# Time series analysis and forecasting

## Overview

Forecasting is done to monitor changes that occur over time. Forecasting methods are divided on to qualitative and quantitative and the last category is represented by time series forecasting and causal forecasting.[1] A time series is a set of observed values for single entity, such as production or sales data, for a sequentially ordered series of time periods.[2]. Where can time series be found? Let’s give some examples:

• Economic indicators: Sales figures, employment statistics, stock market indices.
• Meteorological data: precipitation, temperature, humidity registered on regular basis.
• Environmental monitoring: concentrations of nutrients and pollutants in air masses, rivers, marine basins registered yearly, etc.

Time series analysis is the procedure by which the time-related factors that influence the values observed in the time series are identified and segregated.

## Classical time series models and time series decomposition

The Classical Time Series Model consists of the following parts:

1.Trend (T ): The general long-term movement in the time series values over an extended period of years.
2.Cyclical fluctuations (C ): Recurring up and down movements with respect to trend that have a duration of several years.
3.Seasonal variations (S ): Up and down movements with respect to trend that are completed within a year and recur annually.
4.Irregular variations (I): The erratic variations from trend that cannot be ascribed to the cyclical or seasonal influences.

A trend (T) is an overall long-term upward or downward movement in a time series. Trend analysis can be developed using following procedures: Simple Smoothing (Moving Average approach), Exponential Smoothing (using exponential distribution), and Least-squares Trend Fitting. Analysis of cyclical variations (C) can be easily done using time series with annual values only, because they reflect the effects of only the trend and cyclical components, because the seasonal and irregular components are defined as short-run influences. Seasonal variations (S) analysis is performed with the use of two approaches for the cyclical component identification: trend removing and special seasonal coefficients calculation and exponential modeling with dummy variables. The component, which includes 'irregular variations (I) is unpredictable. Every time series has some unpredictable component that makes it looking like a random variable. Resulting component, time series level (Y) in each particular time period is represented as specific combination of four components specified above. This combination is known as a model, mathematical approximation of time series. Four components (trend, cyclical, seasonal and irregular variations) can be organized in multiplicative and additive time series model. The additive model is useful when the seasonal variation is relatively constant over time. The multiplicative model is usually recommended when the seasonal variation increases over time.

The Classical Multiplicative Model
The model is based on the assumption that for any period in the time series the value of the variable is determined by the four components and the components have a multiplicative relationship:

Y = T × C × S × I

The model is based on the assumption that for any period in the time series the value of the variable is determined by the four components and the components have an additive relationship:

Y = T + C + S + I

A study of time series may focus, therefore, on one or more of the following:

• the overall pattern of change in an indicator over time;
• comparing one time period to another time period;
• comparing one geographic area to another by trend;
• comparing one population (sample) to another in time perspective;
• making future projections.

## Steps in a classical time series analysis

Because the time series decomposition in its easy form do not involve a lot of fundamental mathematics or statistics, these simple results are relatively easy to explain to the user. This is a major advantage because if the user has an appreciation of how the forecast was developed, he or she may have more confidence in its use for decision making.

1. Collect time series data and represent them as a line chart.
2. Describe the variability of the series seen in the plot.
3. Use time series plots to determine whether transformations are necessary.
4. Transform the data if necessary.
5. Use time plots and test statistics to determine if the series is stationary (constant mean and or variance).
6. Make the series stationary if necessary.
7. Fit additive or multiplicative model to series and analyze residuals.
8. If model is statistically significant, make forecasts for the future.

## References

1. Levine, David M., David F. Stephan, Timothy C. Krehbiel, and Mark L. Berenson (2011) Statistics for Managers Using Microsoft Excel. Sixth Edition. Pearson Education, Inc. ISBN: 0136113494]
2. http://stats.oecd.org/glossary