Felix Cheysson

This package implements a spectral approach to the parametric estimation of Hawkes processes from Binned Observations through Whittle likelihood (HawkesBOW). It is based on the results of the article (Cheysson and Lang 2020).

You can install the released version of hawkesbow from GitHub with:

Hawkes processes form a family of models for point processes for which the occurrence of an event temporarily increases the probability of future events occurring (Hawkes 1971). Formally, a Hawkes process

Alternatively, the Hawkes process can be constructed as a poissonian cluster process (Hawkes and Oakes 1974). The process consists of a flow of *immigrants*, the cluster centres, arriving according to a homogeneous Poisson process of intensity *children* according to an inhomogeneous Poisson process of intensity . These in turn independently generate children according to the same process, and so on *ad infinitum*. The processes consisting of an immigrant and all its descendants are therefore branching processes, and are independent of each other. Finally, the process resulting from the superposition of all these branching processes is a Hawkes process of conditional intensity (see the following figure of a realisation of an exponential Hawkes process with ,

This package also supports non-causal Hawkes processes, for which the reproduction kernel

The count sequence of a Hawkes process is the time series generated by the event counts of the process, that is the series obtained by counting the events of the process on intervals of fixed length. Formally, the count sequence with bin size

This package fully supports the parametric estimation of Hawkes processes from their count sequences via minimisation of the Whittle likelihood, and partially supports the estimation from the arrival times via maximisation of the likelihood.

The parameters of a Hawkes process with arrival times on the interval

Note that each step of the optimisation is usually of complexity .

Alternatively, the parameters of a Hawkes process with count sequence can be estimated by minimising the log-spectral (Whittle) likelihood

where denotes the periodogram of

This package supports both Hawkes processes and their count sequences.

Hawkes processes can be simulated by the function `hawkes`

:

where `end`

denotes the end of the interval

First, the immigrants are drawn according to a (potentially inhomogeneous) Poisson process with intensity measure

`fun`

.Second, the number of offsprings of an immigrant are drawn from a Poisson distribution with intensity

`repr`

.Third, these offsprings are distributed according to the

`family`

distribution.Then, further offsprings are generated according to the last two steps.

The argument `fun`

can take a numeric value or be specified as a function, in which case the argument `M`

must be specified as an upper bound on `fun`

(to allow simulation of the immigration process by thinning). The argument `family`

can either be specified as a string `name`

corresponding to a distribution with random generation function `rname`

(for example `exp`

for `rexp`

), or directly as a random generation function. The optional arguments `...`

are passed to the random generation function specified by `family`

.

This returns a list of class `hawkes`

, whose realisations are stored in the member `p`

. Other members of the list are mainly used for support functions.

- Simulate a Hawkes process with immigration intensity , reproduction mean and reproduction kernel on
:

- Simulate a Hawkes process with inhomogeneous immigration process with intensity , reproduction mean and
-triangular reproduction kernel :

```
x <- hawkes(10, fun=function(y) {1+sin(y)}, M=2, repr=0.25,
family=function(n) {1 - sqrt(1 - runif(n))})
```

- Hawkes processes can be plotted with the function
`plot.hawkes`

:

```
plot.hawkes(x, intensity = FALSE, precision = 1e3, fun = NULL, repr = NULL, family = NULL,
M = NULL, ...)
```

If `x`

is of class `hawkes`

, as for objects returned by the function `hawkes`

, arguments `fun`

through `M`

can be ignored. If `intensity`

is set to `FALSE`

, this plots the genealogy of the simulated Hawkes process (as in preceding figure, bottom panel). If it is set to `TRUE`

, this plots the conditional intensity of the process (as in preceding figure, top panel).

Two functions implement the estimation of Hawkes processes: `mle`

from arrival times `whittle`

from count sequences

The maximum likelihood method is implemented by the function

`events`

holds the arrival times `kern`

must be a string (partially) matching one of the reproduction kernels (see below) and `end`

denotes the endpoint

The optimisation of the maximum likelihood function is done by the function `nloptr`

from the package nloptr (Johnson, n.d.) with algorithm L-BFGS-B, where the derivatives of the likelihood are calculated explicitly. By default, parameters are constrained to be positive, and additionally `opts`

and `...`

are passed on to `nloptr`

, so the algorithm, the constraints, or any other parameter of the optimisation procedure can be changed.

Example of use:

The Whittle likelihood method is implemented by the function

`counts`

holds the count sequence `kern`

must be a string (partially) matching one of the reproduction kernels (see below), `binsize`

denotes the bin size `trunc`

is the number of foldings due to aliasing taken into account.

The optimisation of the Whittle likelihood function is done by the function `optim`

, with algorithm L-BFGS-B where the derivatives of the likelihood are approximated by finite differences. By default, parameters are constrained to be positive, and additionally `...`

is passed to `optim`

, so any optimisation parameter can be changed.

Example of use:

```
x = hawkes(1000, fun = 1, repr = .5, family = "exp", rate = 1)
y = discrete(x, binsize = 1)
whittle(y, "Exponential", 1)
```

Note that `discrete`

is a useful function to create the count sequence `x`

of class `hawkes`

.

We introduce the reproduction kernels that are currently implemented in this package. Recall that the Fourier transform of a reproduction kernel is given by

and that it is a Hermitian function .

This is the exponential density function with parameter :

Its Fourier transform is

The exponential kernel can be specified with the string `Exponential`

and the parameter with the usual argument `rate`

. Both maximum and Whittle likelihood methods are fully implemented for exponential kernels. Moreover, the likelihood function is implemented in complexity , using the relations in (Ozaki and Ogata 1979).

This is a symmetrised version of the exponential density function with parameter :

Its Fourier transform is

The symmetric exponential kernel can be specified with the string `SymmetricExponential`

and the parameter `rate`

. Only the Whittle likelihood method is implemented for symmetric exponential kernels. Note that it is a non-causal kernel, as for

This is the gaussian density function with mean and variance :

Its Fourier transform is

The gaussian kernel can be specified with the string `Gaussian`

and its parameters `mean`

and `sd`

respectively. Only the Whittle likelihood method is implemented for gaussian kernels. Note that it is a non-causal kernel, as for

This is a normalised and shifted power law function, with shape and scale

For positive , its Fourier transform is given by

where denotes the integral

With successive integration by parts, this integral can be related to , with .

If or equivalently , the integral `E1_imaginary`

implements this approximation.

If or equivalently , the integral can be related to the incomplete gamma function with imaginary argument

where . We implemented Taylor approximations of this integral, accurate to better than `inc_gamma_imag`

.

The power law kernel can be specified with the string `PowerLaw`

and its parameters and `shape`

and `scale`

respectively. Both maximum and Whittle likelihood methods are implemented for power law kernels.

This is the Pareto density function with shape and scale

For positive , its Fourier transform is given by

see above for the definition of .

Only Pareto kernels with fixed , `Pareto1`

, `Pareto2`

and `Pareto3`

respectively, with parameter `scale`

. Only the Whittle method is available for Pareto kernels.

Improve this vignette: it is currently too sparse and functions of the package could need some better description.

Add some real datasets to the package: real life case-studies with good datasets help understand the functionalities of a package.

Variance and confidence intervals for the estimation with function

`whittle`

: note that currently, the variance-covariance matrix returned by the optimisation method in function`whittle`

is not accurate, as it ignores the dependence within the count sequence. Diagnostics for the estimated model: spectral density based goodness-of-fit tests are to be implemented for the estimated Hawkes processes, based on the work of (Paparoditis 2000).

Custom built-kernels: allow the user to input reproduction kernels that are not already implemented.

Cheysson, Felix, and Gabriel Lang. 2020. “Strong mixing condition for Hawkes processes and application to Whittle estimation from count data.” *arXiv*, March. https://arxiv.org/abs/2003.04314.

Eddelbuettel, Dirk, and Romain François. 2011. “Rcpp : Seamless R and C++ Integration.” *J. Stat. Softw.* 40 (8): 1–18. https://doi.org/10.18637/jss.v040.i08.

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Hawkes, Alan G. 1971. “Spectra of Some Self-Exciting and Mutually Exciting Point Processes.” *Biometrika* 58 (1): 83–90. https://doi.org/10.2307/2334319.

Hawkes, Alan G., and David Oakes. 1974. “A cluster process representation of a self-exciting process.” *J. Appl. Probab.* 11 (03): 493–503. https://doi.org/10.2307/3212693.

Johnson, Steven G. n.d. “The NLopt nonlinear-optimization package.”

Ozaki, T., and Yosihiko Ogata. 1979. “Maximum likelihood estimation of Hawkes’ self-exciting point processes.” *Ann. Inst. Stat. Math.* 31 (1): 145–55. https://doi.org/10.1007/BF02480272.

Paparoditis, Efstathios. 2000. “Spectral density based goodness-of-fit tests for time series models.” *Scand. J. Stat.* 27 (1): 143–76. https://doi.org/10.1111/1467-9469.00184.

Rowe, B. T. P., M. Jarvis, R. Mandelbaum, G. M. Bernstein, J. Bosch, M. Simet, J. E. Meyers, et al. 2015. “GalSim: The modular galaxy image simulation toolkit.” *Astron. Comput.* 10: 121–50. https://doi.org/10.1016/j.ascom.2015.02.002.