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A heteroscedastic linear regression model is developed from plausible assumptions that describe the time evolution of performance metrics for equipment. The inherited motivation for the related weighted least squares analysis of the model is an essential and attractive selling point to engineers with interest in equipment surveillance methodologies. A simple test for the significance of the heteroscedasticity suggested by a data set is derived and a simulation study is used to evaluate the power of the test and compare it with several other applicable tests that were designed under different contexts. Tolerance intervals within the context of the model are derived, thus generalizing well-known tolerance intervals for ordinary least squares regression. Use of the model and its associated analyses is illustrated with an aerospace application where hundreds of electronic components are continuously monitored by an automated system that flags components that are suspected of unusual degradation patterns.

The model and analyses developed in this paper address a problem encountered when analyzing data from service life tests of aerospace hardware packages. Data for as many as 700 performance metrics per part type are automatically stored during surveillance testing and subsequently input into a software program where, up to this point, an ordinary least squares line based on normal-theory has been routinely fit to the data using time-in-service as the explanatory variable. The software program also outputs tolerance intervals based on the ordinary least squares analysis. Engineers monitoring this process are alerted only to those cases where observations in the scatter plot fall outside the tolerance intervals or the tolerance interval crosses a given limit within some specified future time interval (e.g., 60 months). In cases where the alert suggests an increasing accelerated degradation, a proactive corrective action (e.g., part replacement) may be initiated. In cases where the alert suggests less than expected degradation, a cursory investigation to determine if the part is being utilized properly is initiated. Tolerance intervals are often similarly used for monitoring environmental applications [

Compared to having engineers individually examine the data from all the combinations of metrics and part types, the automated monitoring and flagging process is quite cost-effective. However, when the variance of the metric increases with time, the tolerance intervals that result from the ordinary least squares analyses fail in the sense that the intervals become too narrow as time increases. Figure

(Data Set 1) Pointwise 95%-content tolerance intervals with 90% confidence. Dashed lines correspond to an ordinary least squares analysis and solid lines correspond to the estimated weighted least squares analysis.

Figure

Examination of aerospace data for many part types and many performance metrics led us to propose the following model for heteroscedasticity. Let

Equivalently, the model implies that observations are independent and normally distributed with means

Defining

The profile log-likelihood function (e.g., see reference [

A pointwise

If

See Appendix

The pointwise tolerance intervals shown in Figure

Of special importance when analyzing the data for our application is the test of

In Section

The heteroscedastic regression model developed in Section

We begin the derivation of our proposed test for heteroscedasticity by writing the log-likelihood of the general mixed linear model, based on

The distribution of

A locally most powerful invariant (LMPI) test of

From a conceptual perspective, the equivalence between the LMPI test and the

The Breusch and Pagan test for homogeneity is a partial score test derived for a general setting where the observations are independent and normally distributed with means

The White test [

The log-likelihood function for the general mixed linear model was given in (

Figure

(Data Set 2) Pointwise 95%-content tolerance intervals with 90% confidence. Dashed lines correspond to an ordinary least squares analysis and solid lines correspond to the estimated weighted least squares analysis.

Table

Tests for Heteroscedasticity in Figures

Test | Data Set | |||

Data Set 1 | Data Set 2 | |||

Statistic | Statistic | |||

8.92 | 4.83 | .062 | ||

BP | 52.1 | 2.05 | .15 | |

26.6 | 3.29 | .17 | ||

LRT | 34.8 | 1.59 | .10 |

(Data Set 3) Pointwise 95% tolerance intervals with 90% confidence.

(Data Set 4) Pointwise 95% tolerance intervals with 90% confidence.

(Data Set 5) Pointwise 95% tolerance intervals with 90% confidence.

(Data Set 6) Pointwise 95% tolerance intervals with 90% confidence.

It is easy to show that the distributions of the

Column 2 of Table

Power comparison of the tests using

Power of | Ratio of Power Relative to | |||

LRT | BP | |||

0.00 | 0.100 | 0.94 | 0.91 | 0.92 |

0.01 | 0.155 | 0.97 | 0.67 | 0.67 |

0.02 | 0.206 | 0.97 | 0.62 | 0.64 |

0.03 | 0.278 | 1.00 | 0.63 | 0.65 |

0.04 | 0.344 | 0.96 | 0.66 | 0.60 |

0.05 | 0.392 | 0.99 | 0.68 | 0.62 |

0.06 | 0.443 | 1.02 | 0.68 | 0.67 |

0.07 | 0.492 | 1.01 | 0.72 | 0.67 |

0.08 | 0.541 | 1.01 | 0.72 | 0.68 |

0.09 | 0.584 | 1.01 | 0.77 | 0.71 |

0.10 | 0.624 | 1.02 | 0.77 | 0.70 |

0.11 | 0.663 | 1.02 | 0.77 | 0.73 |

0.12 | 0.701 | 1.01 | 0.80 | 0.73 |

0.13 | 0.719 | 1.02 | 0.80 | 0.73 |

0.14 | 0.741 | 1.03 | 0.82 | 0.76 |

0.15 | 0.763 | 1.01 | 0.84 | 0.77 |

0.16 | 0.783 | 1.02 | 0.84 | 0.78 |

0.17 | 0.809 | 1.01 | 0.85 | 0.78 |

0.18 | 0.830 | 1.01 | 0.86 | 0.80 |

0.19 | 0.830 | 1.02 | 0.88 | 0.82 |

0.20 | 0.843 | 1.02 | 0.87 | 0.84 |

0.21 | 0.861 | 1.01 | 0.88 | 0.83 |

0.22 | 0.869 | 1.01 | 0.88 | 0.84 |

0.23 | 0.875 | 1.02 | 0.90 | 0.84 |

0.24 | 0.884 | 1.02 | 0.90 | 0.85 |

0.25 | 0.900 | 1.02 | 0.90 | 0.85 |

0.30 | 0.923 | 1.01 | 0.93 | 0.89 |

0.35 | 0.943 | 1.00 | 0.94 | 0.92 |

0.40 | 0.949 | 1.01 | 0.96 | 0.93 |

Power comparison of the tests using

Power of | Ratio of Power Relative to | |||

LRT | BP | |||

0 | 0.100 | 1.01 | 0.96 | 0.89 |

0.001 | 0.107 | 0.95 | 0.91 | 0.84 |

0.002 | 0.104 | 1.01 | 0.90 | 0.98 |

0.003 | 0.126 | 0.92 | 0.78 | 0.75 |

0.004 | 0.127 | 0.96 | 0.72 | 0.81 |

0.005 | 0.128 | 0.92 | 0.75 | 0.68 |

0.006 | 0.134 | 1.00 | 0.70 | 0.81 |

0.007 | 0.146 | 0.84 | 0.65 | 0.65 |

0.008 | 0.136 | 1.03 | 0.78 | 0.76 |

0.009 | 0.137 | 1.06 | 0.78 | 0.77 |

0.010 | 0.148 | 0.90 | 0.73 | 0.68 |

0.011 | 0.152 | 1.01 | 0.72 | 0.78 |

0.012 | 0.174 | 0.86 | 0.65 | 0.68 |

0.013 | 0.171 | 0.97 | 0.68 | 0.73 |

0.014 | 0.168 | 1.03 | 0.68 | 0.74 |

0.015 | 0.183 | 0.92 | 0.63 | 0.70 |

0.016 | 0.189 | 0.93 | 0.69 | 0.67 |

0.017 | 0.200 | 0.92 | 0.63 | 0.65 |

0.018 | 0.204 | 0.92 | 0.65 | 0.62 |

0.019 | 0.205 | 0.99 | 0.67 | 0.65 |

0.02 | 0.206 | 0.99 | 0.64 | 0.64 |

0.03 | 0.267 | 0.98 | 0.66 | 0.61 |

0.04 | 0.316 | 1.10 | 0.66 | 0.64 |

0.05 | 0.384 | 0.85 | 0.71 | 0.63 |

0.06 | 0.443 | 1.01 | 0.71 | 0.63 |

0.07 | 0.496 | 1.02 | 0.74 | 0.64 |

0.08 | 0.544 | 1.02 | 0.74 | 0.68 |

0.09 | 0.586 | 1.03 | 0.76 | 0.70 |

0.10 | 0.639 | 1.00 | 0.76 | 0.68 |

0.11 | 0.676 | 1.01 | 0.80 | 0.71 |

0.12 | 0.700 | 1.01 | 0.82 | 0.72 |

0.13 | 0.735 | 1.01 | 0.83 | 0.77 |

0.14 | 0.756 | 1.02 | 0.84 | 0.77 |

0.15 | 0.783 | 1.01 | 0.84 | 0.79 |

0.16 | 0.799 | 1.01 | 0.87 | 0.81 |

0.17 | 0.828 | 1.00 | 0.85 | 0.80 |

0.18 | 0.830 | 1.03 | 0.87 | 0.82 |

0.19 | 0.853 | 1.01 | 0.89 | 0.83 |

0.20 | 0.868 | 1.01 | 0.89 | 0.84 |

0.21 | 0.878 | 1.01 | 0.90 | 0.85 |

0.22 | 0.887 | 1.01 | 0.91 | 0.86 |

0.23 | 0.897 | 1.01 | 0.91 | 0.86 |

0.24 | 0.903 | 1.02 | 0.92 | 0.89 |

0.25 | 0.907 | 1.03 | 0.93 | 0.89 |

0.30 | 0.935 | 1.02 | 0.95 | 0.92 |

0.35 | 0.959 | 1.01 | 0.96 | 0.94 |

0.40 | 0.973 | 1.01 | 0.96 | 0.95 |

Our aerospace case study, which pertains to the time evolution of performance metrics for electronic equipment, motivated the derivation of a model for heteroscedastic regression errors. A test for homoscedasticity was proposed and compared with various other tests that are prevalent in the literature. The proposed

Using observations that follow a simple linear regression model

The heteroscedastic regression model described in Section

Referring back to Section

Following [

Let the nonzero eigenvalues of

For

It is shown in [

To prove the equivalence of the LMPI test based on (