-1

Research Article

Simple neural network reveals unexpected patterns of bird species richness

in forest fragments

Claude Monteil*, Marc Deconchat and Gerard BalentForest Dynamics in Rural Environments (DYNAFOR), UMR 1201 INRA-INPT/ENSAT, BP 107, CastanetTolosan Cedex 31326, France; *Author for correspondence (e-mail: monteil@ensat.fr)

Received 24 October 2002; accepted in revised form 15 September 2004

Key words: Birds, Explanatory power, Forest fragmentation, Neural network, Species-area relationships

Abstract

The study of links between bird species richness and forest fragmentation contributes to a better under-standing of landscape biodiversity. Difficulties arise from the necessity to deal with multiple non-linearrelationships between the involved variables. Neural network models provide an interesting solution thanksto their internal set of non-linear neuron-like components. Their ability is well established for prediction,but their complex structure limits the understanding of underlying processes. To open the ‘black box’ andget a more transparent ‘glass box’ model, we selected a simple neural network (2 inputs, 1 hidden layer with3 neurons and 1 output neuron), that improves the prediction of birds species richness (lower root meansquare error) compared to linear, log-linear and logistic models, and simple enough to analyze its internalcomponents and identify patterns in the data. The first hidden neuron provided a sigmoid relationshiprelated to the forest area, the second was like a Boolean operator separating two groups according to thedistance to the nearest source forest larger than 100 ha, and the third acted on the smallest isolatedwoodlots. We revealed a group of isolated woodlots with a higher species richness than less isolatedwoodlots for a given forest area. This result, unexpected according to the literature, was not obvious in theraw data, and could be explained by a regional differentiation in fragmentation history. Our neural networkshowed its ability to improve prediction accuracy in respect to other models, to remain ecologicallyunderstandable and to give new insights into data exploration.

Introduction

Ecological processes explored in the field of land-scape ecology are of prime importance for biodi-versity conservation (Saunders et al. 1991) buttheir study is rather complex because it involvesnumerous correlated parameters acting on differ-ent spatial and temporal scales (Brown and Ma-cleod 1996; Trzcinski et al. 1999). The relationship

between species richness and habitat patch sizeand isolation in the case of forest fragmentation isone of the most studied processes in landscapeecology (Whitcomb et al. 1988; Golley 1989; Op-dam et al. 1993; Bellamy et al. 1996), either as adynamic process combining loss and breakingapart of habitat (Fahrig 2003), or as a static spatialstructure where habitat patches are scattered into amatrix resulting from unknown processes. In

Landscape Ecology (2004) 20: 513–527 � Springer 2005

DOI 10.1007/s10980-004-3317-x

many studies dealing with this latter point of view,species richness increases with the forest area, anddecreases with isolation from large forests con-sidered as sources of species (Lynch and Whigham1984; Opdam et al. 1985; Bellamy et al. 1996;Deconchat and Balent 1996, 2001). Most of thestudies dealing with these questions use linear orlog-linear methods even though there is no eco-logical evidence that the relationships betweenfragmentation variables are linear (Bascompte andSole 1996; Hof and Flather 1996; Jansson andAngelstam 1999; Liu and Ashton 1999).

In the ecological sciences, artificial neural net-works are receiving greater attention as a powerfulstatistical modeling technique (Olden et al. 2004).They have been applied to various fields such asfreshwater (Brosse et al. 1999), impacts of climatechange on a complex landscape of tropical forests(Hilbert and Ostendorf 2001), population andcommunity ecology (Lek and Guegan 1999), andconservation of biological diversity (Ejrnaes et al.2002). Neural networks offer the advantage ofdealing with non-linear relationships withouthaving to define explicitly the relations betweenthe model’s variables (Lek et al. 1996; Jorgensen1999). This is an advantage to improve predictionquality compared with traditional methods such aslinear or logistic regressions (Manel et al. 1999).

On the other hand, the complexity of a neuralnetwork model, related to the number of inputs,the neurons number and the network structure,can increase dramatically far beyond humanunderstanding. This is not a problem when pre-diction is the only expected result, but it is a stronglimitation for ecologists who are looking forunderstandable models based on a theoreticalbackground (Seginer et al. 1994) rather than forsuch ‘black box’ model (Werner and Obach 2001).This may also explain why neural networks havenot received great attention from landscape ecol-ogists.

The aim of this paper is to analyze the effects offorest area and isolation in southwestern Franceon bird species richness by a simple neural net-work with few parameters allowing both sufficientprediction ability, compared to classical basicmodels, and understandable analysis of its internalcomponents in relation with the ecological pro-cesses. This paper aims also to illustrate what kindof help could be provided to landscape ecologistsby neural network methods.

Methods

Study area

The study area (3000 km2) lies between theGaronne and Gers rivers in Southwestern France(Figure 1). It is a hilly region (200–400 m a.s.l.),dissected by south–north valleys, within a sub-Atlantic climate with Mediterranean and moun-tain influences. The forests are fragmented andcover 15% of the area (Balent and Courtiade1992). The most common forest managementsystem is coppice (rotation ranges between 15 and40 years), increasingly often with standard treeschosen to produce high quality wood for sawmills(Guyon et al. 1996). Oaks (Quercus robur and Q.sessiflora), often in association with chestnut(Castanea sativa) in coppice, cherry (Prunus avium)and wild service trees (Sorbus torminalis), are themain tree species in the area. All the bird speciespresent in the area studied can be observed inother French plain regions (Balent et al. 1988;Joachim et al. 1997).

Sampling method

For each point of observation, experiencedobservers recorded all bird species contacted

#

N

0 100 Kilometers

FRANCE

Studiedarea Garonne

Toulouse

Figure 1. Localization map of the study site in Southwestern

France.

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visually or by their vocalizations during 20 minbetween sunrise and 4 h maximum after sunrise(Hutto et al. 1986). There was no statistical dif-ference between observers. The biggest species (i.e.raptors) have been eliminated from statisticalanalyses because they are poorly sampled with thispoint count method. The bird census performedduring the month of May 1993 consisted of 504points distributed in 209 forest patches propor-tionally to patch area (1–30 points). The sampledforest patches were selected to have a large vari-ability of area, from 0.15 to 389 ha, and isolation.Species richness was calculated from the pointspooled at the forest patch level.

Fragmentation variables

Previous analysis has shown that intra-forest fac-tors, associated with vegetation structure, had lessinfluence than fragmentation factors such as sizeand isolation (Icaran 1995). We measured nineforest fragmentation parameters on a binary map(forest/non-forest) obtained from a SPOT imageusing supervised classification (ERDAS software).The variables belonged to three types describingforest fragmentation: morphology of the woodlot,local isolation and isolation from source (Table 1).For the morphology variables, the forest patcheswere extracted from the binary map and shape-corrected with the help of aerial photography, thenwe measured the parameters with ARC/INFOsoftware. For local isolation, inter patch distancesand forest area around selected patches weremeasured with ARC/INFO software. For isola-tion from source, (i.e. largest forest patches con-sidered as potential sources of species), we usedsmall-scale regional maps (1/50000) from the

French National Geographical Institute to esti-mate the distance between selected forest patchesand sources.

Neural networks

We present here a widely used kind of neuralnetwork, the multilayer perceptron (Rumelhartet al. 1986; Bishop 1995; Demuth and Beale 1998),according to the glossary that Stegemann andBuenfeld (1999) developed to assist experts fromother fields in becoming familiar with neural net-work terminology.

StructureA neural network is an information processingsystem whose architecture is intended to mimic thebiological nervous system. It is composed of anumber of cells, called neurons, each of themexchanging information with others in the networkthrough weighted connections. Neurons are ar-ranged in layers, with one or several intermediatelayers of parallel neurons (called hidden layers)lying between the input and output layers. Weused a three-layer architecture with a single hiddenlayer (Figure 2).

The input layer is composed of scaling factors(vector s – vectors are noted in lower-case boldletters) in order to ensure input values of sameorder of amplitude. To preserve the meaning forthe analyst of resulting values of this scaling stage,we used a scaling factor dividing each input valueby the smallest power of ten, strictly greater thanthe maximum of the relevant variable. The hiddenlayer is composed of n neurons. Each hiddenneuron j performs two operations: first, a weightedsum of its inputs xi, including a bias bj; then, a

Table 1. Variables used to describe forest fragmentation.

Name Type Description

AREA Morphology Forest patch area (ha)

PERI Morphology Perimeter of the forest patch (m)

THICK Morphology Radius of the largest circle inscribed in the forest patch (m)

D5FP Local isolation Mean distance to the five nearest forest patches from edge (m)

FOR2 Local isolation Percentage of forest cover within a 2 km radius around barycenter of the forest patch

FOR4 Local isolation Percentage of forest cover within a 4 km radius around barycenter of the forest patch

DIST20 Isolation from source Distance to the nearest forest larger than 20 ha (m)

DIST50 Isolation from source Distance to the nearest forest larger than 50 ha (m)

DIST100 Isolation from source Distance to the nearest forest larger than 100 ha (m)

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transformation of this single outcome using a lin-ear or non-linear activation function F, resultingin an activation value:

aj ¼ Fðwj � xþ bjÞ

where wj is the row-vector of connection weightswij, x the column-vector of inputs xi, and bj thebias of the neuron. The global column-vector a ofall hidden neuron outputs aj may be computed in asingle matrix operation

a ¼ FðW � xþ bÞ

where W is the matrix (noted with upper-case boldletter) of all connection weight wij, and b the col-umn-vector of all biases bj. The activation functionused in the hidden layer is a sigmoid function, suchas the logistic function:

FðxÞ ¼ 1=ð1þ e�xÞ

(outcome in ]0, 1[), as used in our case, or thehyperbolic tangent function:

FðxÞ ¼ 2=ð1þ e�2xÞ � 1

(outcome in ]�1, +1[).For each hidden neuron, this function may

perform a hard or smooth threshold, even a quasi-linear transformation, according to the value of

the weights W, while the bias bj allows an adjust-able setting of the inflection point abscissa.

The output layer is a single neuron that handlesthe single output of the network. Its activationvalue a¢ is computed as:

a0 ¼ F0ðw0 � aþ b0Þ

where F¢ could be a linear or non-linear function(we used the identity function), w¢ the row-vectorof connection weights w¢j linking each hiddenneuron to the output neuron, a the column-vectorof output values aj of each hidden neuron, and b¢the bias of the output neuron.

ResolutionDetermining the neural network parameters(adjustable weights W and w¢, biases b and b¢) is anon-linear optimization process based upon theminimization of an error criterion (sum-squarederror SSE: sum of the squares of differences be-tween predicted and observed outputs) computedon a set of known inputs and outputs. The basicalgorithm used to solve this problem is the back-propagation method (Rumelhart et al. 1986). It isan iterative process: at each step, the current val-ues of weights and biases are used to predict theoutput of each set of input values; the obtained

Figure 2. A three-layered 2-n-1 neural network (2 input nodes, n hidden neurons, 1 output neuron): The first layer (input layer) scales

the 2 input variables x1 and x2 and distributes them to the next layer. The second layer (classically called hidden layer) is composed of a

set of neural nodes performing: (i) a linear combination of all scaled variables through weighting factors wij and a constant bias bj(considered as a weight parameter for an extra input of value 1); (ii) a sigmoıdal transformation of the result. The third layer (output

layer) contains here a single neural node which combines the outputs of the hidden layer through weighting factors w¢j and bias b¢, andperforms an identity transformation. The network output is finally scaled (factor s¢) and rounded to a positive integer value.

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values are compared with the observed values andthe difference is used to correct the weightsthrough a constant multiplicative factor called thetraining rate. The iterative process is stopped whena predefined sum-squared error, or a maximumnumber of steps (for example, 1000), has beenreached. From the SSE may be computed tworelated indicators: the root mean squared error(RMSE), expressed with the same unit as theoutput variable and classically used to handlemodel performance, and the determination coeffi-cient r2 between observed and calculated values ofthe output, classically interpreted in terms of ‘ex-plained variance’ (Hilborn and Mangel 1997). Aminimum value for RMSE entails a maximumvalue for r2 (the reverse is not true because modelsdiffering by a systematic constant bias will havesame r2 but different RMSE).

Because the system is non-linear, the resolutionprocess may converge to a local minimum of SSE,which may be greater than the global minimum.Limiting this risk needs to explore a large domainof input parameters with different random sets ofinitial values (for example, 100). Various algo-rithms may be used to find a good trade-off be-tween resolution speed and input domainexploration, such as momentum, adaptive learningrate or Levenberg-Marquardt optimization(Demuth and Beale 1998). We used this lattermethod with the Matlab� neural network toolbox(Demuth and Beale 1998) and developed someadditional scripts to handle model design andsetting (choice of input variables, number of hid-den neurons, and repetition of calculations).

Number of hidden neuronsSetting the number of hidden neurons is somewhattricky: with too few neurons, the network cannotfit the data very well (underfitting); with too manyneurons, predictions are quite good near thetraining data but may be unreasonable whengeneralizing with other input values (overfitting).Furthermore, the number of parameters growsquickly: for example, a 3-5-1 neural network (3inputs, 5 hidden neurons and 1 output neuron)needs 26 weights and biases, whilst a 3-10-1 needs51. Although some empirical rules have beenproposed, for example using twice the square rootof the sum of input and output nodes (Segineret al. 1994), it is necessary to try various numbersof hidden neurons and examine the generalization

ability of the resulting network. The assessment ofthe prediction error may be performed using var-ious methods, like resubstitution, cross-validationand bootstrapping (Fielding and Bell 1997). Theresubstitution method uses the same data fortraining and testing, which tends to provide anoptimistic measure of prediction. Cross-validationand bootstrapping are resampling methods whichestimate the generalization error with data notused for training (Sarle 2002): the k-fold cross-validation method split the overall set of N datainto k subsets of approximately equal size, andperforms k trainings, each time leaving out one ofthe subsets from training, and using the omittedsubset for testing; the bootstrapping method con-structs training samples with replacement of thedata, so a given data may be used several times forone training. The choice of the k value, the strat-egy of splitting, and the performances and limits ofboth resampling methods are often questioned(Kohavi 1995; Manel et al. 1999; Rivals andPersonnaz 1999; Sarle 2002). Although bothmethods provide a more reliable way to assessprediction error for a given neural network archi-tecture, they compute in each case a set of models,differing from each other with parameter values.As we focused our study on a simple explanatorymodel rather than a complex predictive model,which is more subject to generalization problems,we chose the basic resubstitution method to assessthe quality of the neural network and ensure asufficient confidence, being conscious that thisestimation was optimistic.

Analysis steps

The data analysis aimed to explore the capabilitiesof a simple neural network model to explain theinfluence of some fragmentation parameters on thespecies richness (SR) in a more meaningful waythan the conventional log-linear model linkingforest area to SR. This analysis was conducted inthree steps.

The first step consisted of selecting neural net-work architecture (number of hidden neurons andchoice of input variables) in respect to other clas-sical simple models. Five types of models weretested:(1) univariate linear or log-linear models with any

of the 9 input variables,

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(2) univariate logistic model with the area loga-rithm

(3) bivariate linear models with a combination ofarea logarithm and any of other 8 variables,

(4) multivariate linear model with all the inputvariables (or their logarithm),

(5) bivariate neural networks with the area as firstinput and any of other 8 variables as secondinput with a various number of hidden neu-rons.

The goodness of fit of these models was assessedboth by the RMSE and the determination coeffi-cient (r2) between predicted and observed data, inrelation with their complexity represented by thenumber of parameters defining the model: an-variate linear model has n+1 parameters; alogistic model has three parameters; a neuralnetwork model with n entries, one hidden layer of pneurons and 1 output has (n+2)*p + 1 parame-ters.

For each of the 9 couple of inputs combiningwoodlot area with any other variable, neural net-work architectures were tested with a number ofhidden neurons ranging from 2 to 10. Each casewas trained up to 1000 steps, with 100 repetitionsof initial random parameters set. The neural net-work with the smallest RMSE was retained foreach couple of inputs and each number of hiddenneurons.

The second step consisted of analyzing the outerbehavior of the selected neural network models,i.e. the values of the global output of the model(bird species richness). We compared observed andpredicted values in order to identify potential bias.

The third step consisted of analyzing the innerbehavior of the neural network models, i.e. thevalues of the outputs of hidden neurons. We firstlytried to identify the functional specificity of eachhidden neuron and relate them to an ecologicalsignificance by examining the evolution of eachoutput according to each input variable. We triedthen to assess the relative importance of eachhidden neuron using two methods – a quantitativeone and a qualitative one. For the quantitativemethod, we could think about the weightingcoefficients w¢ between hidden neurons and theoutput neuron (Figure 2), but these raw values arenot a valid indicator because the amplitude of eachhidden neuron output may vary in very largeproportions, for example from 1 to 1000, evenmore (see results section); to better estimate the

contribution of a hidden neuron j to the overalloutput, we therefore computed the product of theamplitude of its output by the corresponding w¢jcoefficient of the output neuron; this absolute co-efficient was then transformed in a ‘relative weight’(percentage), dividing it by the sum of the absolutecoefficients of all hidden neurons. We also tested aqualitative way to rank the importance of eachhidden neuron with a gradual suppression of onehidden neuron: when decreasing by one unit thenumber of hidden neurons, if one or more func-tional specificities of remaining hidden neuronsremain observable, it suggests that these neuronsprovide higher information than the others do.

Results

Inputs correlations and basic models overview

The variables describing the morphology of thewoodlots (AREA, PERI, THICK) were stronglycorrelated (Table 2), but not with the other vari-ables. The mean distance to the five nearest forestpatches (D5FP) and the distance to the nearestforest larger than 20 ha (DIST20) were weaklycorrelated with other variables. The distance toforest larger than 50 ha (DIST50) or 100 ha(DIST100) and the forest area surrounding thewoodlots within 2 km (FOR2) or 4 km (FOR4)presented some high correlations.

With univariate linear or log-linear relations,the best RMSE was obtained with the woodlotarea logarithm (RMSE = 3.03, r2 = 71.4%)(Figure 3). With logistic regression on this latterinput, RMSE slightly decreased to 2.93(r2 = 73.3%). With bi-linear relation combininga second variable to the log of woodlot area, thebest RMSE was observed with the distance tothe nearest forest larger than 20 ha (RMSE= 2.95, r2 = 73.0%). The multiple linearregression of the species richness with all ninevariables (or their logarithm for the morphologyvariables) slightly improved the RMSE to 2.82(r2 = 75.2%).

Neural network architecture selection

Whatever the number of hidden neurons is, theminimal RMSE was obtained with the combina-

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tion of the woodlot area (AREA) and the distanceto the nearest forest patch greater than 100 ha(DIST100). RMSE decreased from 2.72(r2 = 77.0%) with 2 hidden neurons to 2.27(r2 = 84.0%) with 10 hidden neurons, in relationwith an increased number of parameters (Fig-ure 3). We chose to focus the analysis on the 3

hidden neurons network (RMSE=2.62,r2 = 78.5%) because it brought a compromisebetween simplicity, with 13 parameters, and sig-nificant improvement of RMSE compared withnon-neural models.

Neural network outer behavior

The residual values between predicted and ob-served species richness showed a bias (Figure 4):species richness appeared overestimated in thewoodlots with lower species richness (less than

0 10 20 30 40 50

Number of Model Parameters

2.2

2.3

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3.0

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RM

SE

NN10NN9

NN8NN7

NN6NN5

NN4

NN3

Multilinear

NN2

LogisticBilinear

Linear

Figure 3. Relationship between the number of parameters in the

tested models and minimal root mean squared error (RMSE)

between observed and predicted species richness: Linear = uni-

variable linear model with Ln(AREA); Logistic = uni-variable

logistic model with Ln(AREA); Bilinear = bi-variable linear

model with Ln(AREA) and DIST20; Multilinear = multi-

variable linear model with the logarithm of morphology vari-

ables (AREA, PERI, THICK) and the six other variables;

NNi = bi-variable (AREA, DIST100) neural network with i

hidden neurons (number of parameters = 4*i + 1).

Table 2. Correlation matrix between the nine fragmentation variables and with the bird species richness in woodlots (SR)

LnAREA LnPERI LnTHICK D5FP DIST20 DIST50 DIST100 FOR2 FOR4 SR

LnAREA 1.00

LnPERI 0.98 1.00

LnTHICK 0.97 0.90 1.00

D5FP �0.30 �0.30 �0.30 1.00

DIST20 �0.06 � 0.05 � 0.07 0.44 1.00

DIST50 � 0.06 � 0.07 � 0.06 0.39 0.50 1.00

DIST100 � 0.05 � 0.06 � 0.04 0.21 0.45 0.74 1.00

FOR2 0.17 0.17 0.17 � 0.49 � 0.58 � 0.68 � 0.59 1.00

FOR4 0.08 0.08 0.08 �0.50 �0.55 � 0.75 � 0.67 0.88 1.00

SR 0.85 0.83 0.82 � 0.30 0.07 � 0.04 0.05 0.11 0.05 1.00

The morphology variables are log transformed.

0 10 20 30 40Observed Species Richness

-10

-5

0

5

10

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idua

ls

Figure 4. Relationship between the observed species richness

and the residuals (predicted minus observed species richness) in

the AREA-DIST100 2-3-1 neural network.

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10–15 species) and underestimated in the wood-lots with higher species richness. Among thewoodlots larger than approximately 1 ha, twogroups appeared in the values predicted by ourneural network (Figure 5), the most numerousone having a lesser species richness and lowervalues of isolation from source woodlots. Bothgroups had a similar sigmoid-shaped relationshipbetween woodlot area logarithm and speciesrichness.

Neural network inner behavior

Plotting the intermediate outputs from the hiddenlayer (Figure 6) shows how hidden neurons hadbeen specialized by the training process. The 1stneuron implemented a sigmoid-shaped relation-ship according to the woodlot area logarithm. The2nd neuron behaved as a true/false operator sep-arating two groups of forests according to theirdistance to the nearest source forest larger than100 ha, with an exception for three points corre-sponding to forests larger than 200 ha which canbe considered as their own source. The behavior ofthe 3rd neuron was more complex, involving bothinputs: only small area woodlots (area<2 ha) weredistinguished, and the minimum values of theDIST100 plot showed a decreasing linear rela-tionship.

The weighting coefficients w¢ of the outputneuron (Table 3) suggested that the first hiddenneuron was of greatest importance (value near350), the second one played an insignificant part(value near 0.05) and the third one was of lesserimportance than the first one (value near 170).However, their relative weights, in relation withtheir amplitude, were respectively 61, 16 and 23%.Training a 2-hidden neurons neural networkshowed that the behaviors of the two first neurons0 1 10 100 1000

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Figure 5. Comparison between the observed species richness (a)

and the predicted species richness (b) with the AREA-DIST100

2-3-1 neural network. Dot radius is a linear function of distance

to forest source.

Table 3. Parameters of the AREA-DIST100 2-3-1 selected

neural network

Name of parameter Value of parameter

Hidden neuron 1:

Bias b1 7.677E+00

Weight of AREA input w11 6.810E+01

Weight of DIST100 input w21 �5.541E�01Hidden neuron 2:

Bias b2 �1.161E+02

Weight of AREA input w12 4.914E+02

Weight of DIST100 input w22 9.703E+02

Hidden neuron 3:

Bias b3 9.060E+00

Weight of AREA input w13 2.296E+03

Weight of DIST100 input w23 �1.361E+01

Output neuron:

Bias b¢ �5.250E+02

Weight of hidden neuron 1 w¢1 3.527E+02

Weight of hidden neuron 2 w¢2 4.591E�02Weight of hidden neuron 3 w¢3 1.725E+02

AREA and DIST100 are the two inputs used in this 3-hidden

neurons network. The role played by each parameter is

described in Figure 2.

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of the 3-hidden neurons network were present,with relative weights of respectively 81 and 19%;thus, the second hidden neuron had finally moreimportance than the third. We also noted that theparameters of each corresponding hidden neuronwere different in the two neural networks: forexample, the first neuron of the 2-hidden neuronsnetwork had (87.2; �1.08) as weights and 7.53 asbias, while these values were (68.1; �0.55) and 7.68with the 3-hidden neurons network (Table 3).

Discussion

Analyzing the neural network

The neural networks are well known methods toproduce very efficient predictive models that areable to link a set of input variables to a set ofoutput variables (Bishop 1995). In such an ap-proach, the neural network is often seen as a blackbox model (Olden and Jackson 2002) with its innerstructure not of interest. In our case, rather than anetwork with the highest predictive value, we se-lected a neural network with a simple structurethat allows analyzing its inner structure in order toget some insights into the data and the underlyingprocesses. Accuracy is important when the prin-cipal aim is prediction, while simplicity is anadvantage when the aim is understanding (Gilbertand Troitzsch 1999). Because of the ability of theneural network to catch non-linear components ofthe relationships between input and output vari-ables, the difference between the predictive valueof our neural network and a linear regression ispositive, but rather small (RMSE of 2.62 vs. 3.03,r2 of 78.5% vs. 71.3%)., However, higherimprovements are generally found in the literature.For example, the r2 between fish density in FrenchPyrenean rivers and 10 transformed habitat vari-ables on 205 morphology units increases from63% using multiple linear regression to 96% usinga 10-8-1 neural network model (Lek et al. 1996).Such a kind of improvement should be consideredregarding the complexity of the underlying model:in that case, the multi-linear model had 11parameters, whereas the neural network had 97.As stated by Sklar and Hunsaker (2001), there is atrade-off between the complexity of the model andits predictability. Consequently, we consider theinterest of a neural network not only according to

its ability to predict a set of output variables, butfirst of all on its explanatory ability. Some newmethods, such as CART (classification andregression trees), have been developed in thispurpose and allow searching for a tree of rulessplitting the data according to variable thresholds,however they can also become very difficult tointerpret when the number of parameters increases(Munoz and Felicisimo 2004).

Two important questions concerning the inneranalysis of the neural network have to be asked:What part does each of the hidden neurons playqualitatively in the result? In what proportion doeseach of the hidden neurons quantitatively affectthe final result? A simple examination of the neuralnetwork parameter values is not sufficient to pro-vide a quick answer to these questions becausethey are finally combined in complex non-linearequation. A careful observation of the outputsfrom each hidden neuron seems to be an efficientway to identify a hierarchical importance of eachof them. In our case, the output curve of eachhidden neuron was quite easy to identify, and wehave shown that these curves appeared succes-sively in a forward process comparing networkswith increasing number of hidden neurons. Thisforward approach showed that the initial rankingobtained by the relative weights was maintained,but the relative influence of the third neuron waslower than the influence of the second one. Thepossibility to apply this approach in other casesshould be tested to contribute to quantify theimportance of the variables in the neural network(Olden et al. 2004).

Ecological issues

From a general point of view, our analysisemphasizes the influence of forest patch size and ofisolation from large forest patches, sources ofcolonizing species on the species richness (Opdamet al. 1984; Whitcomb et al. 1988). Although theimportance of patch size is widely established, thegreater influence of the distance to a large sourceforest patch compared with the local isolation offorest patches (D5FP, FOR2, and FOR4) deservesto be discussed. Some authors have found a morebalanced influence of both factors (Opdam et al.1985). In the studied landscapes, the percentageof forest is very high everywhere (>25%) and

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consequently, the local isolation of small forestpatches has lower spatial variability than the iso-lation from large forest patches. However, theinfluence of both factors is not very great com-pared with patch size and we suspect that thestrong correlation between them does not make itpossible to clearly separate their respective effects(Opdam et al. 1985). However, local isolation hasbeen shown to have an important correlation withpatch size (Lescourret and Genard 1994).

The output from the first hidden neuron shows astrong relationship between the forest area andspecies richness (Figure 6). The shape of the curvebetween the species richness and the logarithm ofthe area is clearly sigmoid rather than linear, withan inflection point around 10 ha. It suggests that,for large forests, increasing the area no longer hasany effect on the species richness, since the regionalpool of forest bird species has been reached. Themodels most commonly used to fit with species-

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0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

2nd

Neu

ron

Out

put

Figure 6. Hidden neuron outputs versus forest patch area (AREA) and distance to forest source (DIST100). Hidden neurons are in

rows, forest patch area and distance to forest source are in columns.

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area curves (exponential and power) appear to beinadequate for such a shape. However, a logistic orWeibull (Flather 1996) or an extreme value func-tion (Williams 1996) should be more appropriate.

With a more detailed viewpoint, the analysisseems to separate two groups of woodlots: thegroup of most isolated woodlots has a greaterrichness than the group of less isolated ones, with asimilar influence of woodlot area (Figure 5). Theserelationships do not fit with the general theories inlandscape ecology on the influence of isolationfrom sources (Opdam 1990; Opdam et al. 1993).Most of the theories state that the species richnessdecreases when the distance from a source of dis-persion of the species increases. This relation re-sults from the hypothesis that the probability foran individual of a given species of reaching a patchof habitat from a source decreases according to thedistance. Thus the more isolated woodlots arereached by fewer individuals from the differentspecies and, then, should have a lower richness(Opdam et al. 1993).

To explain our unexpected results, we use thedetailed analysis of the second hidden neuronoutput (a2). This neuron acts as a threshold sepa-rating three groups in the data set according toDIST100 (Figure 6). The first group has a lowoutput for low values of DIST100 (<12 km), thesecond group has a high output (a2 > 0.8) forhigh values of DIST100. The third group containsthree forests with very low DIST100 but highoutputs (this group is discussed later). Thethreshold introduced by this neuron in the outputis the source of the two subgroups observed inFigure 5. To test if this separation can be linked toreality or if it is an artifact due to the network, weplotted the groups defined by the second neuronoutput in the raw data, with a linear correlationline for each separate group (Figure 7). The datashow that the three groups have clearly differentcharacteristics. The first one corresponds to themain part of the data set, the second one containsfewer points that have clearly higher species rich-ness than points in the first group for a similarforest area, as shown by the steeper correlationline. The third small group corresponds to thelargest forests in the sample. Thus, we can considerthat the groups identified by the network werealready present in the raw data, but not obviously.This result demonstrates that the neural network isable to extract subtle relationships that are not

obvious to human perception and not identified byother methods, especially linear ones. We lookedfor other factors, not used in the analysis, whichcould explain the division into two groups. Dateand time of census, or identity of the observerfailed to explain this result. The second group ofplots defined by the second hidden neuron wasgeographically clustered in the same zone of thestudied area (Figure 8). This was not obvious inprevious mappings of the raw data or residualsfrom basic log-linear models. Their similaritycould not be associated to a spatial dependencebecause they have more differences with theremaining set of data than between the subsetsseparated by the largest distance. Thus, the speciesrichness should be influenced by a local factor thatis not the same in the other part of the samplingarea. It could be a similar history or a landscapeproperty which has not been identified for themoment (Jokimaki and Huhta 1996; Yahner 1997;Hinsley et al. 1998).

The three large forests separated from the othersby the second hidden neuron, in a third group,

0.1 1.0 10.0 100.0Forest Patch Area (ha)

0

10

20

30

40

Obs

erve

d S

peci

es R

ichn

ess

Figure 7. Relationships between the observed species richness

and the forest patches area: Three groups of woodlots are

identified from the scatter plot representing the relationships

between the second hidden neuron output (a2) and the distance

to forest source (cf. Figure 6). The first group (a2 < 0.8 and

DIST100<12 km) is represented by an oblique cross (x), the

second one (a2 > 0.8 and DIST100>12 km) by a cross (+)

and the third one (a2 > 0.8 and DIST100<12 km) by a circle

(o). The lines represent the linear correlations in each group.

523

behave like the more isolated forests, despite thefact that they were close to other large forests. Thisshows that for these largest forests, isolation doesnot influence bird species richness in the same wayas for smaller ones. We could hypothesize that thelocation of these forests in the region and thebiogeographical influences may have a higherimportance, as it has been demonstrated in thisregion for Fringilla coelebs populations (Joachimand Lauga 1996).

The analysis of the outputs from the 3rd hiddenneuron shows that for areas smaller than 2 ha, theneuron introduces a variability of outputs thatmodulates the result obtained from the combina-tion of the other 2 hidden neurons. For largerwoodlots, the neuron seems to be inactive. Thistherefore means that the neural network model hasidentified a forest area threshold (2 ha), with ahigher variability for a smaller forest than for alarger one. A similar threshold has already beenidentified for bird species richness in England(Bellamy et al. 1996): these authors showed that‘most of the common species were usually present’in this area; their species-area curve also showed ahigher variability of bird species richness underthis 2 ha limit. So, we think that 2 ha represents athreshold: larger forests are always seen as trueforest habitat by birds and their richness is relativeto the wood area, whereas smaller forests do nothave all the characteristics of a true forest and the

species richness depends not only on area but alsoon other factors such as the neighborhood, finescale heterogeneity, stochasticity (Bellamy et al.1996). For DIST100, the meaning of the outputfrom the third hidden neuron is less clear, but itcan be noted that the variability of the output in-creases with the distance DIST100.

Our results let us suspect that the more isolatedwoodlots behave as real forest islands inside amatrix of agriculture, the species richness is thusmainly defined by the forest area, with a highcorrelation between these variables, while in theother landscape, the woodlots are less well sepa-rated, they are connected by hedges and thus theyare not true forest islands and their closenessinfluences the species richness, as well as thewoodlot area does, inducing a lower correlationbetween species richness and forest area (Opdamet al. 1984). The correlations between forest areaand species richness for each of the two maingroups defined by the second neuron according totheir isolation were highly significant separately(r2 = 81.0% n = 27 vs. 71.8% n = 179) but theobserved trend in correlation difference althoughconsistent with our hypothesis, is not statisticallysignificant. A similar statistically not significanttrend was observed among the small woodlots(<2 ha) of these two groups respectivelyr2 = 55.9% n = 17 for the isolated woodlots and22.1% n = 90 for the others.

Sources of uncertainty in the data

The different uncertainties in the data are a po-tential source of error and misinterpretation thatwe may consider to evaluate our result’s robust-ness. The sources of uncertainty arise from (1) theornithological census and (2) the measure of theexplaining variables. The variations in detectionability between the observers along the censusperiod have been managed by a random allocationof the samples and several cross-controls. Thevariability of the detectability of the species inforests is also a crucial point that is very difficult todeal with (Granhlom 1983). However there is noevidence that the detectability of the species varieswith the studied factors (that is to say that the birdspecies of a small woodlot have the same meandetectability than those from a larger one),allowing us considering that the variability does

Figure 8. Map of the woodlots according to the groups gen-

erated by the second hidden neuron output: The three groups

are the same as in Figure 7.

524

not interfere with the general trends we identified.We are confident in the measures of bird speciesrichness because our sampling design of a woodlotallows recording the species associated to the edgesand the infrequent species associated to restrictedforest conditions, thanks to a quite systematiccovering of the forest area. It is well known thatthe inter-annual variability of bird species com-position in woodlots can be very high (Bellamyet al. 1996; Haila et al. 1996), especially in smallwoodlots. We have assumed that the species rich-ness is less sensitive to this variability and could bestudied in a one-year sample.

The measurements of area and isolation aresubject to spatial uncertainty, especially for thesmallest woodlots for which a same boundarywidth error results in a higher area error than forlarger patches (Canters 1997). We delineated thewoodlots from very detailed aerial pictures (scale:1/30,000) allowing an uncertainty lower than 5 m(tree canopy extension over the forest limits), i.e.5.3% of error in area for a 1-ha woodlot with asquare shape. We think that this level of uncer-tainty does not influence the main trends identifiedsince we could not suspect a systematic bias to-ward over or under estimation of forest areas.Thus, the variability introduced by the third neu-rons for the smallest woodlots could result eitherfrom the spatial uncertainty or the ecologicalvariability of the species richness.

Conclusion

The neural network analysis has shown its abilityto help the researchers to discover new informa-tion in their data set and to formulate newassumptions about the underlying ecological pro-cess, provided it is not only used as a black-boxmodeling tool. Thanks to the adjustable non-linearfunctions embedded in the neural network, it al-lows fitting the observed data to complex system ofrelationships between the variables. This ability isgenerally used to produce predictive models with-out consideration for the underlying mechanisms.We showed that with simple neural networks, it ispossible to examine closely the inner structure ofthe network and to identify groups and trends thatwere not obvious in the raw data. The neuralnetwork works has an exploratory tools ratherthan for modeling. We think that neural networks

should be more frequently used by landscapeecologists in order to study their data and to takeinto account the non-linear relationships of spatialdata. This tool is easily available and it can be usedin a broader way than predictive modeling that hasbeen the dominating use until now.

Acknowledgments

This study was funded by the Midi-Pyrenees Re-gional Directorate of the French Ministry ofEnvironment (DIREN). We would also like tothank C. Icaran for her help in the bird census.

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