## Introduction: Time Series Data

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 is a time series.

**Noise:** The noise is an irregular component of variation in a time series.

**Examples of 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.

**Definition:**

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 more or less equally spaced intervals such as minutes, hour, day, month, quarter, or year, etc. 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).

Time series are used in different fields of sciences such as signal processing, pattern recognition, econometrics, mathematical finance, weather forecasting, earthquake prediction, electroencephalography, control engineering, astronomy, and communications engineering among many other fields.

**Continuous-Time Series:**

A time series is said to be continuous when the observation is made continuously in 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 a continuous variable.

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 random variable. The arcane difference between time series and other variables is the use of subscript.

Time series analysis comprises methods for analyzing time-series data in order 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.

Given an observed time series, the first step in analyzing a time series is to plot the given series on a graph taking time intervals ($t$) along X-axis (as an independent variable) and the observed value ($Y_t$) on Y-axis (as dependent variable). Such a graph will show various types of fluctuations and other points of interest.