# Multivariate regression for spatial dataset

I am trying to decide on the correct technique for a multivariate regression with spatial data. I would like to run a regression where the dependent variable is the current snow depth and the independent variables include physiographic parameters (slope, aspect, elevation, etc.) and snow depth for the same site in past years using daily data. The goal is to produce a statistical model with which I can interpolate snow depth across a whole basin based on the physiographic parameters.

Initially I was going to use a standard MVR but came across geographically weighted regression (GWR), which I think is more appropriate since snow depth is very spatially correlated. The third step, after establishing a model and interpolating, would be to distribute the residuals that I'll have at points where I know the snow depth; a common approach in the literature is elevation detrended inverse distance weighting.

1. Would it be incorrect to use MVR instead of GWR?
2. If I use GWR, would it still makes sense to distribute the residuals? From what I read, GWR already includes some correction for the inevitable residuals unlike MVR.

Please correct me if I'm wrong or seem to have misunderstand anything. I'm quite new to spatial statistics. Most of my GWR knowledge comes from Geographically Weighted Regression.

I've only recently started working with GWR, so please keep that in mind, but --

I think the important thing for you to keep in mind is that GWR is most often used as an exploratory tool, that is, to aid hypothesis-formation rather than to carry out hypothesis testing.

The relationship to WLS is important. GWR employs a weighting matrix at each regression point in the space you're analyzing based on a distance-decay function that you specify. Frequently, the function is gaussian; it also has to be (in the GWR nomenclature) calibrated, which may be best understood by analogy to peering at a 2d map through a pinhole. The ~size~ of the pinhole is called the bandwidth and helps to determine how strongly points are weighted within, and without, that pinhole.

In sum, I think that with WLS, you're talking about a scheme for weighting each observation; with GWR, you're talking about a weighting scheme for each observation with respect to each other observation.

The weighting matrices vary throughout space because GWR allows for spatial variation in the parameters of interest. In your case, it may be warranted if you suspect or want to investigate the possibility that the relationship between your independent and dependent variables vary throughout the geographic space you are studying.

Since you've stated that snow depth is very spatially correlated, it sounds like GWR is a promising candidate. If you haven't already, I would make sure to test for autocorrelation with something like Moran's I.

Edit: @Dominik, I think I didn't write the previous paragraph with enough precision. If your suspicion is that the independent variables influence snow depth differently at each location -- in other words, that the regression parameters vary spatially -- then you've got a case that calls for GWR.

To cover our bases, I should say that GWR isn't the only spatial multivariate analysis tool that you could use; it's just the one that I know. In Geographically Weighted Regression: the Analysis of Spatially Varying Relationships (2002), Fotheringham et al. (the leading exponents of GWR) identify the following methods as peers:

1. Casetti's spatial expansion method
2. Spatially adaptive filtering
3. Multilevel modeling
4. Random coefficients model
5. Other spatial regression models, which (unlike GWR), make global parameter estimates but on the basis of a local var-covar matrix.

Having just delved into GWR recently, I do think it's a pretty exciting way to get at the kind of spatial variation you're talking about.

• A point for clarification - I'm inclined to believe that by themselves each ind variable affects snow depth the same way each time. However, it's the interaction between independent variables that changes the effect on snow depth and since no two topographies are the same each geographic location will be influenced differently. So in my mind, the location itself is not influencing the depth but the combination of all the factors at that location are influencing the depth. I think this is a subtle difference, but one that possibly could change my methodology? Apr 22, 2012 at 16:50
• @Dominik, I edited my response to try to answer the question you raised. Apr 22, 2012 at 20:31

As far as I understand the linked info, GWR seems to be a special case and extension of weighted least squares (WLS) regression. With WLS the goal is to minimize variance, most often in order to create accurate predictions, i.e. bs. For ordinary WLS, a pseudoR2 can be calculated (Willett & Singer, 1988). Depending on the variability of the physiographic parameters, you might wish to consider segmented regression if there are clear breakpoints in the data. Given that GWR is somewhat like WLS, I assume that GWR leads to more accurate predictions than MVR. I guess it's easier to implement the MVR, so the answer depends on your resources. If there is only one parameter you would need to take into account (e.g. the one with the most variance), then try WLS. Just be careful on interpreting the meaning of R2.

• I guess I would follow up with what are the main differences between GWR and WLS? I read more about WLS and they seem the same. I want to look at about 30 variables in the regression and there aren't really any clear breakpoints. Apr 21, 2012 at 19:04