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In which situations are statistical assumptions (e.g. multivariate normal distribution within spectral classes) for parametric classifiers (e.g. maximum likelihood ) not valid for land-cover classification?

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  • I'm not sure it belonges here. Maybe cros-validate
    – dof1985
    Apr 12, 2015 at 19:08
  • Sorry, I am not agree... I asked specific situations, not math questions.
    – Kevin
    Apr 12, 2015 at 21:31
  • So good luck, Hope you find your answer here
    – dof1985
    Apr 13, 2015 at 5:38
  • Could you please expand on this a bit further? Assumptions are often violated for parametric classifiers--are you asking how to set a threshold to know when not to use a parametric classifier? You can safely assume that most remotely sensed data are not parametric. Have you looked the random forest classifier?
    – Aaron
    Apr 14, 2015 at 1:20

1 Answer 1

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I'm not 100% sure, but I'll give a tentative answer.

Ideally, if you have an image with "pure" and well defined classes from the spectral point of view, the use of parametric classifiers is safe. You can safely assume that your distribution follows e.g. a multivariate normal, with Gaussian additive noise as in all optical images, and then fit a fixed (usually small) amount of parameters (mean and variance).

However, as soon as you move towards classes which are a "spectral mix" of other spectral subclasses, parametric classifiers may not be the best choice. For instance, the semantic class "urban area" if far from being spectrally pure. This way, a classifier which assumes unimodal multivariate normal distributions will surely fail (think at naive Bayes). Further problems, in the case of spectrally mixed semantic classes, the class support may have weird shapes in the feature space, e.g. requiring nonlinear boundaries. In this case, either you transform the input space so that classes tend to be more recognizable (ideally, "Gaussianizing" them) or you will require some classifier with more capacity, non-parametric and possibly nonlinear (e.g. our beloved random forest).

I would like to make a point, though. The family "parametric classifiers" includes most probabilistic / generative classifiers. However it is not limited to. A nice definition is that parametric classifiers, in general, have a fixed number of parameters, hence in some situations may be not powerful enough to model data (same for regressors). In most cases, fixing the number of parameters makes you assume a specific functional form of your data. A nice example I heard one in a machine learning class, was about neural nets. Once their parameters are fixed (number of hidden layers, number of neurons), they can be seen as parametric classifiers, since you assumed this form (I personally struggle seeing NN as parametric). On the other hand, nonparametric classifiers (e.g. kNN, SVM, RF) makes much fewer assumption about data distribution, since their number of parameters typically depends on the input data (e.g. number of training samples ans in kNN, complexity as in SVM, etc). In general, parameters (weights) of nonparametric models are controlled by the complexity rather than by the functional form. You can find a nice example summing up everything basically in a whole field of machine learning: Gaussian processes and the whole Bayesian nonparametrics (assuming parametric models with infinite number of parameters).

Again, this was tentative, as after many years I still struggle in putting a hard margin (the largest possible) between parametric and nonparametric models... I will be happy to discuss these topics further so if anyone has a comment, it will be very welcome!

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    Very nice answer. One addition is that as we move towards higher resolution imagery the issue becomes the autocovariance (pixels are not independent). These pixels may be spectral pure but are not multivariate normal. A good illustration of this is discrimination of tree crowns, where a given pixel is representative of a specific object. In these cases, parametric classifiers really struggle. May 29, 2015 at 16:33

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