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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 '16 at 22:14
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    $\begingroup$ @Carl, regarding your edit: ARIMA is a model, not an algorithm. $\endgroup$ – Richard Hardy Nov 16 '16 at 6:51
  • $\begingroup$ @RichardHardy Actually all realizable models are algorithms. I tried to explain this many times on this forum, but people are very conservative. A model is a function whose range is a PDF. There are of course unrealizable models such as (even) Gaussian PDF. These are simply mathematical abstractions that simplify things for us, but don't exist in real world. $\endgroup$ – Cagdas Ozgenc Nov 16 '16 at 7:11
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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 '16 at 7:44
  • $\begingroup$ @RichardHardy I said all (realizable) models are algorithms, I didn't say all algorithms are models (at least not statistical ones). The reason for conservatism as far as I can see is because there are bullies in this forum, not for the sake of clarity. $\endgroup$ – Cagdas Ozgenc Nov 16 '16 at 7:46
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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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