Fraser River sockeye salmon: data synthesis and cumulative impacts

Change point analysis of productivity dataDuring the initial phases of this project, it was frequently stated that there had been declines inproductivity of Fraser River sockeye since the early 1990’s. Peterman and Dorner (2011)explored the patterns of productivity in far more detail and now suggest there were in fact 3periods of change. In parallel, we decided it was important to have an objective assessment ofwhen and how strong trends in each of the sockeye stocks were; hypothesizing that there may besubstantial differences among stocks in the timing and strength of declines in productivity. Ifthere were groups of stocks with similar behaviour over time, the similarities among thoseparticular stocks might suggest important factors. We completed these analyses very early in theproject as these data were available long before any of the covariate data from the other technicalreports.Where time-series data are available, a regression model between time and the data set of interestcan be fit to assess whether or not there is a trend over time and to describe the nature of thetrend. Straight line models are easy to interpret as a trend can be estimated by the slope of thestraight line fit to the data (Equation A3.5-1). We fit a straight line model to the log e transformeddata for: a) the entire time series or b) segments of the time-series. A log e transformation is oftenused to linearize exponential growth or stabilize variance typical of population data (Dixon &Pechmann 2005). In many cases it may not be appropriate to fit a single trend line across theentire time-series. There may be periods of either increasing or decreasing trend that do notextend throughout the data set. In particular, it is of interest to understand what the current trendis and when that trend began. Where a single trend line is not appropriate we try fitting a piecewiseregression model where two lines are joined at a single sharp change point (Equation A3.5-2) (Toms & Lesperance 2003). All possible change points are evaluated and the most likely one(i.e., minimizing the sums of squares (SS) of the residuals) based on the data is selected. Morecomplex models are possible (e.g., more than two lines, curved segments rather than straightlines), but are beyond the scope possible for this project given the time and budget. Morecomplex time-series methods that incorporate temporal autocorrelation are also beyond the scopeof this project.Straight linemodel:yt=0+ β 1xtεt(Equation A3.5-1)β +This model fits a single straight line of best fit through the entire timeseries of data.• β1represents the slope of the line of best fitPiece-wiseyt=0+ β 1xtεtfor xt≤ αβ +204