PSI - Issue 19

Edrissa Gassama et al. / Procedia Structural Integrity 19 (2019) 711–718 Gassama et al./ Structural Integrity Procedia 00 (2019) 000 – 000

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2. Data Fitting

In this section, we discuss the iterative procedure used to perform linear regression with censored data points. We first give a brief overview of least-squares regression and discuss the modifications needed to account for censored data points. Finally, we walk through the step-by-step procedure for implementation.

2.1. Least-squares regression

Linear least-squares regression is often used to generate estimates and other statistics when fitting a set of data. For example, the standard linear model to predict a set of values, y , can generally be written as: $ T = X y  (1) where, $ y is the array of predicted values for a set of input values, ( ) ( ) ( )   1 2 , , , m = X K x x x , ( ) i n  ¡ x are arrays containing the input values (or functions of the inputs), n  ¡  is an array containing the parameters of the model, m is the number of observations being fit and n is the number of parameters in the model. For example, for a two parameter model, the array of predicted values is evaluated as:

1 11     1 21 x x

2 12 +  +  +  M 2 22 2 2 m x x x   

     

$ y

=

(2)

1 1 m   x

The objective for least-squares regression is to find the values of  that yield the best fit to the observed data. Explicitly, least-squares regression aims to find the parameters of the model that minimize the 2 L norm of the difference between the observed and predicted values:

$   y y



 T 2 

( ) 

y

− X

f

min

min

=

− =

(3)





2

Now, consider a set of data containing both uncensored and censored values. We can separate the input data as:     1 2 1 2 , and , = = X X X y y y (4) where subscript 1 corresponds to uncensored observations and subscript 2 corresponds to censored observations. The least-squares regression including censored data points can then be written as: ( ) ( )   T T 1 1 2 2 2 2 min f E = − + − X X y y     (5) where ( ) 2 E y corresponds to the expected values of the censored data points. From an intuitive standpoint, if the tests that resulted in the censored data were allowed to run to failure, the true observed lives would be greater than the censored lives. Thus, we can write the expected value of the actual lives for the censored tests given that the lives are longer than the censored values as: ( ) ( ) 2 2 E g = +  y y > y y (6)