Statistics for Data Analyst

Traditional methods of time series analysis are concerned with decomposing a series into a trend, a seasonal variation, and other irregular fluctuations. Although this approach is not always the best but still useful (Kendall and Stuart, 1996).

Components of Time Series

The components, by which a time series is composed, are called the components of time series data. A time series is not necessarily composed of all these four components of Time Series Data. The components of time series are (i) Seasonal Effect, (ii) Cyclic Variation, (iii) Secular Trend, and (iv) Irregular Variation. The four basic components of time series are described below.

Different Sources of Variation are:

Seasonal effect (Seasonal Variation or Seasonal Fluctuations)

Many of the time series data exhibit a seasonal variation which is the annual period, e.g., sales and temperature readings. Seasonal variations are easy to understand and can be easily measured or removed from the data to give deseasonalized data. Seasonal Fluctuations describe any regular variation with a period of less than one year. For example, the cost of various types of fruits and vegetables, clothes, unemployment figures, average daily rainfall, increase in the sale of tea in winter, increase in the sale of ice cream in summer, etc., all show seasonal variations.

The changes which repeat themselves within a fixed period, are also called seasonal variations. For example, traffic on roads in the morning and evening hours, Sales at festivals like EID, etc., an increase in the number of passengers at the weekend, etc. Climate, social customs, religious activities, etc cause seasonal variations. The main causes of seasonal variations are seasons, religious festivals, and social customs.

Other Cyclic Changes (Cyclical Variation or Cyclic Fluctuations)

Time series exhibits Cyclical Variations at a fixed period due to some other physical cause, such as daily variation in temperature. Cyclical variation is a non-seasonal component that varies in a recognizable cycle. These variations are considered a more dangerous effect on business and economic activity. Sometimes series exhibits oscillation which does not have a fixed period but is somewhat predictable. For example, economic data is affected by business cycles with a period varying between about 5 and 7 years.

The cyclical component may describe any regular variation (fluctuations) in time series data in weekly or monthly data. The cyclical variation is periodic and repeats itself like a business cycle, which has four phases (i) Peak/Prosperity (ii) Recession (iii) Trough/Depression (iv) Expansion.

Trend (Secular Trend or Long Term Variation)

It is a longer-term change. Here we take into account the number of observations available and make a subjective assessment of what is long-term. It represents a relatively smooth, steady, and gradual movement of a time series in the same direction. To understand the meaning of the long term, consider the climate variables. These variables sometimes exhibit cyclic variation over a very long time period such as 50 years.

If one just had 20 years of data, this long-term oscillation would appear to be a trend, but if several hundreds of years of data are available, then long-term oscillations would be visible. These movements are systematic where the movements are broad, steady, showing a slow rise or fall in the same direction. The trend may be linear or non-linear (curvilinear). Some examples of secular trends are:

• Increase in prices,
• Increase in pollution,
• an increase in the need for wheat,
• an increase in literacy rate,
• decrease in deaths due to advances in science.

Taking averages over a certain period is a simple way of detecting a trend in seasonal data. Change in averages with time is evidence of a trend in the given series. There are more formal tests for detecting a trend in time series.

Other Irregular Variation (Irregular Fluctuations)

When trend and cyclical variations are removed from a set of time series data, the residual is left, which may or may not be random. Various techniques for analyzing series of this type examine to see “if irregular variation may be explained in terms of probability models such as moving average or autoregressive models, i.e. we can see if any cyclical variation is still left in the residuals. These variations occur due to sudden causes are called residual variations (also called accidental or erratic fluctuations) and are unpredictable. For example, a rise in prices of steel due to strikes in the factory, accidents due to failure of the break, flood, earth quick, and war, etc.

The figure below further explains the components of time series data.

Computer MCQs Online Test

R and Data Analysis

Here we will discuss the graphical representation of time series data, called historigram.

As we have discussed in the introduction to Time Series, 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 the dependent variable: as a function of time). Such a graph will show various types of fluctuations and other points of interest.

A historigram is a graphical representation of a time series that reveals the changes that occurred at different time periods. The first step in the prediction (or forecast) of a time series involves an examination of the set of past observations. In this case, the historigram may be a useful tool. The construction of this involves the following steps described below:

• Use an appropriate scale and take time $t$ along the $x$-axis as an independent variable.
• Use an appropriate scale, and plot the observed values of variable $Y$ as a dependent variable against the given points of time.
• Join the plotted points by line segments to get the required graphical representation.

Historigram Example

Draw a graphical representation of the data to show the population of Pakistan in various census years.

Census Year	1951	1961	1972	1981	1998	2017
Population (Million)	33.44	42.88	65.31	83.78	130.58	200.17

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MCQs General Knowledge

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.

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 more or less equally spaced intervals such as minutes, hours, days, months, quarters, years, 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).

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 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 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 dependent variable). Such a graph will show various types of fluctuations and other points of interest.

R and Data Analysis