Latin hypercube sampling explained code#
The code can produce either random Latin hypercubes or a rather nongeneral set of symmetric ones (I am convinced my second algorithm cannot in principle generate any possible symmetric Latin hypercube, just a specific type). Statistical tests of higher dimensional cases have convinced me it works successfully there as well. I have tested the code below both using automated tests and by visual inspection of the results for the trivial 2D case. Latin hypercube samples are used for the pre-computation instead of a grid, with the idea that they will more efficiently sample the behavior of the function and result in lower interpolation errors. I pre-generate a list of values for this function at a set of points and use interpolation over the parameters to extend this list to any desired point within a set range. The condensed version of why I want to do this is that I want to evaluate a function which is expensive to compute, and depends on many parameters. For 2D and n=4, this is an example of a symmetric Latin hypercube: | |x| | |
, n-1-k), where n is the number of samples. , k), for some integer indices i,j,k, then it also has the sample (n-1-i, n-1-j. One way of expressing this condition, is that (if the samples are zero-indexed) if the hypercube has the sample (i, j. This property essentially means that the hypercube is invariant under spatial inversion. I am also interested in generating a special type of Latin hypercube called a symmetric Latin hypercube. Concatenating these together into a DxN matrix gives a list of coordinates which will form a Latin hypercube. One algorithm for generating this is to generate D random permutations of the integers 0 through N-1, where D is the number of dimensions and N is the desired number of samples. e.g., for 2 dimensions and 4 total samples, this is a Latin hypercube: | | | |x| Latin hypercubes are essentially collections of points on a hypercube that are placed on a cubic/rectangular grid, which possess the property that no two points share any individual coordinate, and every row/column/higher-dimensional-axis is sampled once. Our method may be a useful device to validate the predictions made by FBA-based tools, by describing the optimal flux space associated with these predictions, thus to improve them.I wrote some code to generate Latin hypercube samples for interpolation over high-dimensional parameter spaces. Furthermore, the optimal flux space is subject to change with environmental conditions. These findings imply that the possibilities to reroute greater portions of flux may be limited within metabolic networks of bacteria. In iMO1086, initial variability of 211 reactions could almost entirely be explained by 7 alternative pathway groups. Results show that although many fluxes are identified as variable upon fixing the objective value, majority of the variability can be reduced to several main patterns arising from a few alternative pathways. It is shown to surpass the commonly used Monte Carlo Sampling (MCS) in providing a more uniform coverage for a much larger network in less number of samples. The approach was validated with the elementary mode analysis of a smaller network of Saccharomyces cerevisiae and applied to the GSMR of Pseudomonas aeruginosa PAO1 (iMO1086).
Our method employs Modified Latin-Hypercube Sampling (LHS) to effectively border the optimal space, followed by Principal Component Analysis (PCA) to identify and explain the major sources of variability within it. Herein we present a novel, generic algorithm to characterize the entire flux space of GSMR upon application of FBA, leading to the optimal value of the objective (the optimal flux space). The abundance of alternate flux profiles has led to the evolution of methods to explore the complete solution space aiming to increase the accuracy of predictions. Genome-Scale Metabolic Reconstructions (GSMRs), along with optimization-based methods, predominantly Flux Balance Analysis (FBA) and its derivatives, are widely applied for assessing and predicting the behavior of metabolic networks upon perturbation, thereby enabling identification of potential novel drug targets and biotechnologically relevant pathways.
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