Stockhausen and Fogarty: Removing observational noise from time series data using ARIMA models 



97 



all frequencies. Because the unobserved PSD must 

 be nonnegative, the maximum observation noise (and 

 thus the maximum noise reduction and smoothing that 

 may occur) consistent with additive white noise corre- 

 sponds to the minimum value of the observed process's 

 PSD. Presumed higher values for the observation noise 

 would require that the PSD for the unobserved pro- 

 cess be negative over some frequency range. Box et 

 al.'s (1978) algorithm computes the maximum possible 

 observation noise and uses a time domain formula to 

 calculate a smoothed, "unobserved" time series consis- 

 tent with additive white noise for any noise level up to 

 the maximum. 



The ARIMA-based noise reduction approach was first 

 applied to fisheries trawl survey data by Pennington 

 (1985), who developed an alternative algorithm to that 

 of Box et al. (1978) for estimating the smoothed time 

 series. This algorithm was subsequently used in several 

 studies (Pennington, 1985; Fogarty et al.'; Pennington, 

 1986; Anonymous, 1988, 1993) to smooth time series 

 of abundance indices derived from trawl survey data 

 for the northeast coast of the United States. Of these 

 studies, perhaps only Pennington (1986) constitutes a 

 convincing demonstration of the utility of the ARIMA- 

 based approach to time series noise reduction when the 

 ARIMA model for the underlying process is unknown. 

 Pennington's (1985) demonstration relied on an ARIMA 

 model developed for the (usually unobserved) underlying 

 population that was based on stock assessment results 

 for the particular species considered. In contrast, RW- 

 PUN models were assumed a priori in Fogarty et al.' 

 and Anonymous (1988, 1993) because the short time 

 series (<25 observations per series) considered made 

 identification of the underlying ARIMA models prob- 

 lematic. Only in Pennington (1986) was a model used 

 that was fitted to trawl survey data and tested for ap- 

 propriateness. 



However, substantially longer time series are now 

 available to test the ARIMA-based noise reduction con- 

 cept. Consequently, to revisit the utility of the ARI- 

 MA-based approach for smoothing time series data 

 derived from fisheries-independent trawl survey data, 

 we applied Box et al.'s (1978) approach to smoothing to 

 time series of annual abundance indices derived from 

 the NMFS/NEFSC fisheries-independent bottom trawl 

 survey of nine finfish species during two seasons on 

 Georges Bank. Time series for the fall spanned 40 years 

 (1963-2002), and the spring time series spanned 36 

 years (1968-2003). The species, comprising two elasmo- 

 branchs, three groundfish, two flatfish, and two pelagic 

 schooling species, presented a broad range of life his- 

 tory characteristics (Table 1). 



The noise reduction results we obtained varied among 

 species and between seasons. Despite smoothing at the 

 maximum level of noise reduction consistent with each 

 model, very little smoothing was obtained for haddock 

 (fall), little skate (spring), and silver hake (spring). The 

 models for these time series had among the highest 

 moving average orders (3-5). Examination of the PSD 

 for each model in the frequency domain revealed very 



small minima, indicating no scope for noise reduction. 

 Typically, models that had a MA order >2 exhibited 

 substantially less smoothing than models with a MA 

 order ^2. Models that were of MA order 1 generally 

 resulted in the greatest smoothing. Models that were 

 of MA order did not (and could not) occur. 



Of the 18 time series we considered (Tables 2 and 3, 

 Figs. 2 and 3), only half were adequately represented 

 as random-walk-plus-uncorrelated-noise (RWPUN) 

 models. The ARIMA models we developed were varied 

 in structure, ranging from a simple MA(1) model to 

 rather complicated models with multiple parameters. 

 Thus, our results provide evidence against the appro- 

 priateness of assuming a particular model structure a 

 priori when the objective of the analysis is to identify 

 the underlying dynamic structure of the population. 

 This evidence is further strengthened by the results 

 of Becerra-Munoz et al. (1999), who found only 9 of 

 52 abundance time series for finfish species from the 

 NMFS/NEFSC bottom trawl survey that corresponded 

 to random walk models. 



As an exercise, we also attempted to smooth the nine 

 data sets that were not adequately described as RW- 

 PUN models, using this model structure as an a priori 

 assumption, even though our analysis indicated that 

 other models were more appropriate. We were not able 

 to estimate convergent models for three species: At- 

 lantic cod (fall), winter flounder (spring), and Atlantic 

 mackerel (spring). For the remaining six time series, 

 the smoothed results appeared to be quite reasonable 

 (Fig. 4), although we obtained little noise reduction 

 when we employed the "correct" ARIMA model. The 

 RWPUN-smoothed time series for haddock (Fig. 4, C 

 and F) were similar to that for spawning biomass de- 

 rived from virtual population analysis (see Brodziak 

 et al.'-), but the smoothed time series for silver hake 

 (Fig. 4E) exhibited higher frequency variability than 

 that found for total biomass with a production model 

 (see Brodziak et al.'^). From the standpoint of estimat- 

 ing the unobserved underlying process, these smoothed 

 results should be viewed with some skepticism: the 

 use of the RWPUN model is rather arbitrary in this 

 situation and it may impose artificial structure on the 

 smoothed results. However, it may be that these time 

 series do not meet one of the key assumptions of the 

 noise reduction method: namely that the observation 

 noise is uncorrelated. The ARIMA models for all six 

 time series had MA orders a3, and one effect of cor- 

 related observation noise could be to increase the MA 



- Brodziak, J., M. Traver and L. Col. 2005. Georges Bank 

 haddock. In Assessment of 19 northeast groundfish stocks 

 through 2004 (R. K. Mayo, and M. Terceiro, eds.), section 



2, p. 30-80. 2005 groundfish assessment review meeting. 

 Northeast Fisheries Science Center, Woods Hole, Massachu- 

 setts; 15-19 August 2005. NEFSC Ref Doc. 05-13. NEFSC, 

 166 Water Street, Woods Hole, MA 02543. 



3 Brodziak, J. K. T., E. M. Holmes, K. A. Sosebee, and R. K. 

 Mayo. 2001. Assessment of the silver hake resource in the 

 Northwest Atlantic in 2000, 134 p. NEFSC Ref Doc. 01- 



03. NEFSC, 166 Water Street, Woods Hole, MA 02543. 



