Merge pull request #129 from NOAA-EDAB/andy_edit_lttrend

remove quadratic trend

chapters/Trend_analysis.Rmd

@@ -21,9 +21,9 @@ Summarizing trends for ecosystem indicators is desirable, but the power of stati
 
 In a simulation study [@hardison2019], we explored the effect of time series length and autocorrelation strength on statistical power of three trend detection methods: a generalized least squares model selection approach, the Mann-Kendall test, and Mann-Kendall test with trend-free pre-whitening. Methods were applied to simulated time series of varying trend and autocorrelation strengths. Overall, when sample size was low (N = 10) there were high rates of false trend detection, and similarly, low rates of true trend detection. Both of these forms of error were further amplified by autocorrelation in the trend residuals. Based on these findings, we selected a minimum series length of N = 30 for indicator time series before assessing trend.
 
-We also chose to use a GLS model selection (GLS-MS) approach to evaluate indicator trends in the 2018 (and future) State of the Ecosystem reports, as this approach performed best overall in the simulation study. GLS-MS also allowed for both linear and quadratic model fits and quantification of uncertainty in trend estimates. The model selection procedure for the GLS approach fits four models to each time series and selects the best fitting model using AICc. The models are, 1) linear trend with uncorrelated residuals, 2) linear trend with correlated residuals, 3) quadratic trend with uncorrelated residuals, and 4) quadratic trend with correlated residuals. I.e., the models are of the form
+We also chose to use a GLS model selection (GLS-MS) approach to evaluate indicator trends in the 2018 (and future) State of the Ecosystem reports, as this approach performed best overall in the simulation study. GLS-MS allowed for linear model fits and quantification of uncertainty in trend estimates. The model selection procedure for the GLS approach fits two models to each time series and selects the best fitting model using AICc. The models are, 1) linear trend with uncorrelated residuals, 2) linear trend with correlated residuals. I.e., the models are of the form
 
-$$ Y_t = \alpha_0 + \alpha_1X_t + \alpha_2X_t^2 + \epsilon_t$$
+$$ Y_t = \alpha_0 + \alpha_1X_t +  \epsilon_t$$
 $$\epsilon_t = \rho\epsilon_{t-1} + \omega_t$$
 
 $$w_t \sim N(0, \sigma^2)$$