# Multivariate Function Approximation With a Large Dataset

I have a nice amount of data from a trading strategy I am working on, where I have two different liquidity parameters as x, y variables.

Before entering a trade I am taking the moving average of Volume*SharePrice & if > than my "liquidity requirement" I enter, along with other unmentioned rules. The length of the moving average is my x variable and my liquidity requirement is my y variable. My data includes several outputs over these x and y variables such as P/L, ROI, \$Won, etc.

My question is: how can I use Mathematica to interpolate an approximate function for any output so that I can optimize x, y? For instance say I want to approximate ROI=f(x,y) or P/L=f(x,y)...I dont expect something that describes my data perfectly, but something that approximates it would greatly help.

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type ?*Interpolation* browse the corresponding documentation (which can be evaluated), write a similar code, improve it, and if it works I ll only take 30 % of the net profit for this great advice ;-) –  chris Nov 30 '12 at 8:32
Yeah I saw that documentation and I was doing something wrong before, but now I seem to have it working. Can I use the Interpolating function as a regular function? For instance say I have f1=Interpolation[DollarWon] and f2=Interpolation[DollarLost] can I now do Maximize[f1-f2,{x,y}]? –  Brandon Nov 30 '12 at 16:41
you can do dat = Table[{x, Cos[x]}, {x, 0, 2 Pi, Pi/8.}]; f1 = Interpolation[dat]; NMinimize[{f1[x], x > 0, x < 2 Pi}, {x, 0, 2 Pi}] –  chris Nov 30 '12 at 16:54
I really appreciate the help, but I am still getting hung up on the NMaximize.. I have PL = data[[All, {1, 2, 3}]]; where data is a csv holding all my backtest results. I am able to successfully get the Interpolation with f1 = Interpolation[PL]; But trying NMaximize[{f1[x, y], x >= 30, x <= 90, y >= 1000000, y <= 10000000}, {x, y}] yields NMinimize::nnum: The function value InterpolatingFunction[{{0.,6.28319}},<<3>>,{Automatic}][89.9186,1.83176*10^6] is not a number at {x,y} = {89.9186,1.83176*10^6}. >> –  Brandon Nov 30 '12 at 18:10
I am confused by this bit {x, 0, 2 Pi} in your NMinimize.. isnt this supposed to be the choice variables mathematica searches for? I am having trouble finding documentation on this function regarding multivariate functions... I hope I am not driving you crazy, I really appreciate the help! –  Brandon Nov 30 '12 at 18:16

Let's just consider a test case:

dat = Table[{x, y, -Sin[x]*Sin[y]}, {x, 0, Pi, Pi/8.}, {y, 0, Pi, Pi/8}];

f = Interpolation[Flatten[dat, 1]];


as a check let us plot sections through the interpolation.

Plot[Table[f[x, y], {x, 0, Pi, Pi/8}]//Evaluate, {y, 0, Pi}];


Now let's minimize that interpolated function:

NMinimize[{f[x, y], x > 0, x <= Pi, y >= 0, y <= Pi}, {{x, 0, Pi}, {y, 0, Pi}}]

(* ==> {-1., {x -> 1.5708, y -> 1.5708}} *)

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Awesome! This got me where I needed to be! I really appreciate the write up! –  Brandon Dec 2 '12 at 20:45