Applied Time Series Analysis

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Applied Time Series Analysis

SS 2013 – Week 03

Marcel Dettling

Institute for Data Analysis and Process Design Zurich University of Applied Sciences

marcel.dettling@zhaw.ch http://stat.ethz.ch/~dettling

ETH Zürich, March 4, 2013

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Decomposition

Stationarity is key for statistical learning, but real data often

have trend/seasonality, and are non-stationary. We can (often) deal with that using the simple additive decomposition model:

= trend + seasonal effect + stationary remainder The goal is to find a remainder term , as a sequence of

correlated random variables with mean zero, i.e. a stationary ts.

We can employ: 1) taking differences (=differencing) 2) smoothing approaches (= filtering) 3) parametric models (= curve fitting)

2

Marcel Dettling, Zurich University of Applied Sciences

Applied Time Series Analysis

SS 2013 – Week 03

t t t t

X  m   s R

Rt

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Multiplicative Decomposition

is not always a good model:

Applied Time Series Analysis

SS 2013 – Week 03

t t t t

X  m  s R

Passenger Bookings

Time

Pax

1950 1952 1954 1956 1958 1960

100200300400500600

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Multiplicative Decomposition

Better: , respectively

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Applied Time Series Analysis

SS 2013 – Week 03

t t t t

X  m s R  log(X_t)  m_t  s_t R_t

Logged Passenger Bookings

Time

log(Pax)

1950 1952 1954 1956 1958 1960

5.05.56.06.5

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Applied Time Series Analysis

SS 2013 – Week 03

Smoothing, Filtering: Part 1

In the absence of a seasonal effect, the trend of a non-stationary time series can be determined by applying any additive, linear filter. We obtain a new time series , representing the trend:

- the window, defined by and , can or can‘t be symmetric - the weights, given by , can or can‘t be uniformly distributed - other smoothing procedures can be applied, too.

ˆ

q

t i t i

i p

m a X

_



 

ˆ

_t

m

a

i

p q

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Applied Time Series Analysis

SS 2013 – Week 03

Trend Estimation with the Running Mean

> trd <- filter(SwissTraffic, filter=c(1,1,1)/3)

Time

Index Value

1990 1995 2000 2005 2010

100110120130

Swiss Traffic Index with Running Mean

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Applied Time Series Analysis

SS 2013 – Week 03

Smoothing, Filtering: Part 2

In the presence a seasonal effect, smoothing approaches are still valid for estimating the trend. We have to make sure that the sum is taken over an entire season, i.e. for monthly data:

An estimate of the seasonal effect at time can be obtained by:

By averaging these estimates of the effects for each month, we obtain a single estimate of the effect for each month.

6 5 5 6

1 1 1

ˆ 7,..., 6

12 2 2

t t t t t

m   X _  X _   X _  X _  for t  n 

  

st t

ˆ_t _t ˆ_t s  x m

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Applied Time Series Analysis

SS 2013 – Week 03

Trend Estimation for Mauna Loa Data

> wghts <- c(.5,rep(1,11),.5)/12

> trd <- filter(co2, filter=wghts, sides=2)

Mauna Loa CO2 Concentrations

Time

co2

1960 1970 1980 1990

320330340350360

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Applied Time Series Analysis

SS 2013 – Week 03

Estimating the Seasonal Effects

2 4 6 8 10 12

-3-2-10123

Month

seasn.est

Seasonal Effects for Mauna Loa Data

38

1 13 12 1 12 1

0

ˆ ˆ ˆ ... 1 ( ˆ )

Jan 39 j j

j

s s s x _ m _



    





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Applied Time Series Analysis

SS 2013 – Week 03

Estimating the Remainder Term

ˆ_t _t ˆ_t ˆ_t R  x m  s

Estimated Stochastic Remainder Term

Time

rmain.est

1960 1970 1980 1990

-0.50.00.5

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Applied Time Series Analysis

SS 2013 – Week 03

Smoothing, Filtering: Part 3

• The smoothing approach is based on estimating the trend first, and then the seasonality.

• The generalization to other periods than , i.e. monthly data is straighforward. Just choose a symmetric window and use uniformly distributed coefficients that sum up to 1.

• The sum over all seasonal effects will be close to zero.

Usually, it is centered to be exactly there.

• This procedure is implemented in R with function:

decompose()

12 p 

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Applied Time Series Analysis

SS 2013 – Week 03

Estimating the Remainder Term

> plot(decompose(co2))

320340360

observed 320340360

trend -3-1123

seasonal -0.50.00.5

1960 1970 1980 1990

random

Time

Decomposition of additive time series

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Applied Time Series Analysis

SS 2013 – Week 03

Smoothing, Filtering: STL-Decomposition

The Seasonal-Trend Decomposition Procedure by Loess

• is an iterative, non-parametric smoothing algorithm

• yields a simultaneous estimation of trend and seasonal effect

 similar to what was presented above, but more robust!

+ very simple to apply

+ very illustrative and quick

+ seasonal effect can be constant or smoothly varying - model free, extrapolation and forecasting is difficult

 Good method for „having a quick look at the data“

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Applied Time Series Analysis

SS 2013 – Week 03

STL-Decomposition: Constant Season

stl(log(ts(airline,freq=12)),s.window=„periodic“)

5.05.56.06.5

data -0.2-0.10.00.10.2

seasonal 4.85.25.66.0

trend -0.10-0.050.000.05

2 4 6 8 10 12

remainder

time

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Applied Time Series Analysis

SS 2013 – Week 03

STL-Decomposition: Constant Season

stl(log(ts(airline,freq=12)),s.window=„periodic“)

the seasonal effect here is not time varying

erg$time.series[, 1]

J F M A M J J A S O N D

-0.2-0.10.00.10.2

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Applied Time Series Analysis

SS 2013 – Week 03

STL-Decomposition: Evolving Season

stl(log(ts(airline,freq=12)),s.window=15)

5.05.56.06.5

data -0.2-0.10.00.10.2

seasonal 4.85.25.66.0

trend -0.050.000.05

2 4 6 8 10 12

remainder

time

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Applied Time Series Analysis

SS 2013 – Week 03

STL-Decomposition: Evolving Season

correct amount of

smoothing on the time varying seasonal effect

-0.2-0.10.00.10.2

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Applied Time Series Analysis

SS 2013 – Week 03

STL-Decomposition: Evolving Season

5.05.56.06.5

data -0.2-0.10.00.10.2

seasonal 4.85.25.66.0

trend -0.06-0.020.020.06

2 4 6 8 10 12

remainder

time

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Applied Time Series Analysis

SS 2013 – Week 03

STL-Decomposition: Evolving Season

-0.2-0.10.00.10.2

Monthplot

not enough smoothing

on the time varying

seasonal effect

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Applied Time Series Analysis

SS 2013 – Week 03

Smoothing, Filtering: Remarks

Some advantages and disadvantages:

+ trend and seasonal effect can be estimated

+ , and are explicitly known, can be visualised + procedure is transparent, and simple to implement - resulting time series will be shorter than the original - the running mean is not the very best smoother

- extrapolation of , are not entirely obvious ˆ_t

m sˆ_t

ˆ_t m sˆ_t ˆt

R

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Applied Time Series Analysis

SS 2013 – Week 03

Parametric Modelling

When to use?

 Parametric modelling is often used if we have previous knowledge about the trend following a functional form.

 If the main goal of the analysis is forecasting, a trend in functional form may allow for easier extrapolation than a trend obtained via smoothing.

 It can also be useful if we have a specific model in mind and want to infer it. Caution: correlated errors!

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Applied Time Series Analysis

SS 2013 – Week 03

Parametric Modelling: Example

Maine unemployment data: Jan/1996 – Aug/2006

Unemployment in Maine

Time

(%)

1996 1998 2000 2002 2004 2006

3456

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Applied Time Series Analysis

SS 2013 – Week 03

Modeling the Unemployment Data

Most often, time series are parametrically decomposed by using regression models. For the trend, polynomial functions are widely used, whereas the seasonal effect is modelled with dummy

variables (= a factor).

where

Remark: choice of the polynomial degree is crucial!

2 3 4

0 1 2 3 4 ( )

t i t t

X       t    t    t    t   E

 

1, 2,...,128 ( ) 1, 2,...,12 t

i t



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Applied Time Series Analysis

SS 2013 – Week 03

Polynomial Order / OLS Fitting

Estimation of the coefficients will be done in a regression con- text. We can use the ordinary least squares algorithm, but:

• we have violated assumptions, is not uncorrelated

• the estimated coefficients are still unbiased

• standard errors (tests, CIs) can be wrong Which polynomial order is required?

Eyeballing allows to determine the minimum grade that is required for the polynomial. It is at least the number of maxima the hypothesized trend has, plus one.

E

t

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Applied Time Series Analysis

SS 2013 – Week 03

Important Hints for Fitting

• The main predictor used in polynomial parametric modeling is the time of the observations. It can be obtained by typing time(maine).

• For avoiding numerical and collinearity problems, it is essential to center the time/predictors!

• R sets the first factor level to 0, seasonality is thus expressed as surplus to the January value.

• For visualization: when the trend must fit the data, we have to adjust, because the mean for the seasonal effect is

usually different from zero!

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Applied Time Series Analysis

SS 2013 – Week 03

Trend of O(4), O(5) and O(6)

Unemployment in Maine

Time

(%)

1996 1998 2000 2002 2004 2006

3456

O(4) O(5) O(6)

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Applied Time Series Analysis

SS 2013 – Week 03

Residual Analysis: O(4)

Residuals vs. Time, O(4)

Time

1996 1998 2000 2002 2004 2006

-0.6-0.20.20.6

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Applied Time Series Analysis

SS 2013 – Week 03

Residual Analysis: O(5)

Time

1996 1998 2000 2002 2004 2006

-0.6-0.20.20.6

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Applied Time Series Analysis

SS 2013 – Week 03

Residual Analysis: O(6)

Time

1996 1998 2000 2002 2004 2006

-0.4-0.20.00.20.4

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Applied Time Series Analysis

SS 2013 – Week 03

Parametric Modeling: Remarks

Some advantages and disadvantages:

+ trend and seasonal effect can be estimated

+ and are explicitly known, can be visualised + even some inference on trend/season is possible + time series keeps the original length

- choice of a/the correct model is necessary/difficult - residuals are correlated: this is a model violation!

- extrapolation of , are not entirely obvious ˆ_t

m sˆ_t

ˆ_t m sˆ_t

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Applied Time Series Analysis

SS 2013 – Week 03

Where are we?

For most of the rest of this course, we will deal with (weakly) stationary time series. They have the following properties:

•

If a time series is non-stationary, we know how to decompose into deterministic and stationary, random part.

Our forthcoming goals are:

- understanding the dependency in a stationary series - modeling this dependency and generate forecasts

[ _t]

E X   ( _t) 2

Var X 

( _t, _{t h}) _h Cov X X _  

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Applied Time Series Analysis

SS 2013 – Week 03

Autocorrelation

The aim of this section is to explore the dependency structure within a time series.

Def: Autocorrelation

The autocorrelation is a dimensionless measure for the

amount of linear association between the random variables collinearity between the random variables and .

( , )

( ) ( )

t k t

Cov X X Cor X X

Var X Var X

 



 

X

t k_

X

_t

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Applied Time Series Analysis

SS 2013 – Week 03

Autocorrelation Estimation

Our next goal is to estimate the autocorrelation function (acf) from a realization of weakly stationary time series.

Luteinizing Hormone in Blood at 10min Intervals

Time

lh

0 10 20 30 40

1.52.02.53.03.5-0.20.20.61.0

ACF

Autocorrelation Function

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Applied Time Series Analysis

SS 2013 – Week 03

Autocorrelation Estimation: lag k>1

Idea 1: Compute the sample correlation for all pairs

( , x x

_s _{s k}_

)

1.5 2.0 2.5 3.0 3.5

1.52.02.53.03.5

X_s

X_{s+2}

k=2, cor=0.19

1.5 2.0 2.5 3.0 3.5

1.52.02.53.03.5

X_s

X_{s+3}

k=3, cor=-0.15

1.5 2.0 2.5 3.0 3.5

1.52.02.53.03.5

X_s

X_{s+4}

k=4, cor=-0.19

1.5 2.0 2.5 3.0 3.5

1.52.02.53.03.5

X_s

X_{s+5}

k=5, cor=-0.16

1.5 2.0 2.5 3.0 3.5

1.52.02.53.03.5

X_s

X_{s+6}

k=6, cor=-0.02

1.5 2.0 2.5 3.0 3.5

1.52.02.53.03.5

X_s

X_{s+7}

k=7, cor=-0.01

1.5 2.0 2.5 3.0

1.52.02.53.03.5

X_s

X_{s+8}

k=8, cor=0.01

1.5 2.0 2.5 3.0

1.52.02.53.03.5

X_s

X_{s+9}

k=9, cor=-0.17

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Applied Time Series Analysis

SS 2013 – Week 03

Autocorrelation Estimation: lag k

Idea 2: Plug-in estimate with sample covariance How does it work?

 see blackboard…

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Applied Time Series Analysis

SS 2013 – Week 03

Autocorrelation Estimation: lag k

Idea 2: Plug-in estimate with sample covariance

where

and

1

ˆ( ) 1 ( )( )

n k

s k s

s

k x x x x



n ^ _





 

1

1 ⁿ

t t

x x

n _





Standard approach in time series analysis for computing the acf

ˆ( ) ( , )

ˆ ( )

ˆ(0) ( )

t t k

t

Cov X X k k

Var X

 



^

 

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Applied Time Series Analysis

SS 2013 – Week 03

Comparison Idea 1 vs. Idea 2

 see blackboard for some more information

0 10 20 30 40

-1.0-0.50.00.51.0

acf

Comparison between lagged sample correlations and acf

acf

lagged sample correlations

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Applied Time Series Analysis

SS 2013 – Week 03

What is important about ACF estimation?

- Correlations are never to be trusted without a visual inspection with a scatterplot.

- The bigger the lag k, the fewer data pairs remain for estimating the acf at lag k.

- Rule of the thumb: the acf is only meaningful up to about a) lag 10*log₁₀(n)

b) lag n/4

- The estimated sample ACs can be highly correlated.

- The correlogram is only meaningful for stationary series!!!

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Applied Time Series Analysis

SS 2013 – Week 03

Correlogram

A useful aid in interpreting a set of autocorrelation coefficients is the graph called correlogram, where the are plotted

against the lag k.

Interpreting the meaning of a set of autocorrelation coefficients is not always easy. The following slides offer some advice.

ˆ ( )k



0 5 10 15

-0.20.20.61.0

ACF

Series lh

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Applied Time Series Analysis

SS 2013 – Week 03

Random Series – Confidence Bands

If a time series is completely random, i.e. consists of i.i.d. random variables , the (theoretical) autocorrelations are equal to 0.

However, the estimated are not. We thus need to decide, whether an observed is significantly so, or just appeared by chance. This is the idea behind the confidence bands.

ˆ ( )k



0 5 10 15

-0.20.20.61.0

Lag

ACF

Series lh

Xt



^{( )}^k

ˆ ( )k 0





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Applied Time Series Analysis

SS 2013 – Week 03

Random Series – Confidence Bands

For long i.i.d. time series, it can be shown that the are approximately distributed.

Thus, if a series is random, 95% of the estimated can be expected to lie within the interval

ˆ ( )k



ˆ ( )k

 2 / n



0 5 10 15 20

0.00.40.8

ACF

i.i.d. Series with n=300



^{0,1 /}



N n

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Applied Time Series Analysis

SS 2013 – Week 03

Random Series – Confidence Bands

Thus, even for a (long) i.i.d. time series, we expect that 5% of the estimated autocorrelation coeffcients exceed the confidence

bounds. They correspond to type I errors.

Note: the probabilistic properties of non-normal i.i.d series are much more difficult to derive.

0 5 10 15 20

0.00.40.8

Lag

ACF

i.i.d. Series with n=300