Time Series Short Notes

Overview of Time Series Analysis:

Time series data consists of observations collected over time, often at regular intervals (e.g., daily, monthly, yearly).
The main goals of time series analysis include understanding the underlying patterns and trends, identifying any cyclical or seasonal components, and making forecasts for future values.
Time series analysis techniques are widely used in various fields, such as economics, finance, meteorology, and engineering, among others.

Developed by statisticians George Box and Gwilym Jenkins in the 1970s.
It provides a systematic approach for identifying, estimating, and diagnosing ARIMA models.
The methodology involves the following iterative steps: a. Model Identification: Examine the data patterns, stationarity, and autocorrelation/partial autocorrelation functions to determine the tentative ARIMA model orders (p, d, q). b. Parameter Estimation: Use techniques like maximum likelihood estimation to estimate the model parameters. c. Diagnostic Checking: Analyze the residuals to ensure the model assumptions are met (e.g., white noise, no autocorrelation). d. Forecasting: Use the fitted model to make forecasts for future time periods.

ARIMA stands for Autoregressive Integrated Moving Average.
It is a versatile model that combines three components: a. Autoregressive (AR) component: Models the current value as a linear combination of its past values. b. Integrated (I) component: Handles non-stationarity by taking differences of the time series. c. Moving Average (MA) component: Models the current value as a linear combination of past error terms.
The orders of the ARIMA model are represented as ARIMA(p, d, q), where:
- p is the order of the AR component (number of past values used)
- d is the order of differencing (number of times the series is differenced)
- q is the order of the MA component (number of past error terms used)

The ACF measures the correlation between a time series and its lagged values.
It helps identify the presence and pattern of autocorrelation in the data.
The ACF plot displays the autocorrelation coefficients at different lags.
The pattern of the ACF can provide insights into the appropriate AR and MA orders for an ARIMA model.

In an AR(p) model, the current value is modeled as a linear combination of its p previous values and a random error term.
The AR(p) model is represented as: y_t = c + φ_1y_(t-1) + φ_2y_(t-2) + … + φ_p*y_(t-p) + ε_t
AR models are useful when the time series exhibits autocorrelation in its values.

In an MA(q) model, the current value is modeled as a linear combination of the current and q previous error terms.
The MA(q) model is represented as: y_t = μ + ε_t + θ_1ε_(t-1) + θ_2ε_(t-2) + … + θ_q*ε_(t-q)
MA models are useful when the time series exhibits autocorrelation in its errors.

An ARMA(p, q) model combines both AR(p) and MA(q) components.
The ARIMA(p, d, q) model extends the ARMA model by including an integrated component (I) of order d, which accounts for non-stationarity by taking differences of the time series.
ARIMA models are flexible and can capture a wide range of time series patterns, making them widely applicable.

Stationarity checking: Check if the time series is stationary (constant mean and variance) by examining plots and performing statistical tests. If not, apply differencing or transformations to achieve stationarity.
Model identification: Examine the ACF and Partial Autocorrelation Function (PACF) plots to determine the tentative orders of the AR and MA components.
Parameter estimation: Use methods like maximum likelihood estimation to estimate the model parameters.
Diagnostic checking: Analyze the residuals to ensure they are white noise (no autocorrelation) and meet the model assumptions.
Model selection: Compare different ARIMA models using information criteria (e.g., AIC, BIC) and choose the best-fitting model.
Forecasting: Use the selected ARIMA model to generate forecasts for future time periods, along with prediction intervals.

ARIMA models are widely used due to their flexibility and ability to capture various time series patterns.
They can handle non-stationarity, trends, and seasonal components.
However, ARIMA models have some limitations and assumptions: a. Stationarity assumption: The time series should be stationary or made stationary through differencing or transformations. b. Linear model: ARIMA models assume linear relationships, which may not hold for highly nonlinear data. c. Structural breaks and outliers: ARIMA models can be sensitive to structural breaks, outliers, or changes in the underlying data-generating process. d. Large data requirements: Building accurate ARIMA models typically requires a sufficiently large historical dataset.
Caution should be exercised when interpreting and applying ARIMA models, and diagnostic checks should be performed to ensure the model assumptions are met.