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| {{alta plot types}} | | {{alta plot types}} |
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| ==Residual Plots==
| | {{alta residual plots}} |
| Residual analysis for reliability consists of analyzing the results of a regression analysis by assigning residual values to each data point in the data set. Plotting these residuals provides a very good tool in assessing model assumptions and revealing inadequacies in the model, as well as revealing extreme observations. Three types of residual plots are available in ALTA.
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| ===Standardized Residuals (SR)===
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| The standardized residuals plot for the Weibull and lognormal distributions can be obtained in ALTA. Each plot type is discussed next.
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| <br>
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| ====SR for the Weibull Distribution====
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| Once the parameters have been estimated, the standardized residuals for the Weibull distribution can be calculated by:
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| <br>
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| ::<math>{{\widehat{e}}_{i}}=\widehat{\beta }\left[ \ln ({{T}_{i}})-\ln (\widehat{\eta }(V)) \right]</math>
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| <br>
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| Then, under the assumed model, these residuals should look like a sample from an extreme value distribution with a mean of 0. For the Weibull distribution the standardized residuals are plotted on a smallest extreme value probability paper. If the Weibull distribution adequately describes the data, then the standardized residuals should appear to follow a straight line on such a probability plot. Note that when an observation is censored (suspended), the corresponding residual is also censored.
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| <br>
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| ====SR for the Lognormal Distribution====
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| Once the parameters have been estimated using rank regression, the fitted or calculated responses can be calculated by:
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| ::<math>{{\widehat{e}}_{i}}=\frac{\ln ({{T}_{i}})-{{\widehat{\mu }}^{\prime }}}{{{\widehat{\sigma }}_{{{T}'}}}}</math>
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| <br>
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| Then, under the assumed model, the standardized residuals should be normally distributed with a mean of 0 and a standard deviation of 1 ( <math>\tilde{\ }N(0,1)</math> ). Consequently, the standardized residuals for the lognormal distribution are commonly displayed on a normal probability plot.
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| <br>
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| [[Image:standardized_weibull.gif|thumb|center|400px|Probability plot of standardized residuals.]]
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| <br>
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| ===Cox-Snell Residuals===
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| The Cox-Snell residuals are given by:
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| ::<math>{{\widehat{e}}_{i}}=-\ln [R({{T}_{i}})]</math>
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| <br>
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| where <math>R({{T}_{i}})</math> is the calculated reliability value at failure time <math>{{T}_{i}}.</math> The Cox-Snell residuals are plotted on an exponential probability paper.
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| ::<math></math>
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| <br>
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| [[Image:cox-snell_plot.gif|thumb|center|400px|Probability plot of the Cox-Snell residuals.]]
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| <br>
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| ===Standardized vs Fitted Values===
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| A Standardized vs. Fitted Value plot helps to detect behavior not modeled in the underlying relationship. However, when heavy censoring is present, the plot is more difficult to interpret. In a Standardized vs. Fitted Value plot, the standardized residuals are plotted versus the scale parameter of the underlying life distribution (which is a function of stress) on log-linear paper (linear on the Y-axis). Therefore, in the case of the Weibull distribution, the standardized residuals are plotted versus <math>\eta (V),</math> for the lognormal versus <math>{\mu }'(V),</math> and for the exponential versus <math>m(V).</math>
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| <br>
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| [[Image:standardized_fitted_value.gif|thumb|center|400px|Standardized Residuals vs. Fitted Value.]]
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| <br>
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| {{RS Copyright}}
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