## Introduction

Latin hypercubes (LHs) were designs introduced by McKay, Beckman &
Conover (1979) for computer experiments. An *n*x*k* LH can be
represented by a design matrix X_{nxk} 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 using the algoithm described in Nguyen & Lin (2012). 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 GUI will pop up. Enter 4 in the **Number
of factors** field and 8 in the **Number of runs** field, the LHD window will become:

Now click **START**, LHD will start running and after try 222, the plan of the
constructed LHD for this try pops up in the LHD output window (as the *f* value
reaches 0) and then LHD stops:

**START**button has been changed to the

**RESET**one. If you click this

**RESET**button, the output will disappear and you can use LHD for a new design problem. Note that the default random seed is the one obtained from the system clock and the default number of tries is 1000. You can change these default values if you wish to.

## Output

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

- Try number;
- The number of iterations;
*f*, the objective function.*f*=0 when the constructed LH is an OLH.*|***R***|*, the determinant of the correlation matrix**R**;*r*_{max}, the maximum correlation in terms of absolute value of R;*r*_{ave}, the average of the absolute value of the pair-wise correlations;*Mm*distance, the Euclidean maximin distance (cf. Cioppa & Lucas, 2007). For this space-filling measure, the larger the better.*ML2*, the modified*L*_{2}discrepancy (cf. Cioppa & Lucas, 2007). For this space-filling measure, the smaller the better.- The design plan and the associated random seed;
matrix;**X**'**X****R**, correlation matrix;- The time in seconds LHD used to construct this design.

## Examples

- An OLH of size (
*n,k*)=(8,4) (http://designcomputing.net/gendex/lhd/l1.html). - An OLH of size (
*n,k*)=(9,5) (http://designcomputing.net/gendex/lhd/l2.html). - An OLH of size (
*n,k*)=(17,8) (http://designcomputing.net/gendex/lhd/l3.html). - An OLH of size (
*n,k*)=(33,9) (http://designcomputing.net/gendex/lhd/l4.html). - An OLH of size (
*n,k*)=(33,11) (http://designcomputing.net/gendex/lhd/l5.html). - A near-OLH of size (
*n,k*)=(65,16) (http://designcomputing.net/gendex/lhd/l6.html). - A near-OLH of size (
*n,k*)=(129,22) (http://designcomputing.net/gendex/lhd/l7.html).

**Notes**:

- Example 3: This OLH was constructed by adding an orthogonal column to the
OLH of size (
*n,k*)=(17,7) of size Cioppa & Lucas (2007). - Examples 4-7: The Mm distance criterion was used to construct the near-OLHs in these two examples.

## 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. (2008) A new class of orthogonal Latin hypercubes. *Statistics
and Applications*, **6**, 119-123 (New Series).

Nguyen, N-K. & Lin D.K.J. (2012) A Note on near-Orthogonal Latin Hypercubes with good
space filling properties. *Journal of Statistics Theory & Practice* **6**, 492-500.

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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