I have applied ML to a single band file, specifically a NDWI (McFeeters version). Comparing the results of this classification, with ML applied too a normal 5 band RapidEye Scene, the first one gave me better results (the breaking waves of the Bay of Bengal weren't classified as vegetation, as in the latter).
I'd like to know why this is happening and how does the ML equation changes when it is applied to a single band file?
While an NDWI image is a single-band image, it is the product of subtracting, adding and multiplying multiple bands together--in the McFeeter's case, the values of the Green and Near Infrared (NIR) bands from a previous image are used to calculate a numerical value that is then displayed in a NDWI image (where each pixel value is between -1 and 1).
Your single-band ML classification may have been "better" (or more accurate) because the NDWI version of the image is inherently "classified" a little bit already-- pixels containing water have a drastically different value than dryer things like buildings or dirt, so it is easier for the ML tool to distinguish between the two because ML classifies based on pixel value(s).
As for why breaking waves were classified as vegetation in your RapidEye 5-band image: this may have to do with how well you drew training samples to tell ML what vegetation looks like. The rough surface of the waves may have caused the RGB, Red Edge, and NIR values of the breaking waves pixels to be closer in value to pixels you trained the classifier to recognize as vegetation. When drawing training samples, make sure you're capturing all the variation in the image (i.e., make sure some pixels within these wave areas are assigned as part of the "water" training set). This article gives some helpful tactics when creating training samples; it's written for ArcGIS Desktop but the theory is the same.
You probably mean Gaussian maximum likelihood as this is the default implementation of maximum likelihood in all software. In theory, using more bands brings more information, but there are various reasons that make this statement wrong in practice:
with more dimensions, you need more training samples to estimate the densities, and this number increases exponentially. This issue is true for most classifiers, but particularly with Gaussian ML because you need to inverse a matrix of large dimension.
Gaussian maximum likelihood is a parametric classifier that assumes a gaussian distribution of each class. If you have truncated distribution, or bimodal distributions, etc, then the model does not fit well to your data and you could end up with suboptimal results.
