Template:Gll relationship - Revision history

Richard House: Redirected page to Multivariable Relationships: General Log-Linear and Proportional Hazards

2012-08-15T22:44:36Z

Redirected page to Multivariable Relationships: General Log-Linear and Proportional Hazards

← Older revision		Revision as of 22:44, 15 August 2012
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	~~=General Log-Linear Relationship=~~		#REDIRECT [[Multivariable_Relationships:_General_Log-Linear_and_Proportional_Hazards]]
	~~<br>~~
	When a test involves multiple accelerating stresses or requires the inclusion of an engineering variable, a general multivariable relationship is needed. Such a relationship is the general log-linear relationship, which describes a life characteristic as a function of a vector of <math>n</math> stresses, or <math>\underline{X}=({{X}_{1}},{{X}_{2}}...{{X}_{n}}).</math> ALTA includes this relationship and allows up to eight stresses. Mathematically the relationship is given by:
	~~<br>~~
	~~::<math>L(\underline{X})={{e}^{{{\alpha }_{0}}+\underset{j=1}{\overset{n}{\mathop{\sum }}}\,{{\alpha }_{j}}{{X}_{j}}}}</math>~~

	~~<br>~~
	~~where:~~
	~~<br>~~
	~~• <math>{{\alpha }_{0}}</math> and <math>{{\alpha }_{j}}</math> are model parameters.~~

	~~• <math>X</math> is a vector of <math>n</math> stresses.~~

	This relationship can be further modified through the use of transformations and can be reduced to the relationships discussed previously, if so desired. As an example, consider a single stress application of this relationship and an inverse transformation on <math>X,</math> such that <math>V=1/X</math> or:

	~~<br>~~
	~~::<math>\begin{align}~~
	~~& L(V)= & {{e}^{{{\alpha }_{0}}+\tfrac{{{\alpha }_{1}}}{V}}} =\ & {{e}^{{{\alpha }_{0}}}}{{e}^{\tfrac{{{\alpha }_{1}}}{V}}}~~
	~~\end{align}</math>~~

	~~<br>~~
	~~It can be easily seen that the generalized log-linear relationship with a single stress and an inverse transformation has been reduced to the~~ [[~~Arrhenius Relationship\|Arrhenius relationship]], where:~~
	~~<br>~~
	~~::<math>\begin{align}~~
	~~& C= & {{e}^{{{\alpha }_{0}}}} \\~~
	~~& B= & {{\alpha }_{1}}~~
	~~\end{align}</math>~~

	~~<br>~~
	~~or:~~
	~~<br>~~

	::~~<math>L(V)=C{{e}^{\tfrac{B}{V}}}</math>~~


	~~<br>~~
	Similarly, when one chooses to apply a logarithmic transformation on <math>X</math> such that <math>V=\ln (X)</math> , the relationship would reduce to the [[Inverse Power Law (IPL) Relationship\|Inverse Power Law relationship]]. Furthermore, if more than one stress is present, one could choose to apply a different transformation to each stress to create combination relationships similar to the [[Temperature-Humidity Relationship\|Temperature-~~Humidity~~]] ~~and the [[Temperature-NonThermal Relationship\|Temperature-Non Thermal]]. ALTA has three built-in transformation options, namely:~~

	~~<br>~~
	~~{\|align="center"~~
	\|-
	~~\|None\|\| X=V\|\| Exponential LSR~~
	\|-
	~~\|Reciprocal\|\| <math>V=1/X</math>\|\| Arrhenius LSR~~
	\|-
	~~\|Logarithmic\|\| <math>V=\ln (X)</math>\|\| Power LSR~~
	\|}

	~~<br>~~
	The power of the relationship and this formulation becomes evident once one realizes that 6,651 unique life-stress relationships are possible (when allowing a maximum of eight stresses). When combined with the life distributions available in ALTA, almost 20,000 models can be created.
	~~<br>~~
	~~==Using the GLL Model==~~
	~~<br>~~
	Like the previous relationships, the general log-linear relationship can be combined with any of the available life distributions by expressing a life characteristic from that distribution with the GLL relationship. A brief overview of the GLL-distribution models available in ALTA is presented next.
	~~<br>~~
	~~{{gll exponential}}~~
	~~{{gll weibull}}~~
	~~{{gll lognormal}}~~

	~~===GLL Likelihood Function===~~
	~~<br>~~
	The maximum likelihood estimation method can be used to determine the parameters for the GLL relationship and the selected life distribution. For each distribution, the likelihood function can be derived, and the parameters of model (the distribution parameters and the GLL parameters) can be obtained by maximizing the log-likelihood function. For example, the log-likelihood function for the Weibull distribution is given by:

	~~<br>~~
	~~::<math>\begin{align}~~
	& \ln (L)= & \Lambda =\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left[ \beta \cdot T_{i}^{\beta -1}{{e}^{-T_{i}^{\beta }\cdot {{e}^{-\beta \left( {{\alpha }_{0}}+\mathop{\sum}_{j=1}^{n}{{a}_{j}}{{x}_{i,j}} \right)}}}}{{e}^{-\beta \left( {{\alpha }_{0}}+\mathop{\sum}_{j=1}^{n}{{a}_{j}}{{x}_{i,j}} \right)}} \right] -\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }{{\left( T_{i}^{\prime } \right)}^{\beta }}{{e}^{-\beta \left( {{\alpha }_{0}}+\mathop{\sum}_{j=1}^{n}{{a}_{j}}{{x}_{i,j}} \right)}}+\overset{FI}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\ln [R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime }]
	~~\end{align}</math>~~

	~~<br>~~
	~~where:~~
	~~<br>~~
	~~::<math>\begin{align}~~
	& R_{Li}^{\prime \prime }= & {{e}^{-{{\left( T_{Li}^{\prime \prime }{{e}^{{{\alpha }_{0}}+\underset{j=1}{\mathop{\overset{n}{\mathop{\mathop{}_{}^{}}}\,}}\,{{\alpha }_{j}}{{x}_{j}}}} \right)}^{\beta }}}} \\
	& R_{Ri}^{\prime \prime }= & {{e}^{-{{\left( T_{Ri}^{\prime \prime }{{e}^{{{\alpha }_{0}}+\underset{j=1}{\mathop{\overset{n}{\mathop{\mathop{}_{}^{}}}\,}}\,{{\alpha }_{j}}{{x}_{j}}}} \right)}^{\beta }}}}
	~~\end{align}</math>~~

	~~<br>~~
	~~and:~~
	~~<br>~~
	~~• <math>{{F}_{e}}</math> is the number of groups of exact times-to-failure data points.~~
	~~<br>~~
	~~• <math>{{N}_{i}}</math> is the number of times-to-failure in the <math>{{i}^{th}}</math> time-to-failure data group.~~
	~~<br>~~
	~~• <math>\lambda </math> is the failure rate parameter (unknown).~~
	~~<br>~~
	~~• <math>{{T}_{i}}</math> is the exact failure time of the <math>{{i}^{th}}</math> group.~~
	~~<br>~~
	~~• <math>S</math> is the number of groups of suspension data points.~~
	~~<br>~~
	~~• <math>N_{i}^{\prime }</math> is the number of suspensions in the <math>{{i}^{th}}</math> group of suspension data points.~~
	~~<br>~~
	~~• <math>T_{i}^{\prime }</math> is the running time of the <math>{{i}^{th}}</math> suspension data group.~~
	~~<br>~~
	~~• <math>FI</math> is the number of interval data groups.~~
	~~<br>~~
	~~• <math>N_{i}^{\prime \prime }</math> is the number of intervals in the <math>{{i}^{th}}</math> group of data intervals.~~
	~~<br>~~
	~~• <math>T_{Li}^{\prime \prime }</math> is the beginning of the <math>{{i}^{th}}</math> interval.~~
	~~<br>~~
	~~• <math>T_{Ri}^{\prime \prime }</math> is the ending of the <math>{{i}^{th}}</math> interval.~~
	~~<br>~~
	~~<br>~~

	~~{{Example:GLL}}~~

	~~{{PH Model}}~~

Athanasios Gerokostopoulos: /* Using the GLL Model */

2012-02-20T23:43:59Z

Using the GLL Model

← Older revision		Revision as of 23:43, 20 February 2012
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	The power of the relationship and this formulation becomes evident once one realizes that 6,651 unique life-stress relationships are possible (when allowing a maximum of eight stresses). When combined with the life distributions available in ALTA, almost 20,000 models can be created.		The power of the relationship and this formulation becomes evident once one realizes that 6,651 unique life-stress relationships are possible (when allowing a maximum of eight stresses). When combined with the life distributions available in ALTA, almost 20,000 models can be created.
	<br>		<br>
	===Using the GLL Model===		==Using the GLL Model==
	<br>		<br>
	Like the previous relationships, the general log-linear relationship can be combined with any of the available life distributions by expressing a life characteristic from that distribution with the GLL relationship. A brief overview of the GLL-distribution models available in ALTA is presented next.		Like the previous relationships, the general log-linear relationship can be combined with any of the available life distributions by expressing a life characteristic from that distribution with the GLL relationship. A brief overview of the GLL-distribution models available in ALTA is presented next.
Line 58:		Line 58:
	{{gll lognormal}}		{{gll lognormal}}

	====GLL Likelihood Function====		===GLL Likelihood Function===
	<br>		<br>
	The maximum likelihood estimation method can be used to determine the parameters for the GLL relationship and the selected life distribution. For each distribution, the likelihood function can be derived, and the parameters of model (the distribution parameters and the GLL parameters) can be obtained by maximizing the log-likelihood function. For example, the log-likelihood function for the Weibull distribution is given by:		The maximum likelihood estimation method can be used to determine the parameters for the GLL relationship and the selected life distribution. For each distribution, the likelihood function can be derived, and the parameters of model (the distribution parameters and the GLL parameters) can be obtained by maximizing the log-likelihood function. For example, the log-likelihood function for the Weibull distribution is given by:

Athanasios Gerokostopoulos: /* General Log-Linear Relationship */

2012-02-20T23:43:19Z

General Log-Linear Relationship

← Older revision		Revision as of 23:43, 20 February 2012
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	==General Log-Linear Relationship==		=General Log-Linear Relationship=
	<br>		<br>
	When a test involves multiple accelerating stresses or requires the inclusion of an engineering variable, a general multivariable relationship is needed. Such a relationship is the general log-linear relationship, which describes a life characteristic as a function of a vector of <math>n</math> stresses, or <math>\underline{X}=({{X}_{1}},{{X}_{2}}...{{X}_{n}}).</math> ALTA includes this relationship and allows up to eight stresses. Mathematically the relationship is given by:		When a test involves multiple accelerating stresses or requires the inclusion of an engineering variable, a general multivariable relationship is needed. Such a relationship is the general log-linear relationship, which describes a life characteristic as a function of a vector of <math>n</math> stresses, or <math>\underline{X}=({{X}_{1}},{{X}_{2}}...{{X}_{n}}).</math> ALTA includes this relationship and allows up to eight stresses. Mathematically the relationship is given by:

Athanasios Gerokostopoulos: /* GLL Likelihood Function */

2012-02-20T22:16:39Z

GLL Likelihood Function

← Older revision		Revision as of 22:16, 20 February 2012
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	<br>		<br>
	::<math>\begin{align}		::<math>\begin{align}
	& \ln (L)= & \Lambda =\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left[ \beta \cdot T_{i}^{\beta -1}{{e}^{-T_{i}^{\beta }\cdot {{e}^{-\beta \left( {{\alpha }_{0}}+\mathop{}_{j=1}^{n}{{a}_{j}}{{x}_{i,j}} \right)}}}}{{e}^{-\beta \left( {{\alpha }_{0}}+\mathop{}_{j=1}^{n}{{a}_{j}}{{x}_{i,j}} \right)}} \right] \\		& \ln (L)= & \Lambda =\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left[ \beta \cdot T_{i}^{\beta -1}{{e}^{-T_{i}^{\beta }\cdot {{e}^{-\beta \left( {{\alpha }_{0}}+\mathop{\sum}_{j=1}^{n}{{a}_{j}}{{x}_{i,j}} \right)}}}}{{e}^{-\beta \left( {{\alpha }_{0}}+\mathop{\sum}_{j=1}^{n}{{a}_{j}}{{x}_{i,j}} \right)}} \right] -\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }{{\left( T_{i}^{\prime } \right)}^{\beta }}{{e}^{-\beta \left( {{\alpha }_{0}}+\mathop{\sum}_{j=1}^{n}{{a}_{j}}{{x}_{i,j}} \right)}}+\overset{FI}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\ln [R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime }]
	~~& &~~ -\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }{{\left( T_{i}^{\prime } \right)}^{\beta }}{{e}^{-\beta \left( {{\alpha }_{0}}+\mathop{}_{j=1}^{n}{{a}_{j}}{{x}_{i,j}} \right)}}+\overset{FI}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\ln [R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime }]
	\end{align}</math>		\end{align}</math>

	<br>		<br>
	:where:		where:

	<br>		<br>
	::<math>\begin{align}		::<math>\begin{align}
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	<br>		<br>
	:and:		and:
	<br>		<br>
	• <math>{{F}_{e}}</math> is the number of groups of exact times-to-failure data points.		• <math>{{F}_{e}}</math> is the number of groups of exact times-to-failure data points.

Athanasios Gerokostopoulos: /* Using the GLL Model */

2012-02-20T22:00:37Z

Using the GLL Model

← Older revision		Revision as of 22:00, 20 February 2012
Line 52:		Line 52:
	===Using the GLL Model===		===Using the GLL Model===
	<br>		<br>
	Like the previous relationships, the general log-linear relationship can be combined with any of the available life distributions by expressing a life characteristic from that distribution with the GLL relationship. A brief overview of the GLL-distribution models available in ALTA ~~PRO~~ is presented next.		Like the previous relationships, the general log-linear relationship can be combined with any of the available life distributions by expressing a life characteristic from that distribution with the GLL relationship. A brief overview of the GLL-distribution models available in ALTA is presented next.
	<br>		<br>
	{{gll exponential}}		{{gll exponential}}

Athanasios Gerokostopoulos: /* General Log-Linear Relationship */

2012-02-20T22:00:06Z

General Log-Linear Relationship

← Older revision		Revision as of 22:00, 20 February 2012
Line 48:		Line 48:

	<br>		<br>
	The power of the relationship and this formulation becomes evident once one realizes that 6,651 unique life-stress relationships are possible (when allowing a maximum of eight stresses). When combined with the life distributions available in ALTA ~~PRO~~, almost 20,000 models can be created.		The power of the relationship and this formulation becomes evident once one realizes that 6,651 unique life-stress relationships are possible (when allowing a maximum of eight stresses). When combined with the life distributions available in ALTA, almost 20,000 models can be created.
	<br>		<br>
	===Using the GLL Model===		===Using the GLL Model===

Athanasios Gerokostopoulos: /* General Log-Linear Relationship */

2012-02-20T21:58:53Z

General Log-Linear Relationship

← Older revision		Revision as of 21:58, 20 February 2012
Line 20:		Line 20:

	<br>		<br>
	It can be easily seen that the generalized log-linear relationship with a single stress and an inverse transformation has been reduced to the Arrhenius relationship ~~discussed in Chapter 6~~, where:		It can be easily seen that the generalized log-linear relationship with a single stress and an inverse transformation has been reduced to the [[Arrhenius Relationship\|Arrhenius relationship]], where:
	<br>		<br>
	::<math>\begin{align}		::<math>\begin{align}
Line 35:		Line 35:

	<br>		<br>
	Similarly, when one chooses to apply a logarithmic transformation on <math>X</math> such that <math>V=\ln (X)</math> , the relationship would reduce to the ~~inverse power relation discussed in Chapter 8~~. Furthermore, if more than one stress is present, one could choose to apply a different transformation to each stress to create combination relationships similar to the ~~ones discussed in Chapters 9~~ and 10. ALTA ~~PRO~~ has three built-in transformation options, namely:		Similarly, when one chooses to apply a logarithmic transformation on <math>X</math> such that <math>V=\ln (X)</math> , the relationship would reduce to the [[Inverse Power Law (IPL) Relationship\|Inverse Power Law relationship]]. Furthermore, if more than one stress is present, one could choose to apply a different transformation to each stress to create combination relationships similar to the [[Temperature-Humidity Relationship\|Temperature-Humidity]] and the [[Temperature-NonThermal Relationship\|Temperature-Non Thermal]]. ALTA has three built-in transformation options, namely:

	<br>		<br>

Athanasios Gerokostopoulos: /* General Log-Linear Relationship */

2012-02-20T21:41:58Z

General Log-Linear Relationship

← Older revision		Revision as of 21:41, 20 February 2012
Line 6:		Line 6:

	<br>		<br>
	:where:		where:
	<br>		<br>
	• <math>{{\alpha }_{0}}</math> and <math>{{\alpha }_{j}}</math> are model parameters.		• <math>{{\alpha }_{0}}</math> and <math>{{\alpha }_{j}}</math> are model parameters.
Line 16:		Line 16:
	<br>		<br>
	::<math>\begin{align}		::<math>\begin{align}
	& L(V)= & {{e}^{{{\alpha }_{0}}+\tfrac{{{\alpha }_{1}}}{V}}} \\		& L(V)= & {{e}^{{{\alpha }_{0}}+\tfrac{{{\alpha }_{1}}}{V}}} =\ & {{e}^{{{\alpha }_{0}}}}{{e}^{\tfrac{{{\alpha }_{1}}}{V}}}
	~~& =~~ & {{e}^{{{\alpha }_{0}}}}{{e}^{\tfrac{{{\alpha }_{1}}}{V}}}
	\end{align}</math>		\end{align}</math>

	<br>		<br>
	It can be easily seen that the generalized log-linear relationship with a single stress and an inverse transformation~~, Eqn. (argll),~~ has been reduced to the Arrhenius relationship discussed in Chapter 6, where:		It can be easily seen that the generalized log-linear relationship with a single stress and an inverse transformation has been reduced to the Arrhenius relationship discussed in Chapter 6, where:
	<br>		<br>
	::<math>\begin{align}		::<math>\begin{align}
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	<br>		<br>
	:or:		or:
	<br>		<br>

Athanasios Gerokostopoulos: /* General Log-Linear Relationship */

2012-02-20T21:28:33Z

General Log-Linear Relationship

← Older revision		Revision as of 21:28, 20 February 2012
Line 1:		Line 1:
	==General Log-Linear Relationship==		==General Log-Linear Relationship==
	<br>		<br>
	When a test involves multiple accelerating stresses or requires the inclusion of an engineering variable, a general multivariable relationship is needed. Such a relationship is the general log-linear relationship, which describes a life characteristic as a function of a vector of <math>n</math> stresses, or <math>\underline{X}=({{X}_{1}},{{X}_{2}}...{{X}_{n}}).</math> ALTA ~~PRO~~ includes this relationship and allows up to eight stresses. Mathematically the relationship is given by:		When a test involves multiple accelerating stresses or requires the inclusion of an engineering variable, a general multivariable relationship is needed. Such a relationship is the general log-linear relationship, which describes a life characteristic as a function of a vector of <math>n</math> stresses, or <math>\underline{X}=({{X}_{1}},{{X}_{2}}...{{X}_{n}}).</math> ALTA includes this relationship and allows up to eight stresses. Mathematically the relationship is given by:
	<br>		<br>
	::<math>L(\underline{X})={{e}^{{{\alpha }_{0}}+\underset{j=1}{\overset{n}{\mathop{\sum }}}\,{{\alpha }_{j}}{{X}_{j}}}}</math>		::<math>L(\underline{X})={{e}^{{{\alpha }_{0}}+\underset{j=1}{\overset{n}{\mathop{\sum }}}\,{{\alpha }_{j}}{{X}_{j}}}}</math>

Nicolette Young: /* Example */

2012-02-14T17:58:25Z

Example

@@ Line 105: / Line 105: @@
 <br>
-===Example===
+{{Example:GLL}}
 {{PH Model}}