Bivariate Survival Regression Model for Seasonal Effects in Warranty Data Analysis
Publication Date: 2015-Sep-01
The IP.com Prior Art Database
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Bivariate Survival Regression Model
for Seasonal Effects in Warranty Data Analysis
In a number of essential corporate activities including but not limited to: warranty cost forecasting, field service action (FSA) planning, etc., there arises a need to model/estimate repair probability indexed by vehicle's age and mileage in the presence of seasonal effects. Most often, this task relates to Climate Control, Engine Cooling and other components, whose duty cycle is influenced by the climate and/or seasonal factors. Time-series models, traditionally used in seasonality analysis are not readily applicable to survival data and much less applicable to warranty data, which are subject to bivariate censoring.
We model the seasonal effects as a time-dependent covariate of a bivariate (in age and mileage) survival regression with the underlying life distribution of the location-scale family. This approach improves the estimation and prediction accuracy of the repair probability. Full mathematical details of the proposed solution are discussed below.
There are published solutions separately for bivariate modeling of reliability/warranty data and separately for seasonal effects in reliability/warranty data. However, no solutions have been identified for the simultaneous use of the bivariate model and the seasonal effects.
Mathematical details of the proposed solution
Parameter estimation was performed utilizing the maximum likelihood method applied to the selected bivariate lifetime distribution in time and mileage domains which is based on the Weibull distribution. The following functions are defined for m - mileage:
݂ெሺ݉, ܽ, ܾሻ ൌ ܾ
್ , ݉ 0
ܨெሺ݉, ܽ, ܾሻ ൌ ሺ1 െ ݁ିቀ
್ ሻ, ݉ 0
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The following expressions are for t - Time in Service:
Weibull Hazard Function:
݄ሺݐ, ܽ௧, ܾ௧, Τሻ ൌ ܾ௧
ܽ௧ሺݐ, Τሻ ൬ ݐ
ܽ௧ሺݐ, Dሻ ൌ ߙሺݐ, ݉݊ݐ݄ሺݐ Dሻሻ
D െ Warranty Start Date,
monthሺt ܦሻ െ function that return the calendar month of date ሺt ܦሻ
t - time in service (months)
Cumulative Hazard Function:
ܪሺݐ, ܽ௧, ܾ௧, Τሻ ൌ න ݄ሺݏ, ܽ௧, ܾ௧, Τሻ
Cumulative Distribution Function:
ܨ்ሺݐ, ܽ௧, ܾ௧, Τሻ ൌ 1 െ ݁ିுሺ௧,,,ሻ
Probability Density Function:
்݂ሺݐ, ܽ௧, ܾ௧, Τሻ ൌ ݄ሺݐ, ܽ௧, ܾ...