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The AR(I)MA model or Auto Regressive Integrated Moving Average model is one of the most popular linear models in time series forecasting. In an AR(I)MA model, the future value of a variable is assumed to be a linear function of several past observations and random errors [http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.462.3756&rep=rep1&type=pdf].

However in a lot of other papers I see have the feeling that the data is non-linear. Still they use this method and it produces good results (https://pdfs.semanticscholar.org/9672/ad02c5c7a746658441d6a82221b0420207e0.pdf)?

This (https://i.sstatic.net/CEwLX.jpg) is an example of my own data, which I assume to be non-linear aswell, thus explaining why the ARIMA models produces bad results.

Am I misinterpreting what linear modelling is, or what am I missing here?

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  • $\begingroup$ Could you explain what you mean by "non-linear" data? $\endgroup$
    – whuber ♦
    Commented May 25, 2020 at 15:37
  • $\begingroup$ I assume it to be that there is no linear relationship visible on the graph. As is the case with my examples, since they are very bumpy (up, down, up, down, ...). $\endgroup$
    – Tibo Geysen
    Commented May 25, 2020 at 15:43
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    $\begingroup$ That's a completely different meaning than the sense of "linear model" in your quotation. $\endgroup$
    – whuber ♦
    Commented May 25, 2020 at 16:47
  • $\begingroup$ It relates to the linear relationship between inputs and outputs and not to a 'straight line' in a plot. $\endgroup$
    – Sextus Empiricus
    Commented May 26, 2020 at 7:13

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The linearity here means that your model is a linear combination of the predictors, e.g. in autoregressive model $AP(p)$: $$ X_t = c + \sum_{i=1}^{p}\phi_i X_{t-i} + \epsilon_t $$ Where $\phi_1,...,\phi_p$ are your model parameters, $\epsilon$ represents noise and $c$ is some constant. That equation says that if you want to predict the value of $X_t$ you simply take a linear combination of some $p$ past $X$ values (weighted by your model parameters $\phi$). Given an initial (time series) set of $\{X_{t-p}, ... X_{t-1}\}$ there is no reason to assume that plotting $X_{t}, ...$ will give you a straight line.

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That paper is not a good example of ARIMA models. As stated before, there should be a sum type in the right hand side of the equation, thus a linear relation between the current data point and the previous ones of that variable. That's very different from the relation between the variable and time, which of course won't be linear. If you're starting with ARIMA, beware of papers.

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