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



99 



ent in ARIMA model estimation to reasonable levels for 

 most species. 



Alternative methods, such as locally weighted scat- 

 terplot smoothing (LOESS), moving average filters, 

 exponential smoothing filters. Kalman filters, and fre- 

 quency-domain approaches can be applied to time series 

 to achieve smoother results (e.g., Cleveland and Grosse, 

 1991; Hamilton, 1994). These approaches typically em- 

 ploy at least one user-determined parameter that can 

 be used to change the amount of smoothing that an 

 algorithm achieves. Generally, one "fiddles" with the 

 adjustable parameters until a "nice," smoothed fit is 

 achieved. However, we think it important to distinguish 

 between these smoothing algorithms and the ARIMA- 

 based noise reduction algorithms. It is quite possible to 

 smooth out real fluctuations in the underlying popula- 

 tion process. The principal advantage that we see for 

 the ARIMA-based noise reduction algorithms (used 

 with an appropriate model! over alternative methods 

 is that the former provide a more objective approach 

 to determining an appropriate level of smoothing. As 

 noted previously, Pennington (1985) showed that when 

 a RWPUN model is appropriate, the ARIMA smoothing 

 approach is completely determined by the ARIMA model 

 for the observed time series because it is possible to de- 

 termine the observation noise variance from the model 

 parameters. In the more general case. Box et al.'s (1978) 

 algorithm at least yields a maximum value for the vari- 

 ance of the observational noise and thus sets an upper 

 limit to the amount of noise reduction and smoothing 

 that can be achieved. For trawl survey data, our results 

 from nine time series where RWPUN models were ap- 

 propriate (and we can consequently estimate the actual 

 observation noise variance) indicate that smoothing at 

 ~90% of the maximum possible noise reduction level is 

 not an unreasonable default percentage (Table 4). 



One drawback to the greater application of ARIMA- 

 based noise reduction methods to time series data is 

 the lack of an integrated software package that allows 

 a user 1) to quickly evaluate an appropriate ARIMA 

 model for a given time series, and 2) to calculate the 

 smoothed time series. We used SAS for the first step 

 and MATLAB for the second, but we found this ar- 

 rangement rather awkward and burdensome. However, 

 econometrically oriented software packages such as 

 ForecastPro'' or AutoBox^ that automate model selection 

 may substantially simplify the first step even if they 

 don't address the second step. 



On the whole, ARIMA-based time series models ap- 

 pear to provide the basis for a more objective approach 

 to reducing observation noise in time series data, in- 

 cluding time series of fishery abundance indices derived 

 from trawl survey data, than do more conventional 



'' Business Forecast Systems, Inc. 2006. Website: http://www. 

 forecastpro.com/products/fpfamily/index.html (accessed on 

 29 March 2006). 



^ Automatic Forecasting Systems. 2003. Website: http:// 

 www.autobox.com/autoboxdesc.htm (accessed on 29 March 

 29 2006). 



smoothing approaches. In the absence of additional 

 information regarding the level of observation noise, 

 we recommend smoothing trawl survey data at 909c of 

 the maximum possible noise reduction level. We also 

 suggest that development of an integrated software 

 package for implementing ARIMA-based noise reduction 

 will facilitate future use of this method. 



Finally, if a smoothed time series is desired (e.g., for 

 graphical presentation only), then use of a RWPUN 

 model in lieu of a model-fitting exercise will generally 

 yield a curve pleasing to the eye. Alternatively, other 

 methods such as LOESS could be employed to generate 

 the smoothed results. However, if the resulting time 

 series is to be used for further analysis of the dynami- 

 cal behavior of the fish stock, we strongly recommend 

 that a model-fitting approach be used to identify the 

 most appropriate ARIMA model for the observed time 

 series, from which the time series for the unobserved, 

 underlying process can be computed. Otherwise, real 

 fluctuations in the underlying process may be over- 

 smoothed, resulting in an apparent dynamical behavior 

 that displays little variability. This oversmoothing, in 

 turn, may lead to erroneous conclusions being drawn 

 regarding, for example, the resiliency of a stock to ex- 

 ploitation or environmental change, and to perhaps 

 concomitant errors being propagated in advice provided 

 to fishery managers. 



Acknowledgments 



We would like to thank M. Pennington for his insight- 

 ful comments on an early version of this manuscript. 

 The comments and suggestions from two anonymous 

 reviewers were also very helpful and are much appreci- 

 ated. This work was supported by the National Research 

 Council, which provided funding support for one of us 

 (Stockhausen) as a postdoctoral fellow at the Northeast 

 Fisheries Science Center. 



Literature cited 



Akaike, H. 



1973. Information theory as an extension of the maximum 

 likelihood principal. In Second international symposium 

 on information theory (B. N. Petrov, and F. Csaki, eds.), 

 267-281. Akademiai Kiado, Budapest, Hungary. 

 Anonymous. 



1988. An evaluation of the bottom trawl survey program 

 of the Northeast Fisheries Center. NOAA Tech. Memo. 

 NMFS-F/NEC-52, 83 p. Northeast Fish. Center. 166 

 Water St., Woods Hole, MA 02543. 



1993. Status of fishery resources off the northeastern 

 United States for 1993. NOAA Tech. Memo. NMFS- 

 F/NEC-101, 140 p. Northeast Fish. Center, 166 Water 

 St., Woods Hole, MA 02543. 

 Azarovitz, T. R. 



1981. A brief historical review of the Woods Hole Labora- 

 tory trawl survey time series. In Bottom trawl surveys 

 (W. G. Doubleday, and D. Rivard, eds.), 62-67. Can. 

 Spec. Pub. Fish. Aquat. Sci., vol. 58. 



