```
---
title: "From noise to knowledge: how randomness generates novel phenomena and reveals information"
author:
- name: "Carl Boettiger"
email: "cboettig@berkeley.edu"
affiliations:
- id: a
address: "Dept of Environmental Science, Policy, and Management, University of California Berkeley, Berkeley CA 94720-3114, USA"
attributes:
corresponding: true
abstract: |
Noise, as the term itself suggests, is most often seen a nuisance to ecological insight, a inconvenient reality that must be acknowledged, a haystack that must be stripped away to reveal the processes of interest underneath. Yet despite this well-earned reputation, noise is often interesting in its own right: noise can induce novel phenomena that could not be understood from some underlying determinstic model alone. Nor is all noise the same, and close examination of differences in frequency, color or magnitude can reveal insights that would otherwise be inaccessible. Yet with each aspect of stochasticity leading to some new or unexpected behavior, the time is right to move beyond the familiar refrain of "everything is important" (Bjørnstad & Grenfell 2001). Stochastic phenomena can suggest new ways of inferring process from pattern, and thus spark more dialog between theory and empirical perspectives that best advances the field as a whole. I highlight a few compelling examples, while observing that the study of stochastic phenomena are only beginning to make this translation into empirical inference. There are rich opportunities at this interface in the years ahead.
keywords:
- Coloured noise
- demographic noise
- environmental noise
- quasi-cycles
- stochasticity
- tipping points
date: last-modified
bibliography: refs.bib
format:
elsevier-pdf:
keep-tex: true
journal:
name: Ecology Letters
formatting: preprint
model: 3p
cite-style: authoryear
editor:
markdown:
wrap: 72
knitr:
opts_chunk:
echo: false
warning: false
message: false
---
```{r}
#| include: false
library(dplyr)
library(readr)
library(ggplot2)
library(ggthemes)
```
```{r}
theme_set(theme_grey())
```
# Introduction: Noise the nuisance
To many, stochasticity, or more simply, noise, is just that -- something
which obscures patterns we are trying to infer [@Knape2011]; and an ever
richer batteries of statistical methods are developed largely in an
attempt to strip away this undesirable randomness to reveal the patterns
beneath [@Coulson2001]. Over the past several decades, literature in
stochasticity has transitioned from thinking of stochasticity in such
terms; where noise is a nuisance that obscures the deterministic
skeleton of the underlying mechanisms, to the recognition that
stochasticity can itself be a mechanism for driving many interesting
phenomena [@Coulson2004]. Yet this transition from noise the nuisance to
noise the creator of ecological phenomena has had, with a few notable
exceptions, relatively little impact in broader thinking about
stochasticity. One of the most provocative of those exceptions has
turned the classical notion of noise the nuisance on its head:
recognizing that noise driven phenomena can become a tool to reveal
underlying processes: to become noise the informer. Here I argue that
this third shift in perspective offers an opportunity to better bridge
the divide between respective primarily theoretical and primarily
empirical communities by seeing noise not as mathematical curiosity or
statistical bugbear, but as a source for new opportunities for
inference.
In arguing for this shift, it essential to recognize this is a call for
a bigger tent, not for the rejection of previous paradigms. What I will
characterize as 'noise the nuisance' reflects a predominately
statistical approach, in which noise, almost by definition, represents
all the processes we are not interested in that create additional
variation which might obscure the pattern of interest. By contrast, an
extensive literature has long explored how noise itself can create
patterns and explain processes from population cycling to coexistence.
These broad categories should be seen as a spectrum and not be mistaken
for either a sharp dichotomy nor a reference to a strictly
empirical-theoretical divide. Each paradigm expands upon rather than
rejects the previous notion of noise: the recognition that noise can
create novel phenomena does not mean that noise cannot also obscure the
signal of some process of interest. Likewise, seeking to use noise as a
novel source of information about underlying processes will be informed
by both previous paradigms, as our discussion will illustrate.
Numerical simulations permit poking and prodding investigation
unencumbered by either experimental design or mathematical formalism.
The code and data for the simulations in this paper are maintained at
<https://github.com/cboettig/noise-phenomena>.
To emphasize the underlying trend in the changing roles in which we see
and understand noisy processes, I will also restrict my focus to
relatively simple models primarily from population ecology context.
Simplicity not only makes examples (in equations and in code) more
tractable but also allows us to focus on aspects that are germane to
many contexts rather than unique to particular complexities
[@Levins1966; @Bartlett1960].
Nevertheless, that complexity matters -- few themes have been better
emphasized in the theoretical literature [@Bjornstad2001]. Both the
foundational literature and recent research continue to echo the theme
of understanding the impact different real world complexities have in
stochastic dynamics. As such, we will rely on both textbooks and recent
reviews to provide a proper treatment of these issues, and focus on
broader trends.
This review is structured into three sections: Origins of noise,
emergent phenomena, and noise-driven inference. The first section lays
the conceptual groundwork we will need, while also highlighting a shift
to more and more mechanistically rooted descriptions of noise. We will
see where the common formulation of "deterministic skeleton plus noise
term" comes from, how it is best justified, and when it breaks down. The
second section introduces noise the creator, showing examples of
ecological phenomena generated by stochasticity. These examples will be
familiar to most specialists but illustrate a different way of thinking
than held by most ecologists, where noise is only a nuisance to be
filtered or averaged out. The third and final section, noise the
informer, turns these examples back-to-front, asking what noise can tell
us about a system: such as its underlying resilience or stability, or
the approach of a catastrophic shift. Examples are fewer here, and have
largely yet to benefit from the introduction of either the rigorous
theorems or more complex models so plentiful in the previous sections.
Yet the promise of prediction, of early warning signs before tipping
points, have spurred broad interest in such noise-based inference. This
review is a call to both deepen the connection to mechanism and better
formalize this thinking, but also look more broadly into other ways in
which noisy phenomena can help inform and predict underlying processes.
## Demographic stochasticity
Demographic stochasticity refers to fluctuations in population sizes or
densities that arise from the fundamentally discrete nature of
individual birth and death events. Demographic stochasticity is a
particularly instructive case for illustrating a mechanism for how noise
arises as an aggregate description from a lower-level mechanistic
process. We summarize the myriad lower-level processes that
mechanistically lead to the event of a 'birth' in the population as a
probability: In a population of $N$ identical individuals at time t, a
birth occurs with probability $b_t(N_t)$ (*i.e.* a rate that can depend
on the population size, $N$), which increases the population size to
$N+1$. Similarly, death events occur with probability $d_t(N_t)$,
decreasing the population size by one individual, to $N-1$. Assuming
each of these events are independent, this is a state-dependent Poisson
process. The change in the probability of being in state $N$ is given by
the sum over the ways to enter the state, minus the ways to leave the
state: a simple expression of probability balance known as the master
equation [@vanKampen2007]. Note that in general this approach is equally
applicable to stochastic transitions of any sort, not just step sizes of
+/- 1 and not just birth and death events, but can include transitions
between stage classes or trait values, including mutations to
continuously-valued traits in evolutionary dynamics [e.g.
@Boettiger2010].
The @Gillespie1977 provides an exact algorithmcfor simulating
demographic stochasticity at an individual level.
The algorithm is a simple and direct implementation of the master
equation, progressing in random step sizes determined by the waiting
time until the next event. Free from both the approximations and
mathematical complexity, the Gillespie algorithm is an interesting
example of where we rely on a numerical implementation to check the
accuracy of an analytic approximation, even in the case of simple models
such as we will discuss. Though the algorithm is often maligned as
numerically demanding, it can be run much more effectively even on large
models on today's computers than when it was first developed in the 70s,
and remains an underutilized approach for writing simple and
approximation-free[^1] stochastic ecological models.
[^1]: that is, free from the approximation made by SDE models as we see
in the van Kampen example. All models are, of course, only
approximations.
As our objective is to tie the origins of noise more closely to
biological processes, it will be helpful to make the notion of a master
equation concrete with a specific example. We will focus on the classic
case of @Levins1969 patch model, to illustrate the Gillespie algorithm
and the van Kampen system size expansion
```{=tex}
\begin{align}
\frac{\mathrm{d} n}{\mathrm{d} t} = \underbrace{c n \left(1 - \frac{n}{N}\right)}_{\textrm{birth}} - \underbrace{e n}_{\textrm{death}}, \label{levins}
\end{align}
```
where $n$ individuals compete for a finite number of suitable habitats
$N$. Individuals die a constant rate $e$, and produce offspring at a
constant rate $c$ who then have a probability of colonizing an open
patch that is simply proportional to the fraction of available patches,
$1 - n/N$.
```{r}
#| label: fig-site-simulation
#| fig-cap: "Population dynamics from a Gillespie simulation of the Levins model with large (N=1000, panel A) and small (N=100, panel B) number of sites (blue) show relatively weaker effects of demographic noise in the bigger system. Models are otherwise identical, with e = 0.2 and c = 1 (code in appendix A). Theoretical predictions for mean and plus/minus one standard deviation shown in horizontal re dashed lines."
# create colour palette
colours <- ptol_pal()(2)
# load-data
data <- read_csv(here::here("analysis", "data", "gillespie.csv"),
col_types = "cdiddd")
# recode-data
data <- data %>%
mutate(system_size = recode(system_size, large = "A. 1000 total sites", small= "B. 100 total sites"))
# plot-gillespie
data %>%
ggplot(aes(x = time)) +
geom_hline(aes(yintercept = mean), lty=2, col=colours[2]) +
geom_hline(aes(yintercept = minus_sd), lty=2, col=colours[2]) +
geom_hline(aes(yintercept = plus_sd), lty=2, col=colours[2]) +
geom_line(aes(y = n), col=colours[1]) +
facet_wrap(~system_size, scales = "free_y")
```
@fig-site-simulation shows the results of two exact SSA simulations of
the classic patch model of @Levins1969.
# Conclusions
This review has explored three paradigms in how noise is viewed
throughout the ecological literature, which I have dubbed respectively:
noise the nuisance, noise the creator, and noise the informer. Noise can
be seen as a nuisance almost by definition: in examining the origins of
noise, we have seen how stochasticity is introduced not because
ecological processes are random in some fundamental sense, but rather,
because those processes are influenced by a complex combination of
forces we do not model explicitly. In this view, noise captures all that
additional variation that is separate from the process of interest, and
a rich array of statistical methods allow us to separate the one from
the other in observations and experiments. By examining the origins of
noise, we have seen that despite the complex ways in this noise can
enter a model, that a Gaussian white-noise approximation
[@vanKampen2007; @Black2012] is often appropriate given a limit of a
large system size -- a fact often invokedn implicitly but rarely derived
explicitly from the theorems of @Kurtz1978 and others.
In this context, noise does not act to create phenomena of interest
directly. The sudden transitions we seek to anticipate are still
explained by the deterministic part of the model -- bifurcations. But
nor is noise a nuisance that merely cloaks this deterministic skeleton
from plain view: rather, it becomes a novel source of information that
would be inaccessible from a purely deterministic approach. I believe
more examples of how noise can inform on underlying processes is
possible, but will require greater dialog between these world views.
# Acknowledgements
The author acknowledges feedback and advice from the editor, Tim Coulson
and two anonymous reviewers. This work was supported in part by USDA
National Institute of Food and Agriculture, Hatch project
CA-B-INS-0162-H.
# References
```