I am given a dataset consisting of 10 million molecules. Each row contains:

  • The average value predicted by an ensemble of regression models trained to predict a certain property about chemicals (value of interest)
  • The ensemble variance of the predicted value
  • A heuristic value that characterizes the molecule with respect to some property (value 1)
  • Another heuristic value that characterizes the molecule with respect to some property (value 2)

and the requirements:

  • High values for the value of interest are desired
  • High values for heuristic value 1 are desired
  • Low values for heuristic value 2 are desired

Design a protocol that filters and selects 1000 chemicals from the full dataset, and incorporates the mean value of interest, variance of the value of interest, heuristic value 1, and heuristic value 2. Comment on your rationale for the protocol. How would my protocol be adjusted if there was a constraint of diversity in the selected molecules (ie: selecting for chemicals that are very different from one another in the final set of 1000).

EDIT: I am thinking about PCA if I select only 1 prior component. But is it possible to control that the weights of the features that are desired if higher are positive, while those that are desired if lower negative?


Ignoring for a moment the variance of the first feature, a straightforward approach is to perform a linear combination of the features $x_1$, $x_2$ and $x_3$, with each of the coefficients being a hyper-parameter you set to indicate the relative importance of that feature (perhaps normalizing the features before hand). This will be your molecule's utility measure.

$$u^{(i)} = \beta_1 x_1^{(i)} + \beta_2 x_2^{(i)} + \beta_3 x_3^{(i)} $$

You can then sort according to this utility and select the top 1000. A natural set of coefficients if each feature was of comparable importance would be $\beta_1=1$, $\beta_2=1$ and $\beta_3=-1$ (negative because we want to select for low values of heuristic value 2).

Knowing that we have the variance of the first feature, we can assume that this feature is distributed normally. The utility is now a random variable distributed according to:

$$U^{(i)} \sim \mathcal{N}(\beta_1 \mu_{x_1^{(i)}} + \beta_2 x_2^{(i)} + \beta_3 x_3^{(i)}, \beta_1^2\sigma_{x_1^{(i)}}^{2})$$

From this point, the method for selecting molecules will depend on certain considerations such as your tolerance to variance and if you are more interested in the collective utility of your selection or if each individually selected molecule must meet a certain standard. If you didn't care about variance at all and simply wanted to maximize the collective expected utility you could bypass modelling utility as a random variable all together and stick with the first approach. An alternative if the variance was a factor would be to select the top 1000 molecules according to their mean utility, excluding molecules which didn't have a high probability of having a utility higher than some threshold $P(U > C) > 0.95$. These are just some examples, the exact approach will depend on your criteria which you need to clarify.

As to your question "How would my protocol be adjusted if there was a constraint of diversity in the selected molecules". Diverse in what sense? Have you been provided with any other features to compare the molecules with or do you mean diverse in the three features already given? Again, another simple approach would be to append another feature calculated as the average distance to those molecules already selected.

One of the difficulties with your question is that it is very broad and open ended. There is not a lot of clarity surrounding what constitutes a good selection of molecules from a bad selection. For example, is it better to have a higher heuristic value 1 or a lower heuristic value 2? These sorts of questions need to be handled by a domain expert prior to applying the data science algorithm. If you were given a set A of 3 molecules and a set B of another 3 molecules do you have a clear way of determining which is the better selection? You need to be able to answer this question before you can begin applying any data science or else you are shooting in the dark.

  • $\begingroup$ I agreed with you that this is supposed to be an open-ended question. I really have no clues where I can find the preference between a higher heuristic 1 and a lower heuristic 2. $\endgroup$ Apr 24 at 22:38
  • $\begingroup$ Can I also take the variance as the 4th feature into consideration so that we could assume this feature is more favored if lower? $\endgroup$ Apr 24 at 22:42
  • 1
    $\begingroup$ What 4th feature do you mean? If you are referring to the feature I suggested appending which calculates the distance to molecules already selected, yes, you could potentially use the variance of this as well. $\endgroup$
    – James
    Apr 25 at 3:06
  • $\begingroup$ I mean the variance can be a lower-more-desired feature, which might be applied to this modeling. $\endgroup$ Apr 25 at 16:21
  • $\begingroup$ Ohhh, right, I see. Yes, that would be possible also. Perhaps not as theoretically grounded, but as a simple heuristic it may work well in practice. $\endgroup$
    – James
    Apr 26 at 14:01

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