Re: st: similarity statistic for heatmaps (or any cartesian plane)

[Nick Cox <njcoxstata@gmail.com>]

OK, but it still seems to me that different problems are being conflated:

1. Is y similar to x, compared observation by observation?

2. Is the distribution function of y similar to that of x, pooling
observations?

3. Is the spatial pattern of y similar to that of x, focusing on
locations as well as values?

On #1, people often use correlation techniques, but correlation tests
linearity not agreement. More at e.g.

Cox, N.J. 2006.
Assessing agreement of measurements and predictions in geomorphology
Geomorphology 76: 332-346.

Nick


On Wed, Jun 20, 2012 at 3:26 PM, Raymond Lim <rl2240@columbia.edu> wrote:
> Thanks for the feedback Nick. Yes, my latter statement is dubious. I
> should have said akin to Kolmogorov-Smirnov and not referred to
> spatial 1-dimension. Essentially, I want to test whether the
> distribution function of two surfaces are statistically different. For
> example, say I measure pollution levels in a city in two different
> time periods. I want to test whether the distribution of pollution is
> the same (I don't care about volume of pollution, just distribution of
> it across parts of the city). Alternatively, a correlation measure
> would be good too.
>
> Computing the difference between heatmaps sounds like a good start.
> Thanks again!
>
> - -
> The latter statement is at best dubious and at worst fallacious.
>
> A Kolmogorov-Smirnov test compares distribution functions and says
> nothing whatsoever about spatial distribution, as you can shuffle the
> locations and keep the same distribution functions. Indeed it wouldn't
> surprise me if it were based on an assumption of independent
> measurements likely to be challengeable for most interesting spatial
> data.
>
> Similarly, you can compare heatmaps aspatially and spatially:
> something like concordance correlation compares element by element but
> says nothing about spatial patterns. If I were obliged to choose a
> single single-number statistic to compare heatmaps it might be
> something like a spatial autocorrelation of differences.
>
> But why choose a single statistic any way? Or rather, a map is a
> sample statistic, just map-valued, and the best measure of similarity
> of two heatmaps may be a heatmap of the difference between them (or,
> e.g. their ratio).
>
> Nick
>
> On Tue, Jun 19, 2012 at 4:30 PM, Raymond Lim <rl2240@columbia.edu> wrote:
>> Is there a statistic for computing similarity of two heatmaps (or any
>> sort of Cartesian plane)? I know in 1 dimension, you can compared
>> distributions using a Kolmogorov-Smirnov test, but not sure about
>> 2-dimensions.
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