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| ==T-H Exponential==
| | #REDIRECT [[Temperature-Humidity_Relationship#T-H_Exponential]] |
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| By setting <math>m=L(U,V)</math> in the exponential <math>pdf</math> we can obtain the T-H exponential <math>pdf</math>:
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| ::<math>f(t,V,U)=\frac{1}{A}{{e}^{-\left( \tfrac{\phi }{V}+\tfrac{b}{U} \right)}}\cdot {{e}^{-\tfrac{t}{A}\cdot {{e}^{-\left( \tfrac{\phi }{V}+\tfrac{b}{U} \right)}}}}</math>
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| ===T-H Exponential Statistical Properties Summary===
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| ====Mean or MTTF====
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| The mean, <math>\overline{T},</math> or Mean Time To Failure (MTTF) for the T-H exponential model is given by:
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| ::<math>\overline{T}=\int_{0}^{\infty }t\cdot f(t,V,U)dt</math>
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| Substituting Eqn. (t-h exp pdf) yields:
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| ::<math>\begin{align}
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| & \overline{T}= & \int_{0}^{\infty }t\cdot \frac{1}{A}{{e}^{-\left( \tfrac{\phi }{V}+\tfrac{b}{U} \right)}}{{e}^{-\tfrac{t}{A}{{e}^{-\left( \tfrac{\phi }{V}+\tfrac{b}{U} \right)}}}}dt =\ & A{{e}^{\tfrac{\phi }{V}+\tfrac{b}{U}}}
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| \end{align}</math>
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| ====Median====
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| The median, <math>\breve{T},</math> for the T-H exponential model is given by:
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| ::<math>\breve{T}=0.693\cdot A{{e}^{\tfrac{\phi }{V}+\tfrac{b}{U}}}</math>
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| ====Mode====
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| The mode, <math>\tilde{T},</math> for the T-H exponential model is given by:
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| ::<math>\tilde{T}=0</math>
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| ====Standard Deviation====
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| The standard deviation, <math>{{\sigma }_{T}}</math> , for the T-H exponential model is given by:
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| ::<math>{{\sigma }_{T}}=A{{e}^{\tfrac{\phi }{V}+\tfrac{b}{U}}}</math>
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| ====T-H Exponential Reliability Function====
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| The T-H exponential reliability function is given by:
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| ::<math>R(T,V,U)={{e}^{-\tfrac{T}{A}{{e}^{-\left( \tfrac{\phi }{V}+\tfrac{b}{U} \right)}}}}</math>
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| This function is the complement of the T-H exponential cumulative distribution function or:
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| ::<math>R(T,V,U)=1-Q(T,V,U)=1-\int_{0}^{T}f(T)dT</math>
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| and:
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| ::<math>R(T,V,U)=1-\int_{0}^{T}\frac{1}{A}{{e}^{-\left( \tfrac{\phi }{V}+\tfrac{b}{U} \right)}}{{e}^{-\tfrac{T}{A}{{e}^{-\left( \tfrac{\phi }{V}+\tfrac{b}{U} \right)}}}}dT={{e}^{-\tfrac{T}{A}{{e}^{-\left( \tfrac{\phi }{V}+\tfrac{b}{U} \right)}}}}</math>
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| ====Conditional Reliability====
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| The conditional reliability function for the T-H exponential model is given by:
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| ::<math>R(T,t,V,U)=\frac{R(T+t,V,U)}{R(T,V,U)}=\frac{{{e}^{-\lambda (T+t)}}}{{{e}^{-\lambda T}}}={{e}^{-\tfrac{t}{A}{{e}^{-\left( \tfrac{\phi }{V}+\tfrac{b}{U} \right)}}}}</math>
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| ====Reliable Life====
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| For the T-H exponential model, the reliable life, or the mission duration for a desired reliability goal, <math>{{t}_{R}},</math> is given by:
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| ::<math>R({{t}_{R}},V,U)={{e}^{-\tfrac{{{t}_{R}}}{A}{{e}^{-\left( \tfrac{\phi }{V}+\tfrac{b}{U} \right)}}}}</math>
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| ::<math>\ln [R({{t}_{R}},V,U)]=-\frac{{{t}_{R}}}{A}{{e}^{-\left( \tfrac{\phi }{V}+\tfrac{b}{U} \right)}}</math>
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| or:
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| ::<math>{{t}_{R}}=-A{{e}^{\tfrac{\phi }{V}+\tfrac{b}{U}}}\ln [R({{t}_{R}},V,U)]</math>
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| ===Parameter Estimation===
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| ====Maximum Likelihood Estimation Method====
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| Substituting the T-H model into the exponential log-likelihood equation yields:
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| ::<math>\begin{align}
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| & \ln (L)= & \Lambda =\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left[ \frac{1}{A}{{e}^{-\left( \tfrac{\phi }{{{V}_{i}}}+\tfrac{b}{{{U}_{i}}} \right)}}\cdot {{e}^{-\tfrac{{{T}_{i}}}{A}\cdot {{e}^{-\left( \tfrac{\phi }{{{V}_{i}}}+\tfrac{b}{{{U}_{i}}} \right)}}}} \right] \\
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| & & -\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\frac{T_{i}^{\prime }}{A}{{e}^{-\left( \tfrac{\phi }{{{V}_{i}}}+\tfrac{b}{{{U}_{i}}} \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 }]
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| \end{align}</math>
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| where:
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| ::<math>R_{Li}^{\prime \prime }={{e}^{-\tfrac{T_{Li}^{\prime \prime }}{A}{{e}^{-\left( \tfrac{\phi }{{{V}_{i}}}+\tfrac{b}{U_{i}^{\prime \prime }} \right)}}}}</math>
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| ::<math>R_{Ri}^{\prime \prime }={{e}^{-\tfrac{T_{Ri}^{\prime \prime }}{A}{{e}^{-\left( \tfrac{\phi }{{{V}_{i}}}+\tfrac{b}{U_{i}^{\prime \prime }} \right)}}}}</math>
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| and:
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| • <math>{{F}_{e}}</math> is the number of groups of exact times-to-failure data points.
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| • <math>{{N}_{i}}</math> is the number of times-to-failure data points in the <math>{{i}^{th}}</math> time-to-failure data group.
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| • <math>A</math> is the T-H parameter (unknown, the first of three parameters to be estimated).
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| • <math>\phi </math> is the second T-H parameter (unknown, the second of three parameters to be estimated).
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| • <math>b</math> is the third T-H parameter (unknown, the third of three parameters to be estimated).
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| • <math>{{V}_{i}}</math> is the temperature level of the <math>{{i}^{th}}</math> group.
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| • <math>{{U}_{i}}</math> is the relative humidity level of the <math>{{i}^{th}}</math> group.
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| • <math>{{T}_{i}}</math> is the exact failure time of the <math>{{i}^{th}}</math> group.
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| • <math>S</math> is the number of groups of suspension data points.
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| • .. is the number of suspensions in the <math>{{i}^{th}}</math> group of suspension data points.
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| • <math>T_{i}^{\prime }</math> is the running time of the <math>{{i}^{th}}</math> suspension data group.
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| • <math>FI</math> is the number of interval data groups.
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| • <math>N_{i}^{\prime \prime }</math> is the number of intervals in the <math>{{i}^{th}}</math> group of data intervals.
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| • <math>T_{Li}^{\prime \prime }</math> is the beginning of the <math>{{i}^{th}}</math> interval.
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| • <math>T_{Ri}^{\prime \prime }</math> is the ending of the <math>{{i}^{th}}</math> interval.
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| The solution (parameter estimates) will be found by solving for the parameters <math>A,</math> <math>\phi </math> and <math>b</math> so that <math>\tfrac{\partial \Lambda }{\partial A}=0,</math> <math>\tfrac{\partial \Lambda }{\partial \phi }=0</math> and <math>\tfrac{\partial \Lambda }{\partial b}=0</math> .
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| {{t-h weibull}}
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| {{t-h lognormal}}
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