Objective
Develop an efficient method to compute the statistics s(c) for a given non-linear n-dimensional function f(x, y) with dependent variables x and y, such that F(f(s(x), s(y))) = s(c) while minimizing computational expense.

Background
The non-linear n-dimensional function f(x, y) has dependent variables x and y, and is computationally expensive when directly calculating s(f(x, y)) to obtain s(c). Instead, we want to find an efficient approach to compute F(f(s(x), s(y))) to derive s(c) with reduced computational cost.

Input:

1. A non-linear n-dimensional function f(x, y)
2. Statistics on x and y, denoted s(x) and s(y)

Output:
Statistics on the function output, denoted s(c), such that F(s(x), s(y)) = s(c). For example, given two PDFs P(x) and P(y), we want to approximate F such that F(P(x), P(y)) = PDF(c)

Constraints:
The proposed method should significantly reduce computational cost compared to directly calculating s(f(x, y)) to obtain s(c).

Evaluation Metrics:
The efficiency of the proposed method will be evaluated based on the following criteria:

1. Accuracy: The computed s(c) should be accurate and comparable to the result obtained from s(f(x, y)).
2. Computational cost: The proposed method should demonstrate a significant reduction in computational cost compared to calculating s(f(x, y)) directly.
3. Scalability: The method should be able to handle large-scale problems with high-dimensional functions and large datasets for x and y.
4. Robustness: The method should be robust to variations in the function and input data.

Deliverables

• A mathematical representation or model that captures the relationship between the dependent variables x and y, and the function f(x, y), which allows us to approximate f(s(x), s(y)) without directly computing s(f(x, y)).
• An algorithm or method that efficiently computes F(s(x), s(y)) to obtain s(c) based on the derived mathematical representation or model. This algorithm should be designed to minimize computational cost while maintaining accuracy, scalability, and robustness.
• A method for validating and quantifying the accuracy of the derived s(c) compared to the result obtained from s(f(x, y)). This can be done using techniques like cross-validation, error analysis, or comparisons with benchmark datasets.
• A thorough analysis of the computational cost, scalability, and robustness of the proposed method compared to directly computing s(f(x, y)). This will help demonstrate the practical benefits and efficiency of the derived approach.