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



95 



A Winter skate 



D Silver hake 



E 

 o 



GO 



G Yellowtail flounder 



1965 1970 1975 1980 1985 1990 1995 2000 



B Little skate 



C Atlantic herring 



E Atlantic cod 



II 1 1 1 1 1 1 1 1 1 1 1 iaI,ii II 1 1 1 1 1 1 II I II 1 1 1 1 1 1 

 F Haddock 



1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 



H Winter flounder 



I Atlantic mackerel 



1965 1970 1975 1980 1985 1990 1995 2000 



Year 



1965 1970 1975 1980 1985 1990 1995 2000 



Original 

 Smoothed 



Figure 3 



Comparison of original time series of estimated total biomass on Georges Bank from the fall bottom trawl survey (filled 

 circles) and inverse-transformed, ARIMA-smoothed time series open triangles) for nine finfish species. 



Discussion 



Abundance indices derived from large-scale fishery-inde- 

 pendent surveys typically exhibit interannual variability 

 much higher than one would expect from within-survey 

 variance (Pennington, 1985). Although true variability 

 in the underlying population due to population-dynamic 

 processes is reflected in the variability of an index, so too 

 is observational noise arising both from within-survey 

 sampling variability as well as from environmentally 

 driven factors that affect catchability. Low signal-to- 

 noise ratios in abundance indices due to high obser- 

 vational noise reduce one's ability to detect important 

 changes or trends in actual population abundance. 



To reduce the impact of white observation noise on 

 time series data, Cleveland and Tiao (1976) developed 

 an approach to noise reduction and smoothing that 

 was based on their knowledge of the ARIMA model 

 (Box and Jenkins, 1976) for an associated unobserved 



but underlying stochastic process. Box et al. (1978) 

 extended this approach to address the situation where 

 the ARIMA model for the underlying process was un- 

 known, relying instead on an ARIMA model associated 

 with the observed time series. In general, and certainly 

 in regard to fishery-independent survey data, a model 

 structure for the unobserved, underlying process will 

 not be available. Hence, Box et al.'s (1978) approach 

 will be the norm. 



In the situation where the observed time series is 

 stationary, we found that a frequency domain inter- 

 pretation of Box et al.'s (1978) algorithm is particu- 

 larly enlightening. When the times series is station- 

 ary, observation noise increases the power spectral 

 density (PSD) of the observed process over that of the 

 unobserved process by a fixed amount at all frequen- 

 cies (Fig. 1). Consequently, the PSD of the unobserved 

 process has the same shape as the PSD of the ob- 

 served process, but with a fixed amount removed at 



