Optimal Custom Designs - Revision history

Kate Racaza at 22:52, 10 August 2017

2017-08-10T22:52:45Z

← Older revision		Revision as of 22:52, 10 August 2017
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	{{Template:Doebook\|12}}		{{Template:Doebook\|12}}
	Although two level fractional factorial designs, Plackett-Burman designs, Taguchi orthogonal array and other predefined designs are enough for most applications, sometimes these designs may not be sufficient due to constraints on available resources such as time, cost and factor values. Therefore, in this chapter, we will discuss how to create an optimal custom design. DOE++ has two types of optimal custom designs: regression model-based and distance-based.		Although two level fractional factorial designs, Plackett-Burman designs, Taguchi orthogonal array and other predefined designs are enough for most applications, sometimes these designs may not be sufficient due to constraints on available resources such as time, cost and factor values. Therefore, in this chapter, we will discuss how to create an optimal custom design. A DOE folio has two types of optimal custom designs: regression model-based and distance-based.

	==Regression Model-Based Optimal Designs==		==Regression Model-Based Optimal Designs==
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	===Algorithms for Selecting Model-Based D-Optimal Designs===		===Algorithms for Selecting Model-Based D-Optimal Designs===

	In DOE++, the Federov’s method, the modified Federov’s method and the k-exchange method are used to select test runs from all the candidates. They are briefly explained below.		In a DOE folio, the Federov’s method, the modified Federov’s method and the k-exchange method are used to select test runs from all the candidates. They are briefly explained below.

	'''The Federov algorithm [[DOE References\|[Fedorov, 1972]]]'''		'''The Federov algorithm [[DOE References\|[Fedorov, 1972]]]'''

Kate Racaza: /* Select a D-Optimal Custom Design from a Standard Design */

2014-04-01T21:39:23Z

Select a D-Optimal Custom Design from a Standard Design

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 ===Select a D-Optimal Custom Design from a Standard Design===
-Assume that we need to design an experiment to investigate five factors with two levels each. To run a full factorial design, a total of <math>{{2}^{5}}=32\,\!</math> factor combinations are needed. However, only 11 samples are available. To obtain the most information, which of the 11 factor combinations from the total of 32 should be applied to these 11 samples? To answer this question, we need to first decide what information we want to obtain.
+{{:Optimal_Custom_Design_Example}}
 ===Algorithms for Selecting Model-Based D-Optimal Designs===

Chuck Smith at 20:11, 22 January 2014

2014-01-22T20:11:36Z

← Older revision		Revision as of 20:11, 22 January 2014
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	{{Template:Doebook\|12}}		{{Template:Doebook\|12}}
	~~=Optimal Custom Designs=~~
	Although two level fractional factorial designs, Plackett-Burman designs, Taguchi orthogonal array and other predefined designs are enough for most applications, sometimes these designs may not be sufficient due to constraints on available resources such as time, cost and factor values. Therefore, in this chapter, we will discuss how to create an optimal custom design. DOE++ has two types of optimal custom designs: regression model-based and distance-based.		Although two level fractional factorial designs, Plackett-Burman designs, Taguchi orthogonal array and other predefined designs are enough for most applications, sometimes these designs may not be sufficient due to constraints on available resources such as time, cost and factor values. Therefore, in this chapter, we will discuss how to create an optimal custom design. DOE++ has two types of optimal custom designs: regression model-based and distance-based.

Harry Guo at 16:46, 9 September 2013

2013-09-09T16:46:07Z

← Older revision		Revision as of 16:46, 9 September 2013
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	In DOE++, the Federov’s method, the modified Federov’s method and the k-exchange method are used to select test runs from all the candidates. They are briefly explained below.		In DOE++, the Federov’s method, the modified Federov’s method and the k-exchange method are used to select test runs from all the candidates. They are briefly explained below.

	'''The Federov algorithm [[DOE References\|[10]]]'''		'''The Federov algorithm [[DOE References\|[Fedorov, 1972]]]'''

	Assume there is an initial optimal design with number of runs of n. The initial optimal design can be obtained using the sequential optimization method given in [[DOE References\|[Dykstra 1971, Galil and Kiefer 1980]]]. We need to exchange one of the rows in the initial design with one of the rows from the candidate runs. Let’s call the initial design <math>{{X}_{old}}\,\!</math> and call the design after row exchange <math>{{X}_{new}}\,\!</math>. The determinant of the new information matrix is:		Assume there is an initial optimal design with number of runs of n. The initial optimal design can be obtained using the sequential optimization method given in [[DOE References\|[Dykstra 1971, Galil and Kiefer 1980]]]. We need to exchange one of the rows in the initial design with one of the rows from the candidate runs. Let’s call the initial design <math>{{X}_{old}}\,\!</math> and call the design after row exchange <math>{{X}_{new}}\,\!</math>. The determinant of the new information matrix is:
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	<math>n\times N\,\!</math> deltas (where n is the number of runs in the current design matrix and N is the number of runs in the candidate run matrix) and chooses the best one for exchange. The algorithm stops when the change of the determinate is less than a pre-defined small value.		<math>n\times N\,\!</math> deltas (where n is the number of runs in the current design matrix and N is the number of runs in the candidate run matrix) and chooses the best one for exchange. The algorithm stops when the change of the determinate is less than a pre-defined small value.

	'''The Modified Federov Algorithm [[DOE References\|[5]]]'''		'''The Modified Federov Algorithm [[DOE References\|[Cook and Nachtsheim, 1980]]]'''

	The above Federov algorithm is very slow since it only conducts one exchange after calculating		The above Federov algorithm is very slow since it only conducts one exchange after calculating
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	'''The K-Exchange Algorithm [[DOE References\|[14]]]'''		'''The K-Exchange Algorithm [[DOE References\|[Johnson and Nachtsheim, 1983]]]'''

	This is a variation of the modified Fedorov algorithm. Instead of calculating the deltas for all the design runs in one iteration, it calculates only the deltas for the k-worst runs. First, the algorithm uses the current design matrix to calculate the variance of each run in the design matrix. The k runs with the lowest variances are the runs that need to exchange. Then for each of the k worst runs, it calculates N deltas with all the N candidate runs. If the largest delta is greater than a pre-defined small value, then this row is exchanged with the candidate row, resulting in the largest positive delta. Once all the k points are exchanged, a new design matrix is obtained and the above steps are repeated until no exchange can be conducted. Usually k is set to be:<math>k\le n/4\,\!</math>where n is the number of runs in the optimal design. 2 Distance-Based Optimal Designs		This is a variation of the modified Fedorov algorithm. Instead of calculating the deltas for all the design runs in one iteration, it calculates only the deltas for the k-worst runs. First, the algorithm uses the current design matrix to calculate the variance of each run in the design matrix. The k runs with the lowest variances are the runs that need to exchange. Then for each of the k worst runs, it calculates N deltas with all the N candidate runs. If the largest delta is greater than a pre-defined small value, then this row is exchanged with the candidate row, resulting in the largest positive delta. Once all the k points are exchanged, a new design matrix is obtained and the above steps are repeated until no exchange can be conducted. Usually k is set to be:<math>k\le n/4\,\!</math>where n is the number of runs in the optimal design. 2 Distance-Based Optimal Designs

Harry Guo: /* Algorithms for Selecting Model-Based D-Optimal Designs */

2013-09-09T16:24:27Z

Algorithms for Selecting Model-Based D-Optimal Designs

← Older revision		Revision as of 16:24, 9 September 2013
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	'''The Federov algorithm [[DOE References\|[10]]]'''		'''The Federov algorithm [[DOE References\|[10]]]'''

	Assume there is an initial optimal design with number of runs of n. The initial optimal design can be obtained using the sequential optimization method given in [[DOE References\|Dykstra 1971, Galil and Kiefer 1980]]. We need to exchange one of the rows in the initial design with one of the rows from the candidate runs. Let’s call the initial design <math>{{X}_{old}}\,\!</math> and call the design after row exchange <math>{{X}_{new}}\,\!</math>. The determinant of the new information matrix is:		Assume there is an initial optimal design with number of runs of n. The initial optimal design can be obtained using the sequential optimization method given in [[DOE References\|[Dykstra 1971, Galil and Kiefer 1980]]]. We need to exchange one of the rows in the initial design with one of the rows from the candidate runs. Let’s call the initial design <math>{{X}_{old}}\,\!</math> and call the design after row exchange <math>{{X}_{new}}\,\!</math>. The determinant of the new information matrix is:

	:<math>\left\| X_{new}^{'}{{X}_{new}} \right\|=\left\| X_{old}^{'}{{X}_{old}} \right\|\times \left( 1+\Delta \left( {{x}_{i}},{{x}_{j}} \right) \right)\,\!</math>		:<math>\left\| X_{new}^{'}{{X}_{new}} \right\|=\left\| X_{old}^{'}{{X}_{old}} \right\|\times \left( 1+\Delta \left( {{x}_{i}},{{x}_{j}} \right) \right)\,\!</math>

Harry Guo: /* Algorithms for Selecting Model-Based D-Optimal Designs */

2013-09-09T16:24:07Z

Algorithms for Selecting Model-Based D-Optimal Designs

← Older revision		Revision as of 16:24, 9 September 2013
Line 189:		Line 189:
	'''The Federov algorithm [[DOE References\|[10]]]'''		'''The Federov algorithm [[DOE References\|[10]]]'''

	Assume there is an initial optimal design with number of runs of n. The initial optimal design can be obtained using the sequential optimization method given in [Dykstra 1971, Galil and Kiefer 1980]. We need to exchange one of the rows in the initial design with one of the rows from the candidate runs. Let’s call the initial design <math>{{X}_{old}}\,\!</math> and call the design after row exchange <math>{{X}_{new}}\,\!</math>. The determinant of the new information matrix is:		Assume there is an initial optimal design with number of runs of n. The initial optimal design can be obtained using the sequential optimization method given in [[DOE References\|Dykstra 1971, Galil and Kiefer 1980]]. We need to exchange one of the rows in the initial design with one of the rows from the candidate runs. Let’s call the initial design <math>{{X}_{old}}\,\!</math> and call the design after row exchange <math>{{X}_{new}}\,\!</math>. The determinant of the new information matrix is:

	:<math>\left\| X_{new}^{'}{{X}_{new}} \right\|=\left\| X_{old}^{'}{{X}_{old}} \right\|\times \left( 1+\Delta \left( {{x}_{i}},{{x}_{j}} \right) \right)\,\!</math>		:<math>\left\| X_{new}^{'}{{X}_{new}} \right\|=\left\| X_{old}^{'}{{X}_{old}} \right\|\times \left( 1+\Delta \left( {{x}_{i}},{{x}_{j}} \right) \right)\,\!</math>

Harry Guo: /* Algorithms for Selecting Model-Based D-Optimal Designs */

2013-09-09T16:22:41Z

Algorithms for Selecting Model-Based D-Optimal Designs

← Older revision		Revision as of 16:22, 9 September 2013
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	'''The Federov algorithm [[DOE References\|[10]]]'''		'''The Federov algorithm [[DOE References\|[10]]]'''

	Assume there is an initial optimal design with number of runs of n. The initial optimal design can be obtained using the sequential optimization method given in [Dykstra 1971, Galil and Kiefer 1980]. We need to exchange one of the rows in the initial design with one of the rows from the candidate runs. Let’s call the initial design <math>{{X}_{old}}\,\!</math> and call the design after row exchange <math>{{X}_{new}}\,\!</math>. The determinant of the new information matrix is:		Assume there is an initial optimal design with number of runs of n. The initial optimal design can be obtained using the sequential optimization method given in [Dykstra 1971, Galil and Kiefer 1980]. We need to exchange one of the rows in the initial design with one of the rows from the candidate runs. Let’s call the initial design <math>{{X}_{old}}\,\!</math> and call the design after row exchange <math>{{X}_{new}}\,\!</math>. The determinant of the new information matrix is:

Harry Guo at 16:22, 9 September 2013

2013-09-09T16:22:23Z

← Older revision		Revision as of 16:22, 9 September 2013
Line 186:		Line 186:

	In DOE++, the Federov’s method, the modified Federov’s method and the k-exchange method are used to select test runs from all the candidates. They are briefly explained below.		In DOE++, the Federov’s method, the modified Federov’s method and the k-exchange method are used to select test runs from all the candidates. They are briefly explained below.
	The Federov algorithm [~~Fedorov 1972~~]
			'''The Federov algorithm [[DOE References\|[10]]]'''
	Assume there is an initial optimal design with number of runs of n. The initial optimal design can be obtained using the sequential optimization method given in [Dykstra 1971, Galil and Kiefer 1980]. We need to exchange one of the rows in the initial design with one of the rows from the candidate runs. Let’s call the initial design <math>{{X}_{old}}\,\!</math> and call the design after row exchange <math>{{X}_{new}}\,\!</math>. The determinant of the new information matrix is:		Assume there is an initial optimal design with number of runs of n. The initial optimal design can be obtained using the sequential optimization method given in [Dykstra 1971, Galil and Kiefer 1980]. We need to exchange one of the rows in the initial design with one of the rows from the candidate runs. Let’s call the initial design <math>{{X}_{old}}\,\!</math> and call the design after row exchange <math>{{X}_{new}}\,\!</math>. The determinant of the new information matrix is:

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	<math>n\times N\,\!</math> deltas (where n is the number of runs in the current design matrix and N is the number of runs in the candidate run matrix) and chooses the best one for exchange. The algorithm stops when the change of the determinate is less than a pre-defined small value.		<math>n\times N\,\!</math> deltas (where n is the number of runs in the current design matrix and N is the number of runs in the candidate run matrix) and chooses the best one for exchange. The algorithm stops when the change of the determinate is less than a pre-defined small value.

	'''The Modified Federov Algorithm [~~Cook and Nachtshim 1980~~]'''		'''The Modified Federov Algorithm [[DOE References\|[5]]]'''

	The above Federov algorithm is very slow since it only conducts one exchange after calculating		The above Federov algorithm is very slow since it only conducts one exchange after calculating

Harry Guo at 16:21, 9 September 2013

2013-09-09T16:21:11Z

← Older revision		Revision as of 16:21, 9 September 2013
Line 210:		Line 210:


	'''The K-Exchange Algorithm [[DOE References\|14]]'''		'''The K-Exchange Algorithm [[DOE References\|[14]]]'''

	This is a variation of the modified Fedorov algorithm. Instead of calculating the deltas for all the design runs in one iteration, it calculates only the deltas for the k-worst runs. First, the algorithm uses the current design matrix to calculate the variance of each run in the design matrix. The k runs with the lowest variances are the runs that need to exchange. Then for each of the k worst runs, it calculates N deltas with all the N candidate runs. If the largest delta is greater than a pre-defined small value, then this row is exchanged with the candidate row, resulting in the largest positive delta. Once all the k points are exchanged, a new design matrix is obtained and the above steps are repeated until no exchange can be conducted. Usually k is set to be:<math>k\le n/4\,\!</math>where n is the number of runs in the optimal design. 2 Distance-Based Optimal Designs		This is a variation of the modified Fedorov algorithm. Instead of calculating the deltas for all the design runs in one iteration, it calculates only the deltas for the k-worst runs. First, the algorithm uses the current design matrix to calculate the variance of each run in the design matrix. The k runs with the lowest variances are the runs that need to exchange. Then for each of the k worst runs, it calculates N deltas with all the N candidate runs. If the largest delta is greater than a pre-defined small value, then this row is exchanged with the candidate row, resulting in the largest positive delta. Once all the k points are exchanged, a new design matrix is obtained and the above steps are repeated until no exchange can be conducted. Usually k is set to be:<math>k\le n/4\,\!</math>where n is the number of runs in the optimal design. 2 Distance-Based Optimal Designs

Harry Guo at 16:20, 9 September 2013

2013-09-09T16:20:44Z

← Older revision		Revision as of 16:20, 9 September 2013
Line 210:		Line 210:


	'''The K-Exchange Algorithm [~~Johnson and Nachtsheim 1983~~]'''		'''The K-Exchange Algorithm [[DOE References\|14]]'''

	This is a variation of the modified Fedorov algorithm. Instead of calculating the deltas for all the design runs in one iteration, it calculates only the deltas for the k-worst runs. First, the algorithm uses the current design matrix to calculate the variance of each run in the design matrix. The k runs with the lowest variances are the runs that need to exchange. Then for each of the k worst runs, it calculates N deltas with all the N candidate runs. If the largest delta is greater than a pre-defined small value, then this row is exchanged with the candidate row, resulting in the largest positive delta. Once all the k points are exchanged, a new design matrix is obtained and the above steps are repeated until no exchange can be conducted. Usually k is set to be:<math>k\le n/4\,\!</math>where n is the number of runs in the optimal design. 2 Distance-Based Optimal Designs		This is a variation of the modified Fedorov algorithm. Instead of calculating the deltas for all the design runs in one iteration, it calculates only the deltas for the k-worst runs. First, the algorithm uses the current design matrix to calculate the variance of each run in the design matrix. The k runs with the lowest variances are the runs that need to exchange. Then for each of the k worst runs, it calculates N deltas with all the N candidate runs. If the largest delta is greater than a pre-defined small value, then this row is exchanged with the candidate row, resulting in the largest positive delta. Once all the k points are exchanged, a new design matrix is obtained and the above steps are repeated until no exchange can be conducted. Usually k is set to be:<math>k\le n/4\,\!</math>where n is the number of runs in the optimal design. 2 Distance-Based Optimal Designs

← Older revision		Revision as of 16:24, 9 September 2013
Line 189:		Line 189:
	'''The Federov algorithm [[DOE References\|[10]]]'''		'''The Federov algorithm [[DOE References\|[10]]]'''

	Assume there is an initial optimal design with number of runs of n. The initial optimal design can be obtained using the sequential optimization method given in [[DOE References\|Dykstra 1971, Galil and Kiefer 1980]]. We need to exchange one of the rows in the initial design with one of the rows from the candidate runs. Let’s call the initial design <math>{{X}_{old}}\,\!</math> and call the design after row exchange <math>{{X}_{new}}\,\!</math>. The determinant of the new information matrix is:		Assume there is an initial optimal design with number of runs of n. The initial optimal design can be obtained using the sequential optimization method given in [[DOE References\|[Dykstra 1971, Galil and Kiefer 1980]]]. We need to exchange one of the rows in the initial design with one of the rows from the candidate runs. Let’s call the initial design <math>{{X}_{old}}\,\!</math> and call the design after row exchange <math>{{X}_{new}}\,\!</math>. The determinant of the new information matrix is:

	:<math>\left\| X_{new}^{'}{{X}_{new}} \right\|=\left\| X_{old}^{'}{{X}_{old}} \right\|\times \left( 1+\Delta \left( {{x}_{i}},{{x}_{j}} \right) \right)\,\!</math>		:<math>\left\| X_{new}^{'}{{X}_{new}} \right\|=\left\| X_{old}^{'}{{X}_{old}} \right\|\times \left( 1+\Delta \left( {{x}_{i}},{{x}_{j}} \right) \right)\,\!</math>