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I am working on a problem of predicting event counts based on user history. This is a classical time series analysis problem, and I used the ARIMA model: (wiki).
I also applied a Hawkes point process model for the same purpose. Therein I predict all the points using a univariate Hawkes process and calculate counts.

However, I do not understand the fundamental difference between these two models;
I do know that ARIMA takes previous event counts as input, and the Hawkes model takes event timestamps as input.
Would someone please point out what the other differences between these models are?

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    $\begingroup$ event counts are not so classical, unless the counts are very high so that you could approximate them as continuous variable such as Gaussian. low counts are modeled with discrete processes, such as Poisson process. $\endgroup$
    – Aksakal
    Nov 15, 2016 at 22:14
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    $\begingroup$ @Carl, regarding your edit: ARIMA is a model, not an algorithm. $\endgroup$
    – Richard Hardy
    Nov 16, 2016 at 6:51
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    $\begingroup$ @CagdasOzgenc, I don't want to get too philosophical here, but I think people are "conservative" for a reason. It is practical to use words with narrow meanings when we are talking technical issues. For example, one model can be estimated by many different algorithms, and one algorithm can be used to estimate many different models. If I said an ARIMA algorithm, you would just not know what I am talking about (which of the algorithms that can be used for estimating the model). And if I talked of grid search or Newton-Raphson, would you ever call them models? I wouldn't. $\endgroup$
    – Richard Hardy
    Nov 16, 2016 at 7:44
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    $\begingroup$ @CagdasOzgenc, point taken. But why use a broader term if there is a well-established narrower term that can save some confusion (especially for less educated users than you are)? That is, philosophically you might be right, but in practice I would go for doing things in a convenient, unambiguous way. If algorithm is broader than model, why open up more possibilities than are necessary and potentially distract the discussion from the more narrowly defined topic? $\endgroup$
    – Richard Hardy
    Nov 16, 2016 at 8:02
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    $\begingroup$ @Carl, or probably ANOVA is from multiple models that can be estimated by OLS (where I contrast the model and its estimation technique). E.g. linear regression model may or may not be estimated by OLS. But I don't ANOVA well enough, so I should really read up before commenting :) $\endgroup$
    – Richard Hardy
    Nov 17, 2016 at 22:03

2 Answers 2

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  1. Let's start with AR. AR by default is a time-discrete model, HP or general point process can be defined on continuous space, such as a timeline. In practice, point process data can be discretized using time bins. The data counts in each interval can be either binary or integers.

  2. In AR, $X_t$ is a linear combination of the past values $X_{t-1}, X_{t-2},...$ with additive Normal noise. In HP, the underlying intensity function is a linear combination of past events. To be more specific, $$\lambda_t = \mu+ \alpha_1 X_{t-1} + \alpha_2 X_{t-2} + ... $$ usually people write this as $$\lambda_t = \mu+ \sum_i h(t - i)X_i $$ $h(t - i)$ can be seen as $\alpha_i$. The probability of observing $n$ events in time interval $t$ follows Poisson distribution, $$ \mathbb{P}( X_t =n ) \sim \text{Poisson}(\lambda_t \Delta) $$ $\Delta$ is the width of the time interval.

    In brief, the noise of the observation in HP is not Gaussian.

  3. Come back to ARIMA, HP does not incorporate error terms in the past (no "MA" part), not to mention the temporal difference (no "I" part). You can add something in HP like the "MA" and "I" components, but I've never seen that.
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A time series has time measurements made at regular time intervals, whereas in a Poisson process, including the Hawkes process, the time measurements are distributed in a Poissonian way.

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