Time Series Toolbox
Birgit Erni
25 April 2019
Click on the following for the slides, code and video of this seminar.
Outline: Time Series in Practice
 When and why do we need time series models?
 Basic models and definitions: white noise, AR1, MA, random walk, stationarity.
 3 approaches to time series modelling: ARIMA, Regression, Structural time series / statespace models
understand basic difficulties with time series, construct a few simple but useful models
References

Hyndman, R.J., & Athanasopoulos, G. (2018) Forecasting: Principles and Practice, 2nd edition, OTexts: Melbourne, Australia. https://otexts.com/fpp2

Applied Time Series Analysis for Fisheries and Environmental Sciences. E. E. Holmes, M. D. Scheuerell, and E. J. Ward. (2019). https://nwfsctimeseries.github.io/atsalabs/index.html
More technical:
 Shumway, R.H. and Stoffer, D.S., 2017. Time series analysis and its applications: with R examples. Springer. https://www.stat.pitt.edu/stoffer/tsa4/
Motivating Example: Mauna Loa Atmospheric CO2 Concentration
Shark Attacks in Florida
Source: R package bsts (Bayesian structural time series)
Financial
fpp2 is the package for the book by Hyndman: Forecasting: Principles and Practice, 2nd edition.
Sunspot Area
Electricity Production
Treerings
Electricity Demand
Goals of Time Series Analysis
 prediction / forecast
 impact of single event
 study causal patterns
 detect trends, changes, shifts (in mean, seasonality, variance)
Time Series in R
Time Series in R
Create time series of quarterly data:
ts(rnorm(40), frequency = 4, start = c(2019, 2))
When and why do we need time series models?
When there is autocorrelation in the residuals (after modelling trends, seasonality, effects of explanatory variables).
If we ignore autocorrelation:

standard error estimates are wrong

predictions and prediction intervals are wrong
Terminology / Definitions
time series: \(y_1, ..., y_t\)
autocorrelation: correlation with previous values,
Basic time series processes
AR process (autoregressive)
\[\mbox{AR(p):} \qquad x_t = \phi_1 x_{t1} + \phi_2 x_{t2} + ... + \phi_p x_{tp} + e_t\]
\[\mbox{AR1:} \qquad x_t = \phi x_{t1} + e_t\]
Why would anything behave like this?
I see \(x_{t1}\) as a measure of everything that was not measured explicitly at previous time step.
Random walk
\[x_t = x_{t1} + e_t\]
AR1 processes, different \(\phi\)
Stationarity
 mean, variance, correlations stay constant over time
Why is stationarity important?
There is a single observation per time point. If mean and variance are different for every point, we canâ€™t estimate mean and variance, correlation or model parameters.
AR1 processes are stationary if \(\phi < 1\).
nonstationarity means
mean changes, variance changes, seasonality present, correlation changes
MA(q) process (moving average)
\[y_t = \theta_1 e_{t1} + \theta_2 e_{t2} + \ldots + \theta_p e_{tp} + e_t\]
sum of previous shocks / events
White noise
identically, independently distributed, mean 0, no autocorrelation
ACF and PACF
autocorrelation function, partial autocorrelation function
partical autocorrelation is the correlation between \(z_t\) and \(z_{t+k}\) that is not accounted for by lags 1 to k.
Three approaches to time series modelling
1. ARIMA, very briefly
arima(p, d, q)
p = order of autoregressive process, d = difference order, q = order of moving average process
If \(y_t\) is not stationary then \(y_t  y_{t1}\) sometimes is (first order differences).
ARIMA: autoregressive, integrated, moving average
â€˜integratedâ€™ refers to differencing
Three approaches to time series modelling
2. Regression
Ignore or model autocorrelation in errors like this:
\[y_t = \beta_0 + \beta_1 x_t + \nu_t\]
\[\nu_t = \phi \nu_{t1} + e_t\]
Not this:
\[y_t = \beta_0 + \beta_1 x_t + \phi y_{t1} + e_t\]
problem: \(\beta_1\) is not change in response per unit change in \(x\)
Simulate some data with autocorrelation, fit regression model
acf2
comes from package astsa
, and skips the autocorrelation at lag 1.
ARMA errors
arima
with xreg
models autocorrelation in errors
a1 < arima(y, order = c(0, 0, 0), xreg = x)
a2 < arima(y, order = c(1, 0, 0), xreg = x)
##
## Call:
## arima(x = y, order = c(1, 0, 0), xreg = x)
##
## Coefficients:
## ar1 intercept x
## 0.8238 2.1524 1.9644
## s.e. 0.0544 0.5223 0.0640
##
## sigma^2 estimated as 0.5757: log likelihood = 114.85, aic = 237.7
Compare to true values: \(\beta_0 = 3, \beta_1 = 2, \phi = 0.9, \sigma^2 = 0.64\)
Regression without ARMA errors
##
## Call:
## arima(x = y, order = c(0, 0, 0), xreg = x)
##
## Coefficients:
## intercept x
## 1.4424 1.8305
## s.e. 0.7726 0.1494
##
## sigma^2 estimated as 1.842: log likelihood = 172.43, aic = 350.86
Coefficient estimates are further from true values.