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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Umberto Triacca
Lesson 1: What is a time series
Time series
What is a time series?
Umberto Triacca
Lesson 1: What is a time series
Time series
A time series is a set of observations on a given variable x collected over the time.
Umberto Triacca
Lesson 1: What is a time series
Time series
If we denote with T the set of time points at which observations are made,
Umberto Triacca
Lesson 1: What is a time series
Time series
If we denote with T the set of time points at which observations are made, with xt the observation made at time t,
Umberto Triacca
Lesson 1: What is a time series
Time series
If we denote with T the set of time points at which observations are made, with xt the observation made at time t, then the time series is represented by the set
{xt ; t ∈ T }
Umberto Triacca
Lesson 1: What is a time series
Time series
No restriction is placed on the nature of T .
Umberto Triacca
Lesson 1: What is a time series
Time series
No restriction is placed on the nature of T . However, two important cases are when
Umberto Triacca
Lesson 1: What is a time series
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;
Umberto Triacca
Lesson 1: What is a time series
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.
Umberto Triacca
Lesson 1: What is a time series
Time series
Discrete time series
Umberto Triacca
Lesson 1: What is a time series
Time series
Discrete time series Continuous time series
Umberto Triacca
Lesson 1: What is a time series
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.
Umberto Triacca
Lesson 1: What is a time series
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.
Umberto Triacca
Lesson 1: What is a time series
Time series
Consider a discrete time series.
Umberto Triacca
Lesson 1: What is a time series
Time series
Consider a discrete time series. Let the times at which observations are made be t1 , t2 , ..., tT
Umberto Triacca
Lesson 1: What is a time series
Time series
Consider a discrete time series. Let the times at which observations are made be t1 , t2 , ..., tT satisfying t1 < t2 < ... < tT .
Umberto Triacca
Lesson 1: What is a time series
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
Umberto Triacca
Lesson 1: What is a time series
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),
Umberto Triacca
Lesson 1: What is a time series
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.
Umberto Triacca
Lesson 1: What is a time series
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.
Umberto Triacca
Lesson 1: What is a time series
Time series
When we consider a discrete regularly sampled time series, the observation dates (periods) are denoted by integers t = 1, 2, ..., T
Umberto Triacca
Lesson 1: What is a time series
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 {x1 , x2 , ..., xT }
Umberto Triacca
Lesson 1: What is a time series
Time series
We will consider only discrete regularly sampled time series
Umberto Triacca
Lesson 1: What is a time series
Time series
An important feature of time series data is the frequency at which the data are collected:
Umberto Triacca
Lesson 1: What is a time series
Time series
An important feature of time series data is the frequency at which the data are collected: the sampling frequency
Umberto Triacca
Lesson 1: What is a time series
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
Umberto Triacca
Lesson 1: What is a time series
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,
Umberto Triacca
Lesson 1: What is a time series
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,
Umberto Triacca
Lesson 1: What is a time series
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,
Umberto Triacca
Lesson 1: What is a time series
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,
Umberto Triacca
Lesson 1: What is a time series
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,
Umberto Triacca
Lesson 1: What is a time series
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, ...,
Umberto Triacca
Lesson 1: What is a time series
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.
Umberto Triacca
Lesson 1: What is a time series
Time series
In economics, the most common frequencies are:
Umberto Triacca
Lesson 1: What is a time series
Time series
In economics, the most common frequencies are: annual,
Umberto Triacca
Lesson 1: What is a time series
Time series
In economics, the most common frequencies are: annual, quarterly,
Umberto Triacca
Lesson 1: What is a time series
Time series
In economics, the most common frequencies are: annual, quarterly, monthly,
Umberto Triacca
Lesson 1: What is a time series
Time series
In economics, the most common frequencies are: annual, quarterly, monthly, daily
Umberto Triacca
Lesson 1: What is a time series
Time series
Thus we can have annual, quarterly, monthly, weekly, daily time series.
Umberto Triacca
Lesson 1: What is a time series
Time series
Thus we can have annual, quarterly, monthly, weekly, daily time series. the GDP in a country in successive years
Umberto Triacca
Lesson 1: What is a time series
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
Umberto Triacca
Lesson 1: What is a time series
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
Umberto Triacca
Lesson 1: What is a time series
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 the returns of IBM stock on successive days.
Umberto Triacca
Lesson 1: What is a time series
The time series graph
A important tool in time series analysis is the time series graph.
Umberto Triacca
Lesson 1: What is a time series
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.
Umberto Triacca
Lesson 1: What is a time series
The time series graph Consider, for example, the monthly time series of the international airline passengers Jan 49 -ˆa Dec 60.
Figure : International airline passengers: monthly totals in thousands. Jan 49 - Dec 60
Umberto Triacca
Lesson 1: What is a time series
The time series graph However, it is often convenient to join up consecutive points in time as in the following figure.
Umberto Triacca
Lesson 1: What is a time series
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
Umberto Triacca
Lesson 1: What is a time series
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
This graph is called a time series plot. Umberto Triacca
Lesson 1: What is a time series
Time series
Figure : 1
Umberto Triacca
Lesson 1: What is a time series
Time series
Figure : 2
Umberto Triacca
Lesson 1: What is a time series
Time series
Figure : 3
Umberto Triacca
Lesson 1: What is a time series
Time series
Figure : 4
Umberto Triacca
Lesson 1: What is a time series
The time series graph By plotting data as a function of time, we can gain insight into several data features. These features include:
Umberto Triacca
Lesson 1: What is a time series
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;
Umberto Triacca
Lesson 1: What is a time series
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;
Umberto Triacca
Lesson 1: What is a time series
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;
Umberto Triacca
Lesson 1: What is a time series
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;
Umberto Triacca
Lesson 1: What is a time series
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;
Umberto Triacca
Lesson 1: What is a time series
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;
6
Discontinuities: an unexpected shift, a sudden change of level.
Umberto Triacca
Lesson 1: What is a time series
The time series graph
An increasing trend is apparent in the number of divorces in Italy
Figure : Annual number of divorces in Italy from 1995 to 2007
Umberto Triacca
Lesson 1: What is a time series
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.
Umberto Triacca
Lesson 1: What is a time series
A cyclic component is present in the following time series.
Figure : Sunspots Number. July 1969 to September 2007.
Umberto Triacca
Lesson 1: What is a time series
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.
Umberto Triacca
Lesson 1: What is a time series
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.
Umberto Triacca
Lesson 1: What is a time series
The time series graph
A turning point is present in the unemployment rate in Italy.
Figure : Quarterly unemployment rate in Italy 2000.1-2009.4
Umberto Triacca
Lesson 1: What is a time series
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.
Figure : Annual U.S. price index for gasoline from 1960 to 1995
Umberto Triacca
Lesson 1: What is a time series
The time series graph
In summary, a time series plot is a valuable tool used in order to identify patterns in the data.
Umberto Triacca
Lesson 1: What is a time series
The time series graph
In summary, a time series plot is a valuable tool used in order to identify patterns in the data. bigskip A good picture is better than 1000 words
Umberto Triacca
Lesson 1: What is a time series
Time series
A question:
Umberto Triacca
Lesson 1: What is a time series
Time series
A question:
Why we want to study a time series?
Umberto Triacca
Lesson 1: What is a time series
Time series
The main objective for analyse a time series is to forecast its future values
Umberto Triacca
Lesson 1: What is a time series
Time series
In order to reach this goal we need of a model of the ”mechanism” that has generated the series.
Umberto Triacca
Lesson 1: What is a time series
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).
Umberto Triacca
Lesson 1: What is a time series
