The Autoregressive Integrated Moving Average (ARIMA) method models the next step in the sequence as a linear function of the differenced observations and residual errors at prior time steps. It combines both Autoregression ($$AR$$) and Moving Average ($$MA$$) models as well as a differencing pre-processing step of the sequence to make the sequence stationary, called integration ($$I$$). The notation for the model involves specifying the order for the $$AR(p)$$, $$I(d)$$, and $$MA(q)$$ models as parameters to an ARIMA function, e.g., $$ARIMA(p, d, q)$$. An $$ARIMA$$ model can also be used to develop $$AR$$, $$MA$$, and $$ARMA$$ models. The method is suitable for univariate time series with trend and without seasonal components. Below is an example of ARIMA code implementation

Autoregressive Integrated Moving Average (ARIMA)

The Autoregressive Moving Average (ARMA) method models the next step in the sequence as a linear function of the observations and residual errors at prior time steps. It combines both Autoregression (AR) and Moving Average (MA) models. The notation for the model involves specifying the order for the AR(p) and MA(q) models as parameters to an ARMA function, e.g., ARMA(p, q). An ARIMA model can be used to develop AR or MA models. The method is suitable for univariate time series without trend and seasonal components.

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The autoregression (AR) method models the next step in the sequence as a linear function of the observations at prior time steps.

The notation for the model involves specifying the order of the model p as a parameter to the AR function, e.g. AR(p). For example, AR(1) is a first-order autoregression model.

The method is suitable for univariate time series without trend and seasonal components.

Below is an example of AR code implementation


Autoregression (AR)

A moving average is a method used to find an average of a numerical metric over a specified block of time, n. For example, if you set n = 2 then you wouldn’t have any measurements for the first time period, you would then take the average of the first and second time periods to be moving average for the second time period, then you would take the average of the second and third time period to be the moving average of the third time period, next you would take the average of the third and fourth time period to me the moving average for the fourth time period, so on and so forth for all of the time periods in your data. The moving average can be thought of as a sort of sliding window. Also, note that your first moving average measurement will be at the nth datapoint because that is the first time you will be able to average n points.

Moving Average (MR)

The technique (Shannon, 1951) generates a word in a sequence given the previous words, using cross-entropy as the loss function and perplexity as the evaluation method. The sequence ends once it reaches a predefined length. Each softmax layer is to decide the next word based on the current hidden state in the RNN component. The initial input token(s) would depend on the NLP task's nature (e.g. machine translation, question answering, summarization). 

RNN's Approach to Text Generation in NLP (Autoregressive Generation)

Autoregressive Moving Average (ARMA)

A straightforward method for creating summary vectors for the memory component ($$\mathrm{Mem}$$) is to compute the moving average of past keys and values. For instance, $$\mathrm{Mem}$$ can be constructed from the unweighted moving average of the last $$n_c$$ key and value vectors, providing a simple yet effective way to summarize history.

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