- #LATIN HYPERCUBE SAMPLING AND STRATIFIED FULL#
- #LATIN HYPERCUBE SAMPLING AND STRATIFIED CODE#
- #LATIN HYPERCUBE SAMPLING AND STRATIFIED PLUS#
With low-discrepancy point sets generated by carefully designed deterministic Techniques is that they replace the pseudo-random numbers used in standard Monte Carlo The low-discrepancy sampling techniques introduced inĬhapter 7 are the foundation of a branch of Monte CarloĬalled quasi Monte Carlo. Than uniform random sampling and is often much better. Nevertheless, Latin hypercube sampling is provably no worse Sampling at reducing variance, especially as the number of samples takenīecomes large. Unfortunately, Latin hypercube sampling isn’t as effective as stratified Introduced in Section 7.3), which can generateĪny number of samples independent of the number of dimensions. Left-parenthesis theta comma phi right-parenthesis ray direction-stratifying left-parenthesis x comma theta right-parenthesis and left-parenthesis y comma phi right-parenthesisĪnother solution to the curse of dimensionality that has many of the sameĪdvantages of stratification is to use Latin hypercube sampling (also Lighting example in Section 13.7.1, it is far moreĮffective to stratify the left-parenthesis x comma y right-parenthesis pixel positions and to stratify the Which dimensions are stratified should be done in a way that stratifiesĭimensions that tend to be most highly correlated in their effect on the Fortunately, it is often possible to stratify some of theĭimensions independently and then randomly associate samples from differentĭimensions, as was done in Section 7.3.
#LATIN HYPERCUBE SAMPLING AND STRATIFIED FULL#
Full stratification in upper Dĭimensions with upper S strata per dimension requires upper S Superscript upper D samples, which quicklyīecomes prohibitive. Of dimensionality” as standard numerical quadrature. The main downside of stratified sampling is that it suffers from the same “curse (Compare the edges of the highlights on the ground, for example.) Than (2) when a stratified distribution of sample directions is used There is a reasonable reduction in variance atįigure 13.17: Variance is higher and the image noisier (1) when random sampling is used to Sampling versus a uniform random distribution for sampling ray directionsįor glossy reflection. įigure 13.17 shows the effect of using stratified More variation and will have mu Subscript i closer to the true mean upper Q. If the strata are wide, they will contain This explains why compact strata are desirable if one does not knowĪnything about the function f. Make the strata have means that are as unequal as possible. In fact, for stratified sampling to workīest, we would like to maximize the right-hand sum, so it is best to Mean over each stratum normal upper Lamda Subscript i. It can only be 0 when the function f has the same Inįact, stratification always reduces variance unless the right-hand sum isĮxactly 0.
Second, it demonstrates that stratified sampling can never increase variance. Right-hand sum must be nonnegative, since variance is always nonnegative. There are two things to notice about this expression. See Veach ( 1997) for a derivation of this result. Where upper Q is the mean of f over the whole domain normal upper Lamda.
#LATIN HYPERCUBE SAMPLING AND STRATIFIED PLUS#
Graphical analysis for continuous data and chi-square analysis of categorical data suggested optimal sample size for this study area is approximately 200 to 300 (0.05–0.1%).StartLayout 1st Row 1st Column upper V left-bracket upper F right-bracket 2nd Column equals upper E Subscript x Baseline upper V Subscript i Baseline upper F plus upper V Subscript x Baseline upper E Subscript i Baseline upper F 2nd Row 1st Column Blank 2nd Column equals StartFraction 1 Over upper N EndFraction left-bracket sigma-summation v Subscript i Baseline sigma Subscript i Superscript 2 Baseline plus sigma-summation v Subscript i Baseline left-parenthesis mu Subscript i Baseline minus upper Q right-parenthesis right-bracket comma EndLayout
#LATIN HYPERCUBE SAMPLING AND STRATIFIED CODE#
The cLHS code was run in Matlab™ (Mathworks, 2008) and statistical analysis was performed using the R statistical language (R Development Core Team, 2009). This paper briefly reviews cLHS and investigates different sample sizes for representing five environmental covariates in a 30,000-ha complex landscape in the Great Basin of southwestern Utah. As the smallest possible sample is important for efficient field work, what is the optimal sample size for digital soil mapping? An optimal sample size accurately represents the variability in the environmental covariates and provides enough samples for predictive models. Conditioned Latin Hypercube Sampling (cLHS) is a type of stratified random sampling that accurately represents the variability of environmental covariates in feature space.