Here we will discuss Time Series Data and Time Series Analysis.
The sequence $y_1,y_2,cdots, y_n$ of $n$ observations of a variable (say $Y$), recorded in accordance with their time of occurrence $t_1, t_2, cdots, t_n$, is called a time series. Symbolically, the variable $Y$ can be expressed as a function of time $t$ as
$$y = f(t) + e,$$
where $f(t)$ is a completely determined (or a specified sequence) that follows some systematic pattern of variation, and $e$ is a random error (probabilistic component) that follows an irregular pattern of variation. For example,
Signal: The signal is a systematic component of variation in a time-series.
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Noise: The noise is an irregular component of variation in a time series.
- The hourly temperature recorded at a weather bureau,
- The total annual yield of wheat over a number of years,
- The monthly sales of fertilizer at a store,
- The enrollment of students in various years in a college,
- The daily sales at a departmental store, etc.
Time Series
A time series ${Y_t}$ or ${y_1,y_2,\cdots,y_T}$ is a discrete-time, continuous state process where time $t=1,2,cdots,=T$ are certain discrete time points spaced at uniform time intervals.
A sequence of random variables indexed by time is called a stochastic process (stochastic means random). A data set is one possible outcome (realization) of the stochastic process. If history had been different, we would observe a different outcome; thus, we can think of a time series as the outcome of a random variable.
Usually, time is taken at intervals that are more or less equally spaced, such as minutes, hours, days, months, quarters, or years. More specifically, it is a set of data in which observations are arranged in chronological order (A set of repeated observations of the same variable arranged according to time).
In different fields of science (such as signal processing, pattern recognition, econometrics, mathematical finance, weather forecasting, earthquake prediction, electroencephalography, control engineering, astronomy, and communications engineering, among many other fields), Time-Series Analysis is performed.
Continuous Time Series
A time series is said to be continuous when observations are made continuously over time. The term, continuous is used for a series of this type even when the measured variable can only take a discrete set of values.
Discrete Time Series
A Time-Series is said to be discrete when observations are taken at specific times, usually equally spaced. The term discrete is used for a series of this type even when the measured variable is continuous.
We can write a series as ${x_1,x_2,x_3,cdots,x_T}$ or ${x_t}$, where $t=1,2,3,cdots,T$. $x_t$ is treated as a random variable. The arcane difference between time-series variables and other variables is the use of subscripts.
Time-series analysis comprises methods for analyzing time-series data to extract some useful (meaningful) statistics and other characteristics of the data, while time-series forecasting is the use of a model to predict future values based on previously observed values.
The first step in analyzing time-series data is to plot the given series on a graph, taking time intervals ($t$) along the $X$-axis (as an independent variable) and the observed value ($Y_t$) on the $Y$-axis (as a dependent variable). Such a graph will show various types of fluctuations and other points of interest.
