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I am interested in this problem of learning a machine learning model to take a vector of ranks as input and predict their numerical values.

Let's say I have a matrix $Y$ with shape $m$ (instances) by $n$ (features), I ranked every row of $Y$ using average to handle potential ties and get a matrix of ranks $R$ of the same shape.

My questions are: How could I learn a model $f$ to achieve $y = f(r)$ ? Should I formulate this as a multiple-regression problem? What models fit well for this type of problem?

Here is a python script to generate some toy data:

import numpy as np
from scipy.stats import rankdata

Y = np.random.randn(1000, 20)
R = np.apply_along_axis(rankdata, 1, Y)

print R[0]
print Y[0]

Output:

array([ 10.,   8.,  14.,  18.,   7.,   3.,   1.,   9.,  13.,   4.,  16.,
    20.,   2.,  12.,  19.,   5.,  17.,  11.,  15.,   6.])
array([ 0.06578002, -0.11636595,  0.56441059,  0.7740778 , -0.31002372,
   -0.69271934, -1.83806102,  0.02944196,  0.48905099, -0.68911226,
    0.6119917 ,  1.47756463, -1.65347498,  0.28952666,  1.09095143,
   -0.62324096,  0.7086212 ,  0.21528326,  0.5837112 , -0.35102606])
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  • $\begingroup$ If you know $p(y)$ (e.g. Gaussian in your example), then in principle I believe $p(y|r)$ could be solved as described here. By the way, you should add the order statistics tag to your question. $\endgroup$ – GeoMatt22 Sep 14 '16 at 17:40
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If you have a vector of ranks r, but you don't know the underlying distribution, there is no mapping back to values. If you do have the distribution, you could assume random sampling and use the ranks as the appropriate quantiles and get the value for the distribution.

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This would be an example of multi-output regression. A cursory google search reveals that this is a problem that is both fairly uncommon and is the subject of active research. Compared to multi-label classification, it seems to be understudied. As far as I know, there are no popular software package implementations of such a multi-output regressor.

Two conceptually simple strategies use an ensemble of regressors to predict the output. The first simply constructs a separate regressor for each output column. While simple and quick, it ignores information that might be gained from looking at multiple outputs simultaneously (EG if some outputs are correlated, this method would not be able to use that information to improve its predictions.) Another approach would be to stack an ensemble of regressors, where each regressor predicts its output column looking at the original data and the output of (a) previous regressor(s). This method has the opportunity to use relationships between output values in its predictions. It however, requires more careful design considerations (EG order that you fit the output columns) and has the potential to overfit.

Here is an example of an multi-output regression problem using decision trees with sklearn.

This is a methods survey on multi-output regression. I admit to have only skimmed in in the past, but seems to contain many ideas to begin to work off of.

Edit: If you google search 'Multi-output least-squares support vector regression machines' you can find a paper on research gate that discusses an algorithm that natively outputs multiple values. I sadly don't have enough reputation to post more than two links right now, however.

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