Introduction

A series that changes over time is called time series. For consistency and reliability of time series analysis large number of data is required. It ensures that we have good representative sample and effect of noisy data data can be cut out.

Time series analysis is useful to see how assets and economic variable changes over time. Finance, retail, and economics related areas require time series analysis as currency and sales keep on changing.

Some common use case of time series analysis include:

• Heart rate monitoring
• Stock price prediction
• Weather forecast
• Rainfall measurements

Time frame used in time series can be years, months, weeks, days, hours, minutes and seconds.

When there is need to analyze data over consistent intervals, time series forecasting is used to predict the likelihood of future events. Seasonality or cyclic behavior is commonly observed in time series data.

Components of time series

The various reasons or the forces which affect the values of an observation in a time series are the components of a time series. The four categories of the components of time series are:

• Trend
• Cyclic Variations
• Seasonal Variations
• Random or Irregular movements

Trend

The trend shows the tendency of the data to increase or decrease over a long period of time.  Trends can be linear or non-linear (curvilinear). It is a longer-term change. Some examples of secular trends are:

• Increase in price of daily consumed goods
• Increase in pollution level

Thus over a relatively long period of time, trend shows if the data tends to increase or decrease. Trend is an average, long-term, smooth tendency. Different sections of time within data show varying tendencies in terms of trends that are increasing, decreasing, or stable.  There must, however, be an overall upward, downward, or stable trend.

Cyclic Movements

Cyclic movements are long term oscillations occurring in a time series. These oscillations are mostly observed in economics data and the periods of such oscillations are generally extended from five to ten years or more. These cyclic movements can be studied provided a long series of measurements.  The cyclical variation is periodic in nature and repeats itself like a business cycle. Phase in cyclic variation are:

• Peak or Prosperity
• Recession
• Trough or Depression
• Expansion

 

Seasonal variations

Seasonal variations are caused by climate, social customs, religious activities, etc.  For example, traffic on roads in the office and school hours, sales of light at festivals like Tihar, etc. The main causes of seasonal variations are seasons, religious festivals, and social customs. Thus, seasonal component of a time series is the variation in some variable , lets say like sales, due to some predetermined patterns in its behavior.

Some other examples include the increase in the cost of various types of fruits and vegetables increase in rainy season, increase in average daily rainfall in summer and decrease in winter . The changes which repeat themselves within a fixed period, are also called seasonal variations.

Seasonal effects are different from cyclical effects, as seasonal cycles always occur within one calendar year. But cyclical effects, such as increased unemployment can span time periods shorter or longer than one calendar year.

The main difference between seasonal and cyclical behavior can be figured out by finding how regular the period of change is. A seasonal behavior is always strictly regular. There is a precise amount of time between the peaks and troughs of the data. For instance start of summer and winter season and their span has seasonal behavior.

Cyclical behavior on can drift over time and the time between periods isn't precise. For example, the stock market tends to cycle between periods of high and low values, but there is no set amount of time between period of rise andd fall.

Random or Irregular movements

These are sudden changes occuring in a time series which are unlikely to be repeated. These are the components of a time series which cannot be explained by trends, seasonal or cyclic movements. These variations are sometimes called residual or random components. These variations, though accidental in nature, can cause a continual change in the trends, seasonal and cyclical oscillations during the forthcoming period. Earthquakes, worldwide COVID pandemic are some examples of the root causes irregular movements.

Mathematical representation of time series

Let us represent time by variable t and let y_t  be variable under study.

Since y_t  is function of time , mathematically we can write: y_t = f (t)

Let the value of variable be y₁,y₂,y₃,y₄.....y_n at times t₁,t₂,t₃,t₄.....t_n respectively.

We have three models for time series :

• Additive Model
• Multiplicative Model
• Mixed model

Additive Model

Let us consider T_t, S_t, C_t, and R_t as the trend value, seasonal, cyclic and random fluctuations at time t respectively.

So the value of yt at any time t can be written as: y_t = T_t+ S_t+ C_t+R_t

Additive model assumes that trend value, seasonal, cyclic and random fluctuations act independently of each other.

Multiplicative Model

Let us consider T_t, S_t, C_t, and R_t as the trend value, seasonal, cyclic and random fluctuations at time t respectively.

So the value of yt at any time t can be written as:

y_t = T_t× S_t×C_t×R_t

Thus in multiplicative model, trend value, seasonal, cyclic and random fluctuations in a time series operate proportionately to each other.

Mixed model

It is formed using various combination of additive and multiplicative models.

An example of combination can be :

y_t = T_t× S_t+ C_t×R_t

Another example of combination can be :

y_t = T_t+ S_t+ C_t×R_t