#27: Part 1, Spatial Statistical Models for Stream Networks: Context & Conceptual Foundations

[Dan Isaak]

Having the right tools is a good thing…

Hi Everyone,
This time out we’re embarking on a bit of a diversion from our
regularly scheduled programming as we start a 3 part mini-series on
spatial statistical models for stream networks. It’s out of the
planned sequence since these blogs will ultimately be filed under the
“Climate-Aquatics Cool Stuff Module” but I talked to the blog’s
publisher and he agreed that their content was significant enough that
it warranted an exception & because recent developments are about to
make these models much more accessible for broad application to
streams anywhere, so without further adieu…

For anyone that’s ever tried to build something & get by with poor
quality tools or tools not specifically designed for the job at hand,
you know the frustration. Something can usually be made to work but
it’s often a painful process and the thing built could have been much
better with the right tools. Well, I’d argue that those of us working
with streams and rivers and their critters have long lacked important
tools for analyzing and understanding these systems at one of the
spatial scales that is fundamentally important for planning and
prioritizing conservation efforts. That scale is the network scale—
spanning 100’s – 1000’s of stream kilometers and 10’s – 100’s of
individual streams. It’s this scale at which multiple stakeholders
across a river basin coordinate and make choices about how & where
resources are spent to protect and restore habitat or conserve
populations. From a biological perspective, this scale is also
relevant because it’s one at which metapopulation dynamics and the
exchange of individuals between populations occur to lend populations
some resilience and facilitate the persistence of species within
landscapes. That being the case, the network scale is where “smart
maps,” informed by lots of data to show the status and trends of
important aquatic resources (blog #26), would be powerful tools for
facilitating coordination efforts among resource professionals and
different agencies to make sure we do the smartest things in the
smartest places.

The good news is that there are literally armies of us running around
the woods measuring all manner of stream things at discrete points on
those networks and 100’s – 1000’s of such measurements now exist in
many river basins. Those data are the fundamental elements that could
be stitched together to build the necessary “smart maps” but doing so
also presents a few issues that have been difficult to address until
recently. Those issues involve such things as accurate geo-referencing
of sample locations (now easily overcome with inexpensive GPS units),
computational horsepower for dealing with large databases (more &
cheaper horses all the time), and having standardized protocols to
obtain unbiased measurements of commonly collected stream and
biological attributes (this one’s still evolving but rapidly improving
with development of inexpensive sensors, genetic assay techniques, and
bioassessment protocols. And in practice, having bias in some
measurements may be acceptable if it better informs parts of a network-
scale “smart map” that would otherwise be devoid of information).

The fourth major issue, and it’s a biggie, is having analytical
techniques that can properly deal with those 100’s – 1000’s of
nonrandom, clustered measurements that our uncoordinated army is
generating (graphic 1). Those nonrandom site locations, especially in
areas with high data densities, create a problem called spatial
autocorrelation wherein there’s a certain amount of spatial dependence
in the samples and redundancy in the information they provide. This
redundancy violates one of the basic assumptions inherent to many
statistical methods, that samples are independent of one another.
Analyses that ignore the way sites are located in space and potential
spatial dependence among them run the risk of providing biased
parameter estimates (and poor maps if used to predict) when applied to
these sorts of databases. To deal with the issue, one could simply
discard some of the data until the redundancy was reduced to an
acceptable level, but whose data do you throw away, how much, and
where?

Much better to use it all, if possible, since someone went to the time
and expense of collecting the data. Plus, each sample location
contains some valuable information if the data can be properly
weighted to account for the degree of spatial autocorrelation with
nearby sites. Spatial statistical analyses were developed, in part, to
address this issue, and can be used to derive unbiased parameter
estimates and valid inferences from clustered databases. These
analyses have long been widely available for, and applied to,
terrestrial systems but they simply haven’t existed, other than a few
basic descriptive measures, for data on stream networks. I know
because I searched high & low for them to no avail during my Ph.D. 15
years ago as I struggled to analyze my own autocorrelated datasets.
Although many (myself included) have tried work-arounds that apply
spatial terrestrial analyses to stream data, we’re back to that wrong
tool for the job thing. It works but it’s ultimately fitting a square
peg in a round hole because the way that information moves through
space in a stream network differs fundamentally from a 2-D terrestrial
system.

Key to resolving the issue was that somebody had to develop a
covariance structure that accounted for the unique topology of stream
networks and the spatial configuration of samples measured on those
networks. For example, streams flow in one direction, are linear
features, have flow connected and unconnected sites, and tributary
confluences where big changes occur over small distances. All these
factors have consequences for how spatial patterns in networks
manifest and how nearby sites influence one another. These network
characteristics had to be embedded in the DNA of a new type of
statistical model for streams if accurate renditions of spatial
patterns in data measured on networks were to be made (graphics 2 and
3). I’m happy to say that that atom has finally been cracked and so
this week & in a few subsequent blogs we’re highlighting work by two
pioneers in the field of stream statistics, Jay Ver Hoef and Erin
Peterson. This duo has published steadily on the topic over the last 6
years and have developed a rich literature and solid statistical
foundation for their spatial stream network models (attached are a
couple recent examples, a more complete bibliography is provided in
graphic 4). I can’t do the work proper justice here since most of the
math is over my head but I encourage you to spend some time with it
and for those with especially strong quantitative skills and
statistical bents, you’ll be in hog heaven.

For the rest of us, suffice it to say that a really good tool now
exists for analyzing many types of data measured on stream networks.
Moreover, the Ver Hoef and Peterson models make it possible, even
desirable, to develop integrated, interagency databases because huge
amounts of new and accurate information can often be extracted from
existing data at relatively small costs and over the span of a few
years. Thus, the “smart maps” we need at network scales to describe
historical and future stream temperature patterns could be developed
from databases like NorWeST (blog #25) or others to inform species
conservation efforts (graphic 5). And although these maps won’t be
perfect, they’ll be significant improvements over what’s currently
available, and in the climate change game, we don’t have the luxury of
time, let alone the budgets, to design & execute the perfect
monitoring strategy and then wait decades for the answers. In many
cases, also, the spatial stream models perform significantly better
than their traditional counterparts in terms of raw predictive power
and are enabling new types of analyses and inference about streams
that simply weren’t possible before. Next time out, we’ll explore a
bit of this in more detail but it’s truly the start of a new era when
fundamentally better spatial information about streams and their
biotas is becoming a real possibility. That’s good because it’s still
going to take a lot of work to generate that information and we’re
going to need it as we strive to conserve aquatic biodiversity through
this transitional century.

Until next time, best regards.
Dan