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One way to explain my data is to use the example data below. Here, I use the iris dataset to depict the four independent scores for each instance. My task is to classify each instance into one of the four classes.

> data(iris)
> iris2 <- as.data.frame(scale(iris[,1:4]))
> colnames(iris2) <- c("class_1","class2","class_3","class_4")
> head(iris2)
     class_1      class2   class_3   class_4
1 -0.8976739  1.01560199 -1.335752 -1.311052
2 -1.1392005 -0.13153881 -1.335752 -1.311052
3 -1.3807271  0.32731751 -1.392399 -1.311052
4 -1.5014904  0.09788935 -1.279104 -1.311052
5 -1.0184372  1.24503015 -1.335752 -1.311052
6 -0.5353840  1.93331463 -1.165809 -1.048667

However, the underlying scoring method/logic differs from class to class. Sure, when looking at one class only, a higher score means the instance is more likely of this class, but the difficulty arises when comparing the four scores:

Looking at their distributions, class_1 might have a significant skew, and class_2 a widely different value range. This means I cannot simply use the maximum value, when selecting the final class:

> iris3 <- cbind(
+   iris2,
+   lable_num=max.col(iris2,ties.method="first")
+ )
> head(iris3)
     class_1      class2   class_3   class_4 lable_num
1 -0.8976739  1.01560199 -1.335752 -1.311052         2
2 -1.1392005 -0.13153881 -1.335752 -1.311052         2
3 -1.3807271  0.32731751 -1.392399 -1.311052         2
4 -1.5014904  0.09788935 -1.279104 -1.311052         2
5 -1.0184372  1.24503015 -1.335752 -1.311052         2
6 -0.5353840  1.93331463 -1.165809 -1.048667         2

How should I go about and "level the playing field" in order to select the final class?

Is removing the mean enough? What if I also divide each column by its standard deviation? Or is this taking it one step too far? Am I loosing information by standardizing?

What about the skew issue?

I'm having difficulty discerning between what should be treated as an indication of a popular class (due to a heavy skew, or just larger values), and what traits should be fixed by scaling/transforming.

Are there any other types of approaches I should try?

I cannot change the way the scores have been calculated, and there is no training set I can use to model the final label using the four model scores as input (there are no prior final labels).

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In my mind it's not enought to compare the centered and scaled scores given by all your individual models (each trying to predict if an observation belong to a given class). As you figured out there would still be an issue due to the different type of distribution (in part the skewness) of your different scores.

I think a better idea would be to train a multiple classes classification model on top of the $p$ scores (say you have $p$ classes) you get.

To do this you may for instance use logistic regression, Random Forest or Gradient Boosting Machine (GBM). Also, since this approach would consist in stacking models, be careful about the overfitting.

The simplest way I would do this :

  • Split your data in 3 equal parts (A, B and C)
  • Train your $p$ single class classifiers on A and compute predictions on B and C
  • Train your "on top" model on the predictions made for B, get the predictions of this "on top" model on the dataset C
  • On C, you then can compute the accuracy of your model (test accuracy, free of overfitting)

Remark 1 : if you use a Random Forest for the "on top" model, you may split your data in only 2 parts, and use the OOB (out of bag) scores given by your Random Forest to assess the test accuracy of your final model.

Remark 2 : The kind of problem you are facing is common in the field of neural network since for a $p$ multi-class problem, a neural network usually output $p$ single scores (each corresponding to 1 class classification problem)

Best

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Well I need to say this before I answer your question completely, do not just hurry and start classification. cluster types: hierarchical fuzzy partial center based or what? globular or non - globular?

Cluster Evaluation: (cohesion and separation)

Cohesion: prototype / graph based Silhouette Coefficient and Cophenetic Coefficient

How to determine the number of clusters? -based on natural clustering -Hopkins Statistic

Now specific to your question , http://statweb.stanford.edu/~jtaylo/courses/stats202/kmeans.html follow the link , you will be able to find more learning experience not just an answer for your question.

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