It’s About Time Mark Otto U. S. Fish and Wildlife Service

Introduction Analyzing data through time ARIMA models Regression and time series Trends and differences Interventions Structural models Time series on survey data

Monitoring Background data to assess environmental change Past data at the same site and/or control data at “similar” sites

Monitoring Background data to assess environmental change Past data at the same site and/or control data at “similar” sites Measure the right variables in the right places before the change

Time Series Data Consistently observed Usually equally spaces in time –Annually –Monthly –Daily … No missing observations

Time Series Data-2 Most population and habitat data taken over time

Time Series Data-2 Most population and habitat data taken over time Expect patterns and relationships Data not independent

Time Series Data-2 Most population and habitat data taken over time Expect patterns and relationships Data not independent OLS, ANOVA, GLIM One time series not equal to one observation Non-parametric

Time Series Analysis Data y, n observations n 2 covariance parameters

Time Series Analysis Data y, n observations n 2 covariance parameters Covariance of observation i steps same Less than n covariance parameters

Time Series Analysis-2 Relation of two variables: correlation Relation with of variable with itself i steps ago: autocorrelation Use autocorrelations to decide on model

First Order Autoregressive Data correlated: AR(1) y t = Φy t-1 + a t a t ~N(0,σ 2 a ) Φ(B)y t = a t Variance Var(y t )= σ 2 a /(1-Φ 2 ) Autocorrelations ρ=1, Φ, Φ 2, Φ 3, … Partial autocorrelations drop off

First Order Moving Average Errors correlated: MA(1) y t = a t - θa t-1 a t ~N(0,σ 2 a ) y t = θ (B) a t Variance Var(y t )= σ 2 a (1- θ 2 ) Autocorrelations ρ=1, -θ/(1- θ 2 ), 0, 0, … Partial autocorrelations decay

First Order Moving Average-2 Used in the stock market Made from running averages: early smoothing

ARIMA Models Data show how to model the series –AR: ACFs decay exponentially, PAFCs drop off –MA: ACFs drop off, PACFs decay exponentially Estimate model Check that the residuals are white noise

ARIMA Models Add more lags: MA(3), AR(2) Seasonal lags: Airline Model MA(1)(12) Mix AR and MA: ARMA(1,1) Data usually only support AR(1) or MA(1)

Stationarity Mean and variance constant

Stationarity Mean and variance constant Transform to stationarity –Box-Cox transform –Regression mean –Difference

Stationarity-2 Count data follows a Poisson Log is canonical transformation log(y t )-log(y t-1 )=c+a t Trend in the relative growth mean((y t - y t-1 )/ y t-1 )=e c -1

Regression and Time Series Regression describes the mean ARIMA model describes the variance Regression parameters unbiased Regression standard errors are Most variance explained by regression

Regression Examples Linear trend of logs-average growth Two points Three points Just linear?

Regression Examples Relate to environment –Linear regression –Nonlinear relation Adds explanatory power to model

Interventions Interventions (Box and Tiao) –Outliers Point Level Ramp –Interventions Pulse Level Shift Can model change by knowing its form

Structural Models ARIMA models: data decides the form Structural Models : structure decides the form –Trend –Seasonal –Irregular Use when little data but can assume structure

Time Series and Surveys Time points not just one observation Y t =Y t +e t e t ~N(0,v t ) Time points have mean and variance Could model the survey sample variance with a generalize variance function (GVF) and ARMA model

Periodic Survey Sample Design Point estimates: randomly select each time period Trend estimates: randomly select points and use each period Compromise: rotating panels survey

Time Series and Surveys-2 Statistics on statistics (Link, Bell and Hillmer, Binder) Φ(B)Δ(B)(Y t -x' t β)=u t u t ~N(0,σ 2 u ) Hierarchial model that separates the survey error from the process Estimates are a compromise between the survey estimates and the model

What’s the difference Estimate changes Δ(B)Y t = Y t - Y t-1 =u t u t ~N(0,σ 2 u ) Nonstationary, mean and or variance vary Cannot use linear prediction Use E(Y t |y t ) = y t - E(e t | Δ(B) y t ) Differencing tests not powerful

Multiple Series Measurement error y=Y+e e~N(0,V) c=α 0 +L(α 1 )Y+ε ε ~N(0,Σ) L(α 1 ) is a constraint matrix –Complete annual census –Sum of annual activity Benchmarking

Conclusions Use time series models to describe error Transform to stationarity Regression explains most of the variance Use habitat changes or interventions ARIMA vs. structural models (differencing) Separate survey and process Consider survey design Model relations between multiple series