## Cyclical and Seasonal Variation in Statistics

There is a lot of difference adjust between seasonal variations and cyclical variations.

**Period**

Seasonal fluctuations of a short term normally last for one year or so whereas cyclical variations are long-term and the cyclical completes about 8 to 10 years.

**Regularity**

Both

**seasonal and cyclical variations**are considered regular variations but still, there is a difference between them. the order and period of seasonal fluctuations are regular in nature Businessmen can predict the order of boom and depression during the year and for how many months there will be a boom and for how many months there will be depression.**Effect**

The seasonal variations affect each and every business differently. particular time commerce and industry may experience while some other industries may pass through the phase of depression. But the sickle fluctuations affect the whole economy alike.

Causes the main reason for seasonal fluctuations are customs traditions, habits, and climates while cyclical fluctuations arise due to inflation or deflation actions for employment conditions in the economy.

### CYCLES

Cycle analysis attempts to find recurring major and minor peaks and troughs in price movement for better trade timing. A cycle is a regularly occurring sequence of events. The sun rising every morning and setting in the evening is a cycle. The four seasons are one cycle in financial and commodity markets, as the price movement of a market from a local bottom to a local top and back again. For example, a stock's natural ups and downs may come at regular three-month periods. Every three months, falling prices tend to make a local bottom. In between the bottoms, rising prices tend to make a local top. Knowing where these reversals tend to occur can help time purchases at cycle bottoms and sales at cycle tops.

**How are cycles added?**

**Cycles**are added and return to the example of the sun and the seasons. By adding the annual seasons' cycle to the daily sun cycle, you can forecast a likely temperature for any time of day on any given date. Cycles are tools to be used to help forecast likely turning points in the market and never to try to define the market specifically.

The

**trend**component describes the behavior of the time series in the long term without accounting for seasonal or cyclical effects. Using the trend, we can make broad statements about the time series in the long run, such as the population of the earth increasing or the value of a stock is stagnating. The**seasonality**component explains the systematic and calendar-related movements of a time series. The**cyclical**component for anything else unexplained or irregular with the time series; could be something such as a hurricane driving the number of ice cream trucks down in the short term because it is not safe to be outside. We can use Python to decompose the time series into trend, seasonality, and noise or residuals. The cyclical component is captured in the noise (random, unpredictable data); after we remove the trend and seasonality from the time series, what we are left with is the**residual:**When building models to forecast time series, some common methods include exponential smoothing and

**ARIMA**-family models. ARIMA stands for autoregressive (AR), integrated (I), and moving average (MA). Autoregressive models take advantage of the fact that observation at time t is correlated to a previous observation, for example, at time t - 1.The integrated component concerns the differenced data or the change in the data from one time to another. For example, if we were concerned with a lag (distance between times) of 1, the differenced data would be the value at time t subtracted by the value at time t - 1. Lastly, the moving average component uses a sliding window to average the last x observations, where x is the length of the sliding window. If, for example, we have a 3-period moving average, by the time we have all of the data up to time 5, our moving average calculation only uses time periods 3, 4, and 5 to forecast time 6. The moving average puts equal weight on each time period in the past involved in the calculation. In practice, this isn't always a realistic expectation of our data. Sometimes, all past values are important, but they vary in their influence on future data points. For these cases, we can use exponential smoothing, which allows us to put more weight on more recent values and less weight on values further away from what we are predicting.

Note that we aren't limited to predicting numbers; in fact, depending on the data, our predictions could be categorical in nature—things such as determining which flavor of ice cream will sell the most on a given day or whether an email is a spam or not.

**Measurement of Cyclical Fluctuations**

Croxton and Cowden have mentioned four methods in their book 'Applied General Statistics' for the measurement of cyclical function:

1) Residual Method

2) Direct Method

3) Harmonic Analysis Method

4) Method of Cyclical Averages

### Residual Method

The

**residual**for a specific data point is the difference between the value predicted by the regression and the observed value for that data point. Calculating the residual provides information from a data set. We can calculate residuals we need to find the difference between the calculated value for the independent variable and the observed value for the independent variable.## What is time series?

Time series is an arrangement of statistical data in accordance with its time. It is also a set of observations taken at a specified time (years, months, daily) usually at equal intervals. The analysis of time series time is the most important factor because the variable is related to time which may be either a year, month, week, day, an hour, or even a minute or second.

Now time series is used with reference to economic data and economists are largely responsible for the developments of techniques of time series analysis in Python and R. So time is always an independent variable and another variable is dependent.

### Registration Analysis Used for

### Time series is helpful in forecasting

Statistical techniques have evolved which enable time series to be analyzed in such a way that the influences which have determined the form of that series may be ascertained. Future trends may be predicted to a great extent and as one of the chief responsibilities of management is to forecast the future, the careful, detailed study of time series data is a prerequisite for survival. We will see after some time to use python and forecast stock price for next year(given fig.)

Time series is helpful in understanding past behavior of data such as production, income, turnover or profit, etc. By observing data over a period of time one can easily understand what changes have taken place in the past, Such analysis will be extremely helpful in producing future behavior.

### Performance Evaluation of a Firm

Time series help to improve the performance evaluation of a firm. The objective of the businessman in analyzing time series is to investigate the causes which led the business to develop to the present extent and on the basis of the present operating casual factors to predict future trends. For example, if the expected sale for 2020 was 1,00,000 cars and the actual sale was only 90,000 cars one can investigate the causes of the shortfall in achievement.

### Helpful in cyclical fluctuations

Time series helps to estimate the trade cycle. It can well estimate the cyclical functions in the business such as boom, recession, depression, and recovery. Thus a businessman can save himself from possible loss by properly controlling the trade activities in a planned manner.

### Other utility

Time series helps to analyze sales, cost, price, and many other businesses. The statistical analysis of time series data is of particular interest to the businessman because of a large proportion of the basic statistical data with which he must deal time series data. The quantitative measurement of inventories, cost, price, sales, and many other businesses and economic variables are classified on the basis of intervals of time. One can overcome certain difficulties with the help of analysis of time series like violent fluctuations, periodic changes in prices, probable depressions, and erratic decline of prices. So, Time series analysis has a universal application. Read

### Time series is helpful in a comparative study

Different time series are compared and important conclusions are drawn therefrom.

### How do analyze time series?

The analysis of time series is important for economists, businessmen, scientists, astronomists, geologists, biologists, sociologists, research scholars, etc. It is also used in forecasting on future like ARIMA models

### The module of Time Series

## Long-Term Time Series

The long-term tendency in a time series is known as a trend or secular variation. Secular trend movements are thought of as long-term. For example, we often talk of rising trends in population, price, etc. The trend also called a secular or long-term trend, is the basic tendency of production, sales, income, employment, etc., to grow or decline over a period of time. There may be a series of annual rainfall in centimeters, for a number of years, or the changing population in millions from decade to decade, and data may progressively increase or decline during the whole period considered together. This general rise or decline, as a whole in a time series, is a secular trend.

### 1. Linear or Straight-line trend

### Imagine someone has made the effort of collecting the average price of coffee for 90 years in the city, with intervals of five years and that this has yielded the data.

Let's get this data into Python to see how to plot this linear increasing(straight line) trend.

import pandas as pd

import matplotlib.pyplot as plt

### 2. Non-linear trend

When the period is short, the secular movements can't be expected to reveal themself clearly and the general drift of the series may be unduly influenced by the cyclical fluctuations. The next point is that for concluding whether the data is showing an uptrend tendency or a downtrend tendency, it is not necessary that rise or fall must continue in the same direction through the period. We have to observe the general tendency of data. We can say that the period as a whole was characterized by an upward movement or by a downward movement, we say that a secular trend was present.

For example, if we observe the trend of prices over a period of 50 years and find except for a year or two the prices are continuously rising, we could call for a secular rise in prices.

### Cyclical Variation

Cyclical represents variations in time series that usually last longer than a year and regular,, neither in amplitude nor in length. Cyclical fluctuations of business cycle movements are recurrent up and down movements around secular trend levels which have a duration anywhere from 2 to 16 years. Most of the time series relating to economics and business show some kind of cyclical variation.

There are four periods or phases in the business cycle:

1. Prosperity

2. Decline

3. Depression

4. Recovery

There is no definite period of cyclical fluctuation, this period varies from 3 to 10 years. The concept of cyclical function can be understood more precisely by the examples that after every 3 years there is the tendency of bumper crops of mangoes or after every four years production of sugarcane reaches the peak, etc.

### Random or Irregular Variation

Random variation is intended to include all types of variations than those accounting for the trend, seasonal and cyclical fluctuations. Irregular variations are caused by such isolated special occurrences as floods, earthquakes, and wars.

### Seasonal Variation

Seasonal variation refers to annual periodicity in the business of economic theory, the ideas involved can be extended to include periodicity over any interval of time such as daily, hourly, weekly, etc., depending on the type of data available.

There are some factors that cause seasonal variation:

#### Climate and Weather Conditions

The most important factor causing seasonal variation is the climate. Changes in the climate and weather

conditions such as rainfall, humidity, heat, etc. act as different products and industries.

#### Customs, Traditions, and Habits:

Nature is primarily responsible for seasonal variation in time series, customs, traditions, and habits also have their impact. For example, on certain occasions like Deepawali, Dussehra, Christmas, etc. there is a big demand for sweets and also there is a large demand for cash before the festivals because people want money for shopping and gifts.

The study and statistical measurement of seasonal patterns constitute a very important part of the analysis of a time series. The analysis of seasonal variations is mainly done to serve some objectives:

a. To analyze the past seasonal behavior

b. To predict seasonal variations for short-term planning.

## Mathematical Models in Time Series

For the purpose of the analysis of time series, mainly two mathematical models are used:

1. Additive Model

2. Multiplicative Model

### Additive Model

There is an additive relationship among all the components of the time series. Hence a time series is the sum total of all these components.

In other words

**O=T +S+C+I**

O means original data

T means trend

S means seasonal variation

I mean irregular variation

But the same model is used in the measurement and segmentation of short-term variations.

O-T=S+C+I

O-T-S=C+I

O-T-(S+C)=I

### Multiplicative Model

A time series is the multiplication of all these four components.

**O=T*S*C*I**

This is used mainly to measure and segregate the short-term variations.

The decomposition of time series may be understood as:

O/T=S*C*I

#### Difference between the Additive and Multiplicative Models

In a multiplicative model, the only trend is shown as a unit and all other components are expressed as the ratio of the trend, wears in an additive model components are shown as units of actual data. Hence the seasonal, cyclical, and irregular variations of the multiplicative model are computed in the form of indexes.

## Measurement of Seasonal Variation

We see seasonal patterns in economics and business. When data are expressed annually there is no seasonal variation. But monthly or quarterly data frequently exhibits strong movements and considerable interest attaches to devising a pattern of average seasonal variation.

**Relationship between trend and seasonal variation:**In the additive model, seasonal variations remain stable in the majority of cases despite an increase or decrease in trend while multiplicative model the ratio of seasonal variation to the trend remains stable. In practice, the multiplicative model is considered more appropriate in the analysis of time series in economic and business fields because the various factors influencing these series are influenced by one another.

#### There are some methods used for measuring seasonal variations:

**Simple Average Method**

This is the simplest method of getting a seasonal index.

a.Arrange the unadjusted data by years and months.

b. The total of the values of the quarters is calculated.

c. The total of each quarterly value is divided by the number of years to find out the quarterly average.

Quarterly Average = Total of each quarter / No. of years

d. The total of all quarterly averages is divided by their number to find out the general average:

General Average=Total of Quarterly Average / 4

e. Quarterly average divided by general average to find out the seasonal index:

Seasonal Index =( Quarterly Average*100) / General Average

### Ratio to trend method

The ratio-to-trend method is the simplest of all methods of measuring seasonality. This method of calculating a seasonal index also called the percentage to trend method is relatively simple and yet an improvement over the method of simple average.

#### How to calculate the ratio to trend method?.

step1. Yearly Average = Yearly Total / No. of Quarters in a year

step 2. Applying the least square method, trend values are calculated.

step3. Quarterly increment = Annual increment / 4

step4 Then quarterly trend values are calculated.

step 5. Applying the multiplicative model the original data of all the periods are divided by the concerning trend.

Ratio to Trend =( Original Data*100) / Quarterly Trend Value.

step 6. Quarterly Average = Total of Quarterly Trend Ratio / No. of Years

step 7. General Average = Total of Quarterly Average / 4

Last, step 8.Seasonal Index = (Quarterly Average*100) / General Average

### Ratio to Moving Average Method

The computation by ratio to moving average method is identical to the computation of the ratio to trend seasonal index, except that a moving average trend is substituted for the list square trend used in the previous calculation. The steps necessary for determining seasonal patterns are:

Step1) On The basis of the moving average method, rent values are calculated since data are quarterly hence four quarterly moving averages are calculated.

Step 2) Then applying the multiplicative model, ratios to moving averages are calculated for all the periods. For the purpose of their computation original data is divided by trend values and multiplied by a hundred.

Ration to moving average= (O x 100)/T

Here: O = original data

T = trend value by moving average method

Step 3) The ratio of moving averages is arranged in a table.

Step 4) Quarterly averages are calculated by dividing all the quarterly values by the number of years:

Quarterly Average = Total of each quarter/number of years

Step 5) General average is calculated by dividing the total of quarterly averages by their number:

General average= Total of quarterly average/ 4

Step 6) Finally calculate the seasonal index:

Seasonal index= (Quarterly average x 100)/General average

### Link Relative method

Amongst all the methods of measuring seasonal variations, the link relative method is the most difficult one when this method is adopted the following steps are taken to calculate the seasonal variation indices

1) Calculate the link relative to seasonal figures.

Link relative = (current season's figure x 100)/ previous seasons' figures.

2) Calculate the average of the link relative to them each month

3) Convert these averages into chain relatives on the base of the first season

average of link relative= Total of link relative of each quarter/ No. of link relative in each quarter.

4) Calculate the chain relatives of the first season on the basis of the last season

Chain relative= (Average of LR of the current season's figure x chain relative of the previous season's figure)/100

5) For correction, the chain relative of the first season calculated by the first method is deducted from the chain relative (of the first season) calculated by the second method. The difference is divided by the no. of seasons. The resulting figure multiplied by 1,2,3 (and so on) is deducted res[ectively from the chain relatives of the 2nd, 3rd, 4th (and so on) seasons. These are correct chain relatives.

Chain relative of the first term = (chain relative of the last season's figure x average of link relative of 1st season)/100

6) Express the correct chain relatives as percentages of their averages. There provide the required seasonal indices by the method of link relative.

The difference per Quarter= Difference between chain relative/4

7) General average is calculated by dividing the total of all corrected chain relatives by their no.

General average = total of corrected chain relatives/4

8) In the last step seasonal indices are calculated by dividing the corrected chain relatives by the general average.

Seasonal Index= (corrected chain relatives x 100)/ General average

Learn Forecasting

### METHOD OF MEASURING TRENDS

### 1. Free Hand Curve Method:

The free Hand Curve Method is very simple and if drawn carefully, the trend fitted

by this method will be a close proximation to a mathematically fitted trend.

The free Hand Curve Method is the simplest method of studying trends. There is

no need for any mathematical calculation which saves time and labor. It is

a flexible method in the sense that it can be used to represent both linear and

non-linear.

The free Hand Curve Method has demerits:

The subjective method is highly subjective there is all possibility of drawing

different curves by different persons for the same set of data. It will be

dangerous to use for forecasting or making predictions.

### 2. Semi-Average Method:

Semi Average Method is very simple to use and saves a lot of time and labor.

It does not depend on individual estimates:

a. Division of time series into two equal parts.

b. Calculation of two averages.

c. Plotting of originaldata on graph paper

d. Plotting of both semi-average

e.Trend line

### 3. Moving Average Method:

The trend is time series may be isolated by the method of moving average.

This consists in averaging out seasonal and other short-term

functions from the series so that one is left with only the trend in the series.

The number of items taken for averaging will be the number required to cover the

period over the fluctuations that occur. The period of the moving average is to be

decided in light of the length of the cycle. Since the moving average

method is most commonly applied to data that is characterized by cyclical

variations, it is necessary to select a period for the moving average which

coincides with the length of the cycle, otherwise, the cycle will not be removed.

The moving average period is divided into two parts:

A) Odd Period Moving Average

B) Even Period Moving Average

### 4. Least Square Method:

The Least Square Method can be found with the help of a straight line.

The formula for a straight line is Y=a+bx. This formula describes any

one of an infinite number of lines. Therefore, it is necessary to decide

which line best describes the data. The principle of least square

helps in determining the line that best describes the data and its states a

line of best fit to a series of values is a line of the sum of the squares

whose deviations will be minimum.

### Conversion of Annual Trend Equations into Monthly Trend Equations

If the annual trend equation can be converted into a monthly trend equation by dividing the computed constant 'a' by 12 and the value of 'b' by 144.'a' is divided by 12 because is the sum of 12 months, while 'b' is divided by 144 because firstly it is converted into monthly value and then again is converted into monthly trend equations:

Y=(a/12)+(b/144)X

Example: Convert the annual trend equation on monthly basis:

Y=30+3.6X

Solution:

Y=(30/12)+(3.6/144)X

=2.5+0.025X

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