In regard to the first part of your question:

Each variable should be normalized, this does not alter the data, but transforms it in order to make the numerical algorithms work more efficiently. You can say normalizing is sufficient, since the results would still be the same if another, 'better', transformation was used (this may exist, though I do not know of any).

Now the rest of the question gets mixed up with the second question:

How to interpret the results?

When you get a new sample and need to classify it, you should first apply the transformations on it. This is, the exact same transformations that you applied on your training data. The reason is that those were the values used for training the classifier, and so it is 'calibrated' to give an accurate result for the normalized value.

Same goes, when you want to test the effect of varying a single variable, you modify it and then apply the transformation, this will give you the new output.

Cheers,

Ariel

Thank you, Naceur and Ariel!

If I log-transform it I understand that it's enough and I won't have to normalize it. I don't know why but I feel more comfortable normalizing than log-transforming (in terms of the "shape" of the resulting data). I guess I will have to normalize it. However, there are several methods for "normalization". The 2 most common methods I find for this type of data are to subtract the minimum and divide by the range of the variable (so all variables vary from 0 to 1) and the other is to subtract the mean and divide by the standard deviation (also known as "standardizing"). I see both of them being used before calculating Euclidean distances, depending on the study... I understand that having a range of 0-1 when you have outliers can compress your "normal" data too much, but I will use non-parametric methods as much as possible because the distribution of ecological data tends to be far from normal, so "standardizing" seems to be a bit "artificial", because standard deviations are not symmetrical from the mean. Eliminating outliers or changing their values will make me lose valuable training data or will not reflect the real value measured in the field (that I want to use as a predictor), so that will not be an option.

What do you think is the best option? Should I be so afraid of standardizing or is it ok? Or should I just have equal range for all variables, regardless of eventual outliers?

It is really sad to study biology for a decade and then figure out I spend more time with statistics... :p

I would go for Standardizing, but really don't worry too much about it, as I said, the only real difference is performance. I don't think you intend to run this over and over again or do you?

You are defintely correct when asking the question. Pre-treatment may substantially change all the results. The key point is the definition of "distance" ( or similarity) of units. Using the Euclidean distance you actially postulate that all variables have the same influence on the "distance" and a lot of other properties. For example, if two units have the same values of all parameters except one (x), then adding the same value( say, 1) to this x will increase the distance by the value (say, 1) irrespectively to the initial value of x. This is not true in (almost) any cases . Another consequence of the simmetry of the Euclidean distance is that if you have two parameters (x1 and x2) first with range (0.0 to 0.1) and second with range (0.0 100.0) then adding 2 to x2 can not be compensated by any admissible change in x1. Thus the choice of the distance is very important. The Standardizing transform all parameters into the same scale (with SD=1). However, you should think if the new parameters are equally important and (if not) what is the realtive importance of change the parameters by 1. Squared values of these ralative importance(versus one of them) will serve as the coefficients in the definition of distance. Do not think that doing this you introduce some arbitratry values in your definition. Using standard formula (without coefficients) you declare that all coefficients are 1 that may be very bad choice.

There are other difinitions of the distance and, for example, maximum difference (with coefficients) may be more appropriate for some problems.

I do not know how you are going to proceed but this point is importnat for (almost) any analysis

Yours is a very practical issue which I resolved as follows when I had a similar problem with MV statistics. I found someone who was teaching Biostatistics and I went to his office and invited him to lunch at a moderately nice restaurant. Over lunch I posed my problem. Your questions are very clear so I would just stick to those. I then made the point very clear that I did not want him to do my work for me, but to help me understand how to manage the nuances and the follow-up statistical concerns as a concerned researcher who wants, not to contribute more to the state of apparent statistical chaos in your field but to help raise it to a higher level with his help. I did this in other studies later and all of these people were very helpful. Trying to do this first time around on your own I think will just add to the confusion state. Best get some solid help from someone who knows and contribute to taking your field to a higher level.

Henry

As already pointed out, you don't lose your data when you normalize (by log transformation) or standardize them. Standardization is advisable when running a cluster analysis, for example. When you standardize, you simply remove the effects of different variable units and use a single scale based on SD units.