Post on 03-Jan-2016
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
• 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
• n2 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)yt = Φyt-1 + at at~N(0,σ2
a)
Φ(B)yt = at
• Variance Var(yt)= σ2a/(1-Φ2)
• Autocorrelations ρ=1, Φ, Φ2 , Φ3 , …
• Partial autocorrelations drop off
First Order Moving Average
• Errors correlated: MA(1)yt = at - θat-1 at ~N(0,σ2
a)
yt = θ (B) at
• Variance Var(yt)= 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
• Transform to stationarity– Box-Cox transform– Regression mean– Difference
Stationarity-2
• Count data follows a Poisson
• Log is canonical transformationlog(yt)-log(yt-1)=c+at
• Trend in the relative growthmean((yt- yt-1)/ yt-1 )=ec-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 Yt=Yt+et et~N(0,vt)
• 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)(Yt-x'tβ)=ut ut~N(0,σ2
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)Yt= Yt- Yt-1=ut ut~N(0,σ2
u)
• Nonstationary, mean and or variance vary
• Cannot use linear prediction
• Use E(Yt |yt) = yt- E(et | Δ(B) yt)
• Differencing tests not powerful
Multiple Series
• Measurement error y=Y+e e~N(0,V)
c=α0+L(α1)Y+ε ε ~N(0,Σ)
• L(α1) is a constraint matrix
– Complete annual census– Sum of annual activity
• Benchmarking