Moving Average Model with TSFP and Analyze Boston (November 20th).

The Moving Average Model: or MA(q), is a time series model that predicts the current observation by taking into account the influence of random or “white noise” terms from the past. It is a part of the larger category of models referred to as ARIMA (Autoregressive Integrated Moving Average) models. Let’s examine the specifics:

• Important Features of MA(q):

1. 1. Order (q): The moving average model’s order is indicated by the term “q” in MA(q). It represents the quantity of historical white noise terms taken into account by the model. In MA(1), for instance, the most recent white noise term is taken into account.
   2. White Noise: The current observation is the result of a linear combination of the most recent q white noise terms and the current white noise term. White noise is a sequence of independent and identically distributed random variables with a mean of zero and constant variance.
   3. Mathematical Equation: An MA(q) model’s general form is expressed as follows:
      Y t = μ + Et + (θ(1) ​E(t−1) )​+( θ(2) E(t-2) ​)+…+(θq​Et−q)
      Y t is the current observation.
      The time series mean is represented by μ.
      At time t, the white noise term is represented by Et.
      The weights allocated to previous white noise terms are represented by the model’s parameters, which are θ 1​, θ 2​,…, θ q​.

• Key Concepts and Considerations:

1. 1. Constant Mean (μ): The moving average model is predicated on the time series having a constant mean (μ).
   2. Stationarity: The time series must be stationary in order for MA(q) to be applied meaningfully. Differencing can be used to stabilise the statistical characteristics of the series in the event that stationarity cannot be attained.
   3. Model Identification: The order q is a crucial aspect of model identification. It is ascertained using techniques such as statistical criteria or autocorrelation function (ACF) plots.

• Application to Time Series Analysis:

1. 1. Estimation of Parameters: Using statistical techniques like maximum likelihood estimation, the parameters θ 1, θ 2, …, θ q, are estimated from the data.
   2. Model Validation: Diagnostic checks, such as residual analysis and model comparison metrics, are used to assess the MA(q) model’s performance.
   3. Forecasting: Following validation, future values can be predicted using the model. Based on the observed values and historical white noise terms up to time t−q, the forecast at time t is made.

• Use Cases:

1. 1. Capturing Short-Term Dependencies: When recent random shocks have an impact on the current observation, MA(q) models are useful for detecting short-term dependencies in time series data.
   2. Complementing ARIMA Models: To create ARIMA models, which are strong and adaptable in capturing a variety of time series patterns, autoregressive (AR) and differencing components are frequently added to MA(q) models.

Let’s try to fit an MA(1) model to the ‘logan_intl_flights’ time series from Analyze Boston. But before that it’s important to assess whether the ‘logan_intl_flights’ time series is appropriate for this type of model. The ACF and PACF plots illustrate the relationship between the time series and its lag values, which aids in determining the possible order of the moving average component (q).