# Multi-level methods and approximating distribution functions

## Abstract

Biochemical reaction networks are often modelled using discrete-state, continuous-time Markov chains. System statistics of these Markov chains usually cannot be calculated analytically and therefore estimates must be generated via simulation techniques. There is a well documented class of simulation techniques known as exact stochastic simulation algorithms, an example of which is Gillespie’s direct method. These algorithms often come with high computational costs, therefore approximate stochastic simulation algorithms such as the tau-leap method are used. However, in order to minimise the bias in the estimates generated using them, a relatively small value of tau is needed, rendering the computational costs comparable to Gillespie’s direct method. The multi-level Monte Carlo method (Anderson and Higham, Multiscale Model. Simul. 10:146–179, 2012) provides a reduction in computational costs whilst minimising or even eliminating the bias in the estimates of system statistics. This is achieved by first crudely approximating required statistics with many sample paths of low accuracy. Then correction terms are added until a required level of accuracy is reached. Recent literature has primarily focussed on implementing the multi-level method efficiently to estimate a single system statistic. However, it is clearly also of interest to be able to approximate entire probability distributions of species counts.more »

- Authors:

- Mathematical Institute, University of Oxford, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG (United Kingdom)

- Publication Date:

- OSTI Identifier:
- 22611442

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: AIP Advances; Journal Volume: 6; Journal Issue: 7; Other Information: (c) 2016 Author(s); Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY; ACCURACY; APPROXIMATIONS; COMPARATIVE EVALUATIONS; CORRECTIONS; DISTRIBUTION; DISTRIBUTION FUNCTIONS; MARKOV PROCESS; MONTE CARLO METHOD; PROBABILITY; REDUCTION; SIMULATION

### Citation Formats

```
Wilson, D., E-mail: daniel.wilson@dtc.ox.ac.uk, and Baker, R. E.
```*Multi-level methods and approximating distribution functions*. United States: N. p., 2016.
Web. doi:10.1063/1.4960118.

```
Wilson, D., E-mail: daniel.wilson@dtc.ox.ac.uk, & Baker, R. E.
```*Multi-level methods and approximating distribution functions*. United States. doi:10.1063/1.4960118.

```
Wilson, D., E-mail: daniel.wilson@dtc.ox.ac.uk, and Baker, R. E. Fri .
"Multi-level methods and approximating distribution functions". United States.
doi:10.1063/1.4960118.
```

```
@article{osti_22611442,
```

title = {Multi-level methods and approximating distribution functions},

author = {Wilson, D., E-mail: daniel.wilson@dtc.ox.ac.uk and Baker, R. E.},

abstractNote = {Biochemical reaction networks are often modelled using discrete-state, continuous-time Markov chains. System statistics of these Markov chains usually cannot be calculated analytically and therefore estimates must be generated via simulation techniques. There is a well documented class of simulation techniques known as exact stochastic simulation algorithms, an example of which is Gillespie’s direct method. These algorithms often come with high computational costs, therefore approximate stochastic simulation algorithms such as the tau-leap method are used. However, in order to minimise the bias in the estimates generated using them, a relatively small value of tau is needed, rendering the computational costs comparable to Gillespie’s direct method. The multi-level Monte Carlo method (Anderson and Higham, Multiscale Model. Simul. 10:146–179, 2012) provides a reduction in computational costs whilst minimising or even eliminating the bias in the estimates of system statistics. This is achieved by first crudely approximating required statistics with many sample paths of low accuracy. Then correction terms are added until a required level of accuracy is reached. Recent literature has primarily focussed on implementing the multi-level method efficiently to estimate a single system statistic. However, it is clearly also of interest to be able to approximate entire probability distributions of species counts. We present two novel methods that combine known techniques for distribution reconstruction with the multi-level method. We demonstrate the potential of our methods using a number of examples.},

doi = {10.1063/1.4960118},

journal = {AIP Advances},

number = 7,

volume = 6,

place = {United States},

year = {Fri Jul 15 00:00:00 EDT 2016},

month = {Fri Jul 15 00:00:00 EDT 2016}

}