Time Series Analysis: Concept, Additive and Multiplicative Models

A time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. Examples of time series are heights of ocean tides, counts of sunspots, and the daily closing value of the Dow Jones Industrial Average.

Time series are very frequently plotted via line charts. Time series are used in statistics, signal processing, pattern recognition, econometrics, mathematical finance, weather forecasting, earthquake prediction, electroencephalography, control engineering, astronomy, communications engineering, and largely in any domain of applied science and engineering which involves temporal measurements.

Time series analysis comprises methods for analyzing time series data in order to extract meaningful statistics and other characteristics of the data. Time series forecasting is the use of a model to predict future values based on previously observed values. While regression analysis is often employed in such a way as to test theories that the current values of one or more independent time series affect the current value of another time series, this type of analysis of time series is not called “time series analysis”, which focuses on comparing values of a single time series or multiple dependent time series at different points in time. Interrupted time series analysis is the analysis of interventions on a single time series.

Time series data have a natural temporal ordering. This makes time series analysis distinct from cross-sectional studies, in which there is no natural ordering of the observations (e.g. explaining people’s wages by reference to their respective education levels, where the individuals’ data could be entered in any order). Time series analysis is also distinct from spatial data analysis where the observations typically relate to geographical locations (e.g. accounting for house prices by the location as well as the intrinsic characteristics of the houses). A stochastic model for a time series will generally reflect the fact that observations close together in time will be more closely related than observations further apart. In addition, time series models will often make use of the natural one-way ordering of time so that values for a given period will be expressed as deriving in some way from past values, rather than from future values.

Additive model:

1. Data is represented in terms of addition of seasonality, trend, cyclical and residual components
2. Used where change is measured in absolute quantity
3. Data is modeled as-is

Additive model is used when the variance of the time series doesn’t change over different values of the time series.

On the other hand, if the variance is higher when the time series is higher then it often means we should use a multiplicative models.R

eturni=pricei−pricei−1=trendi−trendi−1+seasonali−seasonali−1+errori−errori−1returni=pricei−pricei−1=trendi−trendi−1+seasonali−seasonali−1+errori−errori−1I

f error’s increments have normal iid distributions then returni has also a normal distribution with constant variance over time.

Multiplicative model:

1. Data is represented in terms of multiplication of seasonality, trend, cyclical and residual components
2. Used where change is measured in percent (%) change
3. Data is modeled just as additive but after taking logarithm (with base as natural or base 10)

If log of the time series is an additive model then the original time series is a multiplicative model, because:l

og(pricei)=log(trendi⋅seasonali⋅errori)=log(trendi)+log(seasonali)+log(errori)log(pricei)=log(trendi⋅seasonali⋅errori)=log(trendi)+log(seasonali)+log(errori)

So the return of logarithms:

log(pricei)−log(pricei−1)=log(pricei/pricei−1)