"an autoregressive integrated moving average (ARIMA) model is a generalization of an autoregressive moving average (ARMA) model."
An extension of ARIMA that includes seasonal components.
Time Series Analysis: A statistical method that analyzes and models time series data to determine patterns, trends, and behaviors. It involves identifying the underlying structure and relationships between the observed data points and forecasting future values.
Stationarity: A property of time series data that is constant over time and does not exhibit any systematic or structural changes. It is an important assumption for many models in time series analysis, including ARIMA.
Autoregressive Integrated Moving Average (ARIMA): A powerful and widely used time series model that captures the dependence among observations by incorporating three components: autoregression, differencing, and moving average. ARIMA models are used for forecasting and identifying trends, cycles, and outliers in time series data.
Seasonality: A common pattern of variation in time series data that repeats at regular intervals, such as weekly, monthly, or yearly. Seasonality can affect the accuracy of forecasts if it is not properly accounted for in the modeling process.
Autocorrelation Function (ACF): A statistical measure that shows the degree of correlation between a time series and its lagged values at different time intervals. ACF is used to identify patterns in the data that can be used to inform the selection of ARIMA models.
Partial Autocorrelation Function (PACF): A statistical measure that shows the degree of correlation between a time series and its lagged values after controlling for the effects of other intervening variables. PACF is used to identify the order of the autoregressive and moving average components in ARIMA models.
Seasonal Autoregressive Integrated Moving Average (SARIMA): A variation of ARIMA that incorporates seasonality into the time series model. SARIMA models are used to capture the seasonal patterns in time series data and improve the accuracy of forecasts.
Fourier Analysis: A mathematical method that decomposes time series data into a sum of sine and cosine waves with different frequencies. Fourier analysis can be used to identify the seasonality and frequency components in time series data and inform the selection of SARIMA models.
Box-Jenkins Methodology: A popular approach to time series modeling that involves a systematic process of selecting and fitting ARIMA models to the data. The Box-Jenkins methodology emphasizes diagnostic testing and model selection based on ACF and PACF plots.
Bayesian Time Series Analysis: A statistical approach to time series modeling that uses Bayesian inference to estimate the parameters of the model and predict future values. Bayesian time series analysis allows for the incorporation of prior knowledge and the updating of predictions based on new data.
Non-seasonal ARIMA: A non-seasonal ARIMA model is used when the time series does not exhibit any seasonal patterns. It is represented as ARIMA(p,d,q), where p is the number of autoregressive terms, d is the degree of differencing, and q is the number of moving average terms.
Seasonal ARIMA: A seasonal ARIMA model is used when the time series exhibits seasonal patterns. It is represented as ARIMA(p,d,q)(P,D,Q)m, where P is the number of seasonal autoregressive terms, D is the degree of seasonal differencing, Q is the number of seasonal moving average terms, and m is the number of periods in each season.
Hybrid ARIMA: A hybrid ARIMA model combines both non-seasonal and seasonal ARIMA models. It is usually used when the time series exhibits both non-seasonal and seasonal patterns.
Transfer Function ARIMA: A transfer function ARIMA model is used when there are additional external variables that can affect the time series. It is represented as ARIMA(p,d,q)X, where X represents the external variables.
"ARIMA models are applied in some cases where data show evidence of non-stationarity in the sense of mean (but not variance/autocovariance)."
"an initial differencing step (corresponding to the 'integrated' part of the model) can be applied one or more times to eliminate the non-stationarity of the mean function (i.e., the trend)."
"When the seasonality shows in a time series, the seasonal-differencing could be applied to eliminate the seasonal component."
"we are motivated to make stationary a non-stationary time series, e.g., by using differencing, before we can use the ARMA model."
"the ARMA model, according to the Wold's decomposition theorem, is theoretically sufficient to describe a regular (a.k.a. purely nondeterministic) wide-sense stationary time series"
"The AR part of ARIMA indicates that the evolving variable of interest is regressed on its own lagged (i.e., prior) values."
"The MA part indicates that the regression error is actually a linear combination of error terms whose values occurred contemporaneously and at various times in the past."
"The I (for 'integrated') indicates that the data values have been replaced with the difference between their values and the previous values."
"The purpose of each of these features is to make the model fit the data as well as possible."
"Non-seasonal ARIMA models are generally denoted ARIMA(p,d,q) where parameters p, d, and q are non-negative integers."
"p is the order (number of time lags) of the autoregressive model."
"d is the degree of differencing (the number of times the data have had past values subtracted)."
"q is the order of the moving-average model."
"Seasonal ARIMA models are usually denoted ARIMA(p,d,q)(P,D,Q)m"
"where m refers to the number of periods in each season."
"the model may be referred to based on the non-zero parameter, dropping 'AR', 'I' or 'MA' from the acronym describing the model."
"ARIMA(1,0,0) is AR(1)."
"ARIMA(0,1,0) is I(1)."
"ARIMA(0,0,1) is MA(1)."