[BUG] CBPM not aligned with paper

[samihamdan]

The CBPM method we implemented has slight variations from the original: using the mean instead of sum for summation; the weighting might also not be aligned.
While this should not make a fundamental difference, users should at least have the option to exactly follow the paper.

[kaurao]

the code from authors is available here: https://www.nitrc.org/projects/bioimagesuite/ (see file codeshare_behavioralprediction.m)
the only difference I can notice is what @samihamdan mentioned, using sum instead of mean.

[NevenaK]

Thanks for the issue @samihamdan! I am using the CBPM implementation and would like to see if/what difference sum vs mean makes. And would also like to use it as the authors intended :) If I can help adjusting that, let me know :)

[kaurao]

The mean versus sum should not make any difference to predictions as the difference is same linear transformation across observations (just multiplying features with a constant). The linear model used for learning will take care of this and the predictions will be the same. The difference will be the weights assigned to the two features (pos and neg) but they will be again proportional so that should not be an issue as well. But I still agree that sum should be available to users.

Here is empirical demonstration of this using R ;)

# generate data
set.seed(1)
x1 = rnorm(100)
x2 = rnorm(100)
y = x1 + x2
# get a linear model
l1 = lm(y~., data=data.frame(y=y,x1=x1,x2=x2))
# get a linear model with scaled x
l2 = lm(y~., data=data.frame(y=y,x1=10*x1,x2=5*x2))
# plot
plot(l1$fitted.values, l2$fitted.values, xlab='l1', ylab='l2')

[NevenaK]

Nice, thank you!

[samihamdan]

This PR: #221 introduces sum as default option for cbpm, but still allows other aggregation functions.