Inspection of the plot of the time series of CPUE by year leads to an interesting observation. The behaviour of the series in the early years of the fishery exhibits different behaviour to that in the latter years, implying the presence of a breakpoint around the year 2003-4 (Figure 22).
Figure 22 CPUE plotted by year with an apparent breakpoint around 2004. This suggests that an LM that includes an annual linear trend that incorporates a breakpoint could lead to an improved explanation of CPUE.
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Visual inspection of Figure 22 suggests the location of a breakpoint in the CPUE measure. In order to quantify this breakpoint, fifteen potential models were fitted, allowing for each of the fifteen years in the dataset to be tested as candidate breakpoints. The AIC values of these fifteen candidate models were then compared to determine the best year to use for the breakpoint (Figure 23).
The year with the lowest AIC is the most suitable breakpoint for the CPUE data. In this case, the best choice of breakpoint is 2004. As the year decreases or increases either side of 2004, the model AIC values increase which provides support for the identification of 2004 as the optimal breakpoint. The ΔAICs for the candidate breakpoint models are all greater than 80. This gives the model with 2004 as the breakpoint an AIC weight of one and the remaining candidate models an AIC weight of zero.
After identifying the optimal breakpoint, a selection of LMs were fitted with a linear trend in year with a breakpoint at 2004. This new breakpoint model is fitted with all of the month terms including the harmonics (Table 15). The extension of these models involves the inclusion of interactions and the species variable (Table 15 - Table 18). A direct comparison can be made between these models and those in Table 9 to Table 12 which treat CPUE as undergoing abrupt annual fluctuations or having a linear trend with a constant slope.
SUMMARY:
Modelling CPUE with a yearly linear trend that incorporates a breakpoint leads to significantly better models than modelling with a constant linear trend. However, modelling CPUE with
abrupt annual fluctuations leads to a model which is significantly better than those which incorporate a breakpoint in the linear trend. This indicates that although the presence of a breakpoint is significant, the annual variability in CPUE is better explained by a model allowing
abrupt annual fluctuations.
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Figure 23 A plot of AIC values for candidate breakpoint years for modelling CPUE with a segmented linear trend. The breakpoint models are LMs fitted with an annual linear trend in CPUE where the slope of the trend is allowed to differ either side of the breakpoint. Fifteen separate models are fitted with the candidate breakpoint ranging from 1998 to 2012. The minimum AIC leads to identification of the year 2004 as the optimal breakpoint.
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Annual linear trend with a breakpoint in 2004 𝜶 + 𝜷𝟏𝒚𝒆𝒂𝒓 + 𝜷𝟐(𝒚𝒆𝒂𝒓 − 𝟐𝟎𝟎𝟒)𝑰{𝒚𝒆𝒂𝒓≥𝟐𝟎𝟎𝟒}CPUE undergoes an annual linear decrease where the rate of decrease is different either side of 2004.
Species is not included in these models indicating that the CPUE behaves the same for each species Month
NULL
A monthly pattern for CPUE is absent
19080 (148, 0) Month_F
A monthly pattern in CPUE results in fluctuations which repeat annually
18932 (0, 0.88)
Month_N
There is a linear trend in CPUE within each year which repeats annually
19070 (138, 0)
Month_N1
Within each year there is a seasonal pattern for CPUE with one peak and trough which repeats annually
19015 (83, 0)
Month_N2
Within each year there is a seasonal pattern for CPUE with two peaks and troughs which repeats annually
18998 (66, 0)
Month_N3
Within each year there is a seasonal pattern for CPUE with three peaks and troughs which repeats annually
18948 (16, 0.0003)
Month_N4
Within each year there is a seasonal pattern for CPUE with four peaks and troughs which repeats annually
18938 (6, 0.044)
Month_N5
Within each year there is a seasonal pattern for CPUE with five peaks and troughs which repeats annually
18937 (5, 0.072)
Table 15 AIC values for models of CPUE which incorporate a linear trend with a breakpoint in 2004 and variants of month as explanatory variables. Variable interactions and species are not included in these models. The 𝛥AIC and AIC weights are included in brackets to the right of the model AIC. The best of these breakpoint models is in bold and includes abrupt monthly fluctuations in CPUE.
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*
Annual linear trend with a breakpoint in 2004 𝜶 + 𝜷𝟏𝒚𝒆𝒂𝒓 + 𝜷𝟐(𝒚𝒆𝒂𝒓 − 𝟐𝟎𝟎𝟒)𝑰{𝒚𝒆𝒂𝒓≥𝟐𝟎𝟎𝟒}CPUE undergoes an annual linear decrease where the rate of decrease is different either side of 2004.
Species is not included in these models indicating that the CPUE behaves the same for each species Month
Month_F
A monthly pattern in CPUE results in fluctuations which vary annually
18792 (0, 0.69)
Month_N
There is a linear trend in CPUE within each year which varies annually
19036 (244, 0)
Month_N1
Within each year there is a seasonal pattern for CPUE with one peak and trough which varies annually
18922 (130, 0)
Month_N2
Within each year there is a seasonal pattern for CPUE with two peaks and troughs which varies annually
18875 (83, 0)
Month_N3
Within each year there is a seasonal pattern for CPUE with three peaks and troughs which varies annually
18809 (17, 0.00014)
Month_N4
Within each year there is a seasonal pattern for CPUE with four peaks and troughs which varies annually
18794 (2, 0.25)
Month_N5
Within each year there is a seasonal pattern for CPUE with five peaks and troughs which varies annually
18797 (5, 0.057)
Table 16 AIC values for models of CPUE which incorporate a linear trend with a breakpoint in 2004 and variants of month as explanatory variables including interactions. The 𝛥AIC and AIC weights are included in brackets to the right of the model AIC. The model with the lowest AIC is in bold and includes abrupt monthly changes in CPUE. The model including month with four harmonics has the second lowest AIC with a weight of 0.25 making it a candidate for the best model in this table.
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three_species
Annual linear trend with a breakpoint in 2004 𝜶 + 𝜷𝟏𝒚𝒆𝒂𝒓 + 𝜷𝟐(𝒚𝒆𝒂𝒓 − 𝟐𝟎𝟎𝟒)𝑰{𝒚𝒆𝒂𝒓≥𝟐𝟎𝟎𝟒}
CPUE undergoes an annual linear decrease where the rate of decrease is different either side of 2004.
Adding species to the model implies that each species has the same CPUE pattern but with different magnitudes Month
NULL
A monthly pattern for CPUE is absent
17191 (120, 0) Month_F
A monthly pattern in CPUE results in fluctuations which repeat annually
17071 (0, 0.96)
Month_N
There is a linear trend in CPUE within each year which repeats annually
17182 (111, 0)
Month_N1
Within each year there is a seasonal pattern for CPUE with one peak and trough which repeats annually
17142 (71, 0)
Month_N2
Within each year there is a seasonal pattern for CPUE with two peaks and troughs which repeats annually
17124 (53, 0)
Month_N3
Within each year there is a seasonal pattern for CPUE with three peaks and troughs which repeats annually
17085 (14, 0.00087)
Month_N4
Within each year there is a seasonal pattern for CPUE with four peaks and troughs which repeats annually
17080 (9, 0.011)
Month_N5
Within each year there is a seasonal pattern for CPUE with five peaks and troughs which repeats annually
17078 (7, 0.029)
Table 17 AIC values for models of CPUE which incorporate a linear trend with a breakpoint in 2004 and variants of month and species as explanatory variables. The 𝛥AIC and AIC weights are included in brackets to the right of the model AIC. The model with the lowest AIC is in bold and includes abrupt monthly changes in CPUE. The models including month with three, four or five harmonics also have non-zero weights.
89
*
three_species
Annual linear trend with a breakpoint in 2004 𝜶 + 𝜷𝟏𝒚𝒆𝒂𝒓 + 𝜷𝟐(𝒚𝒆𝒂𝒓 − 𝟐𝟎𝟎𝟒)𝑰{𝒚𝒆𝒂𝒓≥𝟐𝟎𝟎𝟒}
CPUE undergoes an annual linear decrease where the rate of decrease is different either side of 2004
The interaction with species implies that each species has its own CPUE pattern Month
Month_F pattern for CPUE with three peaks and troughs which varies annually
16582 (14, 0)
Month_N4
Within each year there is a seasonal pattern for CPUE with four peaks and troughs which varies annually
Table 18 AIC values for models of CPUE which incorporate a linear trend with a breakpoint in 2004 and variants of month and species as explanatory variables including interactions. The 𝛥AIC and AIC weights are included in brackets to the right of the model AIC. The model with the lowest AIC is in bold and includes abrupt monthly changes in CPUE. The models including month with four or five harmonics also have non-zero AIC weights.
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