Lesson 1: What is a time series - Scuola Superiore Sant'Anna · 2014-02-14 · Lesson 1: What is a time series Umberto Triacca Dipartimento di Ingegneria e Scienze dell’Informazione

Lesson 1: What is a time series

Umberto Triacca

Dipartimento di Ingegneria e Scienze dell’Informazione e MatematicaUniversita dell’Aquila,

[email protected]

Umberto Triacca Lesson 1: What is a time series

Time series

What is a time series?


Time series

A time series is a set of observations on agiven variable x collected over the time.


Time series

If we denote with T the set of time points at which observationsare made, with xt the observation made at time t, then the timeseries is represented by the set

{xt ; t ∈ T }


Time series

No restriction is placed on the nature of T . However, twoimportant cases are when

T is a finite set of time points;

T is a closed time interval.


Time series

Discrete time series

Continuous time series


Time series

Discrete time series. A discrete time series is one in which theset of times at which observations are made is a finite set.

Continuous time series. A continuous time series is obtainedwhen the observations are made continuously over someclosed time interval.


Time series

Consider a discrete time series. Let the times at whichobservations are made be

t1, t2, ..., tT

satisfyingt1 < t2 < ... < tT .

If the time points are equally spaced (i.e. ti+1 − ti = ∆ for alli = 1, ...,T − 1, where ∆ > 0 is a constant), we call the timeseries regularly sampled. Otherwise, the sequence of observationsis called irregularly sampled time series.


Time series

When we consider a discrete regularly sampled time series, theobservation dates (periods) are denoted by integers t = 1, 2, ...,T

Thus a discrete regularly sampled time series on one variable xwith T observations, will be represented by the sequence

{x1, x2, ..., xT}


Time series

We will consider only discrete regularly sampled time series


Time series

An important feature of time series data is the frequency at whichthe data are collected:

the sampling frequency

Series can be recorded annually, quarterly, monthly, weekly, daily,..., every minutes.


Time series

In economics, the most common frequencies are:

annual,

quarterly,

monthly,

daily


Time series

Thus we can have annual, quarterly, monthly, weekly, daily timeseries.

the GDP in a country in successive years

the money supply in a country in successive quarters

the number of international airline passengers in successivemonths

the returns of IBM stock on successive days.


The time series graph

A important tool in time series analysis is the time series graph.

A time series graph is a scatter plot with the observationvalues on the y-axis and the corresponding dates on thex-axis.



Consider, for example, the monthly time series of the internationalairline passengers Jan 49 -a Dec 60.

Figure : International airline passengers: monthly totals in thousands.Jan 49 - Dec 60



However, it is often convenient to join up consecutive points intime as in the following figure.

Figure : International airline passengers: monthly totals in thousands.Jan 49 - Dec 60

This graph is called a time series plot.


Time series

Figure : 1


Time series

Figure : 2


Time series

Figure : 3


Time series

Figure : 4



By plotting data as a function of time, we can gain insight intoseveral data features. These features include:

1 Trend: a tendency, a long term change in the mean level;

2 Seasonality: an intra-year fluctuation that is more or lessstable year after year with respect to timing, direction andmagnitude;

3 Cycle: an inter-year periodic fluctuation around the trend,revealing a succession of phases of expansion and contraction;

4 Outliers: values that do not appear to be consistent with therest of the data;

5 Turning points: different trends within a series;

6 Discontinuities: an unexpected shift, a sudden change of level.



An increasing trend is apparent in the number of divorces in Italy

Figure : Annual number of divorces in Italy from 1995 to 2007



A clear seasonal component is contained in the series presented inthe following Figure.

Figure : Industrial production - total industry (excluding construction) inItaly, Index (2005 = 100) from January 1990 to February 2010.


A cyclic component is present in the following time series.

Figure : Sunspots Number. July 1969 to September 2007.



Figure : Daily returns of the Dow Jones Industrials index from 80/01/02to 89/12/29

Direct inspection of figure suggests the series of daily returns ofthe Dow Jones Industrials index contains some outliers. Inparticular, the crash of October 19, 1987.



A turning point is present in the unemployment rate in Italy.

Figure : Quarterly unemployment rate in Italy 2000.1-2009.4



The annual U.S. price index for gasoline is shown in this Figure. Itappears from the graph the presence of a structural break in thedata.

Figure : Annual U.S. price index for gasoline from 1960 to 1995



In summary, a time series plot is a valuable tool used in order toidentify patterns in the data.bigskip

A good picture is better than 1000 words


Time series

A question:

Why we want to study a time series?


Time series

The main objective for analyse a time series is to forecast itsfuture values


Time series

In order to reach this goal we need of a model of the ”mechanism”that has generated the series.A model of the so-called Data Generator Process (DGP).


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Lesson 1: What is a time series - Scuola Superiore Sant'Anna · 2014-02-14 · Lesson 1: What is a time series Umberto Triacca Dipartimento di Ingegneria e Scienze dell’Informazione