Time Series Modeling and Forecasting

A time-series is a sequence of data points representing the state of a “system” as it evolves over time. Each data point represents the system’s state at the corresponding time step. The definition of a system in this context is very broad; it could be any physical system or even an abstract phenomenon. The state is composed of a set of attributes of the system modeling its behavior.

For example, one can represent the state of a vehicle in terms of its location, speed, fuel status, engine temperature, and so on. The state of a financial market can be modeled in terms of its trend (bear or bull) and values of various indices. Some of these attributes may not be directly observable. If we say that a vehicle is healthy, the attribute “health” is not directly observable, but is inferred from other observables such as engine temperature, noise, and emissions. Similarly, financial market trends are not directly observable but can be inferred from other observables such as the values of various indices and stock prices of major companies.

Modeling a time-series problem involves deciding on what attributes constitute a state and then defining state transition and observation models. The state transition model infers future state from the current. In other words, we can use such a model to predict market trends, for example. The observation model indicates what one expects to observe at a time step.

Here is a Markov modeling example taken from the book Computational Business Analytics (see Ref below). The book discusses various AI and machine learning approaches to time series modeling, including hidden Markov models, dynamic Bayesian networks, and Kalman filters.

Consider a three-state Markov model for financial market trends over short time frames. The terms used here for describing the states are the following:

Upward market trend (bull)
Downward market trend (bear)
No change (static)
 
The model and the transition probability from one state to another are given in the figure below. As an example, the probability of the observation sequence {bear, bull, static, static, bear} is 0.001. Such predictive evolutions of sequences will allow one to predict market given a sequence of precursor states.

Continuing with our financial market prediction example, the set of hidden states is {bull, bear, static} and the observables are the trend of the market index value considered from the set {up, down flat}. The prior and transition probabilities are as before, and the observable likelihoods are shown in the figure below.

Consider an evolution of the above HMM in the figure below. The probability of the sequence {up, down, down} can be computed as 0.000432.

For more details on various techniques and algorithms: Subrata Das. (2012). Computational Business Analytics, Chapman & Hall/CRC Press.