Mathematical statement of the composition of Time Series, Business Mathematics and Statistics

# Mathematical statement of the composition of Time Series, Business Mathematics and Statistics - Business Mathematics and Statistics - B Com

Mathematical Statement of the Composition of Time Series

A time series may not be affected by all type of variations. Some of these type of variations may affect a few time series, while the other series may be effected by all of them. Hence, in analysing time series, these effects are isolated. In classical time series analysis it is assumed that any given observation is made up of trend, seasonal, cyclical and irregular movements and these four components have multiplicative relationship.

Symbolically :

O = T × S × C × I

where O refers to original data,
T refers to trend.
S refers to seasonal variations,
C refers to cyclical variations and
I refers lo irregular variations.

This is the most commonly used model in the decomposition of time series.

There is another model called Additive model in which a particular observation in a time series is the sum of these four components.

O = T + S + C + I

To prevent confusion between the two models, it should be made clear that in Multiplicative model S, C, and I are indices expressed as decimal percents whereas in Additive model S, C and I are quantitative deviations about trend that can be expressed as seasonal, cyclical and irregular in nature. If in a multiplicative model. T = 500, S = 1.4, C = 1.20 and I = 0.7 then

O = T × S × C × I

By substituting the values we get

O = 500 × 1.4 × 1.20 × 0.7 = 608

In additive model, T = 500, S = 100, C = 25, I = –50

O = 500 + 100 + 25 – 50 = 575

The assumption underlying the two schemes of analysis is that whereas there is no interaction among the different constituents or components under the additive scheme, such interaction is very much present in the multiplicative scheme. Time series analysis, generally, proceed on the assumption of multiplicative formulation.

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## FAQs on Mathematical statement of the composition of Time Series, Business Mathematics and Statistics - Business Mathematics and Statistics - B Com

 1. What is Time Series in business mathematics and statistics?
Ans. Time series refers to a sequence of data points collected over a specific time period. In business mathematics and statistics, time series analysis is used to analyze and forecast trends, patterns, and behavior of a variable over time. It helps in understanding the past performance and predicting the future values of a particular variable.
 2. How is Time Series useful in business decision-making?
Ans. Time series analysis provides valuable insights for business decision-making. It helps in identifying seasonality, trends, and cyclic patterns in data, enabling businesses to make informed decisions related to production, inventory management, sales forecasting, and financial planning. By analyzing historical data, businesses can anticipate future demand, optimize resources, and improve overall operational efficiency.
 3. What are the components of a Time Series?
Ans. Time series can be decomposed into four main components: trend, seasonality, cyclical variations, and random variations. - Trend represents the long-term upward or downward movement in data. - Seasonality captures the regular and predictable patterns that occur within a specific time frame, such as daily, weekly, or yearly. - Cyclical variations refer to fluctuations that are not predictable and occur over an extended period. - Random variations, also known as irregularities or residuals, are unpredictable fluctuations that cannot be attributed to any specific component.
 4. How can Time Series analysis help in financial forecasting?
Ans. Time series analysis plays a crucial role in financial forecasting. By analyzing historical financial data, businesses can identify patterns, trends, and seasonality in their financial performance. This information can be utilized to forecast future revenues, expenses, profits, and cash flows. Financial forecasting based on time series analysis helps businesses in budgeting, investment planning, risk management, and making strategic financial decisions.
 5. What statistical methods are commonly used in Time Series analysis?
Ans. Several statistical methods are used in Time Series analysis, including: - Moving averages: This method calculates the average of a specific number of consecutive data points to smoothen out fluctuations and highlight trends. - Exponential smoothing: It assigns different weights to different data points, giving more importance to recent observations while forecasting future values. - Autoregressive Integrated Moving Average (ARIMA): ARIMA models consider the correlation between an observation and a previous observation, as well as the difference between consecutive observations, to forecast future values. - Seasonal decomposition of time series (STL): This method decomposes a time series into trend, seasonality, and remainder components, allowing for separate analysis and forecasting of each component. - Box-Jenkins method: It involves identifying and fitting an appropriate ARIMA model to the time series data, considering autocorrelation and partial autocorrelation functions.

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