Asymmetric measure of non-linear dependence/correlation? I am definitely not a statistician/mathematician so feel free to tell me I'm an idiot if I am.
As far as I can tell from my Wikipediaing all of the main measures of dependence are symmetric and measure the shared information between dimensions. I am looking for an asymmetric method of measuring dependence. 
I mean asymmetric in the sense that f(x,y) might return the degree to which x depends on y where f(y,x) would return the degree to which y depends on x.
I have significant control over what form the data will take so I'm not too worried about exactly how I would handle the n-dimensional stuff right now - I intended to leave that as open as possible so as not to restrict the possibilities. Continuing my previous notation perhaps the possibility to implement f(x,y,z) such that the degree to which x could be determined given y and z is returned.
I'm not confident in my use of terminology so I'll provide an example.
Say I had a bunch of x,y pairs where y=1 when x>0 and y=0 otherwise. Basically a step function. Measuring the degree that y depends on x should give a high output (ideally 1) because y can be entirely determined given x. However the degree to which x depends on y is lower - the output should indicate that a relationship is present but that x cannot be fully determined given y.
Bare in mind that the example provided is a simplification - in the real world I'm interested in doing this with a bunch of different datasets with potentially fairly complex non-linear non-monotonic relationships. I will however have my data in x,y pair form.
I have put together a list of a few non-essential properties that an ideal solution would have.
Ideal property wishlist:


*

*Output between 0, 1

*Computationally cheap

*Capable of handling n-dimensional data

*Relatively easy to implement in a non-specialised software environment

 A: The $R^2$ of a multivariate regression model is such an asymmetric measure. The regression model of $Y$ on $X$ leads to a different $R^2$ than the regression model of $X$ on $Y$. This is because the value is computed using the proportion of vertical distance from the mean accounted for by the line of best fit using the conditional mean of $Y$ on $X$ to predict average potential outcomes for $Y$.
EDIT: In the discourse below, it was shown that the $R^2$ may be conserved and "symmetric". The $R^2$ has a particular application and is useful for summarizing high dimensional predictive models. In general, dependence is a sophisticated mathematical concept, and you can rarely inform much about the dependence between two variables without making strong (and often untestable) assumptions. I think for conveying aspects of the interrelationship between two variables in an applied setting, the term "association" is much better.
For smaller models, simply using the coefficient and its 95% confidence interval from a linear regression model is sufficient for reporting the first order trend in those data. These are well established association measures. Even if the trend is possibly nonlinear, a linear regression model has a coefficient that is taken to be a "rule-of-thumb" difference in outcomes for a unit difference in some regressor. These will necessarily be different for regression models treating a $Y$ variable as an outcome or a regressor. I see models of this form presented often in the literature with as many as 20 adjustment variables in large sample sizes.
A: It is possible to define nonsymmetric dependence measure R(X,Y) such that R(X,Y)=0 if and only if Y is independent of X, R(X,Y)=1 if and only if Y is a function of X. X and Y can be vector of random variables, continuous or discrete. An example would be a circle X^2 + Y^2 = 1, where neither X nor Y is a function of the other. Traditional symmetric dependence measures such as mutual information or Hellinger distance produce maximum dependence, but the new measure gives value R(X,Y)=R(Y,X)=0.5. Another example is Y=X^2. Again traditional measures give maximum value, but the new measure gives values R(X,Y)=1, R(Y, X)=0.5. Linear correlation gives 0 in both examples. The measure satisfies a new set of conditions different from Renyi's axioms for symmeyric dependence measures. In the continuous case, the measure is based on copula, thus nonparametric. Please check out my recent work on arXiv: 1502.03850, 1511.02744, 1512.07945 on bivariate, multivariate and discrete nonsymmetric dependence measures.
Here are some more details: The nonsymmetric dependence measure R(X,Y) is defined as distance between the cumulative distribution of Y conditional on X and the unconditional cumulative distribution of Y. It equals zero when the two distributions are the same, which implies Y is independent of X. It takes maximum value when Y is a function of X, or the cumulative distribution of Y conditional on X has a single jump from zero to one. This can be extended to n-dimensions, where R(X1,...,Xn,Y) is defined as distance between the cumulative distribution of Y conditional on X1,...,Xn and the unconditional cumulative distribution of Y.  It takes minimum value (zero) when Y is independent of X1,...,Xn. It takes maximum value when Y is a function of X1,...,Xn.
