Stochastic Modelling of Time Series - Hydrologic Time Series Analysis: Theory and Practice - page 84

5
Stochastic Modelling of Time Series
In practice, hydrologists often deal with a limited amount of recorded data
(i.e., a sample ) while analyzing a hydrologic time series. This sample consists
of a limited number of realizations of the population of same hydrologic
process. When a hydrologic time series is characterized with statistical and
probabilistic parameters, it represents a probability of occurrence of one of its
possible stages. This probabilistic occurrence of the hydrologic time series is
considered as one realization. All possible realizations of the hydrologic process
constitute a population . The concept of terms sample and population has
already been explained in Chapter 2. The main intent of the most hydrologic
studies is to understand and quantitatively describe the population as well as
the process that generates it based on a limited number of samples . Also,
future predictions and/or simulations about the hydrologic time series can be
made by applying statistical tools and techniques using probabilistic or
stochastic models based on the historical data. When a hydrologic time series
is analyzed in this manner, the technique is known as ' stochastic modelling '
of time series and the parameters described with statistic and probabilistic
terms are called ' stochastic parameters '.
Stochastic models are used to model a time series without considering
physical nature of the time series (Box and Jenkins, 1976; Shahin et al.,
1993). In hydrology, common stochastic models are: pure random (or white
noise) model, autoregressive (AR) model, moving average (MA) model,
autoregressive moving average (ARMA) model, and autoregressive integrated
moving average (ARIMA) model. In this chapter, common stochastic processes
are discussed with a major emphasis on autoregressive integrated moving
average process. Step-by-step procedure for stochastic modelling of the time
series is explained.
5.1 Common Stochastic Processes
Different stochastic models mainly follow distinct stochastic processes. The
stochastic processes associated with stochastic models are briefly described in
this section.