Degradation Data Analysis - Revision history

Lisa Hacker: /* Using Extrapolated Intervals */

2023-07-12T15:58:17Z

Using Extrapolated Intervals

← Older revision		Revision as of 15:58, 12 July 2023
Line 78:		Line 78:

	===Using Extrapolated Intervals===		===Using Extrapolated Intervals===
	The parameters in a degradation model are estimated using available degradation data. If the data is large, the uncertainty of the estimated parameters will be small; otherwise, the uncertainty will be large. Since the failure time for a test unit is predicted based on the estimated model, we sometimes would like to see how the parameter uncertainty affects the failure time prediction. Let’s use the exponential model as an example. Assume that the critical degradation value is <math>{{y}_{crit}}\,\!</math>. The predicted failure time will be		The parameters in a degradation model are estimated using available degradation data. If the data is large, the uncertainty of the estimated parameters will be small; otherwise, the uncertainty will be large. Since the failure time for a test unit is predicted based on the estimated model, we sometimes would like to see how the parameter uncertainty affects the failure time prediction. Let’s use the exponential model as an example. Assume that the critical degradation value is <math>{{y}_{crit}}\,\!</math>. The predicted failure time will be:

	::<math> \hat{x}=\frac{\ln ({{y}_{crit}})-\ln (\hat{b})}{{\hat{a}}}\,\!</math>		::<math> \hat{x}=\frac{\ln ({{y}_{crit}})-\ln (\hat{b})}{{\hat{a}}}\,\!</math>

Lisa Hacker: /* Using Extrapolated Intervals */

2023-07-12T15:57:13Z

Using Extrapolated Intervals

← Older revision		Revision as of 15:57, 12 July 2023
Line 78:		Line 78:

	===Using Extrapolated Intervals===		===Using Extrapolated Intervals===
	The parameters in a degradation model are estimated using available degradation data. If the data is large, the uncertainty of the estimated parameters will be small; otherwise, the uncertainty will be large. Since the failure time for a test unit is predicted based on the estimated model, we sometimes would like to see how the parameter uncertainty affects the failure time prediction. Let’s use the exponential model as an example. Assume that the critical degradation value is <math>{{y}_{crit}}\,\!</math>. The predicted failure time will be:		The parameters in a degradation model are estimated using available degradation data. If the data is large, the uncertainty of the estimated parameters will be small; otherwise, the uncertainty will be large. Since the failure time for a test unit is predicted based on the estimated model, we sometimes would like to see how the parameter uncertainty affects the failure time prediction. Let’s use the exponential model as an example. Assume that the critical degradation value is <math>{{y}_{crit}}\,\!</math>. The predicted failure time will be

	::<math> \hat{x}=\frac{\ln ({{y}_{crit}})-\ln (\hat{b})}{{\hat{a}}}\,\!</math>		::<math> \hat{x}=\frac{\ln ({{y}_{crit}})-\ln (\hat{b})}{{\hat{a}}}\,\!</math>

Lisa Hacker: /* Example */

2023-05-23T17:52:17Z

Example

← Older revision		Revision as of 17:52, 23 May 2023
Line 76:		Line 76:

	These times-to-failure can now be analyzed using traditional life data analysis to obtain metrics such as the probability of failure, B10 life, mean life, etc. This analysis can be automatically performed in the Weibull++ degradation analysis folio.		These times-to-failure can now be analyzed using traditional life data analysis to obtain metrics such as the probability of failure, B10 life, mean life, etc. This analysis can be automatically performed in the Weibull++ degradation analysis folio.

	~~<div class="noprint">~~
	~~{{Examples Box\|Weibull++ Examples\|<p>More degradation analysis examples are available! See also:</p>~~
	~~{{Examples Both\|http://www.reliasoft.com/Weibull/examples/rc4/index.htm\|Degradation Analysis\|http://www.reliasoft.tv/weibull/appexamples/weibull_app_ex_4\|Watch the video...}}<nowiki/>~~
	}}
	~~</div>~~

	===Using Extrapolated Intervals===		===Using Extrapolated Intervals===

Cruz Daniel: /* Using Extrapolated Intervals */

2021-03-23T17:31:00Z

Using Extrapolated Intervals

← Older revision		Revision as of 17:31, 23 March 2021
Line 102:		Line 102:


	In Weibull++, the ~~default~~ confidence level is 90%.		In Weibull++, the confidence level used is 90%.

	====Example====		====Example====

Nicolette Young: /* Example */

2015-12-23T21:43:48Z

Example

← Older revision		Revision as of 21:43, 23 December 2015
Line 219:		Line 219:


	[[Image:Destructive_Degradation_Example_Plot1.png\|center\|~~500px~~\| ]]		[[Image:Destructive_Degradation_Example_Plot1.png\|center\|550px\| ]]

Nicolette Young: /* Destructive Degradation Analysis */

2015-12-23T21:43:29Z

Destructive Degradation Analysis

← Older revision		Revision as of 21:43, 23 December 2015
Line 127:		Line 127:
	The Destructive Degradation Analysis applies to cases where the sample has to be destroyed in order to obtain a degradation measurement. As a result, degradation measurements for multiple samples are required at different points in time. The analysis performed is very similar to Accelerated Life Testing Analysis (ALTA). In this case, the "stress" used in ALTA becomes time, while the random variable becomes the degradation measurement instead of the time-to-failure. Given a defined level of failure (or the degradation level that would constitute a failure), the probability that the degradation measurement will be beyond that level at a given time can be obtained. The following plot shows the relationship between the distribution of the degradation measurement and time. The red shaded area of the last two pdfs represents the probability that the degradation measurement will be less than the critical degradation level at the corresponding times.		The Destructive Degradation Analysis applies to cases where the sample has to be destroyed in order to obtain a degradation measurement. As a result, degradation measurements for multiple samples are required at different points in time. The analysis performed is very similar to Accelerated Life Testing Analysis (ALTA). In this case, the "stress" used in ALTA becomes time, while the random variable becomes the degradation measurement instead of the time-to-failure. Given a defined level of failure (or the degradation level that would constitute a failure), the probability that the degradation measurement will be beyond that level at a given time can be obtained. The following plot shows the relationship between the distribution of the degradation measurement and time. The red shaded area of the last two pdfs represents the probability that the degradation measurement will be less than the critical degradation level at the corresponding times.

	[[Image:Destructive_Degradation_1.1.png\|center\|~~450px~~\| ]]		[[Image:Destructive_Degradation_1.1.png\|center\|500px\| ]]

	===Destructive Degradation Models===		===Destructive Degradation Models===
Line 177:		Line 177:
	The relationship between the distribution of degradation measurement and the distribution of failure time is illustrated in the following plot.		The relationship between the distribution of degradation measurement and the distribution of failure time is illustrated in the following plot.

	[[Image:Destructive_Degradation_2.png\|center\|~~450px~~\| ]]		[[Image:Destructive_Degradation_2.png\|center\|500px\| ]]

	====Example====		====Example====
Line 219:		Line 219:


	[[Image:Destructive_Degradation_Example_Plot1.png\|center\|~~450px~~\| ]]		[[Image:Destructive_Degradation_Example_Plot1.png\|center\|500px\| ]]

Nicolette Young: /* Non-Destructive Degradation Analysis */

2015-12-23T21:42:44Z

Non-Destructive Degradation Analysis

← Older revision		Revision as of 21:42, 23 December 2015
Line 11:		Line 11:
	The Non-Destructive Degradation Analysis applies to cases where multiple degradation measurements over time can be obtained for each sample in the test. Given a defined level of failure (or the degradation level that would constitute a failure), basic mathematical models are used to extrapolate the degradation measurements over time of each sample to the point in time where the failure will occur. Once these extrapolated failure times are obtained, it is merely a matter of analyzing the extrapolated failure times in the same manner as conventional time-to-failure data. As with conventional life data analysis, the amount of certainty in the results is directly related to the number of samples being tested. The following figure combines the steps of the analysis by showing the extrapolation of the degradation measurements to a failure time and the subsequent distribution analysis of these failure times.		The Non-Destructive Degradation Analysis applies to cases where multiple degradation measurements over time can be obtained for each sample in the test. Given a defined level of failure (or the degradation level that would constitute a failure), basic mathematical models are used to extrapolate the degradation measurements over time of each sample to the point in time where the failure will occur. Once these extrapolated failure times are obtained, it is merely a matter of analyzing the extrapolated failure times in the same manner as conventional time-to-failure data. As with conventional life data analysis, the amount of certainty in the results is directly related to the number of samples being tested. The following figure combines the steps of the analysis by showing the extrapolation of the degradation measurements to a failure time and the subsequent distribution analysis of these failure times.

	[[Image:Non-Destructive Degradation.png\|center\|~~450px~~\|]]		[[Image:Non-Destructive Degradation.png\|center\|500px\|]]

	===Non-Destructive Degradation Models===		===Non-Destructive Degradation Models===
Line 118:		Line 118:
	Use the exponential degradation model for the degradation analysis, and the Weibull distribution parameters with MLE for the life data analysis. The following report shows the estimated degradation model parameters.		Use the exponential degradation model for the degradation analysis, and the Weibull distribution parameters with MLE for the life data analysis. The following report shows the estimated degradation model parameters.

	[[Image:Model Parameters and Stds.png\|center\|~~450px~~\| ]]		[[Image:Model Parameters and Stds.png\|center\|500px\| ]]

	The following report shows the extrapolated failure time intervals.		The following report shows the extrapolated failure time intervals.

	[[Image:Extraploated Failure Time Intervals.png\|center\|~~450px~~\| ]]		[[Image:Extraploated Failure Time Intervals.png\|center\|500px\| ]]

	==Destructive Degradation Analysis==		==Destructive Degradation Analysis==

Kate Racaza: /* Example */

2015-12-01T18:11:30Z

Example

← Older revision		Revision as of 18:11, 1 December 2015
Line 77:		Line 77:
	These times-to-failure can now be analyzed using traditional life data analysis to obtain metrics such as the probability of failure, B10 life, mean life, etc. This analysis can be automatically performed in the Weibull++ degradation analysis folio.		These times-to-failure can now be analyzed using traditional life data analysis to obtain metrics such as the probability of failure, B10 life, mean life, etc. This analysis can be automatically performed in the Weibull++ degradation analysis folio.

			<div class="noprint">
	{{Examples Box\|Weibull++ Examples\|<p>More degradation analysis examples are available! See also:</p>		{{Examples Box\|Weibull++ Examples\|<p>More degradation analysis examples are available! See also:</p>
	{{Examples Both\|http://www.reliasoft.com/Weibull/examples/rc4/index.htm\|Degradation Analysis\|http://www.reliasoft.tv/weibull/appexamples/weibull_app_ex_4\|Watch the video...}}<nowiki/>		{{Examples Both\|http://www.reliasoft.com/Weibull/examples/rc4/index.htm\|Degradation Analysis\|http://www.reliasoft.tv/weibull/appexamples/weibull_app_ex_4\|Watch the video...}}<nowiki/>
	}}		}}
			</div>

	===Using Extrapolated Intervals===		===Using Extrapolated Intervals===

Kate Racaza: /* Destructive Degradation Analysis */

2015-05-21T16:01:24Z

Destructive Degradation Analysis

← Older revision		Revision as of 16:01, 21 May 2015
Line 124:		Line 124:

	==Destructive Degradation Analysis==		==Destructive Degradation Analysis==
	The Destructive Degradation Analysis applies to cases where the sample has to be destroyed in order to obtain a degradation measurement. As a result, degradation measurements for multiple samples are required at different points in time. The analysis performed is very similar to ~~the~~ Accelerated Life Testing Analysis (ALTA). In this case, the ~~“stress”~~ used in ALTA becomes time while the random variable instead of time-to-failure ~~becomes the degradation measurement.~~ . Given a defined level of failure (or the degradation level that would constitute a failure), the probability that the degradation measurement will be beyond that level at a given time can be obtained. The following plot shows the relationship between the distribution of the degradation measurement and time. The red shaded area of the last two pdfs represents the probability that the degradation measurement will be less than the critical degradation level at the corresponding times.		The Destructive Degradation Analysis applies to cases where the sample has to be destroyed in order to obtain a degradation measurement. As a result, degradation measurements for multiple samples are required at different points in time. The analysis performed is very similar to Accelerated Life Testing Analysis (ALTA). In this case, the "stress" used in ALTA becomes time, while the random variable becomes the degradation measurement instead of the time-to-failure. Given a defined level of failure (or the degradation level that would constitute a failure), the probability that the degradation measurement will be beyond that level at a given time can be obtained. The following plot shows the relationship between the distribution of the degradation measurement and time. The red shaded area of the last two pdfs represents the probability that the degradation measurement will be less than the critical degradation level at the corresponding times.

	[[Image:Destructive_Degradation_1.1.png\|center\|450px\| ]]		[[Image:Destructive_Degradation_1.1.png\|center\|450px\| ]]
Line 139:		Line 139:
	Similar to accelerated life testing analysis, the assumption is that the location or log-location parameter of the degradation measurement distribution will change with time while the shape parameter will remain constant. For each distribution:		Similar to accelerated life testing analysis, the assumption is that the location or log-location parameter of the degradation measurement distribution will change with time while the shape parameter will remain constant. For each distribution:

	:* Weibull: <math>\ln (\eta )</math> is set as a function of time while <math>\beta </math> remains constant		:* Weibull: <math>\ln (\eta )\,\!</math> is set as a function of time while <math>\beta \,\!</math> remains constant.
	:* Exponential: <math>\ln (MTTF)</math> is set as a function of time		:* Exponential: <math>\ln (MTTF)\,\!</math> is set as a function of time.
	:* Normal: <math>\mu </math> is set as a function of time while <math>\sigma </math> remains constant		:* Normal: <math>\mu \,\!</math> is set as a function of time while <math>\sigma \,\!</math> remains constant.
	:* Lognormal: <math>{\mu }'</math> is set as a function of time while <math>{\sigma }'</math> remains constant		:* Lognormal: <math>{\mu }'\,\!</math> is set as a function of time while <math>{\sigma }'\,\!</math> remains constant.
	:* Gumbel: <math>\mu </math> is set as a function of time while <math>\sigma </math> remains constant		:* Gumbel: <math>\mu \,\!</math> is set as a function of time while <math>\sigma \,\!</math> remains constant.

	Finally, given the selected distribution, a degradation model is used to represent the change of the location (or log-location) parameter with time. The following degradation models can be used:		Finally, given the selected distribution, a degradation model is used to represent the change of the location (or log-location) parameter with time. The following degradation models can be used:

	:* Linear: <math>\mu (t)=b+a\times t</math>		:* Linear: <math>\mu (t)=b+a\times t \,\!</math>
	:* Exponential: <math>\mu (t)=b\times {{e}^{a\times t}}</math>		:* Exponential: <math>\mu (t)=b\times {{e}^{a\times t}}\,\!</math>
	:* Power: <math>\mu (t)=b\times {{t}^{a}}</math>		:* Power: <math>\mu (t)=b\times {{t}^{a}}\,\!</math>
	:* Logarithm: <math>\mu (t)=a\times \ln (t)+b</math>		:* Logarithm: <math>\mu (t)=a\times \ln (t)+b\,\!</math>
	:* Lloyd-Lipow: <math>\mu (t)=a-\frac{b}{t}</math>		:* Lloyd-Lipow: <math>\mu (t)=a-\frac{b}{t}\,\!</math>

	The distribution and degradation models parameters are then calculated using Maximum Likelihood Estimation (MLE).		The distribution and degradation models parameters are then calculated using Maximum Likelihood Estimation (MLE).

	For example, if a normal distribution is used to represent the degradation measurement and a linear degradation model is assumed, then the standard deviation, <math>\sigma </math>, will be assumed constant with time and the mean, <math>\mu </math>, will be:		For example, if a normal distribution is used to represent the degradation measurement and a linear degradation model is assumed, then the standard deviation, <math>\sigma \,\!</math>, will be assumed constant with time, and the mean, <math>\mu \,\!</math>, will be:

	:: <math>\mu (t)=b+a\times t</math>		:: <math>\mu (t)=b+a\times t\,\!</math>

	~~The CDF of the degradation measurement <math>x(t)</math> is:~~

	:: <math>~~\Pr (~~x(t)~~<X)=\Phi \left( \frac{X-\mu (t)}{~~\~~sigma }~~ \~~right)~~</math>		The CDF of the degradation measurement <math>x(t)\,\!</math> is:

	~~Given the CDF, the parameters~~ <math>\sigma ~~</math>~~, ~~<math>b</math> and <math>a~~</math> ~~are estimated using Maximum Likelihood Estimation.~~		:: <math>\Pr (x(t)<X)=\Phi \left( \frac{X-\mu (t)}{\sigma } \right)\,\!</math>

	Assuming that the requirement is that the measurement needs to be greater than a critical degradation value <math>{{D}_{crit}}</math> for the product to fail, the probability of failure at time t will be:

	:: <math>F(t)=\Pr (x(t)>{{D}_{crit}})=1-\Phi \left( \frac{{{D}_{crit}}-\mu (t)}{\sigma } \right)</math>		Given the CDF, the parameters <math>\sigma \,\!</math>, <math>b\,\!</math> and <math>a\,\!</math> are estimated using Maximum Likelihood Estimation.

			Assuming that the requirement is that the measurement needs to be greater than a critical degradation value <math>{{D}_{crit}}\,\!</math> for the product to fail, the probability of failure at time <math>t\,\!</math> will be:

			:: <math>F(t)=\Pr (x(t)>{{D}_{crit}})=1-\Phi \left( \frac{{{D}_{crit}}-\mu (t)}{\sigma } \right)\,\!</math>


	It should be noted that the failure threshold could be specified as a degradation measurement less than (for the case of decreasing degradation) or greater than (for the case of increasing degradation) a critical degradation value.		It should be noted that the failure threshold could be specified as a degradation measurement less than (for the case of decreasing degradation) or greater than (for the case of increasing degradation) a critical degradation value.
Line 201:		Line 204:


	1. Estimate parameters using the Linear degradation model and the 2-Parameter Weibull distribution		Do the following:
	2. Plot the degradation curve vs. time		# Estimate parameters using the Linear degradation model and the 2-Parameter Weibull distribution
	3. Calculate the reliability at 5 years.		# Plot the degradation curve vs. time
			# Calculate the reliability at 5 years.


	'''Solution'''		'''Solution'''

	Enter the data into a ~~Destructive Degradation~~ folio in Weibull++. Select '''Linear''' under the Degradation Model and '''2P-Weibull''' under the Measurement Distribution. Set the critical degradation to '''150'''. Click '''Calculate'''.		Enter the data into a destructive degradation folio in Weibull++. Select '''Linear''' under the Degradation Model and '''2P-Weibull''' under the Measurement Distribution. Set the critical degradation to '''150'''. Click '''Calculate'''.

	:1. The estimated parameters are <math>\hat{\beta }=3.618336\,\!</math>, <math>\hat{a}=-0.331695\,\!</math> and <math>\hat{b}=7.154643\,\!</math>.		:1. The estimated parameters are <math>\hat{\beta }=3.618336\,\!</math>, <math>\hat{a}=-0.331695\,\!</math> and <math>\hat{b}=7.154643\,\!</math>.

Kate Racaza at 15:43, 21 May 2015

2015-05-21T15:43:11Z

← Older revision		Revision as of 15:43, 21 May 2015
Line 3:		Line 3:


	In some cases, it is possible to ~~either~~ directly measure the degradation of a physical characteristic over time, as with the wear of brake pads ~~or with~~ the propagation of crack size or the degradation of a performance characteristic over time such as the voltage of a battery or the luminous flux of an LED bulb. These cases belong to the Non-Destructive Degradation Analysis category. In other cases, direct measurement of degradation might not be possible without invasive or destructive measurement techniques that would directly affect the subsequent performance of the product ~~and~~ therefore only one degradation measurement is possible. Examples are the measurement of corrosion in a chemical container or the strength measurement of an adhesive bond. These cases belong to the Destructive Degradation Analysis Category. In either case, however, it is necessary to be able to define a level of degradation or performance at which a failure is said to have occurred.		In some cases, it is possible to directly measure the degradation of a physical characteristic over time, as with the wear of brake pads, the propagation of crack size, or the degradation of a performance characteristic over time such as the voltage of a battery or the luminous flux of an LED bulb. These cases belong to the Non-Destructive Degradation Analysis category. In other cases, direct measurement of degradation might not be possible without invasive or destructive measurement techniques that would directly affect the subsequent performance of the product; therefore, only one degradation measurement is possible. Examples are the measurement of corrosion in a chemical container or the strength measurement of an adhesive bond. These cases belong to the Destructive Degradation Analysis Category. In either case, however, it is necessary to be able to define a level of degradation or performance at which a failure is said to have occurred.


Line 25:		Line 25:
	where <math>y\,\!</math> represents the performance, <math>x\,\!</math> represents time, and <math>a,\,\!</math> <math>b\,\!</math> and <math>c\,\!</math> are model parameters to be solved for.		where <math>y\,\!</math> represents the performance, <math>x\,\!</math> represents time, and <math>a,\,\!</math> <math>b\,\!</math> and <math>c\,\!</math> are model parameters to be solved for.

	Once the model parameters <math>{{a}_{i}}\,\!</math>, <math>{{b}_{i}}\,\!</math> and <math>{{c}_{i}}\,\!</math> are estimated for each sample <math>i\,\!</math>, a time, <math>{{x}_{i}}\,\!</math>, can be extrapolated, which corresponds to the defined level of failure <math>y\,\!</math>. The computed <math>{{x}_{i}}\,\!</math> values can now be used as our times-to-failure for subsequent life data analysis. As with any sort of extrapolation, one must be careful not to extrapolate too far beyond the actual range of data in order to avoid large inaccuracies (modeling errors).		Once the model parameters <math>{{a}_{i}}\,\!</math>, <math>{{b}_{i}}\,\!</math> and <math>{{c}_{i}}\,\!</math> are estimated for each sample <math>i\,\!</math>, a time <math>{{x}_{i}}\,\!</math> can be extrapolated, which corresponds to the defined level of failure <math>y\,\!</math>. The computed <math>{{x}_{i}}\,\!</math> values can now be used as our times-to-failure for subsequent life data analysis. As with any sort of extrapolation, one must be careful not to extrapolate too far beyond the actual range of data in order to avoid large inaccuracies (modeling errors).

	====Example====		====Example====
	'''Crack Propagation Example (Point Estimation)'''		'''Crack Propagation Example (Point Estimation)'''

	Five turbine blades are tested for crack propagation. The test units are cyclically stressed and inspected every 100,000 cycles for crack length. Failure is defined as a crack of length 30mm or greater. The following table shows the test results for the five units at each cycle:		Five turbine blades are tested for crack propagation. The test units are cyclically stressed and inspected every 100,000 cycles for crack length. Failure is defined as a crack of length 30mm or greater. The following table shows the test results for the five units at each cycle:

	::<math>\begin{matrix}		::<math>\begin{matrix}
Line 40:		Line 40:
	500 & 29 & 30 & 33 &26 & 33 \\		500 & 29 & 30 & 33 &26 & 33 \\
	\end{matrix}\,\!</math>		\end{matrix}\,\!</math>


	Use the exponential degradation model to extrapolate the times-to-failure data.		Use the exponential degradation model to extrapolate the times-to-failure data.
Line 56:		Line 57:
	\end{matrix}\,\!</math>		\end{matrix}\,\!</math>

	~~Substituted~~ the values into the underlying exponential model, ~~solved~~ for <math>x\,\!</math> or:
			Substituting the values into the underlying exponential model, solve for <math>x\,\!</math> or:

	::<math>x=\frac{\text{ln}(y)-\text{ln}(b)}{a}\,\!</math>		::<math>x=\frac{\text{ln}(y)-\text{ln}(b)}{a}\,\!</math>


	Using the values of <math>a\,\!</math> and <math>b\,\!</math>, with <math>y=30\,\!</math>, the resulting time at which the crack length reaches 30mm can then found for each sample:		Using the values of <math>a\,\!</math> and <math>b\,\!</math>, with <math>y=30\,\!</math>, the resulting time at which the crack length reaches 30mm can then found for each sample:
Line 70:		Line 73:
	Unit E & \text{452,832} \\		Unit E & \text{452,832} \\
	\end{matrix}\,\!</math>		\end{matrix}\,\!</math>


	These times-to-failure can now be analyzed using traditional life data analysis to obtain metrics such as the probability of failure, B10 life, mean life, etc. This analysis can be automatically performed in the Weibull++ degradation analysis folio.		These times-to-failure can now be analyzed using traditional life data analysis to obtain metrics such as the probability of failure, B10 life, mean life, etc. This analysis can be automatically performed in the Weibull++ degradation analysis folio.
Line 79:		Line 83:

	===Using Extrapolated Intervals===		===Using Extrapolated Intervals===
	The parameters in a degradation model are estimated using available degradation data. If the data is large, the uncertainty of the estimated parameters will be small~~. Otherwise~~, the uncertainty will be large. Since the failure time for a test unit is predicted based on the estimated model, we sometimes would like to see how the parameter uncertainty affects the failure time prediction. Let’s use the exponential model as an example. Assume the critical degradation value is <math>{{y}_{crit}}\,\!</math>. The predicted failure time will be:		The parameters in a degradation model are estimated using available degradation data. If the data is large, the uncertainty of the estimated parameters will be small; otherwise, the uncertainty will be large. Since the failure time for a test unit is predicted based on the estimated model, we sometimes would like to see how the parameter uncertainty affects the failure time prediction. Let’s use the exponential model as an example. Assume that the critical degradation value is <math>{{y}_{crit}}\,\!</math>. The predicted failure time will be:

	::<math> \hat{x}=\frac{\ln ({{y}_{crit}})-\ln (\hat{b})}{{\hat{a}}}\,\!</math>		::<math> \hat{x}=\frac{\ln ({{y}_{crit}})-\ln (\hat{b})}{{\hat{a}}}\,\!</math>


	The variance of the predicted failure time will be:		The variance of the predicted failure time will be:
Line 94:		Line 99:
	::<math>{{x}_{U}}=\hat{x}+{{K}_{1-\alpha /2}}\sqrt{Var(\hat{x})}\,\!</math>		::<math>{{x}_{U}}=\hat{x}+{{K}_{1-\alpha /2}}\sqrt{Var(\hat{x})}\,\!</math>
	::<math>{{x}_{L}}=\hat{x}-{{K}_{1-\alpha /2}}\sqrt{Var(\hat{x})}\,\!</math>		::<math>{{x}_{L}}=\hat{x}-{{K}_{1-\alpha /2}}\sqrt{Var(\hat{x})}\,\!</math>


	In Weibull++, the default confidence level is 90%.		In Weibull++, the default confidence level is 90%.
Line 101:		Line 107:

	Using the same data set from the previous example, predict the interval failure times for the turbine blades.		Using the same data set from the previous example, predict the interval failure times for the turbine blades.


	'''Solution'''		'''Solution'''