# Presentation: 281252

## 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,σ2a)
- 	 Φ(B)y*t* = a*t*
- Variance  Var(y*t*)= σ2a/(1-Φ2)
- Autocorrelations ρ=1, Φ, Φ2 , Φ3 , ...
- Partial autocorrelations drop off

## First Order Moving Average

- Errors correlated: MA(1)
- y*t* = a*t* - θa*t-1        *a*t* ~N(0,σ2a)
- 	 y*t* = θ (B) a*t*
- Variance  Var(y*t*)= σ2a(1- θ2)
- Autocorrelations ρ=1, -θ/(1- θ2), 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
-     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,σ2u)
- 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,σ2u)
- 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