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Structured Time-Series Models

soupa_soundararajan1

Srinidhi

May 04, 2016
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Modelling & Forecasting Time-Series data has been one of the cornerstones of Predictive Analytics in the era of Big Data. There are a plethora of forecasting techniques available today whose context can be a pain to understand and as we know, in the war on noise, context serves as a crucial ammunition. To that end, we, Hemanth Sindhanuru & Srinidhi K, from LatentView Analytics are presenting this series of articles where we will be discussing a structured methodology to understand, analyse & forecast time-series data

TIME-SERIES COMPONENTS
Our current scientific literature provides us with numerous techniques for modelling time-series that have originated from different fields of study, ranging from Classical Statistics to Complex Signal Processing Algebra. A major family of these techniques, named Structured Time-Series Models build on the idea that a time-series can be separated into certain components to which we can attribute a time-series’ properties.

• SEASONAL COMPONENT
Seasonal component or Seasonality of a time-series refers to the periodic variations inherent in the data. These periodic deviations generally arise due to systemic factors such as effect of weathers, marketing campaigns etc. For e.g. increased sales every holiday season, a change in stock market trading activity every three months on account of Quarterly Earnings Reports etc.

• TREND COMPONENT
Trend Component refers to the pattern followed by the mean of the series with time over a long term, either increasing or decreasing & the nature of this change, either linear or non-linear or ambiguous.

• CYCLIC COMPONENT
Cyclic component refers to a recurrent though not periodic variation inherent in the series. This usually is correlated to the business cycle of the entity. The business cycle refers to the fluctuations in the macro-economic factors defined in terms of booms and recessions. It differs from the Seasonal component in the way that the average length of the cyclic patterns is much longer than the seasonal pattern and the fluctuations are of varying duration.

• REMAINDER COMPONENT
As the name suggests, this refers to the leftover component once we have siphoned off the aforementioned Seasonality, Cycle & Trend. Ideally this component should be devoid of any residual Seasonality, Cycle & Trend and must be exhibit perfect stationarity.

TIME-SERIES DECOMPOSITION
The decomposition of a time-series refers to the separation of a time-series into its aforementioned components, namely Seasonality, Trend, Cycle & Remainder. We can classify the methods we use for decomposing a time-series under two broad approaches.

Model Based Approach:
Under this approach, explicit models are built for each of the time-series’ components to explain their behaviour. The context of this approach can be clearly understood by considering the following example

• Holt Winters Seasonal Exponential Smoothing Model

The aforementioned model contains three separate formulations for the Seasonality, Trend & Remainder components. Modelling the time-series’ components involves estimating the respective parameters α, β and γ.

• Fourier Transformation modelling of the Seasonality

The Fourier Transformation is built on the insight that any time signal (in our case, a time-series) can be interpreted as a linear combination of sinusoidal waves with different amplitudes & frequencies. Consider a time-series with N observations. A generic element Y(t) of this series can be modelled by Fourier Transformation (i.e. a linear along with an error component e(t)

Filter Based Approach:
Filter based methods do not assume any individual models for the components. The extraction of the components is done using ad-hoc filters to the raw time series. These filters decompose the time-series into components generally through successive iterations of smoothing. These methods are non-parametric.

We will be using the STL algorithm for decomposing a time-series as we move forward in this series owing to its versatility to time-series with any seasonality and its simplicity in application.

Keep watching this space for more

BACK TO THE FUTURE – A Beginner’s Guide to Forecasting
1. A Primer on Time-Series Forecasting
2. Structured Time-Series Models
3. Periodicity of a Seasonal Time-Series