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# Bivariate Survival Regression Model for Seasonal Effects in Warranty Data Analysis

IP.com Disclosure Number: IPCOM000242940D
Publication Date: 2015-Sep-01
Document File: 3 page(s) / 51K

## Publishing Venue

The IP.com Prior Art Database

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Page 01 of 3

Defensive Publication

Bivariate Survival Regression Model
for Seasonal Effects in Warranty Data Analysis

Problem

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.

Solution

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.

Prior Art

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:

Weibull PDF:

݂ሺ݉, ܽ, ܾሻ ൌ ܾ
ܽ ቀ݉

ܽ ቁ

௔ ቁ

௕ିଵ

݁ିቀ

௔ ቁ

, ݉ ൒ 0

Weibull CDF:

ܨሺ݉, ܽ, ܾሻ ൌ ሺ1 െ ݁ିቀ

ሻ, ݉ ൒ 0

Page 02 of 3

The following expressions are for t - Time in Service:

Weibull Hazard Function:

݄ሺݐ, ܽ, ܾ, Τሻ ൌ ܾ
ܽሺݐ, Τሻ ൬ ݐ

ܽሺݐ, Τሻ൰

ܽሺݐ, 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, ܽ, ܾ, Τሻ ൌ න ݄ሺݏ, ܽ, ܾ, Τሻ

Cumulative Distribution Function:
ܨሺݐ, ܽ, ܾ, Τሻ ൌ 1 െ ݁ିுሺ௧,௔,௕,஋ሻ

Probability Density Function:

݂ሺݐ, ܽ, ܾ, Τሻ ൌ ݄ሺݐ, ܽ, ܾ...