Time Series & Arima Model of Forecasting

Time Series & ARIMA Model (Forecasting)

What is Time Series?

A time series is a sequence of data points recorded at evenly spaced time intervals, such as:

• hourly temperature readings

• daily stock prices

• annual population

• monthly sales

Each point represents a value measured over time.

Importance of Time Series Analysis

• Predict future trends

• Detect patterns and anomalies

• Risk mitigation

• Strategic planning

• Competitive advantage

Components of Time Series

1. Trend

• Long-term movement or direction of data

• Can be increasing, decreasing, linear, or nonlinear

2. Seasonality

• Patterns repeating at fixed intervals

• Examples: yearly festivals, monthly sales peaks, weekly cycles

3. Cyclical Variation

• Long-term fluctuations without a fixed period

• Usually related to economic or business cycles

4. Irregular/Noise

• Random, unpredictable variations

• Not explained by trend, seasonality or cycles

Time Series Forecasting

It uses historical data to predict future values.
Example: Forecasting population of India for 2037, 2047, 2057 using past population data.

ARIMA Model

ARIMA stands for:
A – Autoregressive (AR)
I – Integrated (I)
MA – Moving Average (MA)

It is a popular forecasting technique for time series data that changes with time (population, temperature, sales etc.).

1. Autoregressive (AR)

• Current value depends on past values of the series.

• Linear relationship.

Formula:

2. Integrated (I) $({\Delta }^d{x}_t)$

• Uses differencing to make the series stationary.

• Stationary means: mean, variance, covariance do not change over time.

1. $x_t$

• Original time series data

• Example: temperature, stock price, population, sales, etc.

2. $\Delta$

• The differencing operator

• It removes trend and makes the series stationary.

• First difference:

  $$\Delta x_t=x_t-x_{t-1}$$

• Second difference:

  ${\Delta }^2{x}_t=\Delta (\Delta x_t)$

3. $d$

• Order of differencing

• Tells how many times the data is differenced

• Purpose: make the series stationary

• Usually $d=0,1,\text{or }2$

4. $y_t$

• The stationary transformed series after differencing

• This $y_t$​ is then used by the AR and MA parts of ARIMA.

Hyperparameter Summary

If you want, I can also explain with a

3. Moving Average (MA)

• Current value depends on past forecast errors.

ARIMA = AR + I + MA

ARIMA combines all three components to build accurate time-based predictions.

Model Parameters (p, d, q)

These are hyperparameters and are tuned (trial & error) to get the best forecasting model.