4.6  Stochastic Processes

Measure time t in appropriate units—days, months, years. A time series is a series {^–αx, … , ^–1x, ⁰x} of n-dimensional vectors observed iteratively over a period of time [–α, 0].  The natural number n is called the dimensionality of the time series. If n = 1, the time series is univariate; otherwise it is multivariate. Vectors ^tx are recorded with a frequency corresponding to a single unit of time. These concepts are illustrated in Exhibits 4.5 and 4.6 with a two-dimensional time series of daily high and low temperatures at the summit of Mt. Washington during the month of January 2015.

Exhibit 4.5: Daily high and low temperatures at the summit of Mt. Washington during the month of January 2015. Temperatures are in degrees Fahrenheit. Time t = –30 corresponds to January 1, 2015. Source: Mt. Washington Observatory.

Exhibit 4.6: Graph of daily high and low temperature data from Exhibit 4.5.

Often, each value ^tx of a time series x corresponds to an observation made precisely at time t, but this is not always the case. In our Mt. Washington example, each day’s high and low temperatures ^tx₁ and ^tx₂ are realized at different times during a given day, but we associate them both with the specific integer point in time t.

Presumably, the process that generated a time series will continue into the future. We are interested in future values, which we treat as random. To model these, we specify a model called a stochastic process based upon the time series. A stochastic process—or process—is a sequence of random vectors ^tX with t taking on integer values.3 Values t extend back to –∞ and forward to ∞. Modeling all these terms may seem excessive, especially for practical work. We do so as a mathematical convenience. It saves us having to artificially model some initial or terminal behavior for the process.

Given a time series {^–αx, … , ^–1x, ⁰x}, we construct a stochastic process {… , ^–1X, ⁰X, ¹X, …} by treating the time series as a single realization of the corresponding segment {^–αX, … , ^–1X, ⁰X} of the stochastic process. We apply statistical techniques to specify a stochastic process that is consistent with such a realization. The undertaking is called time series analysis.