(b) The std devs: using two sample points gives lower values, i.e., reduces noise, as desired. (a) An instantaneous snapshot of the estimates: using two sample points per grid box and time step ( n = 2, squares) reduces the noise somewhat. Here n denotes the number of sample points per grid box and time step n t denotes the total number of points in the Latin hypercube sample. (37) and (40) (thick solid line), traditional Monte Carlo ( n = 1, n t = 1) (circles), Latin hypercube sampling with ( n = 1, n t = 12) (triangles), and Latin hypercube sampling with ( n = 2, n t = 12) (squares). Here F Gt( s) is a truncated Gaussian, and C denotes cloud fraction.Įstimates of grid box average Kessler autoconversion for the BOMEX cumulus simulation. Then the cumulative distribution function F G is used to map V into a point X from a single Gaussian PDF.Ī depiction of a single Gaussian PDF, G( s), its corresponding cumulative distribution function, F G( s), and the corresponding cumulative distribution function associated with the cloudy part ( s > 0), F Gt( s). First a point V is chosen randomly from a uniform distribution. All squares are associated with equal probability (figure not drawn to scale).Īn illustration of the inverse distribution function method for a single Gaussian PDF. (c) The third point is selected randomly from within the only square that remains. (b) The second sample point is chosen, and its row and column are excluded (vertical hatching). The square’s column and row (light shading) are excluded when selecting subsequent squares.
![a latin hypercube sampling r a latin hypercube sampling r](https://media.springernature.com/original/springer-static/image/chp%3A10.1007%2F978-3-319-55914-8_1/MediaObjects/428669_1_En_1_Fig1_HTML.gif)
(a) A “square” is chosen at random (dark shading) and within it is randomly chosen the first sample point (dot).
#A latin hypercube sampling r pdf
The PDF has been divided into three “rows” i.e., strips of nearly constant r t, and three “columns,” i.e., strips of roughly constant w. I successfully used this method with sample sizes as low as 25.A three-point Latin hypercube sample taken from the cloudy portion ( r t > 10) of a bivariate ( w × r t) single Gaussian PDF. Also, you're not restricted to sample sizes of $6n$. This has the advantage over LHC that you can decide to add more samples later, while retaining relatively even sample coverage, and low correlations between variables. Basically, you take a sequence over the real space $^5$, and then map each dimension to your variables (so if you get something in the lower half of your $$ dimension for your first 2-level variable, then you choose level 1, etc.).
![a latin hypercube sampling r a latin hypercube sampling r](https://www.ajjacobson.us/recombining-tree/images/2389_84_62-latin-hypercube-sampling.jpg)
If you expect that your effect size is going to be small relative to noise, then choose a larger sample size.Īnother method that might be sensible is to use a Low-discrepancy sequence, like the Sobol sequence.
![a latin hypercube sampling r a latin hypercube sampling r](https://image.slidesharecdn.com/lhs-170217020840/95/latin-hypercube-sampling-3-638.jpg)
The number of samples you choose is up to you, but more samples will give you more reliable results, and will also help avoid correlation between variables (you should check this when you decide what your samples are, before you actually take them). However, you can do LHC if you use $6n$ (lowest common multiple of 3 and 2) levels, and then map that to your 2- and 3-level spaces. You can't technically do standard LHC sampling, or orthogonal sampling, because it requires each dimension to have the same number of levels. Depending on your experiment (and the difficulty of taking samples), you should ideally just sample everything. The total number of sample combinations you have is $2\times 3 \times 2 \times 3 \times 3 = 108$ (or what ever).