Easy: do cokriging. If you need references, please ask. Cheers.

I'm not sure why a standard binary logit or probit would not work, since this is the typical approach. It may be possible to use a principal component of v,w,x,y, but this isn't necessarily better. A lasso or even stepwise procedure would help select among candidate models, in terms of which covariates to include. This of course presumes a linear relationship. If not, a nearest neighbor or nonparametric regression might work better, but the basic principles are the same.

What is the prize for winning the challenge?

Thanks, @Fred. I agree with you. I think that most people reading this know the concepts that you have mentioned and would agree that they are generally useful. I asked this question because I am curious what people really do once they get a dataset. Do you start plotting it? Does inspecting the plots or summary statistics influence the choice of model? What model will you try first? I am curious about the model priors of people.

@Ravi, the prize is, of course, eternal glory on researchgate!

The dinosaurs way: Exlude case with v=0 (which predict z=-1 perfectly). Apply log to y.

Start with logistic regression z~x + ly + w.

drop w. Inspect graphically. Include v1=(x/SD(x)-ly/SD(ly))^2

Then z~x+ly+v1 separates perfectly.

The CV-accuracy should be 100% again.

The data generating mechanism?

The ten's root of y and x should be independently normal, w and v independent both

of y and x.

Then z is defined determanistically via z=1 iff f(x)*g(ly)>c for some simple smooth f and g. - Maybe, perhaps ...

I'd be interested in hearing more about cokriging from Emilio; I don't see why this would be an obvious choice here. In any case, I don't think it makes sense to expend a lot of analysis time on something like this without knowing the context, i.e., what does the response represent, what are the covariates, what models do subject-matter experts have for this kind of data . . .

Lacking all that, I just plotted each variable against z in Excel. Nothing obvious there except that v seems to be just a binary 0/1 version of z, thus of no use as a predictor.

I agree with Fred, the next obvious thing to do is a binary regression, e.g., Logistic regression in R:

challengeData

I'm also interested in why co-kriging would be first choice. Especially since I imagined co-kriging to be something like multi-task Gaussian process regression. While GP regression is surely a good starting point here (at least for me, just out of habit), I cannot see how to sensibly use multiple tasks when only a single response variable is available.

I'd be happy if Emilio could shed more light on this issue.

What to do first with the data? ALWAYS: plot the data. In any and all ways you can think of.

About the analysis, I'd use a GLM with binomial distribution and a logit link to start with. You can fit the full model first, remove the least significant predictor among those with p < 0.05 (or whatever alpha you like) and repeat, until only significant predictors remain. Alternatively, you can use the deviance of the model to decide what variable to remove at each step (if any). With your final model, you can produce a predicted value (a proportion) for each observation and assign a 1 to those with predicted value > 0.5 and 0 otherwise. Then, use these assigned values to calculate the percentage of correct predictions.

Sorry, I meant p > 0.05 (obviously)

After a cursory analysis I got a ~96% correct predictions of z with the model I suggested above. All predictors were significant.

Simple logistic regression, restricted to data with v=1, and using x, log(y) and their squares leads to a 100 % accurate classifier, even if non convergency is signalized because of perfect separation.

It appears that even simple regression models do a good job with this, but for fun I plugged the data into a nested CV script for SVM classifiers that I'm using for another project. I used 80% of the data for training and 20% for testing in the nested CV framework. An SVM with a Gaussian kernel had an ROC score of 1.000 and 100% accuracy on the training set and selected x,y and v as features. For the test set, it had an ROC score of 0.994 and 95% accuracy.

Dear Mark, that's exactly what I expected after plotting x*y colored by z within (v=1).

My first choice is to plot an overview of the data. This tells me that z==0 if v==0. This is the first part of my classifier. Then looking at the data restricted to v==1, plotting x and log y colored by z (it was too bunched together when y was linear) we get the second plot, which looks very separable as Jochem mentioned - a simple polynomial SVM or logistic regression can classify this subset perfectly.

If this were a scientific problem, I'd continue looking at transformations of Y, because it looks like a simple transformation of a Normal distribution (see the third plot) - so the transformation would give information about the process that produced the data. X already looks Normal.

A simple decision tree analysis has only 1 misclassification, and the model ignores w as a classifier. This is only slighter more complex than a plot; beyond this model I would look to other information about the system represented by the data if I wanted to attach meaning to a model.