What is a Time Series
A?time series?is a special kind of statistical data. Specifically, it is a collection of numerical measurements, called?observations
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that are indexed by a time stamp t= 1, 2,3, ……., n, these time stamps are regularly spaced in time with equal intervals between any two adjacent stamps.
To gain some intuition for time series, consider the following examples:
In all these examples, as well as in a general time series, data take the form of?discrete?measurements of a real-world phenomenon that evolves continuously in time. A general probabilistic model to describe such phenomena is called a?stochastic process, which is simply a collection of random variables {Xt} indexed by a time parameter.
Dependence in Time Series
The most important characteristics of time series data is that we make no assumption about independence of the random variables {Xt}?. In fact, most time series data are?dependent, typically because past realizations influence future observations through the nature of the real-world phenomenon that produces these data.
?Deterministic Dependence: Trend and Seasonality
Some of the main characteristics of dependence in a time series are trends and seasonal variations. These are?deterministic?dependencies of the random variables Xt?of our series on the time stamp t?.
The?trend?can be linear or nonlinear, and is sometimes monotone in time.
The?seasonal variation?is the cyclical component of the time series. It is a periodic function whose values repeat at a fixed time interval. The period of seasonal variation may be assumed known or unknown depending on the data and application.
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Note that many of the time series examples above exhibit deterministic dependence. For instance, heart rate is periodic, and temperature has seasonal variations with a twelve months period.
Stochastic Dependence
Beyond deterministic trends and seasonal variations, the random variables Xt?at different time points may also be dependent on each other. (Recall that dependence of a collection of random variables means that their joint distribution is not equal to the product of the marginal distributions.)
For the examples of time series above, think of whether?Xt+1?may be dependent or independent of the preceding.
For example, the price of a stock tomorrow may be above or below the price of this stock today, and our expectation may be that it will be equal to the price today. (Because markets are efficient, the best forecast of the price tomorrow is the price today.) In other words, the?change?between the stock price today and the stock price tomorrow may be completely random and independent of the price today, the price tomorrow, which is the sum of the price today with this change, does depend on the price today.
Very often, time series observations that are close to each other in time are strongly correlated. For some time series this correlation decays as the time distance between observations increases, while the variation of stays constant over time. For other time series, the correlation of with future observations stays constant, while the total variation of the series accumulates and increases with time.
The goal of time series analysis is to understand and exploit the deterministic and stochastic dependencies of the stochastic process that generated the data. Specifically, we want to develop statistical tools, algorithms, and models to
In many applications of time series analysis, the main objective is to forecast future observations of the series. This requires estimating the deterministic component and fitting a statistical model that captures the correlation structure between adjacent observations. We can then form a prediction of the future from the past by extrapolating on all the dependencies in the series that we learn from data.