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Example 

To illustrate dynamic factor analysis, a small data set was used. The data set consisted of 11 time series of  CPUE (catch per hour trawling) of the lobster species Nephrops at 11 stations in the Atlantic Ocean south of Iceland between 1960 and 1999. The data were available on an annual basis. Details on this data set can be found in Eiríksson (1999). It is shown how dynamic factor analysis can be used to detect common trends in the time series, interactions between the series  and relationships with explanatory variables like sea surface temperature and the NAO index.

 

Plotting the Time Series

Every analysis should always start with a simple time plot. The 11 standardised time series are plotted in Figure 1. The figure indicates that there is a reasonably amount of congruency in the series.

Figure 1. Time plot of 11 Nephrops CPUE time series

In first instance we ignored the explanatory variables. To determine how many common trends to use, the AIC was calculated for models containing 1, 2, 3 and 4 common trends. The values are given in Table 1.

Table 1. Values of AIC using a dynamic factor model with M common trends.

Number of common trends (M) AIC
1 716.03
2 688.50
3 689.96
4 694.41
11 univariate trends 973.99

Results indicated that the model containing 2 common trends is the most appropriate model (smallest AIC value). The AIC for the model containing 3 common trends was only marginal larger. A validation on this model did not give any reason to prefer it above the model with two common trends. Results for the model containing two common trends are presented below. 

Results for the dynamic factor model with 2 common trends

The estimated common trends are presented in Figure 2 and the corresponding factor loadings are given in Figure 3. 

 

  Figure 2. First and second common trends and 95% confidence interval.

 

Figure 3. Factor loadings corresponding to the first two common trends.

The factor loadings indicated that there are three groups of stations:

group 1 stations 8, 9, 10 and 11
group 2 stations 1, 2, 3, 5, 6 and 7
group 3 station 4

 

 

The time series at stations 8, 9, 10 and 11 are clearly related to the second common trend. Station 4 is mainly determined by the first common trend. Both the first and second common trends are important for time series of  group 2. An explanation of this grouping might be that the stations of group 2 are located south-west of Iceland and the stations of group 1 are south-east of Iceland. Eiríksson (1999) found a similar pattern in these data and argued that the distinction between the stations might be due to differences in  temperature and soil type. To illustrate the differences between the time series of the groups, fitted values for all stations are presented in Figure 4. Blue lines in this figure correspond to stations of group 1. The fitted curve of station 4 is the curve that had the high values between 1968-1973, 1982-1990, and the dips in 1990 and 1997. Differences between stations of group 1 and 2 were mainly between 1960-1965, 1980-1983, 1985-1986, 1990-1993 and 1996-1998.

Figure 4. Fitted values for all stations. Blue lines correspond to stations 8, 9, 10 and 11.

 

Explanatory variables.

So far, explanatory variables were ignored. Available explanatory variables were sea surface temperature and the NAO index. Both explanatory variables were scalar variables. This means that we only had one value for each explanatory variable in year t. Unfortunately, the sea surface temperature series  was only available up to 1994. Dynamic factor models containing 1, 2 and 3 common trends and the 2 explanatory variables were applied on the CPUE data up to 1994. Results indicated that neither sea surface temperature nor the NAO index was significantly related to the Nephrops series. In these analyses, time lags were not taken into account. 

Conclusions

Dynamic factor analysis applied on the 11 Nephrops time series indicated that there are two underlying common trends. The factor loadings indicated that the distinction between the series is probably based on geographical differences. Neither the sea surface temperature nor the NAO index was significantly related to any of the time series. To understand which biological mechanisms are driving the two common trends, further information is needed (e.g. soil type).

References

Eiríksson, H. (1999). Spatial variability of CPUE and mean size as possible criteria for unit stock demarcations in analytical assessments of Nephrops at Iceland. Rit Fiskideildar, 16, 239-245.