Hi,
I remember in the Wartenberg paper there being a big problems; 1) The moran matrix is not symetric 2) The matrix can be non-positive definite, and I think not even non-negative definite. I believe Robin Reich worked out a bivariate Moran's I, and I think there is a publication in Environmental and Ecological Statistics, I don't remember which issue. I worked on the probelm a bit and came up with a combination grand-tour and Moran Scatter plot. In otherwords, project into 1-dimension using a random orthonormal basis, calculate Moran's I on the projected data. The derivatives can be easily derived (quadratic form) and the function can be optimized ocver projections to maximize/minimize Moran's I. I had some other thoughts on this at the time, but I would have to dig out my notes. There is an interactive implementation of this in java nested in the ORCA libraries. Also along these lines William Christensen did some really nice work on spatial factor analysis: Christensen, W. F., and Amemiya, Y. (2001). "Generalized shifted-factor analysis method for multivariate geo-referenced data," Mathematical Geology, 33, 801-824 Christensen, W. F., and Amemiya, Y. (2002). "Latent variable analysis of multivariate spatial data," Journal of the American Statistical Association, 97, 302-317. I think ther solution was in the cases where the spatial covariance was not pd they nudged the matrix until it was. I think aysymptotically there is no bias. Anyway if i am wrong about the multivariate Moran's I, I am sure Luc will set me straight. My 2c Nicholas |
no, you're right about the Wartenberg approach :)
The implementation in GeoDa doesn't follow Wartenberg, only in spirit. It is simply Wy on x where both y and x have been standardized (mean zero, sd = 1). I'm not sure it's always that meaningful in a general bivariate sense, but for space-time analysis it is. Here, y and x would be the same variable but one time period apart (how is x at i correlated with its neighbors at another point in time). Best not to make too much of these bivariate associations thought, they often hide more complex multivariate relationships. Thanks for the pointer on the grand tour Moran scatterplot idea. That is indeed very interesting. L. On Tuesday, November 4, 2003, at 09:59 AM, Nicholas Lewin-Koh wrote: > Hi, > I remember in the Wartenberg paper there being a big problems; > 1) The moran matrix is not symetric > 2) The matrix can be non-positive definite, and I think not even > non-negative definite. > > I believe Robin Reich worked out a bivariate Moran's I, and I think > there > is a > publication in Environmental and Ecological Statistics, I don't > remember > which issue. > I worked on the probelm a bit and came up with a combination grand-tour > and Moran Scatter plot. > In otherwords, project into 1-dimension using a random orthonormal > basis, > calculate Moran's I on > the projected data. The derivatives can be easily derived (quadratic > form) and the function can be optimized ocver projections to > maximize/minimize Moran's I. I had some other thoughts on this at the > time, but I would have to dig out my notes. There is an interactive > implementation of this in java nested in the ORCA libraries. > > Also along these lines William Christensen did some really nice work on > spatial factor analysis: > > Christensen, W. F., and Amemiya, Y. (2001). "Generalized > shifted-factor > analysis method for multivariate geo-referenced data," Mathematical > Geology, 33, 801-824 > > Christensen, W. F., and Amemiya, Y. (2002). "Latent variable analysis > of > multivariate spatial data," Journal of the American Statistical > Association, 97, 302-317. > > I think ther solution was in the cases where the spatial covariance was > not pd they nudged the matrix until it was. I think aysymptotically > there > is no bias. > > Anyway if i am wrong about the multivariate Moran's I, I am sure Luc > will > set me straight. > > My 2c > > Nicholas > > _______________________________________________ > R-sig-Geo mailing list > R-sig-Geo at stat.math.ethz.ch > https://www.stat.math.ethz.ch/mailman/listinfo/r-sig-geo > |
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