the farmsteads. SDTs provide an integrated way to analyze and simulate crop allocation processes within a single integrated framework. The ease of constructing decision trees suggests potential couplings of SDT to various landscape-scale ecological models requiring a detailed description of the land use mosaic as input data. Ó 2010 Elsevier Ltd. All rights reserved.

1. Introduction Agricultural landscapes usually consist of a mosaic of fields where land use and land management practices are located nonrandomly in space and time. Landscape heterogeneity is an important factor driving ecological processes, energy and matter fluxes, and, consequently, agriculture environmental impacts on water, soil, and biodiversity. A number of landscape-scale models have been developed to assess agriculture’s impact on the environment, e.g., hydrological models (e.g., Beaujouan et al., 2002), gene flow models (e.g., Angevin et al., 2008), and soil organic matter models (e.g., Gabrielle et al., 2002), which require explicit descriptions of field patterns and land use and land management allocation in space and time. However, these data are often scarce and not available over the whole studied area or for the whole studied period. The acquisition of these data may be time- and money-consuming and sometimes impossible. Spatio-temporal models for crop allo⇑ Corresponding author. Address: INRA, Agrocampus Ouest, UMR1069 SAS, 65 rue de Saint-Brieuc, CS 84215, F-35042 Rennes, France. Tel.: +33 223485142; fax: +33 223485430. E-mail address: [email protected] (V. Viaud). 0308-521X/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.agsy.2010.08.003

cation over landscapes can help produce realistic comprehensive inputs for the aforementioned landscape-scale ecological models (Agarwal et al., 2002). The spatio-temporal allocation of crops over a field pattern is designed at the farm level and results from a combination of driving factors. Crops are chosen and allocated to fields according to farming system characteristics, farm production objectives, and to the characteristics of each field (e.g., soil quality, size, distance to the farmstead) relative to those of the other fields in the farm (Rounsevell et al., 2003; Thenail and Baudry, 2004). Crop succession in time is mainly driven by agronomic principles (breaking pest and weeds cycles, accounting for crop needs and soil nutrients). Several approaches have been developed to design crop allocation models, but the handling of both spatial and temporal constraints raises technical and scientific challenges (Verburg, 2006). One of the simplest models is based on first-order Markov Chains or transition matrices that are easy to implement and require few data (Logofet and Lesnaya, 2000; Walter et al., 2003; Coppedge et al., 2007). The crop grown in a field is only determined from the preceding crop in this model. However, the model does not account for any of the drivers of the spatio-temporal allocation of

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crops. More complex approaches based on stratified Markov chains (Weaver and Perera, 2003; Ladet et al., 2005; Castellazzi et al., 2007; Pocewicz et al., 2008) require a large amount of data to compute the transition probabilities (Usher, 1979; Baker, 1989) and hardly deal with spatial variability in the processes of land cover change (Huang et al., 2007). The calibration and validation of models such as cellular automata (Bockstael et al., 1995; Veldkamp and Fresco, 1996; Wu and Webster, 1998; Verburg et al., 2004) and explicitly rule-based models require strong expertise (Thornton and Jones, 1998; Largouët and Cordier, 2001; Gaucherel et al., 2006a) and can involve arbitrary decisions (Li and Yeh, 2002; Houet and Hubert-Moy, 2006). Generalized linear models (Aspinall, 2004; Huang et al., 2007) rely on parametric hypotheses and can be hard to interpret. On the contrary, decision trees are non-parametric methods known to be efficient without parameter tuning, accessible and useful for non-experts (Quinlan, 1981; Breiman et al., 1984). In the field of land cover modeling, decision trees are mainly used as classification tools based on remotely-sensed data, pedological and geomorphological factors or spatial descriptors of the farm territory (Friedl et al., 2002; Lawrence et al., 2004; Thenail and Baudry, 2004). Although decision trees have been used to simulate dynamic processes in other research fields (Gladwin, 1989; Heidenberger, 1996; Jordan et al., 1997; Hazen, 2002; Tarim et al., 2006), to our knowledge, they have not yet been applied to crop allocation modeling. The aim of this work was to compare a crop allocation model accounting for crop allocation driving factors made with decision trees to a reference approach based on transition matrices. A case study comparing both modeling approaches was performed in the Naizin catchment (Western France), where crop allocation over the field pattern was known for the period 1993–2006. We focused on modeling main crop successions (one main crop grown per year). We identified the potential relevant driving factors of the spatiotemporal allocation of crops in this catchment. Transition matrices and crop allocation models based on stochastic decision trees were then calibrated and used to simulate the spatio-temporal allocation of crops over 40 years. The structure of both models is discussed, and their performances are compared.

2. Materials and methods

is below 5%), its highest point is at 136 m above the sea level, while its outlet is at 60 m (Walter and Curmi, 1998). 2.2. Potential explanatory variables of crop allocation at farm and field level To be able to model the spatio-temporal allocation of crops over the field pattern in the Naizin catchment, we identified variables or field attributes that we assumed to be potentially explanatory of the crop observed for a given combination of field and year. These variables correspond to two decision levels, the farm and the field. 2.2.1. Farm level 2.2.1.1. Farm type. Farms having the same activity are subjected to the similar constraints regarding crop management (Thenail and Baudry, 2004) and farm types may be characterized by specific production objectives (Thenail and Baudry, 1994). This is particularly true for fodder production (grass and silage maize), which fulfills the needs of cattle. Four farm type groups were considered in this study: dairy farm, pig farm, dairy-pig farm, and other farm types. Each field was classified according to the type of farm it belonged to. 2.2.2. Field level 2.2.2.1. Preceding crop. Crop rotation is known to play a key role in the management of soil fertility and nutrients dynamics of the farm as a whole (Thornton and Jones, 1998). The preceding crop was thus considered as one of the potential explanatory factors of the crop in a given field and a given year. 2.2.2.2. Distance to farmstead. To optimize the use of the farm labor time, crops requiring a lot of operations (tillage, fertilization, cattle feeding) are usually located closer to the farmstead (Baudry et al., 2006). Given the locations of the farmsteads and of their fields (Bordenave et al., 2005; Payraudeau et al., 2006), we associated a distance to farmstead to each field, computed as the Euclidean distance between the farmstead and the centroid of the field (in m). 2.2.2.3. Field area. Field size may be limiting for tillage operation and may explain why some fields are preferentially used for pasture than for other crops. Each field was characterized by its area (in m2).

2.1. Study area We studied the Naizin catchment (Brittany, France: 48.0°N, 2.8E), with an area of 12 km2, of which 85% is usable agricultural area (UAA). The Naizin catchment has been extensively studied since 1971 by various French research organizations (Cheverry, 1998) and is part of the long-term research observatory ORE AgrHys. Farming activities and agricultural land cover are known from farm surveys for the period 1993–2002 (Bordenave et al., 2005; Payraudeau et al., 2006). The farms are of mixed-crop livestock type, with 13 specialized dairy farms (33% UAA), six specialized pig farms (34% UAA), 10 mixed dairy-pig farm (19% UAA) and 10 farms with other products (14% UAA). The original agricultural land cover data were homogenized, and the following crops were considered: permanent pasture, temporary pasture, fallow, seed ray-grass, vegetable (mainly beans), rapeseed, maize, wheat (this class also includes other cereals such as barley), and potato (Fig. 1a). The relative area occupied by each crop weakly changes from 1 year to another (Fig. 1b). The soils are silty loam, and they present mainly variation in their drainage characteristics; the soil system comprises an upland well-drained domain and a poorly-drained bottom land (Fig. 1c). The topography of the catchment is relatively smooth (mean slope

2.2.2.4. Soil characteristics. Soils of the Naizin catchment mainly differ in their drainage condition, whereas texture and depth are quite homogeneous. Some crops are sensitive to soil waterlogging (Dogliotti et al., 2003; Houet and Hubert-Moy, 2006). Based on the catchment soil map (Walter and Curmi, 1998), each field was characterized by its dominant soil waterlogging class. Three classes were distinguished: the well-drained soils (43% UAA) where no redoximorphic feature was observed within the first 80 cm, the intermediate soils (44% UAA) where weak soil redoximorphic features are observed at less than 80 cm, and the poorly-drained soils (13% UAA) where redoximorphic features are observed from the surface. Topography was not included as a significant driving factor because of the observed gentle slopes that were not limiting for the preparation and cultivation of crops in this area. We assumed these potential explanatory variables were time-invariant for the 1993– 2002 period. 2.3. Crop allocation models Our objective was to simulate crop succession over a field pattern stochastically for 40 years, starting from the crop pattern observed in 2000. An implementation of temporal first-order Markov

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Fig. 1. (a) Agricultural land cover in the Naizin catchment in 2000; (b) change in the area occupied by each crop over the period 1993–2002; and (c) map of the soil waterlogging classes in the Naizin catchment.

chains (transition matrices) served as a reference model for comparison with a model based on stochastic decision trees. Both model types were implemented either with consideration of the farm type (specific models) or without (generic models).

2.3.1. Temporal first-order Markov chains: transition matrices First-order Markov chains are transition probability models, which stochastically describe processes that move a system through a set of states in a sequence of time steps, with the property that the next step depends on the current state. The number of

potential states is assumed to be constant with time (Yemshanov and Perera, 2002). Modeling crop succession with a Markov chain means that the presence of a crop on a field in a current year is determined solely by the previous crop (Logofet and Lesnaya, 2000). The system studied is the set of fields. The states of the Markov chain are the different crops. The transition probabilities are defined as a change from one crop to another. In the Markov chain approach, the prediction of future state Xt+1 can be calculated by solving the matrix equation:

X tþ1 ¼ X t  Pt

ð1Þ

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where Xt is a 1  n state vector at the current time (i, j 6 n, where n is the number of crops), Pt is an n  n matrix of transition probabilities made of pij values that describe the probability of transition from discrete state i to state j between time t and t + 1. At any time t, the transition probabilities of Pt matrix satisfy the following conditions:

8 > < 0 6 pij 6 1; n P > : pij ¼ 1

8 i; j ð2Þ

j¼1

The P matrix can be built experimentally from a several-year dataset by counting all the observed annual transitions from a crop i to a crop j and converting these transition frequencies into probabilities. Starting from an observed crop pattern corresponding to the initial year of the simulation, the transition matrix was used to assign iteratively the next crops to fields with a random allocation algorithm. For each crop i (i in 1. . .n) sequentially, the fields occupied by the crop i in the current time step were selected, and a next crop was assigned to each of them according to the probabilities given by the transition matrix P.

infoðTÞ ¼ 

m X j¼1

nj=T pj=T log2 ðpj=T Þ with pj=T ¼ Pm j¼1 nj=T

where info(T) is the information entropy of data partition T, log2 is the logarithm with base 2, m is the number of crops, and nj/T is the number of records in T belonging to the class ‘‘the next crop j”. Given a node or test X that divides dataset T into n subsets Ti (i in 1. . .n), the total information content after applying X is the sum of the information of the n subsets weighted by the number of records belonging to the class ‘‘the next crop is j” in each subset:

infoX ðTÞ ¼

n X i¼1

!

Pm

j¼1 nj=T i Pm j¼1 nj=T

infoðT i Þ

ð4Þ

The information gain due to the attribute tested in X is given by:

gainðT; XÞ ¼ infoðTÞ  infoX ðTÞ

ð5Þ

The gain criterion selects the test X that maximizes gain(T, X). The information gain favors tests with numerous subdivisions. The gain ratio criterion sidesteps this problem by also accounting for the potential information from the partition itself:

splitinfoX ðTÞ ¼ 

n X i¼1

2.3.2. Model based on stochastic decision tree (SDT) As formalized by Quinlan (1981) and Breiman et al. (1984), decision trees (DTs) are classification algorithms mainly used as large dataset exploration methods to model the class of a record on the basis of a set of other attributes and possibly to predict the class of a record on the basis of its attributes. Our objective was to assign the next crop to the fields of the Naizin catchment on the basis of the other field descriptors (Section 2.2). A decision tree is constructed from a training dataset which consists of records. Each record is completely described by a set of attributes and by a class label. Attributes can have discrete or continuous values. Trees are a way to classify the records by a sequence of simple tests relative to their attributes. In this study, the training dataset consisted of all fields throughout the calibration period. The record consisted of a field per given year and was characterized by the following attributes: current crop, area, distance to farmstead, and soil waterlogging class. The class label to be predicted was the next crop (i.e., the crop in the following year). A tree is made of a sequence of nodes that partition the records into mutually exclusive groups. The first node is the root node. At all internal nodes, an attribute to be tested is identified, and a test splits the records into disjoint sets. The splitting continues until a terminal node, or leaf, forms that indicates the class assignment. All internal nodes have two or more child nodes. All paths start at the root node and end at a leaf. Each path represents a decision rule, joining all of the tests along that path, so that a decision tree performs multistage hierarchical decision making. All paths are mutually exclusive (Murthy, 1998). To build DTs from our dataset, we used the C4.5 algorithm (Quinlan, 1993) (10-fold cross-validation in this study) that could handle both discrete and continuous attributes. The algorithm was implemented in the Weka data-mining software (Witten and Frank, 2005). The algorithm induces a tree by recursively selecting and subdividing attributes. At each internal node, it is necessary to find a suitable test for splitting the data into subsets. In the C4.5 algorithm, two splitting criteria derived from information theory are used to find the most useful attribute for discriminating the data: the information gain and the gain ratio. The information contained in a dataset partition is evaluated by an indicator derived from the Shannon entropy index (Shannon, 1948):

ð3Þ

Pm

j¼1 nj=T i Pm j¼1 nj=T

!

Pm log2

j¼1 nj=T i Pm j¼1 nj=T

!! ð6Þ

Therefore, the final splitting metric is the gain ratio:

gain ratioðXÞ ¼

gainðXÞ splitinfoX ðTÞ

ð7Þ

A better gain ratio thus indicates that the records in the new partitions are more homogeneous with regard to the ‘‘next crop” class. After a DT is built, C4.5 goes through a pruning phase to obtain a correctly-sized tree. Knowing when to stop and obtaining the right sized tree are important to avoid noise and overfitting effects on model generalization and use for prediction (Murthy, 1998). Pruning of DT was done by replacing a whole subtree by a leaf that was labeled with the most frequent class. The replacement took place if a decision rule established that the rate of misclassified records in the subtree was greater than in the single leaf. The originality of our approach was to expand the leaves of the deterministic pruned DT into stochastic nodes by converting the frequencies of the different values of class into probability estimates (Quinlan, 1990; Provost and Domingos, 2003). The tree obtained was referred to as a stochastic decision tree (SDT). Similarly to the transition matrix, the SDT model was used to predict the next crops, starting from an observed crop pattern corresponding to the initial year of the simulation. Each field was characterized by its attributes (current crop, area, distance to farmstead, soil waterlogging class). Each field was classified according to the SDT model and was assigned a specific leaf. A random number was drawn from zero to one (uniform law) and was compared to the cumulated probabilities of the possible classes of the leaf (which sum up to 1). The class (next crop) corresponding to the interval in which the random number falls was assigned to the field. 2.3.3. Model comparison Our goal was to reproduce the spatial and temporal patterns of crops observed in the study area. The models were therefore compared with respect to these two criteria. 2.3.3.1. Prediction of expected annual crop transitions. Transition matrices and SDT models were calibrated using the crop allocations observed during the period 1993–1998 as a training dataset. Crop allocations observed during the period 1999–2002 were used as a validation dataset. For each crop transition of the validation

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dataset, 30 transitions were simulated by each model as statistical repetitions. Then, contingency tables between the observed and the simulated crop provided a way to compare the prediction efficiency of transition matrices and SDT models. 2.3.3.2. Role of agronomic driving factors. Once trained, transition matrices and SDT models were used to simulate crop allocation over 40 years, starting from the crop allocation pattern observed in 2000. To evaluate the models’ ability to comply with agronomic constraints, the simulation results (2000–2040) were compared to the observed dataset (1999–2002) according to the three following criteria: – Production objectives per farm type. The annual proportion areas occupied by each crop within each farm type and simulated by the models over 40 years were compared to the annual observed proportions. – Spatial organization of crops around the farmstead. We used the mean distance weighted by the field area (area-weighted distance: AWD) as a synthetic metric.

PFðft;c;yÞ AWDðft; c; yÞ ¼

field¼1 dist field  areafield PFðft;c;yÞ field¼1 areafield

ð8Þ

where AWD(ft, c, y) is the AWD (distance in m) calculated with the set of fields F(ft, c, y) belonging to farm type ft, occupied by crop c at year y, distfield (in m) is the distance of the field to the farmstead, and areafield is its area (in m2). The distribution of the AWDs computed per farm type on the simulated data was compared to the distribution of the AWDs computed on the observed data. – Preferential allocation of crops related to soil waterlogging class. For each crop, the distribution of the proportion areas allocated to poorly-drained soil for the simulated dataset was compared to the observed proportions without consideration of the farm type. The significance of the comparisons mentioned above for the 30 statistical repetitions was tested with non-parametric Wilcoxon distribution tests (P < 0.05). 3. Results 3.1. Structure of the crop succession models 3.1.1. Transition matrices One generic and four specific transition matrices were constructed from the training dataset (1993–1998). The generic transition matrix is represented in Table 1. Maize and wheat played a major role in crop rotations because they had major cumulative

probabilities of becoming the next crop. Conversely, fallow, potato, rapeseed and seed ray-grass were marginal and intermittent crops (Fig. 1b), as they had minor cumulative probabilities of becoming the next crop. Permanent pastures had a 100% self-replacement probability and a small probability of becoming the next crop after a vegetable. This means that the area occupied permanent pastures and that was simulated by the Markov model could only increase with time. The probabilities of specific transition matrices differed mainly for maize and pastures (as suggested by higher probabilities for the dairy farm type and lower probabilities for the pig farm type) and secondarily for vegetables (as suggested by the higher probabilities for the pig farm type).

3.1.2. SDT models For the sake of clarity, Fig. 2a represents only the general structure of the generic and farm type specific SDTs. The root node of all SDTs involved the ‘‘current crop” attribute, which makes the SDT very similar to a first-order Markov chain. The other nodes stratify the transition probabilities by waterlogging classes, area and distance splits. However, more than half of the tree branches lead directly to a leaf without any other intermediate node. The SDT specific for dairy-pig farms contained only one branch involving an attribute other than the current crop. Depending on the farm type, the ‘‘current crop is wheat” branch involved subtrees with different combinations of tested attributes. The sample branches from the dairy SDT (Fig. 2b) illustrate how various attributes can be successively tested and lead to different sets of transition probabilities. The SDT also shows that once a node has partitioned the tree by testing a discrete attribute such as waterlogging, the attribute will not be tested further down the tree structure. On the contrary, continuous attributes, such as field area and distance, can be tested at any stage of the SDT because nodes perform binary splits of the dataset.

3.2. Comparative analysis of the simulations 3.2.1. Prediction of expected annual crop transitions Table 2 shows the ability of the transition matrices and SDT models to predict the next crop on a validation dataset representing all the annual crop transitions observed in Naizin between 1999 and 2002. Marginal crops (fallow, potato, rapeseed, vegetable), which represent a small proportion of the catchment area (Fig. 1b) exhibited the lowest success rates. For the other crops, success rates ranges from 30% to 97%. Successful prediction rates of specific models for non-marginal crops were above success rates of the generic models by 2–10%. The performances of SDTs and transition matrices were similar with a difference of 1% for the overall prediction rate.

Table 1 Generic transition matrix computed for the crops of the Naizin catchment over the period 1993–1998. Current crop

Permanent pasture Temporary pasture Fallow Seed ray-grass Vegetable Rapeseed Maize Wheat Potato

Next crop: transition probabilities Permanent pasture

Temporary pasture

Fallow

Seed ray-grass

Vegetable

Rapeseed

Maize

Wheat

Potato

1.000 0.000 0.000 0.000 0.021 0.000 0.000 0.000 0.000

0.000 0.709 0.000 0.000 0.103 0.000 0.202 0.000 0.082

0.000 0.005 0.333 0.000 0.008 0.000 0.015 0.006 0.000

0.000 0.000 0.000 0.583 0.000 0.000 0.000 0.000 0.000

0.000 0.050 0.133 0.000 0.107 0.200 0.093 0.133 0.082

0.000 0.003 0.000 0.000 0.000 0.000 0.006 0.006 0.000

0.000 0.149 0.266 0.25 0.276 0.300 0.274 0.477 0.469

0.000 0.079 0.268 0.167 0.452 0.400 0.404 0.200 0.377

0.000 0.005 0.000 0.000 0.033 0.100 0.006 0.000 0.000

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(a) Wh Mz

distance

hydromorphie

hydromorphy

generic

Pot

Wh

random<= 0.05

(b)

0.05 < random<= 0.24

area

0.24 < random<= 0.25

area

hydromorphy=1 0.25 < random<= 0.26

distance

0.26 < random<= 0.99

distance

Mz hydromorphy dairy

Rape Pot TempP Veg

random> 0.99 area Wh distance

Mz

Wh

random<= 0.03 area

TempP hydromorphy

0.03 < random<= 0.06

area

distance

distance<=141 0.06 < random<= 0.08 random> 0.08

Veg area

Mz Rape TempP Wh

random<= 0.03

hydromorphy

Wh

dairy_pig

0.03 < random<= 0.10

distance

dairy

TempP

area <= 8925

hydromorphy=2

0.10 < random<= 0.14 0.14 < random<= 0.93

Wh

random> 0.93

distance area distance

pig

distance

Mz

area

Veg

distance

hydromorphy

Wh

0.10 < random<= 0.23

Mz

0.23 < random<= 0.95

distance

other

random<= 0.10

distance<=744

area > 8925

Wh

TempP Veg

distance > 141

area

Mz hydromorphy distance TempP area Pot

Mz Pot

distance>744 distance

area

random<= 0.14 hydromorphy=3

0.14 < random<= 0.83 random> 0.83

Mz

TempP

random> 0.95

Veg

random<= 0.08

Wh

0.08 < random<= 0.83 random> 0.83

Mz TempP

TempP Veg

Fig. 2. (a) Simplified structures of the generic and specific stochastic decision trees (distances and areas are expressed in m and m2, respectively. Waterlogging classes are: 1 = well-drained, 2 = intermediate and 3 = waterlogged. Crop classes are: PermP = permanent pasture, TempP = temporary pasture, Fal = fallow, SRG = seed ray-grass, Mz = maize, Wh = wheat, Veg = vegetable, Rape = rapeseed, Pot = potato). Only those branches are represented for which attributes other than the current crop appeared and (b) sample of the stochastic decision tree for the dairy farm type, corresponding to temporary pasture as the current crop.

Table 2 Success rate of the transition matrices and the stochastic decision trees by expected next crop on a validation dataset. Next crop

Permanent pasture Temporary pasture Fallow Seed ray-grass Maize Wheat Vegetable Rapeseed Potato All crops

Successful prediction rates (%) Transition matrices

Stochastic decision trees

Generic

Specific

Generic

Specific

97 52 21 63 30 36 13 0 3 38

97 54 31 54 32 39 10 0 5 40

97 52 22 63 34 38 12 1 4 39

97 55 32 59 34 41 12 0 7 41

3.2.2. Role of agronomic driving factors Whatever the model used, the area occupied by each crop remained overall constant over the 40-year simulation and similar to the proportions observed in 2000, except for the permanent pastures. Their area slightly increased over the simulation period (0.5 ha1 year1) (results not shown).

3.2.2.1. Farm production objectives. Errors in crop proportions were between 0% and 18% on average (Table 3); the simulations result-

ing from SDTs and transition matrices modified significantly the simulated crop proportions compared to the observed proportions. Major differences occurred for maize and temporary pastures in the pig and dairy farm types. Errors associated with generic transition matrices were larger than those associated with specific matrices for most of the crops except for temporary pastures in dairy farms and permanent pastures in pig farms. The differences with the observed proportions were similar for SDTs and transition matrices. However, the significant differences in proportions simulated by SDTs were generally higher by 2–5% than the errors associated with the transition matrices. The sign of significant differences was generally the same among transition models for a given crop. 3.2.2.2. Spatial organization of crops around the farmstead. Fig. 3 shows the errors in the spatial organization of crops resulting from the simulations for the dairy farm type, which exhibited the most spatially-structured crop pattern in the observed dataset; differences between the observed and simulated AWD ranged between 50 m and 300 m. The Wilcoxon tests (results not shown) showed that the differences between the observed and simulated AWD were significant for all crops except maize and wheat. SDTs generally better reproduced the spatial organization of crops than transition matrices; the patterns simulated by SDTs for wheat, temporary pastures and permanent pastures were very similar to the observed one. Major differences occurred for maize and vegetables. For a given crop, a specific SDT caused smaller AWD differences than the generic SDT. The figure also shows that the

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Table 3 Difference of crop proportions of the farm type area between the 40-year simulations and the 1999–2002 observation period for the transition matrices and the stochastic decision trees (SDTs). Farm type

*

Crop

Observed proportion (%)

Difference in crop proportion (%) Transition matrix generic

Transition matrix specific

SDT generic

SDT specific

Dairy

Perm. pasture Temp. pasture Maize Wheat

11 40 19 17

2* 12* 9* 6*

3* 4* 2* 0

4* 16* 11* 7*

NAa 5* 4* 4*

Dairy-pig

Perm. pasture Temp. pasture Maize Wheat

8 23 13 28

0 3* 0 1

NA 8* 4* 2

7* 2 2* 0

NA 10* 2* 0

Pig

Perm. pasture Temp. pasture Maize Wheat

1 5 36 38

4* 18* 13* 6*

15* 10* 11* 10*

9* 13* 11* 6*

13* 1 1 8*

Wilcoxon test, 30 statistical repetitions, P < 0.05. Value not available because the number of fields was not sufficient to test the statistical significance of the results with the Wilcoxon test.

a

Wheat

Maize

Vegetable

Farmstead

Legend 200m

Mean AWD (in m) observed generic transition matrix specific transition matrices generic stochastic decision tree specific stochastic decision tree

400m

Dairy AWD

600m

800m

Permanent pasture

1000m

Temporary pasture

Fig. 3. Simulated and observed spatial distribution of the crops around the farmstead for the dairy farm type, for each crop allocation model. The metric used is the areaweighted distance (AWD) of the crop pattern. The grey dotted circle shows the average distance of the fields to the farmsteads in the dairy farm type.

Table 4 Difference of crop area proportion located on the waterlogged soil class between the 40-year simulation period and the 1999–2002 observation period for the transition matrices and the stochastic decision trees (SDTs). Crop

Perm. pasture Temp. pasture Fallow Seed Ray-grass Maize Wheat Vegetable Rapeseed Potato *

Observed proportion (%)

25 8 52 5 3 4 5 0.3 5

Difference of crop proportion (%) Transition matrix generic

Transition matrix specific

SDT generic

SDT specific

6* 1 43* NAa 2 3* 1 7 2

6* 3* 39* 8 2* 3* 1 10 6*

8* 1 24* 2 1 1 2 6 1

6* 4* 9 1 1 3* 1 4 6*

Wilcoxon test, 30 statistical repetitions, P < 0.05. Value not available because the number of fields was not sufficient to test the statistical significance of the result with the Wilcoxon test.

a

transition matrices tend to fix all the crops’ AWDs to the mean AWD of the dairy farms (636 m), i.e., the spatial organization of crops tends to be independent of the distance to the farmstead. Similar results were found for the other farm types (results not shown).

3.2.2.3. Preferential allocation of crops with respect to the soil waterlogging class. Table 4 shows the ability of the crop allocation models to reproduce the proportion area of a crop on the poorlydrained soil class. The absolute differences with the observed proportions were similar for the SDT and transition matrices.

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Besides permanent pastures, which stayed on the same fields by definition, observed data in the Naizin catchment showed that poorly-drained soils were mainly occupied by fallow, which exhibited the most striking differences in results between simulations in favor of specific rather than generic implementations, and, more generally, SDT over transition matrices. This result shows that the explanatory variables in the SDT were not sufficient to fully explain the spatio-temporal allocation of fallows over the field pattern.

4. Discussion and conclusion 4.1. Comparing observations and simulations The analysis of data collected in the Naizin catchment over the period 1993–2002 allowed us to quantify some features of the spatial allocation of crops in this agricultural landscape. The dairy farm type exhibited a highly-structured spatial organization of crops around the farmstead; pastures grazed by dairy cows were located as close to the farmstead as possible because dairy cows move daily from the farmstead into the fields, whereas permanent grasslands grazed by heifers, which require little management, are located further away from the farmstead. Pigs are bred in buildings associated with the farmstead, thus, this farm type required farmstead field trips only for crop management and explains why the pig farm type had a larger AWD (798 m). Thenail and Baudry (2004) report a similar spatial organization of crops according to the farm type. All field descriptors (soil waterlogging, field area, and field distance to farmstead) were actually incorporated into the SDTs, but the difference in simulation performances between SDTs and transition matrices varied according to the driving factor considered. Each driving factor is discussed hereafter.

4.1.1. Farm production objectives The root node of SDTs corresponds to the current crop attribute. SDTs are thus very similar to transition matrices, whose mathematical properties (no negative term in the matrix, ergodicity) are known to induce or maintain stationarity of crop proportions when used to simulate crop successions (Usher, 1992; Logofet and Lesnaya, 2000; Castellazzi et al., 2007). In our study, the transition matrices had such properties except for the permanent pasture, and the observed stability of crop proportions over the 1993– 2002 period suggests that the system already was approximately at a stationary state. The structural similarity of the SDTs to the transition matrices explains why SDTs and transition matrices showed similar performances in terms of crop production objectives. Stratifying the training dataset by farm type allowed us to build specific models, which simulated crop proportions closer to the observed ones. However, because the structure of SDTs and transition matrices happened to be common, SDT simulations were also sensitive to ‘‘absorbing” crops (Usher, 1979); the area dedicated to permanent pastures increased at a very slow speed because the only transition leading to permanent pastures was from the vegetables, which were marginally represented in the catchment.

4.1.2. Spatial organization of crops around the farmstead SDTs performed better than transition matrices when trying to reproduce the spatial organization of crops around the farmstead. Fig. 2 shows that the major next crops were explained by the distance attribute and the distance criterion, and Table 1 shows that they had also the highest probability of becoming the next crop.

4.1.3. Preferential allocation of crops over the poorly-drained soil class SDTs and transition matrices exhibited similar performances, except for fallows for which the use of specific SDTs enabled us to allocate them on waterlogged soils, as observed in reality. The fact that matrices were able to show the same performances as SDTs on several crops is not so surprising for two reasons. First, soil waterlogging is a constant property of the landscape over the observation period (1993–2002). Therefore, it is very unlikely to be observed as transitions between resistant and sensitive crops, and this may have resulted in null or marginal transition probabilities in the matrices. Second, the drainage systems set up by the farmers may have caused local discrepancies in the training dataset, with sensitive crops actually able to grow on soils mapped as waterlogged. 4.2. Validation and generalization of SDT simulations Compared to the construction of crop transition models, such as ROTAT (Dogliotti et al., 2003) and ROTOR (Bachinger and Zander, 2007), our approach replaced expert knowledge by a data-mining phase to grow the decision trees. The construction of decision trees is known to be sensitive to the quality and the noisiness of the training dataset (Breiman et al., 1984), therefore a validation of the decision tree is necessary. For this purpose, we compared the simulation results with an observed dataset with tests relying on non-parametric hypotheses, but still assuming time-invariance of the agronomic driving factors. This assumption of time-invariance of the driving factors is strong but realistic if the socio-economic factors such as crop market values and agricultural policies do not change (Aspinall, 2004). The use of a decision tree on a catchment when it has been trained on a different catchment may also be biased if the construction of the DT was subject to overfitting of the training dataset (Breiman et al., 1984; Murthy, 1998). Local configurations of fields, farmsteads and the road network may be a significant source of bias that can be avoided by adjusting the stop rule in the DT growing phase and by parameterizing the pruning phase. 4.3. Interests and limits of the SDT with regard to transition matrices In our study, we found few differences in the simulation performance between transition matrices and SDTs. This result can be partly explained by the fact that, in the dataset we used, crops are highly organized in space, and crop rotations are followed without deviation. It is likely that any pattern found in 1 year will persist the following year and may be captured by transition matrices. Yet, SDTs provide a way to model, i.e., to explain and formalize the spatio-temporal allocation of crops and to simulate them over space and time within a single integrated framework and with a relatively-low computational cost. On the contrary, transition matrices are able to simulate patterns by bringing information about the driving factors of crop allocation into the model. Additionally, SDTs may be more relevant than transition matrices to simulate the consequences of crop patterns and changes in farming systems (conversion of one farm type to another, substitution of one crop by another) by following, for example, individual farmer decisions, changes in market forces, or new environmental regulations. This first application opens doors to potential new uses of decision tree methods in the field of land cover change modeling. Most of the land cover change models require an initial land cover state to simulate the land cover successions, but stochastic decision trees could also be trained by removing the ‘‘current land cover” attribute in the training dataset and used to model the initial state of a landscape where land cover is unknown or incomplete. This

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