Lecture 1
See also the Week 7 Course module guide on Vula with
information on what elements of the content to engage
with each day.
Introduction to
Note that this is just a guide – all the content is available
for you to engage with at the pace you want to do so.
time series analysis
(fpp) Chapter 1:
Sections 1.1 – 1.7
1 3
Introduction to Time Series Analysis
FREE companion online resource:
Time Series vs. Cross-Sectional Data
Forecasting: Principles and Practice by Rob
J Hyndman and George Athanasopolous
§ Cross-sectional data is data observed or measured at one point in time
e.g. Final STA2020 marks
Any variable that is regularly measured over time in sequential order
(https://www.otexts.org/book/fpp2)
§
(e.g. hourly, daily, weekly, monthly, quarterly, or yearly) is called a time series
I highly recommend this textbook as Definition: A time series is a sequence of observations collected at regular
excellent supplementary reading that delves equally spaced intervals over a period of time
a bit deeper into topics that we discuss here • Time series data, i.e. records that are measured sequentially over time, are extremely
(Parts of chapters 1, 2, 3, 6, 7, 8) common. They arise in virtually every application field, such as e.g.:
• Business
– Sales figures, production numbers, customer frequencies
Whenever you see ‘(fpp)’ in the slides, it is referring • Economics
to the Forecasting: Principles and Practice online – Stock prices, exchange rates, interest rates
textbook 2
• Official Statistics
4
– Census data, personal expenditures, road casualties
,Introduction to Time Series Analysis Introduction to Time Series Analysis
5 7
Introduction to Time Series Analysis Introduction to Time Series Analysis
6 8
,Why Time Series Analysis?
§ Economic conditions vary over time. The success of many businesses is dependent on Two fundamental
time series concepts:
being able to foresee future events and plan accordingly.
§ Time series analysis is a collection of statistical techniques that attempt to isolate and
quantify the influence of these events and changes in conditions in order to build a model
that utilizes this information to forecast future values of the time series
Autocorrelation and
§ Standard inferential techniques which assume independence of observations (e.g.
regression) do not work well when data is collected at regular or equally spaced time
intervals because the observations are likely to be dependent. A single chance event may
Stationarity
affect all later observations.
§ So we cannot assume that the data constitute a random sample. Much of the
methodology in time series analysis is aimed at explaining this correlation using
appropriate statistical models
9 11
Basic Assumption Underlying Time Series 1. Autocorrelation
Forecasting • Suppose we have a data !! , !" … !# where T is the length (number of observations) in
the time series. The most basic assumption in inferential statistics is that !$ are
The factors that influenced patterns of activity in the past and present will independent, i.e. we have a random sample. The independence is a nice property, since
continue to do so in more or less the same manner in the future (i.e. we using it we can derive a lot of useful results.
assume that past patterns will continue into the future).
• The problem is that frequently this property does not hold for time series data since
the observations are all from the same variable and recorded at equally spaced
intervals of time. This often results in the values of a time series being correlated with
Thus the overall purpose of time series analysis is to identify and isolate
one another. This is referred to as autocorrelation or serial correlation
these influencing factors from the past in order to better understand the
process underlying the time series, for predictive purposes. • The assumption of independence of observations in a multiple linear regression model
commonly fails when the sample data have been collected over time and the
We can conduct a time series analysis to: regression model fails to effectively capture any trends. In such circumstances, the
random errors in the model are often positively correlated over time, so that each
a) Develop a better understanding of the pattern of behavior present and random error is more likely to be similar to the previous random error than it would be
the factors/components that have given rise to that pattern if the random errors were independent of one another.
b) Develop a model that captures as much information from the time series
as possible to forecast future values of the time series • The majority of time series are dependent i.e. they exhibit significant autocorrelation
at some lag g, where ‘lag’ refers to the number of time periods between observations
c) … and other uses which we don’t consider in this course at which we measure the autocorrelation.
10 12
,2. Stationarity Time series plot
In order to forecast or make predictions, we need to ensure that time series data are
stationary. A time series is said to be stationary if its statistical properties are constant
over time i.e. they are independent of time. This implies that: 1) The first step in a time series analysis is to plot the data and observe any
patterns that have occurred over time using a time series plot/line graph
1) The process generating the data (the “Data Generating Process” (DGP)) has a
constant mean --- a time series plot is a line graph of the observed data !! against time ".
2) The variability of the time series is constant over time
2) The time series plot enables us to detect and to describe
patterns/factors/components (of the past behavior) of the series. The
“A stationary time series is one whose properties do not depend on the time at which the
successive changes in values are comparable because they all relate to a
series is observed. Thus, time series with trends, or with seasonality, are not stationary —
the trend and seasonality will affect the value of the time series at different times… common time interval between observations.
In general, a stationary time series will have no predictable patterns in the long-term.
3) The identified components help in finding a suitable statistical model to
Time plots will show the series to be roughly horizontal (although some cyclic behaviour is
describe the data, which enables us to forecast future values of the time
possible), with constant variance.” (fpp)
series, on the basis that past patterns will continue into the future.
Note that most time series we encounter are non-stationary and we often need to
transform them so that they exhibit stationarity (see later in course)
13 15
Components of a Non-Stationary Time Series
Time Series 4 components:
Graphics
1) TREND
2) CYCLICAL variation
3) SEASONAL variation and the
(fpp) Chapter 2: 4) IRREGULAR/RANDOM variation
Sections 2.1–2.3, 2.8-2.9
14 16
See also the Week 7 Course module guide on Vula with
information on what elements of the content to engage
with each day.
Introduction to
Note that this is just a guide – all the content is available
for you to engage with at the pace you want to do so.
time series analysis
(fpp) Chapter 1:
Sections 1.1 – 1.7
1 3
Introduction to Time Series Analysis
FREE companion online resource:
Time Series vs. Cross-Sectional Data
Forecasting: Principles and Practice by Rob
J Hyndman and George Athanasopolous
§ Cross-sectional data is data observed or measured at one point in time
e.g. Final STA2020 marks
Any variable that is regularly measured over time in sequential order
(https://www.otexts.org/book/fpp2)
§
(e.g. hourly, daily, weekly, monthly, quarterly, or yearly) is called a time series
I highly recommend this textbook as Definition: A time series is a sequence of observations collected at regular
excellent supplementary reading that delves equally spaced intervals over a period of time
a bit deeper into topics that we discuss here • Time series data, i.e. records that are measured sequentially over time, are extremely
(Parts of chapters 1, 2, 3, 6, 7, 8) common. They arise in virtually every application field, such as e.g.:
• Business
– Sales figures, production numbers, customer frequencies
Whenever you see ‘(fpp)’ in the slides, it is referring • Economics
to the Forecasting: Principles and Practice online – Stock prices, exchange rates, interest rates
textbook 2
• Official Statistics
4
– Census data, personal expenditures, road casualties
,Introduction to Time Series Analysis Introduction to Time Series Analysis
5 7
Introduction to Time Series Analysis Introduction to Time Series Analysis
6 8
,Why Time Series Analysis?
§ Economic conditions vary over time. The success of many businesses is dependent on Two fundamental
time series concepts:
being able to foresee future events and plan accordingly.
§ Time series analysis is a collection of statistical techniques that attempt to isolate and
quantify the influence of these events and changes in conditions in order to build a model
that utilizes this information to forecast future values of the time series
Autocorrelation and
§ Standard inferential techniques which assume independence of observations (e.g.
regression) do not work well when data is collected at regular or equally spaced time
intervals because the observations are likely to be dependent. A single chance event may
Stationarity
affect all later observations.
§ So we cannot assume that the data constitute a random sample. Much of the
methodology in time series analysis is aimed at explaining this correlation using
appropriate statistical models
9 11
Basic Assumption Underlying Time Series 1. Autocorrelation
Forecasting • Suppose we have a data !! , !" … !# where T is the length (number of observations) in
the time series. The most basic assumption in inferential statistics is that !$ are
The factors that influenced patterns of activity in the past and present will independent, i.e. we have a random sample. The independence is a nice property, since
continue to do so in more or less the same manner in the future (i.e. we using it we can derive a lot of useful results.
assume that past patterns will continue into the future).
• The problem is that frequently this property does not hold for time series data since
the observations are all from the same variable and recorded at equally spaced
intervals of time. This often results in the values of a time series being correlated with
Thus the overall purpose of time series analysis is to identify and isolate
one another. This is referred to as autocorrelation or serial correlation
these influencing factors from the past in order to better understand the
process underlying the time series, for predictive purposes. • The assumption of independence of observations in a multiple linear regression model
commonly fails when the sample data have been collected over time and the
We can conduct a time series analysis to: regression model fails to effectively capture any trends. In such circumstances, the
random errors in the model are often positively correlated over time, so that each
a) Develop a better understanding of the pattern of behavior present and random error is more likely to be similar to the previous random error than it would be
the factors/components that have given rise to that pattern if the random errors were independent of one another.
b) Develop a model that captures as much information from the time series
as possible to forecast future values of the time series • The majority of time series are dependent i.e. they exhibit significant autocorrelation
at some lag g, where ‘lag’ refers to the number of time periods between observations
c) … and other uses which we don’t consider in this course at which we measure the autocorrelation.
10 12
,2. Stationarity Time series plot
In order to forecast or make predictions, we need to ensure that time series data are
stationary. A time series is said to be stationary if its statistical properties are constant
over time i.e. they are independent of time. This implies that: 1) The first step in a time series analysis is to plot the data and observe any
patterns that have occurred over time using a time series plot/line graph
1) The process generating the data (the “Data Generating Process” (DGP)) has a
constant mean --- a time series plot is a line graph of the observed data !! against time ".
2) The variability of the time series is constant over time
2) The time series plot enables us to detect and to describe
patterns/factors/components (of the past behavior) of the series. The
“A stationary time series is one whose properties do not depend on the time at which the
successive changes in values are comparable because they all relate to a
series is observed. Thus, time series with trends, or with seasonality, are not stationary —
the trend and seasonality will affect the value of the time series at different times… common time interval between observations.
In general, a stationary time series will have no predictable patterns in the long-term.
3) The identified components help in finding a suitable statistical model to
Time plots will show the series to be roughly horizontal (although some cyclic behaviour is
describe the data, which enables us to forecast future values of the time
possible), with constant variance.” (fpp)
series, on the basis that past patterns will continue into the future.
Note that most time series we encounter are non-stationary and we often need to
transform them so that they exhibit stationarity (see later in course)
13 15
Components of a Non-Stationary Time Series
Time Series 4 components:
Graphics
1) TREND
2) CYCLICAL variation
3) SEASONAL variation and the
(fpp) Chapter 2: 4) IRREGULAR/RANDOM variation
Sections 2.1–2.3, 2.8-2.9
14 16
