LHD: Program for Constructing Latin Hypercube Designs

  1. Introduction
  2. Using LHD
  3. Output
  4. Examples
  5. References

Introduction

Latin hypercubes (LHs) were designs introduced by McKay, Beckman & Conover (1979) for computer experiments. An nxk LH can be represented by a design matrix Xnxk with n rows (runs) and k columns (factors), each of which includes n uniformly spaced levels. An LH is called an orthogonal LH (OLH) if each pair of columns of this LH has zero correlation. Examples of OLHs can be found in Ye (1998), Cioppa & Lucas (2007) and Nguyen (2008). OLHs are generally inflexible with respect to the numbers of runs and factors and poor with respect to the space-filling property. The OLHs of Steinberg & Lin (2006), for example, are available for nearly n-1 columns in n runs only when n=16, 256, 65 or 536.

LHD is a Gendex module for constructing near-OLHs with good space-filling property. The near-orthogonal OLHs constructed by LHD are quite good compared to those of Cioppa & Lucas (2007) and those in http://www.ams.sunysb.edu/~kye/olh.html with respect to both orthogonality and space-filling properties.

Using LHD

Let's assume all Gendex class files are in the directory c:\gendex and suppose you want to construct a resolvable LHD of size (n,k)=(8,4) (Example 1). At the working directory, type the following command at the Command Prompt (case is important):

java -cp c:\gendex lhd

The LHD window will pop up. Enter 4 as the Number/Additional number of factors and 8 as the Number of runs and 1000 as the No. of tries, the LHD window will become:

Now click START, the LHD window will become:

and the OUTPUT window showing the constructed orthogonal LHD of size (n,k)=(8,4) will pop up. Note that the START button has been changed to the STOP one. If you close the pop-up window, the STOP button will become a RESET one. If you click this RESET button, the output will disappear and you can use LHD for a new design problem. Also note that the default random seed is the one obtained from the system clock and the default number of tries is 100. You can change these default values if you wish to.

Output

The output window contains the result of the best try and is also saved in the file ibd.htm in the working directory. This file can be read by a browser such as IE or Firefox. Information for this try includes:

  1. try number;
  2. the random seed used;
  3. the number of iterations;
  4. f, the objective function. f=0 when the constructed LH is an OLH.
  5. |R|, the determinant of the correlation matrix R;
  6. rmax, the maximum correlation in terms of absolute value of R;
  7. Mm distance, the Euclidean maximin distance (cf. Cioppa & Lucas, 2007). For this space-filling measure, the larger the better.
  8. ML2, the modified L2 discrepancy (cf. Cioppa & Lucas, 2007). For this space-filling measure, the smaller the better.
  9. the factor levels of the constructed LH;
  10. X'X matrix;
  11. R, correlation matrix;
  12. the time in seconds LHD used to construct this design.

Examples

  1. An OLH of size (n,k)=(8,4) (http://designcomputing.net/gendex/lhd/l1.html).
  2. An OLH of size (n,k)=(9,5) (http://designcomputing.net/gendex/lhd/l2.html).
  3. An OLH of size (n,k)=(17,8) (http://designcomputing.net/gendex/lhd/l3.html).
  4. A near-OLH of size (n,k)=(65,16) (http://designcomputing.net/gendex/lhd/l4.html).
  5. A near-OLH of size (n,k)=(129,22) (http://designcomputing.net/gendex/lhd/l5.html).

Notes:

References

Cioppa, T. M. & Lucas, T.W. (2007) Efficient nearly orthogonal and space-filling Latin Hypercubes. Technometrics 49, 45-55.
McKay, M. D., Beckman, R.J., and Conover, W. J. (1979) A comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics 21, 239-245.
Nguyen, N-K. (2009) A new class of orthogonal Latin hypercubes. Statistics and Applications, 6, 96-100 (New Series).
Ye, K. Q. (1998) Orthogonal Latin Hypercubes and their application in computer experiments. J. of the American Statistical Association 93, 1430-1439.

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