Time Series Analysis

Time Series Analysis

Time series is an ordered sequence of values of a variable at equally spaced time intervals. Its applications are in two folds:

  • Get to understand the underlying forces and structure that produced the observed data.
  • Come up with a fit model for forecasting, monitoring or even feedback and feedforward control.

Time series analysis is usually used in a myriad of fields including:

  • Census analysis
  • Budgetary analysis
  • Economic forecasting
  • Workload projections
  • Stock market analysis
  • Inventory studies and many more

Also, there is a wide range of methods that can be used to model and forecast time series. Some of them include the following:

  • Box-Jenkins ARIMA models
  • Box-Jenkins Multivariate models
  • Holt-Winters exponential smoothing (can be single, double or triple)

The selection of the appropriate time-series technique will depend on the user’s application and preference.

Univariate time series models

Univariate time series consists of single (scalar) observations recorded sequentially over equal time increments. Some real-world examples of univariate time series models are southern oscillations to predict el Nino effects and monthly CO2 concentration.

A univariate time series data set is normally given as a single column of numbers. However, time is an implicit variable in the time series. The time variable or index does not need to be explicitly given if the data is equi-spaced. Also, sometimes the time variable may be explicitly used for plotting the series. But it is not used in the time series model itself.

Common approaches to univariate time series

There are a plethora of approaches used to model univariate time series. Our time series online experts have outlined them below:

  • Trend seasonal residual decomposition

This approach decomposes the time series into a trend seasonal and residual component. An example of this approach is the triple exponential smoothing.

  • Frequency-based methods

This approach is often used in scientific and engineering applications to analyze the time series in the frequency domain. An example of the frequency-based method is modeling a sinusoidal type data set. The fundamental tool used for frequency analysis of time series is the spectral plot.

  • Autoregressive (AR) models

It is a linear regression of the current value of the series against one or more prior values of the series, Standard linear least squares techniques are used to analyze AR models.

  • Moving average models

A moving average model is a linear regression of the current value of the series against the random shocks of one or more prior values of the series.

  • Box-Jenkins Approach

This approach was popularized by Box and Jenkins in their book “Time Series Analysis Forecasting and Control”. It combines the moving average and autoregressive approaches.

Multivariate Time Series Models

Multivariate time series models are made up of multiple time-series that can usefully contribute to forecasting.  The choice of the model used is often guided by both economic theory and empirical experience. The vector autoregression is the multivariate time series expansion of univariate autoregression.


Time series observations which are a vector of numbers can be modeled using a multivariate form of the Box-Jenkins model. This model is sometimes called the ARMAV model for autoregressive moving average vector or simply the vector ARMA process.

Parameters and covariance matrix estimation

It is complicated and difficult to estimate the matrix parameters and covariance matrix without software. Estimating the moving average matrices is especially an uphill task. Suppose we choose to ignore the MA component(s), we will be left with the ARV model. Multivariate least squares can be used to estimate the parameter matrices. There are also other methods that can be used such as maximum likelihood estimation.

Modeling techniques in Multivariate time series models

  • VARMA models

We can use these models to study the dynamic relationships among age groups, place of treatment, disposition, and triage category as well as improving the accuracy of predictions. VARMA modeling technique allows multiple dependent time series to be modeled together. The time series can also be accounted for both cross and within correlations of the series.

The applications and mathematical theory of forecasts associated with the analysis of VARMA is quite complex. If you need exceptional help, then take our time series analysis homework help.

  • ARMA models

These models are derived from univariate time series modeling methods. Their modeling process is similar to that of the VARMA method. However, ARMA only allows one time series to be modeled at a time. The ARMA method was developed by Box and Jenkins to provide a general framework for forecasting a non-stationary observed time series data.