© 2003 by ICES/CIEM International Council for the Exploration of the Sea/Conseil International pour l'Exploration de la Mer
Age at maturation predicted from routine scale measurements in Norwegian spring-spawning herring (Clupea harengus) using discriminant and neural network analyses
a Institute of Marine Research PO Box 1870 Nordnes, N-5817 Bergen, Norway
b Adaptive Dynamics Network, International Institute for Applied Systems Analysis Schlossplatz 1, A-2361 Laxenburg, Austria
*Correspondence to G. H. Engelhard; tel: +47 55238643; fax: +47 55238687. e-mail: georg.engelhard{at}imr.no.
We evaluate two methods allowing the prediction of age at maturation from the widths of the annual growth layers in scales (or otoliths) in a case study on Norwegian spring-spawning herring. For this stock, scale measurements have been made routinely for many decades. We compare the performance in classifying age at maturation (at 3, 4,...,9 years) between conventional discriminant analysis (DA) and the new methodology of artificial neural networks (NN) trained by back-propagation against a ‘control’ of historical estimates of age at maturation obtained by visual examination of scales. Both methods show encouraging, and about equally high, classification success. The marginal differences in performance are in favour of DA, if the proportion of correctly classified cases is used as criterion (DA 68.0%, NN 66.6%), but in favour of NN if other criteria are used, including prediction error (error >1 year: DA 5.2%, NN 2.9%), and the average degree of under- or overestimation (underestimation 1.1% of mean with DA; overestimation 0.2% of mean with NN). Evidence is provided that both methods approach the a priori limits to maximal classification success, limits set by overlapping combinations of predictor variables between maturation groups. The methods allow studies on age at maturation in this stock over a very long timespan, including periods well before, during, and after its collapse to commercial extinction. Similar techniques might well be applicable to any other fish stock with long-term data on scale or otolith growth layers.
Keywords: age at maturation, classification, discriminant analysis, growth layers, neural network, Norwegian spring-spawning herring, scale measurements
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Introduction |
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 Top Introduction Material and methodsResultsDiscussionAppendixReferences |
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Age at maturation, or the age at which animals attain the capacity to reproduce, is tightly linked to lifetime reproductive success of individuals (Stearns, 1992; Bernardo, 1993). At a population level, age at maturation is therefore a key factor influencing stock productivity. Large fluctuations in many commercially important fish stocks may be better understood if long-term trends in age at maturation can be established (e.g. Godø, 2000, on the dynamics of maturation in Northeast Arctic cod, Gadus morhua). Unfortunately, for many fish stocks, such knowledge is relatively limited or absent. By contrast, data on length and age composition are often collected routinely, normally through studies of the annual growth layers in either scales or otoliths.
Can routine growth layer measurements be used to predict age at maturation with a satisfactory degree of confidence? This question is addressed here for Norwegian spring-spawning herring, Clupea harengus, a large herring stock of primary commercial significance and characterized by substantial fluctuations in biomass (Hjort, 1914; Toresen and Østvedt, 2000). From high abundance in the 1940s, the stock declined drastically during the 1950s and 1960s, probably through a combination of overfishing and unfavourable climatic conditions, until it collapsed to near extinction commercially in the late 1960s. Since then it has recovered, slowly during the 1970s, but more rapidly since the late 1980s; it is now considered fully recovered (Toresen and Østvedt, 2000). Stock size fluctuations were often accompanied by pronounced changes in patterns of growth and maturation (Runnström, 1936; Toresen, 1990). The long-term patterns of maturation in the stock, however, have not yet been properly described, and are particularly poorly known for the most recent three decades.
Age at maturation varies considerably in Norwegian spring-spawning herring, largely because of the wide range of environmental conditions experienced by juveniles (Dragesund et al., 1980). The spawning areas are distributed over a wide latitudinal gradient along the west coast of Norway between 58 and 69°N (Devold, 1963; Hamre, 1990). After hatching, juveniles drift northeastwards with the Norwegian coastal current, and so either reach the fjords of western and northern Norway, or the Barents Sea (Dragesund, 1970). Those ending up in temperate waters along the Norwegian west coast generally grow rapidly, and usually spend only 1–2 years in nursery areas before migrating to the Norwegian Sea to mix with shoals of older fish. By contrast, those ending up in Arctic water masses of the Barents Sea grow slowly, usually remaining for some 3–5 years in nursery areas (Barros and Holst, 1995). After, usually, one or two summers, during which the herring forage and live more pelagically in the Norwegian Sea, they mature. This results in ages at maturation that can vary between 3 and 9 years, with most fish maturing at ages 4–8 (Runnström, 1936).
Since the early 1900s, extensive collections of Norwegian spring-spawning herring scales have been made by the Institute of Marine Research (IMR), Bergen, Norway (Hjort, 1914). In herring scales, the formation of gonads, characterizing the process of maturation, is reflected by a subtle change in width and microstructure of the corresponding annual growth layer (Lea, 1928; Runnström, 1936). This, however, can only be observed in well-preserved scales from the lateral side of the body. Prior to ca. 1970, the sampled herring were mainly caught by driftnet and/or purse-seine techniques that result in relatively minor scale loss, allowing the collection of scales in good condition. Experienced scale readers routinely counted the number of spawning rings on the basis of visual evaluation of scale structure, so the age at maturation could be derived directly by observation. Since the 1970s, however, samples have not only been taken from purse-seiners, but also to an increasing extent from pelagic trawlers. This has resulted in greater scale loss and, hence, a relatively large proportion of scales of poorer quality. Direct observation of age at maturation from scales was, therefore, discontinued in 1974.
Information exists on the widths of annual growth layers in Norwegian spring-spawning herring scales over a long period, starting in the 1930s and extending to the present day. If it were possible to predict age at maturation from such data, it would allow the establishment of a long time-series on this important life history characteristic. Such information would be particularly valuable in the light of changes in the stock when it collapsed in the 1960s. The goal of the present paper is to describe and to evaluate the efficiency of two methods, discriminant analysis (DA) and artificial neural network (NN) analysis, at classifying age at maturation in adult Norwegian spring-spawning herring, based on the widths of annual growth layers in scales.
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Material and methods |
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Data collection
The two classification methods were evaluated using mature individuals of Norwegian spring-spawning herring collected by the IMR between 1935 and 1973. Samples of 100–200 herring were collected from driftnet, beach-seine, purse-seine, or trawl catches, caught by both commercial and research vessels. For each fish, standard measurements were taken, including body mass, total length, sex, and maturity stage. When available, up to four scales were collected from the skin just behind the operculum, along the lateral body line. Scales were mounted on microscopic glass plates coated with gelatine and conserved for later analysis. By microscopically examining the scales shortly after preparation, scale readers determined the age, based on the total number of growth layers (Lea, 1911), and the age at maturation, based on observations on each of the growth layers (Lea, 1928, 1929; Runnström, 1936). This implied a distinction between (1) ‘coastal’ rings corresponding to the juvenile stage (rather narrow to very wide summer zones, divided by either diffuse or sharp winter rings), (2) ‘oceanic’ rings corresponding to the late immature stage when the animals live in the Norwegian Sea (wide summer zones, divided by diffuse winter rings), and (3) ‘spawning’ rings corresponding to years during which the herring spawned (narrow to very narrow outer summer zones, divided by sharp winter rings; Figure 1).
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For many of these historical scale samples (collected before 1974), measurements on annual growth layers were carried out recently (during the 1990s). Those measurements followed the new methodology of scale examination that has been in practice at the IMR since 1974 and has replaced the older method described earlier, which distinguished between coastal, oceanic, and spawning rings. The new method was described by Barros and Holst (1995); it involves the measurement of the total radius of the scale and of the radius of each annual growth layer up to the ninth along a line running from the focus to the edge of the scale, using a stereomicroscope fitted with an ocular micrometer (Figure 1).
The data
The data used here included 45 386 herring that satisfied the following conditions: (1) they had been classified as mature on evaluation of their gonads; (2) age and age at maturation had been interpreted by observation of the scales; and (3) the widths of annual scale increments had been quantified by direct measurement (unit: mm).
For classifying age at maturation, data must be stratified by age, for two reasons. First, the number of measured scale increments and, consequently, that of explanatory variables is dependent on age. Second, while a mature fish caught at, say, age 4 can only have matured at an age of either 3 or 4, a fish caught at, say, age 11 may have matured at any age between 3 and 9. An overview of the age-stratified data set is given in Table 1.
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Discriminant analysis (DA)
DA is a well-known statistical procedure used to build predictive models of group membership on the basis of observed characteristics of each individual. Two-group DA generates one discriminant function based on linear combinations of the predictor variables that result in the best discrimination between two groups. Multiple DA allows discrimination between more than two groups by generating a set of discriminant functions. These functions are generated from animals with known group membership; the functions can then be applied to other animals of unknown group membership, given that measurements for the predictor variables are available.
For our application to predict age at maturation from scale measurements, it was practical to stratify the data according to age (Table 1). This implied that, for each age group (age 3, 4,..., 8, 9+), distinct discriminant analyses were carried out. Predictor variables were the widths of annual scale increments up to the ninth, after log-transformation of the data to obtain normality and to increase the homogeneity of variance. The dependent variable was age at maturation (at 3, 4,..., 9 years), defined a priori on the basis of direct observations by scale readers. The discriminant analyses were carried out using the SPSS 10.0.7 Windows package (SPSS Inc., 1989–1999).
Artificial neural network (NN)
Artificial NNs were also tried as a method of predicting age at maturation from scale measurements in Norwegian spring-spawning herring. Such NNs imitate human neuron functioning, transforming an activating variable into a non-linear response. They can be applied as an alternative to various statistical procedures, and are particularly useful in cases of non-linear relationships between predictor and dependent variables (Fausett, 1994; Basheer and Hajmeer, 2000). Several recent studies have shown that, for the purpose of classification, artificial NNs often have superior predictive performance to conventional statistical procedures, i.e. DA and logistic regression (Edwards and Morse, 1995; Lek et al., 1996; Simmonds et al., 1996; Manel et al., 1999a,b; Özesmi and Özesmi, 1999).
We constructed artificial NNs written in the programming language C. For each age group (age 3, 4,..., 8, 9+), separate NNs analyses were carried out (as was the case with DA). These networks could be characterized as three-layer feed-forward NNs; their architecture is described in Appendix. The networks were trained by means of the back-propagation learning algorithm (Rumelhart et al., 1986). The prediction of age at maturation from scale measurements by use of NNs occurred in two major phases. First, during the training phase, internal parameters within the network (weights) were adjusted iteratively, such that the performance of the network, equivalent to predicting age at maturation accurately, was maximized; this stage continued until there was no further increase in network performance, or classification success (see Appendix for more details on the training procedure). Second, during the prediction phase, the final, optimal network obtained during the training phase was used to predict age at maturation for all fish in the database.
Performance of classification methods
Three indicators were used to judge the quality of the results obtained using DA and NNs:
- Classification success, defined as the proportion of correctly classified individuals, assessed per age group, as a general indicator of classification success.
- Prediction error, defined as absolute differences between observed and predicted values for age at maturation, and expressed either as a mean prediction error averaged over all cases, or as the proportion of cases where age at maturation was misclassified by more than 1 year.
- Degree of under- or overestimation. Using a Wilcoxon Signed Rank test, we examined the extent to which there was a tendency for estimated values for age at maturation, predicted using either DA or NN, to be either higher or lower than observed values.
Quantification of overlap between maturation groups
If the different maturation groups show considerable overlap in (combinations of) explanatory variables (i.e. scale measurements), then it can be expected that classification success will be a priori limited to some extent, regardless of the classification method used. We examined the a priori limitations to classification success for both discriminant and NN analyses, by quantifying the degree of overlap in the multidimensional character space between the different maturation groups.
Overlap was quantified as follows. For all fish belonging to a certain age group (age 3, 4,..., 9+), the coordinates describing their locations in the multidimensional character space were determined by their values for the scale measurements. Next, for each individual (1) the number of ‘neighbouring’ fish situated closer to it than a certain maximal distance in the multidimensional character space was calculated; and (2) of these, the fraction of similar neighbours was computed (i.e. other fish characterized by the same age at maturation as the focal one). The fraction of similar neighbours within a certain multivariate distance was then averaged over all fish in a given age group. This procedure, simplified to a two-dimensional character space, is visualized in Figure 2.
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The mean fraction of similar neighbours situated within a very small multivariate distance will provide an indication of the hypothetical limitations to maximal classification success. Unfortunately, when the distance defining neighbourhood decreases, the total number of neighbours will also decrease. As a result, the estimation of the fraction of similar neighbours will become less accurate when distance decreases. Therefore, the procedure was repeated for a range of different neighbourhood-defining distances (0.05, 0.10, 0.15,..., 2.50 mm). The results were plotted and examined graphically.
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Results |
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TopIntroductionMaterial and methodsResultsDiscussionAppendixReferences |
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Classification success
The overall proportion of correct classifications of age at maturation in herring based on scale measurements was 68.0% if DA was used, and 66.6% if NNs were used. The overall success rate was only marginally, but nevertheless significantly (p<0.0001) higher for DA than for NN (Table 2).
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An assessment of classification success per age group (Table 2) shows that the proportion of successful classifications decreased with age, as expected, from 100% in mature fish caught at age 3 (only maturation at age 3 is possible) to 56.4% (DA) or 54.9% (NN) in mature fish caught aged 9 or older (maturation at all ages from 3 to 9 is possible). For all age classes, there were only minor differences in classification success based on the two methods. In particular, methodological differences were negligible for fish caught at ages 3–6; for older fish, classification success was marginally higher using DA than NN.
Prediction error
Performance as indicated by prediction error was marginally better for NN than for DA (Table 3). The overall mean prediction error, defined as the absolute difference between observed and predicted values for age at maturation, was 0.382 years with DA and 0.367 years with NN. The proportion of cases where the prediction error was >1 year was higher with DA (5.2%) than with NN (2.9%). The difference in prediction error between the two methods was small, but significant (p<0.0001; Table 3).
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Degree of misclassification, as expected, increased with age at which caught (Table 3). In fish caught at age 4, prediction error averaged 0.017 years with both methods (range of error, 0–1 year). In fish caught at age 9 or more, prediction error averaged 0.521 years if DA was used, and 0.503 years if NNs were used (range of error, 0–5 years). Although differences in the degree of misclassification between the two proposed methods were negligible for fish caught young (3–5, 7), the differences were more pronounced for fish caught older (6, 8, and higher).
Prediction error resulting from DA was significantly correlated with that resulting from NN (Table 3). This implied that if age at maturation for a given fish was misclassified using DA it was also likely to be misclassified to a similar extent using NN.
Degree of under- or overestimation
Over the whole data set, age at maturation was slightly underestimated with DA, but on average only by 0.06 years or <1.1% of the mean age at maturation (Table 4). When NN were used for prediction, age at maturation was on average overestimated, but to an even lesser extent of just 0.01 years (0.2% of mean age at maturation; Table 4). For the different age groups, age at maturation predicted with DA or NN was on average similar to, slightly lower, or slightly higher than the averages for the observed values. The degree of under- or overestimation was always very small, never accounting for >2.3% of the average observed values.
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Overlap in scale-measurement data between maturation groups
There was considerable overlap in data on scale measurements between the different maturation groups. This was shown by the analysis, per age group, of the average fraction of neighbours similar in age at maturation as a function of multivariate neighbourhood-defining distance (Figure 3). If a relatively low neighbourhood-defining distance was chosen (range 0.05–0.20 mm), there was close agreement between the percentage of neighbours similar in age at maturation per age group (Figure 3), and classification success using both DA and NN (Table 2). For example, for fish aged 5, within a multivariate distance of 0.1 mm, the mean fraction of neighbours similar in age at maturation was 81.0%, in accordance with classification successes of 83.3% with DA and 82.9% with NN. For fish aged 9 or more, the mean fraction of similar neighbours within the same distance was 53.3%, in accordance with classification successes of 56.4% with DA and 54.9% with NN.
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Discussion |
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TopIntroductionMaterial and methodsResultsDiscussionAppendixReferences |
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The study shows that age at maturation in Norwegian spring-spawning herring can be estimated from scale measurements with reasonably high prediction success. Two entirely different classification methods, DA and NN analysis, predicted age at maturation correctly in about 67–68% of all cases (Table 2). These success rates are especially encouraging if one considers that maturation in Norwegian spring-spawning herring may occur at no less than seven different ages, ranging from 3 to 9 (Runnström, 1936). For adult herring caught at age 4, where there are only two possible ages at maturation (3 and 4), classification success was as high as 98.3% regardless of the method used. There was only a very small fraction of fish where the difference between observed and predicted age at maturation was more than 1 year; this fraction was 5.2% with DA and 2.9% with NN (Table 3). Over the whole data set there was a significant, but only minor, degree of underestimation of age at maturation if DA was used (by 1.1% of mean age at maturation), and a significant but even lower degree of overestimation if NN were used (by 0.2% of mean age at maturation; Table 4). The significant levels of under- and overestimation using DA and NN, respectively, may be explained by the very large sample size (n=45 386), rendering it extremely likely that levels of significance would be reached even with very small differences, in particular in the case of paired comparisons. These combined indicators of classification performance underscore the fact that DA and NN are promising tools to predict age at maturation in herring from routine scale measurements.
The methods described here will allow analyses of long-term trends in age at maturation in one of the world's economically most important fish stocks (ICES, 2001), and in addition will allow studies on the effect of age at maturation on a range of important parameters, including size and condition at age, recruitment, and the total number of reproductive events throughout the lifespan of the herring under different fishing pressures (Beverton, 1992; Beverton et al., 1994; Godø, 2000). Moreover, such data will allow analysis of reaction norms for age and size at maturation, with the possibility of disentangling genetic from phenotypic aspects of maturation (Heino et al., 2002). It is also important to note that similar methods might well be applicable to other commercially significant fish stocks, where either scales or otoliths are being collected routinely and the annuli examined for age and growth studies.
In this case study on Norwegian spring-spawning herring, the quality of the results obtained with DA or NN was equally high, and the question as to which method performed better in classifying age at maturation depended on the performance criterion used. If classification success, or the total number of cases classified at the exact, correct age at maturation, were used as performance criterion, then DA gave marginally but significantly better results than NN (Table 2). The two other performance criteria, however, indicated better results with NN. First, prediction error was slightly, though significantly, lower with NN (Table 3), implying that, if a case was misclassified, the error was likely to be smaller with that method than with DA. Second, using NN, age at maturation was on average overestimated to a (even) lower degree, than it was underestimated using DA (Table 4). Combined, these differences in classification performance between the two methods are of such small magnitude that DA and NN may be considered equally successful procedures at deriving age at maturation from scale measurements in Norwegian spring-spawning herring.
In fact, it appears that both methods approach the maximal, hypothetical limits to classification as imposed a priori by the overlap in the predictor variables between the different groups to be classified. A clear indication of this was given by the analysis quantifying the degree of overlap in scale measurement data in the multidimensional character space between maturation groups (Figure 3). At small neighbourhood-defining distances, the mean fraction of neighbours similar in age at maturation did not increase to approximately 100% as would be expected were there no overlap in explanatory variables between the different groups. By contrast, for each of age groups 4–9, this fraction, with decreasing neighbourhood-defining distances, approached percentages similar to those characterizing classification success using either DA or NN (Table 2). This implies that DA and NN approach the limit to classification success imposed by the nature of the predictor data, and that no considerable improvement may be expected from any classification method.
To some extent, the overlap in scale data between maturation groups could be attributed to measurement inaccuracy when the radii of the scale annuli were measured, because the smallest unit of measurement was approximately 0.05 mm. Moreover, although several skilled readers in a team with many years of experience were involved in the original process of observing age at maturation, some judgement was often necessary in the distinction between typical oceanic rings in the scale, corresponding with the late immature stage (relatively wide summer zones, separated by diffuse winter rings) and spawning rings, corresponding with the mature stage (narrow summer zones, separated by sharp, fine winter rings: Lea, 1928; Runnström, 1936). The observations may therefore not always have provided the correct classifications for age at maturation. In particular, the annulus corresponding to the process of maturation is often intermediate in width to spawning and previous oceanic rings. Although generally the processes of maturation, such as gonad formation and the first spawning migration, lead to a considerable reduction in body growth rate, many fish in their year of maturation form only (very) small gonads (Slotte and Fiksen, 2000), or may migrate to less distant spawning grounds (Slotte, 1999). This will result in a lesser reduction in the rate of body growth and a less clear ‘mark’ of maturation in the scale structure (Runnström, 1936). Environmental influences in the year of maturation, and natural variation in maturation between fish, are other possible causes of overlapping scale data among groups.
The similar performances of NN and DA in predicting age at maturation did not agree with several recent studies, that report better, sometimes far better, classification results obtained with the relatively newly developed NN technique than with conventional, statistical methods of classification, i.e. DA and logistic regression (e.g. Lek et al., 1996; Manel et al., 1999a,b; Özesmi and Özesmi, 1999). In solving other biological problems, NNs also appear to be a superior alternative when compared with parametric modelling techniques such as multiple linear regression (e.g. Lek et al., 1995; Baran et al., 1996; Mastrorillo et al., 1997; Chen and Ware, 1999). In general, the high predictive power of NNs can be attributed to their ability to handle non-linear relationships between predictor and dependent variables particularly well, through the presence of many intervening information-processing units that each use the binary logistic activation function (Fausett, 1994; Basheer and Hajmeer, 2000). This advantage of NNs over parametric statistics, however, probably does not apply to our data, because the relationship between maturation and growth (and hence, scale structure) appears to be rather linear, or log-linear (Holst, 1996; Slotte, 1999). A further advantage of NNs is the small impact of extreme values on prediction success, and the absence of any specific assumptions on the distribution of the data (although data transformation may improve computational speed), while Gaussian data are an important assumption to be met in DA. Indeed, the data on scale increments, if log-transformed, were normally distributed, with virtually no extreme values.
Hence, in this study on Norwegian spring-spawning herring, the data fulfilled very well the conditions required for the conventional statistical procedure of DA, and the extra advantages provided by NNs were relatively small. Such conditions might, however, be different in the case of other fish stocks for which investigations on age at maturation based on growth layer data are being considered. NNs could still be the preferred option, in particular if data assumptions for DA are not fulfilled, and/or maturation is related with scale growth, or otolith growth, in an irregular, non-linear fashion.
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Appendix |
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Architecture of a neural network to derive age at maturation from scale measurements
The architecture of the artificial NNs applied here can be described as three-layer feed-forward NNs trained by back-propagation. The first layer (input layer) consists of 3–9 input units containing the information on the predictor variables (i.e. data on annual scale increments, after log-transformation); the number of input units is therefore equivalent to the number of measured increments. The second layer (hidden layer) consists of 14–19 hidden units; the total number of hidden units is chosen on the basis of a trade-off between limiting computation time and obtaining sufficiently satisfactory classification results. The third layer (output layer) consists of a single output unit determining the output of the network, representing the variable to be predicted (i.e. age at maturation). All units in the input layer are connected with all units in the hidden layer, and these in turn are all connected with the output unit. Specific, modifiable weights are attributed to each of the connection links between units of successive layers. These weights are the link between the problem and the solution and are therefore said to contain the ‘knowledge’ of the NN about the overall problem (Baran et al., 1996). At the start of the training phase, all weights in the network are initialized to small random values (–0.1,..., 0.1); over the course of the training phase, the weights are gradually modified using the back-propagation learning algorithm until network performance is optimized (Rumelhart et al., 1986).
Each unit has an activation that determines its output signal. The activations of the input units are equal to the values for the predictor variables of a given case. The activations of the hidden units are computed in two steps. First, each input unit emits a weighted output signal to all hidden units, equal to its activation multiplied by the weight associated with the specific connection link. Each of the hidden units summarizes the weighted input signals to compute its net input, as follows (Fausett, 1994):
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Next, each hidden unit sends its weighted output signal to the output unit o, which summarizes its incoming signals to compute its own net input, as follows:
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The back-propagation learning rule (Rumelhart et al., 1986) to train the network implies that weights are modified in a backward sweep, according to the generalized delta rule (Rumelhart et al., 1986; McClelland and Rumelhart, 1988). First, an error information term 
o is computed for the output unit by comparison of the actual and desired output of the net:
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wh is the weight correction term for the link from hidden unit h to the output unit, and 
is the learning rate parameter (here, a value of 0.1 was chosen). Next, for each hidden unit, an error information term h is computed:
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The training phase was terminated based on the mean-squared-error convergence criterion (McClelland and Rumelhart, 1988). The data set was divided into a training subset of 67% of the cases, and a testing subset of 33% of the cases. During training epochs, the network is first adapted based on all cases in the training subset; its performance is then monitored on the basis of the independent testing subset. A measure of performance is the mean-squared-error (E) calculated over all cases in the testing subset, as follows:
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Acknowledgements |
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We thank Sébastien Barot, Mikko Heino, Jens Christian Holst, Ingolf Røttingen, Aril Slotte, and Reidar Toresen, for valuable discussions during the study, and Oddvar Dahl, Annlaug Haugsdal, and Jan Henrik Nilsen, for providing extensive information on various aspects of the analysis of herring scales. The study was supported by the European Research Training Network ModLife (Modern Life History Theory and its Application to the Management of Natural Resources), funded through the Improving Human Potential Programme of the European Commission (Contract Number HPRN-CT-2000-00051).
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References |
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