I have been reading Zuur, Ieno and Smith (2007) Analyzing ecological data, and on page 262, they try to explain how nMDS (non-metric multidimensional scaling) algorithm works. As my background is in biology and not math or statistics per se, I'm having hard time understanding a few points and would ask you if you could elaborate on them. I'm reproducing the entire algorithm list for clarity, and I hope I'm not breaking any laws by doing so.
- Choose a measure of association and calculate the distance matrix D.
- Specify m, the number of axes.
- Construct a starting configuration E. This can be done using PCoA.
- Regress the configuration on D: D_ij = (alpha) + (beta)E_ij + (epsilon)_ij.
- Measure the relationship between the m dimensional configuration and the real dinstances by fitting a non-parametric (monotonic) regression curve in the Shepard diagram. A monotonic regression is constrained to increase. If a parametric regression line is used, we obtain PCoA.
- The discrepancy from the fitted curve is called STRESS.
- Using non-linear optimization routines, obtain a new estimation of E and go from step 4 until convergence.
Questions: In 4., we regress the configuration to D. Where do we use the estimated parameters (alpha), (beta) and (epsilon)? Are these used to measure distance from the regression (Shepard diagram) in this new configuration
In regard to number 7, can you talk a little about non-linear optimisation routines? My internet search came up pretty much empty in terms of a layman's explanation. I'm interested in knowing what this routine tries to achieve (in nMDS). And I guess the next question depends on knowing these routines: what represents convergence? What converges to where?
Can someone add "nmds" tag? I can't create new tags yet...