Time Series

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Lesson 1: What is a time series

Umberto Triacca

Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica Universit`a dell’Aquila,

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Time series

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Time series

A time series is a set of observations on a

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Time series

If we denote withT the set of time points at which observations are made,

with xt the observation made at timet, then the time

series is represented by the set

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Time series

If we denote withT the set of time points at which observations are made, withxt the observation made at timet,

then the time series is represented by the set

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Time series

If we denote withT the set of time points at which observations are made, withxt the observation made at timet, then the time

series is represented by the set

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Time series

No restriction is placed on the nature of T .

However, two important cases are when

T is a finite set of time points; T is a closed time interval.

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Time series

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

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Time series

T is a finite set of time points;

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Time series

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Time series

Discrete time series

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Time series

Discrete time series Continuous time series

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Time series

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

Continuous time series. A continuous time series is obtained when the observations are made continuously over some closed time interval.

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Time series

Discrete time series. A discrete time series is one in which the set of times at which observations are made is a finite set. Continuous time series. A continuous time series is obtained when the observations are made continuously over some closed time interval.

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Time series

Consider a discrete time series.

Let the times at which observations are made be

t1, t2, ..., tT satisfying

t1< t2< ... < tT.

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

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Time series

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

t1, t2, ..., tT

satisfying

t1< t2< ... < tT.

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Time series

t1< t2< ... < tT.

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Time series

t1< t2< ... < tT. If the time points are equally spaced

(i.e. ti +1− ti = ∆ for all i = 1, ..., T − 1, where ∆ > 0 is a constant), we call the time series regularly sampled. Otherwise, the sequence of observations is called irregularly sampled time series.

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Time series

t1< t2< ... < tT.

If the time points are equally spaced (i.e. ti +1− ti = ∆ for all i = 1, ..., T − 1, where ∆ > 0 is a constant),

we call the time series regularly sampled. Otherwise, the sequence of observations is called irregularly sampled time series.

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Time series

t1< t2< ... < tT.

If the time points are equally spaced (i.e. ti +1− ti = ∆ for all i = 1, ..., T − 1, where ∆ > 0 is a constant), we call the time series regularly sampled.

Otherwise, the sequence of observations is called irregularly sampled time series.

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Time series

t1< t2< ... < tT.

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Time series

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

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

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Time series

When we consider a discrete regularly sampled time series, the observation dates (periods) are denoted by integers t = 1, 2, ..., T Thus a discrete regularly sampled time series on one variable x with T observations, will be represented by the sequence

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Time series

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Time series

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

the sampling frequency

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

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Time series

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Time series

Series can be recorded

annually, quarterly, monthly, weekly, daily, ..., every minutes.

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Time series

Series can be recorded annually,

quarterly, monthly, weekly, daily, ..., every minutes.

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Time series

Series can be recorded annually, quarterly,

monthly, weekly, daily, ..., every minutes.

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Time series

Series can be recorded annually, quarterly, monthly,

weekly, daily, ..., every minutes.

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Time series

Series can be recorded annually, quarterly, monthly, weekly,

daily, ..., every minutes.

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Time series

Series can be recorded annually, quarterly, monthly, weekly, daily,

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Time series

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

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Time series

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Time series

In economics, the most common frequencies are:

annual, quarterly, monthly, daily

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Time series

In economics, the most common frequencies are: annual,

quarterly, monthly, daily

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Time series

quarterly,

monthly, daily

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Time series

quarterly, monthly,

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Time series

quarterly, monthly, daily

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Time series

Thus we can have annual, quarterly, monthly, weekly, daily time series.

the GDP in a country in successive years

the money supply in a country in successive quarters the number of international airline passengers in successive months

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Time series

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Time series

the money supply in a country in successive quarters

the number of international airline passengers in successive months

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Time series

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Time series

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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 observation values on the y -axis and the corresponding dates on the x -axis.

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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 observation values on the y -axis and the corresponding dates on the x -axis.

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The time series graph

Consider, for example, the monthly time series of the international airline passengers Jan 49 -ˆa Dec 60.

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The time series graph

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

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

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The time series graph

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

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The time series graph

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Time series

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Time series

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Time series

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Time series

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The time series graph

By plotting data as a function of time, we can gain insight into several 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 less}

stable year after year with respect to timing, direction and magnitude;

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 the

rest of the data;

5 Turning points: different trends within a series;

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The time series graph

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

2 _{Seasonality: an intra-year fluctuation that is more or less}

rest of the data;

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The time series graph

rest of the data;

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The time series graph

rest of the data;

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The time series graph

rest of the data;

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The time series graph

rest of the data;

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The time series graph

rest of the data;

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The time series graph

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

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The time series graph

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

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

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A cyclic component is present in the following time series.

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The time series graph

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

Direct inspection of figure suggests the series of daily returns of the Dow Jones Industrials index contains some outliers.

In particular, the crash of October 19, 1987.

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The time series graph

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

Direct inspection of figure suggests the series of daily returns of the Dow Jones Industrials index contains some outliers. In

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The time series graph

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

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The time series graph

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

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The time series graph

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

bigskip

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The time series graph

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

bigskip

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Time series

A question:

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Time series

A question:

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Time series

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

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Time series

In order to reach this goal we need of a model of the ”mechanism” that has generated the series.

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Time series

In order to reach this goal we need of a model of the ”mechanism” that has generated the series.