Lets say you are optimizing three parameters, by grid searching, you obtain: SharpeRatios as a function of parameters p1, p2, p3, where p1 takes value on a set of n1 values, p2 takes value on a set of n2 values, and p3 takes value on a set of n3 values. Therefore, you obtain the SharpeRatios which is of size n1 x n2 x n3... What's the best way to visualize these data, suppose our goal is to check the smoothness of the surface? Thanks a lot!

Color coding the 4. dimension? e.g. 0..255=shades of red, 256..511 shades of blue ....., just to give a rough idea. Can take this to any level of sophistication.

For 4D, which you mention in your thread title, use 3D surfaces in "families". Each 3D surface of the family would be an iso of that 4th dimension. Since you are only measuring performance, there is not much yield to be expected from the "studies". Someday in the future you may get to the understanding that the optimum is surrounded by less than optimum and the primary reason is the limitation of market granularity as it relates to market capacity. Soros uses "reflexivity" to describe some market characteristics. Tailoring performance to the psychology of market operation is a neat ballpark; one branchpoint takes one from the offer to the imbalance in the market. There is no choice in the end. Defining performance is best done by consideration of the optimum multiple of the market's capacity. For me it came down to not being "stranded" as a consequence of the dynamics.

http://www.sciencegl.com/index.html However, you will not be able to figure out anything by looking at the data, I highly doubt it. Fast bifurcation modes in your case are suppressed by method of optimization and can be missed all together. You can have a system that looks smooth but in reality it becomes chaotic at certain regions with no heteroclinic orbit available http://en.wikipedia.org/wiki/Heteroclinic_orbit Fighter planes are inherently unstable systems due to aerodynamic shape (small airfoil area) but become stable with the introduction of control systems that manage heteroclinic orbits and constrain the dynamics from one equilibrium point to another. This is the principle I used in my first system 10 years ago that made me just 100K trading commodity futures. Since then the number of heteroclinic orbits available to systems has been reduced dramatically due to the introduction of ETF instruments. You have no chance with your three parameters. IT will take just a few bars to lock your system in a bifurcation mode.

Why not reduce it to a search problem - shortest path first should give you some optimized paths to use given the different values, instead of optimizing the actual values themselves.... If the data has a time and allocation component, this should work. This is my interpretation, but I'm assuming you have several strategies in place here that you are trying to find the best sharpe values for...

Thanks a lot IntradayBill... could you please elaborate on this "Since then the number of heteroclinic orbits available to systems has been reduced dramatically due to the introduction of ETF instruments."? Thank you!

Thanks but could you please give an example about how to use shortest path first search here? Thank you!

Great question. Animate it. Display 3D, then give yourself a slider for the fourth (and fifth) dimension. Or, if a dimension is discretely (rather than 'continuously') indexed, display a collection of '3D' objects in a line, or an array for 5D data. You might search "information visualization" on amazon for books. I have Chaomei Chen's book stashed away somewhere, which is thought provoking, but I can't recall anything on 4D. As an aside, a SF novel called Diaspora by Greg Egan has a chapter or two towards the end where a character ends up in the richness of a 5D+ universe, faced with the prospect of returning to 3D. I can't say this will solve your problem but it might give you some ideas. Ask mathematicians/physicists. Interested to hear what you figure out.

ETFs brought a large number of unskilled traders to the market, people who did not understand futures and forex but with ETFs they could trade it like stocks. This was one reason for marketing these instruments anyway. This created more triangle arbitrage situations between ETF and futures and cash indices for large capitalization firms and market makers. This arbitrage limits the ability of systems for maintaining stability in equity growth because markets become more efficient with continuous 3-way arbitrage and price annomalies are rare. I have found a way to deal with this but it requires high capitalization, I mean way higher from what I can put at risk.

Okay I have tried, but it's too hard - due to my poor space imagination... Lets reduce the difficulty a bit. Essentially we want to find a few good local maxima in the sampled function: f(p1, p2, p3) What are good local maxima? 1. Of course they have to be local peaks; 2. But they cannot isolated points, i.e. the peaks surrounded by abyss. They should be the peaks that have nice smoothly decaying neighborhood... Any thoughts? esp. in Matlab? Thanks!