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MAJOR EDIT:

I attempted to better describe my problem here. My original post (more or less) is below, if for whatever reason that is needed.

$\underline x$ is a linearly spaced vector of arbitrary element spacing (and thus size). Let's call the length of $\underline x$ the scalar quantity $l$. Written in terms of its elements, $\underline x$ is defined as:

$\underline x=[x_1,\space x_2,\space ...,\space x_i,\space ...,\space x_l]$

Now, I have a set of $n$ vectors $\underline y_j$ which are all of length $l$ as well. Each element $i$ for all $\underline y_j$ is defined as some function of all $x_i$ as shown below:

$\underline y_1=[y_{11},\space y_{12},\space ...,\space y_{1i},\space ...,\space y_{1l}]=[f_1(x_1),\space f_1(x_2),\space ...,\space f_1(x_i),\space ...,\space f_1(x_l)]$ ... $\underline y_j=[y_{j1},\space y_{j2},\space ...,\space y_{ji},\space ...,\space y_{jl}]=[f_j(x_1),\space f_j(x_2),\space ...,\space f_j(x_i),\space ...,\space f_j(x_l)]$ ... $\underline y_n=[y_{n1},\space y_{n2},\space ...,\space y_{ni},\space ...,\space y_{nl}]=[f_n(x_1),\space f_n(x_2),\space ...,\space f_n(x_i),\space ...,\space f_n(x_l)]$

It is unknown how any $f_j$ is defined, and this will be important later. Now is when it gets fun. I have a set of $m$ vectors $\underline z_k$ that are once again of length $l$. Each element $i$ for all $\underline z_k$ is defined as some function of all $\underline y_j$ as shown below:

$\underline z_1=[z_{11},\space z_{12},\space ...,\space z_{1i},\space ...,\space z_{1l}]=g_1([\underline y_1,\space \underline y_2,\space ...,\space \underline y_i,\space ...\space \underline y_n])$ $\underline z_k=[z_{11},\space z_{12},\space ...,\space z_{1i},\space ...,\space z_{1l}]=g_k([\underline y_1,\space \underline y_2,\space ...,\space \underline y_i,\space ...\space \underline y_n])$ $\underline z_m=[z_{11},\space z_{12},\space ...,\space z_{1i},\space ...,\space z_{1l}]=g_m([\underline y_1,\space \underline y_2,\space ...,\space \underline y_i,\space ...\space \underline y_n])$

Something to note is $m$ and $n$ are not necessarily equal quantities. Now for the crux of the problem: I have data analogous to all $\underline z_k$ as a function of $\underline x$. Let's call these vectors $\underline z_{k,\space data}$. Essentially, I want to determine all $f_j$ by iteratively changing all $\underline y_j$ until the black box programs $g_k$ output $\underline z_k$ approximately equal to $\underline z_{k,\space data}$. Does that make sense? What I am looking for is, I think, analogous to finding $f(x)$ given data for $g(f(x))$ versus $x$.

In any case, I would love a solution in Python using SciPy's optimization functions. I would be happy with freely available resources like textbook or lecture PDFs as well.

ORIGINAL POST:

I am not a mathematician but here goes.

x is a linearly spaced vector with arbitrary element spacing. I have a set of n vectors yi that each have the same length as x (call this length l). The elements of each vector yi are (theoretically) functions of x's elements. That is to say, yij = f(xj) for all elements j in x and yi (each comprising l elements).

I also have a set of m vectors zk, the elements of which are defined by some function acting on all n vectors yi. That is to say, zk = g(y1, ... yi, ... yn) for all k. I want to be very clear that m and n are not necessarily equal, but each zk also comprises l elements. I will be referring to g(y1, ... yi, ... yn) as the black box for the remainder of this post.

I have data for each zk plotted against x, and cannot get data for each zk against all yi. Is there a way to programmatically construct each yi such that by passing all yi to the black box results in plots that correctly match the data for zk versus x in a reasonable amount of time? Basically, can I find all yi(x) given data between x and all zk(x) = zk(y1(x), ... yi(x), ... yn(x))?

A general solution in Python using SciPy would be incredible. My particular problem regards an n = 2 and m = 4 (albeit z2 = z4, if that changes anything) case. But I will gladly take links to freely available texts and lectures, as well. Thank you!

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  • $\begingroup$ What is a line space? $\endgroup$
    – Ted Black
    Apr 2 at 19:48
  • $\begingroup$ @TedBlack linearly spaced vector, like what you'd get from np.linspace in Python using NumPy. $\endgroup$
    – kriegersan
    Apr 2 at 19:56
  • $\begingroup$ Did you mean to say $z_k = g_k(y_1 \dots y_n)$? The way you have it written, every $z_k$ is the same. $\endgroup$
    – Alex K
    Apr 2 at 20:36
  • $\begingroup$ @AlexK You are correct. I am going to rewrite this entire post once I get home, bear with me. I wrote it all on mobile as that is all I have at the minute. $\endgroup$
    – kriegersan
    Apr 2 at 20:40
  • $\begingroup$ @AlexK It is done. $\endgroup$
    – kriegersan
    Apr 2 at 23:18

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