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ICES Journal of Marine Science: Journal du Conseil 2005 62(3):353-359; doi:10.1016/j.icesjms.2004.12.009
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© 2004 International Council for the Exploration of the Sea

An application of two techniques for the analysis of short, multivariate non-stationary time-series of Mauritanian trawl survey data

Karim Erzinia,*, Cheikh A. O. Inejihb and Kim A. Stobberupc

a Centro de Ciências do Mar (CCMAR), Universidade do Algarve 8000-117 Faro, Portugal
b Institut Mauritanien de Recherches Océanographiques et des Pêches (IMROP) B.P. 22, Nouadhibou, Mauritania
c Instituto Nacional de Investigação Agrária e das Pescas (IPIMAR) Av. Brasília, 1449-006 Lisboa, Portugal

*Correspondence to K. Erzini: tel: +351 289 800100; fax: +351 289 818353. e-mail: kerzini{at}ualg.pt.

Min/max autocorrelation factor analysis (MAFA) and dynamic factor analysis (DFA) are complementary techniques for analysing short (>15–25 y), non-stationary, multivariate data sets. We illustrate the two techniques using catch rate (cpue) time-series (1982–2001) for 17 species caught during trawl surveys off Mauritania, with the NAO index, an upwelling index, sea surface temperature, and an index of fishing effort as explanatory variables. Both techniques gave coherent results, the most important common trend being a decrease in cpue during the latter half of the time-series, and the next important being an increase during the first half. A DFA model with SST and UPW as explanatory variables and two common trends gave good fits to most of the cpue time-series.

Keywords: dynamic factor analysis, indicators, Mauritania, metrics, min/max autocorrelation factor analysis, multispecies, time-series

Received 1 April 2004; accepted 10 November 2004.


   Introduction
Top
 Introduction
Material and methods
 Results
 Discussion
 References
 
The main use of ecosystem indicators is to summarize trends in ecosystem structure or functioning that can be related to fishing and environmental variability. A variety of metrics based on size and trophic structure, relative abundance, and diversity have been developed (Bianchi et al., 2000; Pauly et al., 2000; Rice, 2000; Rochet and Trenkel, 2003; Trenkel and Rochet, 2003), many of which are based on time-series derived from regular monitoring surveys.

Rice (2000) discusses the merits and limitations of different classes of metrics as indicators of ecosystem state. Aggregate, community-level indicators have clear merits, because they integrate over the individual components, but their disadvantage is that effects on individual species cannot be traced. This may be a crucial shortcoming, because the primary response of a species to exploitation (as well as to environmental change) is largely determined by its life history characteristics (Rochet, 1998). Also, the effects of fishing and environmental variables cannot easily be evaluated using aggregate indicators (Rice, 2000). While techniques such as canonical correspondence analysis, redundancy analysis, and canonical correlation may be used to analyse multivariate data with explanatory variables, their utility for analysing trends and changes over time is limited (Bull et al., 2001; Zuur et al., 2003b).

Trend, seasonality, and explanatory variables can be handled by alternative techniques such as vector auto-regressions, auto-regressive integrated moving average (ARIMA) models, multivariate ARIMA, and dynamic regression models (Stergiou, 1991; Rothschild et al., 1996; Stergiou and Christou, 1996; Stergiou et al., 1997; Lloret, 2003). However, such techniques require long, stationary, and complete time-series, and are not efficient for handling common trends (Stergiou and Christou, 1996). Solow (1994), Zuur et al. (2003a, b) and Zuur and Pierce (2004) propose two statistical techniques that are particularly suitable for relatively short (>15–25 y), non-stationary multivariate time-series: min/max autocorrelation factor analysis (MAFA) and dynamic factor analysis (DFA). They can be used to extract and identify common trends from multiple time-series, to estimate index functions, to evaluate interactions between response variables, and to determine the effects of explanatory variables. Moreover, the effects of introducing lags in the explanatory variables can also be evaluated (Zuur et al., 2003b; Zuur and Pierce, 2004).

Fisheries applications of DFA have been limited to multiple time-series of catch per unit effort (cpue) for Nephrops norvegicus (Zuur et al., 2003b) and Loligo forbesi (Zuur and Pierce, 2004), with sea surface temperature (SST) and NAO index as explanatory variables, and an analysis of landings from southern Portugal with NAO index, river run-off, and an upwelling index as explanatory variables (Erzini, 1995). Here, we apply the two techniques to multispecies cpue time-series (1982–2001) of 15 fish and two cephalopod species taken in trawl surveys off Mauritania, with indices of NAO, upwelling, SST, and fishing effort as explanatory variables. The aim is to identify important underlying patterns in the data and to evaluate their relative importance, as well as to investigate the effects of introducing lags in the explanatory variables for individual species.


   Material and methods
Top
 Introduction
 Material and methods
Results
 Discussion
 References
 
Response and explanatory variables
Mauritanian trawl survey data were available for the years 1982–2001 (IMROP, 2002). Species were selected on the basis of their overall abundance and importance to local demersal fisheries. Pelagic species were excluded because their catches by bottom trawls are likely not representative. The 15 fish species (Table 1) selected, along with octopus (Octupus vulgaris) and cuttlefish (Sepia officinalis), accounted for 43% of survey catches over the entire period.


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  		Table 1 Names, codes, and information on the 15 fish species selected (TL, total length; SL, standard length).

 
Trawl surveys were stratified by season (warm, June–October; cold, November–May), zone (north, >19.25°N; south, <19.25°N), and depth (stratum 1, 0–20 m; 2, 20–50 m; 3, 50–100 m; 4, 100–227 m). Deeper stations were eliminated because they were sampled less frequently and species composition was very different. Generalized Linear Models (GLM; McCullagh and Nelder, 1992) were used to remove the effects of these factors and thus to estimate standardized annual abundance. GLM analysis was carried out on log-transformed mean survey catch (kg per 30 min) by species, accounting for the factors year, season, zone, and depth. Zero values were eliminated, because the objective was to determine relative change over time, not absolute abundance.

Four explanatory variables were used: mean SST during the cold season (November–May), based on data available on a 1° x 1° grid from COADS (Comprehensive Ocean-Atmosphere Data Set; http://www.cdc.noaa.gov/coads/), an upwelling index based on wind data at latitude 20.5°N (Sedik, 1975), the annual mean NAO index based on the difference of normalized sea level pressures between Ponta Delgada, Azores, and Stykkisholmur/Reykjavík, Iceland (http://www.cgd.ucar.edu/~jhurrell/nao.html; Hurrell, 1995), and an index of fishing effort for the cephalopod trawl fleet in Mauritanian waters (IMROP, 2002). To facilitate the interpretation of factor loadings and canonical correlations, the time-series of cpue and explanatory variables were standardized by dividing annual values by the overall mean.

Min/max autocorrelation factor analysis (MAFA)
MAFA (Solow, 1994) is a type of principal component analysis (PCA) in which the axes represent a measure of autocorrelation and give an indication of the association between variable Yt and Yt+k, where k is a time-lag (k = 1, 2,...). Unlike PCA, in which the first axis explains most of the variance, the first MAFA axis has the highest autocorrelation, and because trends are associated with high autocorrelation, it represents the main trend in the data. MAFA can be used to extract trends from multiple time-series, to estimate index functions, and for smoothing (Solow, 1994).

The loadings determine the relationship of individual response variables to particular MAFA axes. Cross-correlations between MAFA axes and response variables, also known as canonical correlations, are a measure of the relationship between Yt and Xt–k, and allow identification of significant relationships between trends and explanatory variables. We used the Brodgar (http://www.brodgar.com) software package to carry out MAFA of the time-series of response variables and explanatory variables.

Dynamic factor analysis (DFA)
DFA is a dimension reduction technique that can be used to identify underlying common patterns in a multivariate time-series, evaluate interactions between response variables, and determine effects of explanatory variables (Zuur et al., 2003a). Time-series are modelled as a function of a linear combination of common trends, a constant level parameter, one or more explanatory variables, and noise (Zuur and Pierce, 2004).

Using the Brodgar software package, we fitted a series of DFA models, ranging from the simplest (with one common trend plus noise) to the most complex (with four common trends, two explanatory variables, plus noise). Models were fitted with both a diagonal covariance matrix and a symmetric positive-definite covariance matrix. Models with explanatory variables were fitted at lag = 0, 1, and 2, and Akaike's information criterion (AIC) was used as a measure of goodness-of-fit and to compare models (Zuur et al., 2003b). Factor loadings were used to make inferences regarding the importance of particular trends, representing underlying common patterns over time, both to specific response variables, and to different groups of response variables.


   Results
Top
 Introduction
 Material and methods
 Results
Discussion
 References
 
The cpue time-series showed considerable variability, and trends cannot easily be identified (Figure 1a). This also largely applies to the explanatory variables, although fishing effort clearly peaked in 1987 and 1996, and (as expected) SST and UPW were negatively correlated.


Figure 1
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  		Figure 1 Standardized series for (a) the 17 species (cpue), and (b) the four explanatory variables.

 
Although autocorrelations were high (0.97 and 0.92, for first and second MAFA axis, respectively), neither was significant (p > 0.59). The scores for the first axis showed a steady increase from 1982 to 1992, followed by a sharp decline, whereas the second axis displayed a sharp increase to 1988, followed by a slight decrease and levelling off thereafter (Figure 2a). The canonical correlations illustrating the relationships between species and the first axis (Figure 3) indicate positive correlations for 15 of the 17 species, but only four of these were significant (p < 0.05; Brotula barbata, Citharus linguatula, Epinephelus aeneus, and Umbrina canariensis), corresponding to a main trend of decreasing cpue in recent years. The significant (p < 0.05) negative correlation for Mustelus mustelus indicates an opposite trend. In addition, four species were significantly (p < 0.05) and positively correlated with the second axis (Brachydeuterus auritus, C. linguatula, Pomadasys incisus, and Pseudopeneus prayensis). None of the correlations between the first two MAFA axes and the explanatory variables was significant.


Figure 2
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  		Figure 2 (a) Scores for the first two MAFA axes, and (b) the two main common trends for the DFA model with SST and UPW as explanatory variables.

 


Figure 3
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  		Figure 3 Canonical correlations for the first MAFA axis (for species codes, see Table 1).

 
The best DFA fits were obtained for non-lagged explanatory variables and a symmetric, non-diagonal matrix (Table 2). Of the models with no explanatory variables, the best fit was for three common trends (AIC = 740). When explanatory variables were included, AIC values were smaller, the model with two common trends and SST and UPW being the best (AIC = 573). The main common trend showed an increase to a maximum in 1993, followed by a decline, whereas the second common trend depicted a general decrease from 1982 to 2001 (Figure 2b). The two main common trends were similar to those for the first two MAFA axes (Figure 2a), especially for the second half of the period.


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  		Table 2 Values of Akaike's information criterion (AIC) for DFA models with a variable number of common trends (ct; M = 1–4) and different sets of explanatory variables (exp; SST, sea surface temperature; NAO, North Atlantic Oscillation index; UPW, upwelling index; FE, fishing effort), based on a symmetric, non-diagonal matrix (emboldened, lowest AIC value within series with M = 1–4 ct).

 
The estimated t-values for the regressions for individual species and SST and UPW, respectively, were relatively large for Diplodus bellottii (SST) and Brotula barbata (UPW), indicating strong relationships (Table 3). Relatively large diagonal elements of the error covariance matrix (R > 0.55) were obtained for M. mustelus, O. vulgaris, S. officinalis, Sparus caeruleostictus, and U. canariensis, indicating that these cpue series did not fit well. However, the relatively small covariance values (range: –0.19 to +0.42; with only 9 of 136 >0.3) indicate that the common pattern generally fitted well.


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  		Table 3 Estimated t-values for the explanatory variables SST and UPW (for species codes see Table 1).

 
The biplot of factor loadings (Figure 4) shows that, with one exception (M. mustelus), all species are positively correlated with at least one of the two principal trends, often with two. Observed and fitted cpue series for the seven species best correlated with one of the two main common trends (B. auritus, B. barbata, C. linguatula, E. aeneus, Plectorhyncus mediterraneus, P. prayensis, and U. canariensis) and the one negatively correlated with both (M. mustelus) are given in Figure 5. For all 17 species, the observed cpue and fitted cpue were significantly positively correlated ({alpha} = 0.05), indicating that the model adequately described the trend, despite considerable interannual variability in cpue. With the exception of M. mustelus showing an increase in later years, all other species showed a marked decline in cpue during the second half of the period.


Figure 4
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  		Figure 4 Biplot of DFA factor loadings (for species codes, see Table 1).

 


Figure 5
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  		Figure 5 Observed and fitted cpue for the DFA model with two common trends and two explanatory variables (SST and UPW): (a) Brachydeuterus auritus, (b) Brotula barbata, (c) Citharus linguatula, (d) Epinephelus aeneus, (e) Mustelus mustelus, (f) Plectorhyncus mediterraneus, (g) Pseudopeneus prayensis, and (h) Umbrina canariensis.

 

   Discussion
Top
 Introduction
 Material and methods
 Results
 Discussion
References
 
The two techniques applied, MAFA and DFA, apparently provided coherent results, indicating that the two most important trends in the response variables are (i) a decline in cpue during the second half of the time-series, and (ii) a general decrease in cpue over time. What drove these trends is less clear, but the available evidence suggests that the environment (SST, and to a lesser extent UPW; the two factors are obviously related, in that a strong UPW should lead to lower SST) has been more important than fishing. The importance of the pelagic environment is not surprising, given the life history characteristics of the species studied (Table 1). Both octopus and cuttlefish live for about one year (Sobrino et al., 2002), and approximately half the fish species are relatively small, short-lived, and fast-growing, and recruit to the fishery in their first year of life. Moreover, most species spend part of their early life off the bottom, and thus are subjected to the pelagic environment. In the northwestern Mediterranean, catch and cpue of most fish species is significantly positively correlated with river flow and a wind-mixing index for lags <1 year (Lloret et al., 2001).

Notwithstanding, the index of trawl effort used may not be the best to characterize exploitation in the area. The cephalopod fishery is only one of several trawl fisheries operating in Mauritanian waters, and may not be relevant for some of the species caught in a survey covering a large area and depth range.

In conclusion, we have demonstrated two useful techniques for evaluating the state of multispecies fisheries, and the environmental and fisheries factors affecting them. The techniques have a number of advantages compared with other methods used to evaluate the effects of fishing on ecosystems (Rice, 2000), especially in terms of their ability to handle trends, explanatory variables, missing values, and interactions between trends, response, and explanatory variables.


   Acknowledgements
 
We thank Alain Zuur for statistical advice.


   References
Top
 Introduction
 Material and methods
 Results
 Discussion
 References
 

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