A time series can be considered as a value set {Xt}
representing measurements which are taken at different periods of time t =
1,2,3,4…..n. We can easily appreciate
the fact that time itself is a continuous variable. However the measurements
for time in many cases are made at specific intervals or points in time and
thus they appear as discrete variables. Usually a single variable or value is analysed
at each point in time which is known as univariate analysis. In such cases the
single variable is analysed even if multiple variables are recorded for that
point for instance the different variables might be daily weather conditions at
the meteorological station or stock prices or traded volume data on hourly
bases. The data are mostly though not always defined for equal time intervals.
For instance, hourly, daily, monthly or weekly basis. There is also bivariate
or multivariate time series analysis.
As we covered the temporal autocorrelation in the
previous post, we discussed an example of a time series namely that of the
price of AAPL stock over a period of time. We illustrated that the time series
data from such example could be analysed to identify patterns using the ACF and
PACF functions represented using correlograms. Herein we take the analysis
further and study more ways of examining and analysing patterns in this kind of
data. Further from this analysis we arrive at predictive models that are based
on a predictive or explanatory function and devise forecasting techniques based
on the same. First of all we examine the statistical methods that can be
applied to time series data by examining their behaviour over a period of time.
We then have a look at time series data with varying periodicities or
frequencies. Analysis of such series is particularly useful for prediction of
phenomena that show multiple periodicities over a period of time.
There are many different types of time series such as
1) economic datasets like share prices, GDP or inflation, income datasets 2)
Physical time series such as river flow data, meteorological data, pollution
monitoring data 3) Marketing time series data such as sales figures,
advertising response data, 4) demographic time series like population levels
over time, 5) manufacturing data such as process output and control charts 6)
binary processes such as digital data sequences in switching systems and data
transmission systems) and 7) temporal point processes like 1 D point processes,
2 D point processes and spatial point datasets. In case of some of these data
series types, the data measurement can be continuous such as in a barograph
measuring air pressure or a data communications channel through which the traffic
flows are recorded continuously. Other data could be measured or recorded only
at specific points in time such as the daily closing price of a stock on the
stock exchange. A considerable proportion of data series techniques address the
time series problems pertaining to the latter type which is termed as discrete
data series and is usually recorded at fixed intervals.
There are many different reasons why an analyst
would undertake the analysis of a time series data. The reasons include simple
descriptive requirements of analysing data in order to identify the main aspects
of the data such as means, peaks and troughs in the data or periodicity or
critical points of change in trends. Another purpose to analyse time series
might be to predict trends in future and yield estimates of the given quantity
or phenomena being measured based on historic data. Earlier, prediction or
estimation had been a static procedure. However now an increasingly real time
predictive modelling are being used to assess the future trends continuously
for a given period of time based on current data being generated. The
applicability of such modelling is especially more in high pressure or
emergency control situations such as disaster management, earthquake or tsunami
predictions or infrastructure management like communications and power
management etc. as well as in the financial markets. Time series predictions
use raw data to make forecasts about phenomena and are useful in shorter term
forecasting. However for long term forecasting or for forecasting for data with
complex behaviour, an explanatory analysis is used to arrive at accurate
forecasts and the data on many variables might be used to provide future values
of multiple time series variables underlying a given construct such as GDP
which requires measurement of many different variables to arrive at accurate
estimates of the same.
Another example is the UK Treasury model which is
used for econometric forecasting and as of now uses 30 main equations and about
100 independent variables that are used as input variables to arrive at the
model predictions.
Forecasting is dependent on what can be termed as a
well behaved data. This amounts to use of historic data and related information
to predict future values of data variables. In many cases this can be very
effective, but often does not take into account the very unexpected and sudden
changes. We need to be cautious of such unexpected changes and account for them
in our predictive models.
Any time series almost always has some degree of
autocorrelation and the analysis of autocorrelation is usually one of the first
tasks carried out after cleansing of data and basic visual inspection though.
Apart from this data sometimes also exhibit some level of periodicity and the length and magnitude of such patterns requires
closer examination.
This so as no matter how many instances of a
predictable time series data patterns one might have observed, there will still
be some patterns that might consist of sudden and unexpected changes. This
unexpected scenario can be treated as a valuable paradigm contributing to
analysis of vulnerable situations where sudden and drastic changes such as
major wars, famines or banking crises occur almost out of the blue. It must be
noted that while analysing a time series, if at the specific point at which the
series is examined does not affect the results, then the series can be said to
be stationary. In more formal terms, a stationary time series is one whose
joint probability distribution is not affected by a shift in time or in space.
This implies that the mean and variance of the data are constant across the
time and/space. However, this condition is seldom fully achieved. In cases
where the series include trend and/or periodic behaviour it is common for these
components to be identified and accounted for or decomposed prior to further
analysis. There are many models used in time series analysis, which include
simple autoregressive (AR) models, moving average (MA) models and also combined
ARMA models. These models assume stationarity. There are more complex models
however, such as ARCH and GARCH that allow for heteroskedasticity and are also
supported in specialized software packages, such as in econometric modelling.
