Abstract
In this paper, the survival function of the Weibull distribution was estimated by the Classical Maximum Likelihood Estimate Method for the scale and shape parameters, and then the efficiency of the estimated parameters was calculated based on the mean square error and compared with the proposed method that deals with the contamination problem before estimating the parameters of the survival function for Weibull distribution through the use of Wavelets (Daubechies2), (Symlet3), and (Coiflit4) with several different methods of estimating the level of thresholding depending on the rule of soft. For the purpose of estimating and comparing the efficiency of the proposed method with the classical method, simulations were carried out for several different cases of the values for scale and shape parameters of Weibull distribution, contamination percentages, and different sample sizes as well as real data based on a MATLAB code designed for this purpose, the statistical program (SPSS) and the (Easy Fit) program. The study showed the efficiency of the precision parameters estimate for Weibull distribution when there was a data contamination problem when using the proposed method compared to the classical method.
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Highlights
All the proposed methods have better efficiency than the classical method in estimating scale and shape parameters for Weibull Distribution depending on the average of criterion (MSE) for all cases.
Full Text
Introduction
Survival analysis (also called timetoevent analysis or duration analysis) is a branch of statistics aimed at analyzing the expected duration of time until one or more events happen called survival times or duration times such as death in biological organisms and failure in mechanical systems. This topic is called reliability theory or reliability analysis in engineering. Survival analysis is a collection of statistical procedures for data analysis for which the outcome variable of interest is time until an event occurs. It is the study of time between entry into observation and a subsequent event. Now the scope of the survival analysis has become wide. Survival analysis is a set of statistical techniques used to describe and specify time to accident data. We use the term ‘failure’ in survival analysis to describe the occurrence of the task event (even though the event may actually be a ‘success’ such as recovery from therapy). The term ‘survival time’ specifies the length of time taken for failure to occur. (David, 2012. Singh, 2011).
Weibull distribution is a very beneficial distribution in survival analysis and reliability analysis. Several methods have been demonstrated to estimate the parameters of different distributions such as the method of moments method, maximum likelihood, etc. The Weibull distribution has gained much weight in the real world and is increasingly used in reliability and lifetime analysis or survival analysis. The 2parameter Weibull distribution has the shape parameter ( β ) and the scale parameter (α ). Most distributions such as normal, gamma, inverse gamma, and some other common distributions have two parameters which are of immense interest. (Nketiah, 2021).
The contamination of the estimation of the intercept have only a small impact on the estimation of the regression coefficients. Good leverage points are observations that are on the outskirts of the design space but are near to the regression line. They have a minor impact on the estimate of both the intercept and the regression coefficients, but they have an impact on inference. In contrast, bad leverage points are observations that are far off the regression line. Signals are typically contaminated by random noise, hence, several methods have been used to smooth noisy signals including the Fourier transform, the Svitzky Goloylocal polynomial, the mean filters, and Gussian function and so on. However, these methods usually smooth the signal to reduce the noise, but, in the process, also blure the signal. In recent years, a new method has been introduced to the method of denoising known as wavelet shrinkage. (Taha and Saleh, 2022)
Wavelet shrinkage make denoising a method of reducing noise in signals. Wavelet shrinkage is a signal denoising technique based on the idea of thresholding the wavelet coefficients. Donoho et al. (1995a) have introduced the method of wavelet shrinkage for general curve estimation problems. There are several good reasons why wavelet shrinkage can be used for estimation function. (Mustafa, and Taha, 2013)
Survival Analysis
Survival analysis is always treated with the analysis of data in times of accidents in life. The survival analysis and modeling the time it takes events occur, i.e. this typical event is death which is derived from the name ' survival ' analysis. Let T be a random variable that represents failure time of an event with probability of density function f (t) and cumulative distribution function
F(t) = Pr(T £ t) , the survival function S(t) is defined as: (David, 2012)
S(t) = Pr (T t) = 1F(t)
Weibull Distribution
The probability density function and the cumulative distribution function of a two parameter Weibull distribution with scale parameter, α > 0 and shape parameter, β > 0, are given by, (Nketiah, 2021)
) (1)
The cumulative distribution function is,
(2)
Scale and shape parameters estimated by using Maximum Likelihood Estimation (MLE).
The method of MLE is a commonly used procedure for estimating parameters. Assume be a random sample of size obtained from a population with pdf, where is a hidden vector of parameter, likelihood function is given as,
(3)
The MLE of is the value of that maximizes the likelihood function or the loglikelihood function where
(4)
By applying Eqn.(3) to the Weibull probability density function 1n Eqn.(1)
the likelihood function will be:
(5)
Taking the algorithm of both sides, we get:
Differentiating β, α , we obtain the estimating equations as follows:
Equations 8 and 9 are solved numerically to obtain the estimated parameters.
The data come from two types of distributions the first of which is called Basic Distribution that generates good data while the second of which is called Contamination Distribution and P is a ratio of contamination then the distribution of an arbitrary observation is (Hawkins, 1980):
In literature on data mining and statistics, outliers are sometimes known as abnormalities, discordant, deviants or anomalies. In the majority of applications, the data is produced by one or more producing processes which may either reflect system activity or observations made about entities. Noisy data is data that have been made due to the presence of too much variation. It is presumed that the signal or observation is presented and disguised by noise. The difficulty of separating the noise from the signal or observation has long been a focus in statistics. So, the useful data need to be used to inform researchers. However, the percentage of noisy data that is relevant is frequently too small to be useful. (Taha and Saleh, 2022)
Wavelet Shrinkage
Wavelet shrinkage is a wellestablished technique for removing the noise present in the observation while preserving the significant features of the original data (Donoho, 1994). The wavelet shrinkage is based on thresholding of the wavelet coefficients. The wavelet shrinkage has several good properties that gained this popularity in statistics nearly minimax for a wide range of loss function and for general function classes; simple, practical and fast; adaptable to spatial and frequency in homogeneities; readily extendable to high dimensions; applicable to various problems such as density estimation and inverse problems. In statistics, applications of wavelets arise mainly in the tasks involving nonparametric regression, density estimation, assessment of scaling, functional data analysis and stochastic processes. (Donoho and Johnstone , 1995)
Wavelets are small waves that can be grouped together to form larger waves or different waves (Ali, et al, 2022). A few fundamental waves were used, i.e. they were stretched in infinitely many ways, and moved in infinitely many ways to produce a wavelet system that could make an accurate model of any wave. Consider generating an orthogonal wavelet basis for functions (the space of square integrable real functions), starting with two parent wavelets: the scaling function
(also called farther wavelet) and the mother wavelet . Other wavelets are then generated by and (Donald et al., 2004). The dilation and translation of the functions are defined by formulas (11) and (12).
(11)
(12)
The discrete wavelet transform (DWT) is a broadly applicable observation of processing algorithm which is benefit in several applications, for e.g. science, engineering, mathematics and computer science. DWT decomposes an observation by using scaled and shifted versions of a compact supported basis function (mother wavelet), and provides multiresolution representation of the observation. It gives a vector of observations y consisting of 2^{k} observations where k is an integer and the DWT of y due to formula (13). (Ali, et al, 2022)
(13)
Where w is wavelet matrix with (n × n) dimension, W is a vector with (n × 1) dimension including both scaling and wavelet coefficients. The vector of wavelet coefficients can be organized into (k+1) elements. W= [W_{1}, W_{2}, …, W_{k},V_{k0 }]^{T }at each DWT, the approximate coefficients are divided into bands using the same wavelet as before with the result that the details are appended with the details of the latest decomposition as in the following formula (Taha and Saleh, 2022):
(14)
At each level (k) the observations can be reconstructed from the denoise data (reducing the contamination) by the inverse DWT (Ramazan et al., 2002).
The simplest method of nonlinear wavelet denoising is thresholding in which the wavelet coefficient is sub divided into two sets one of which represents signal while the other represents noise. To apply the thresholds of the wavelet coefficients, there are different rules and several different methods for choosing a threshold value exist such as:
The SURE threshold proposed by Donoho and Johonstone (1994), which is based upon the minimization of Stein's risk estimator. In SURE threshold method specifies a threshold estimate of at each level k for the wavelet coefficients and then for the soft threshold estimator, we have.
(15)
Where be a wavelet coefficient in the kth level, and then select that minimizes SURE
(16)
The optimal minimax threshold method is submitted by Donoho and Johonstone (1994) as an improvement to the universal threshold method. Minimax is based on an estimator that attains to the minimax risk as:
(17)
Where
(18)
Where and , denote the vectors of true and estimated sample values. The threshold minimax estimator is different from universal counter parts in which the minimax threshold method concentrates on reducing the overall mean square error (MSE) but the estimates are not oversmoothing.
Donoho and Johnstone (1994) proposed universal threshold which is given by
(19)
Where N is the data length series and is the estimator of standard deviation of details coefficients, which is estimated as:
(20)
MAD is the median absolute deviation of the wavelet coefficients at the finest scale
There are two main thresholding rules:
It was proposed by Donoho & Johnostone in 1995. Soft thresholding zeros all the signal values smaller than δ followed by subtracts δ from the values larger than δ which is defined as follows:
(21)
Where
(22)
and
(23)
Donoho and Johnstone proposed Hard thresholding which is the simplest thresholding technique based on the premise of (keep or kill). Hard thresholding zeroes out all the signal values smaller than δ. The wavelet coefficient is set to the vector with element. “Quotation” (Donoho, and Johnstone , 1995)
(24)
Proposed Method
The proposed method included dealt with the contamination problem of Weibull distribution in survival analysis using Wavelet Shrinkage. First, compute the DWT coefficients for a wavelet (Daubechies, Symlets, and Coiflets wavelets). Second, the threshold level is estimated by one of the methods (e.g. SURE, Minimax, and Universal threshold). Third, Thresholding rules (Soft) is used to keep or kill the discrete wavelet coefficients. Thus, we get the modified DWT coefficients , then it is used to compute the inverse of the modified DWT (Taha and Jwana, 2022) as in formula (25).
(25)
Finally, the data for Weibull distribution which have less contamination are used to estimate the shape and scale parameter of the Weibull distribution using the method of maximum likelihood and then analyze the survival function on this basis.
Evaluation Criterion
To measure the accuracy of the estimated parameters (scale and shape) of the Weibull distribution, the mean squared error (MSE) can be used as in the following formula:
(26)
m: number of samples.
Experimental and Application
To compare between the classical and the proposed method in terms of efficiency and accuracy of the estimated parameters for Weibull distribution and reliability function, an experimental aspect was done by simulating the Weibull distribution, then an applied aspect of the real data based on MSE criterion and by designing a program in MATLAB (version 2020a) dedicated to this purpose (Appendix).
Four cases were selected for scale parameter (0.5 and 1) and shape parameter (5 and 10), the sample size (50 and 100) and the addition of contamination percentages (10% and 20%) has a Cauchy distribution (α = 0 and β = 0.5). For the first experimental with n = 100, figure (1) is shown.
Figure (1): The Original data (*), Contamination data (.), and Denoise data ()
Figure (1) shows the scatter plot of the data generated from the Weibull distribution (*) at scale parameter (0.5) and shape parameter (5), and the values of the scatter of the contamination data (.) at 10% contamination, and then the data processed from the contamination () using the (Sym3) wavelet with universal threshold and soft rule. The Survival function of the Weibull distribution for the contaminated and treated data is shown in Fig. 2 and 3 respectively.
For the purpose of the comparison between the proposed and classical method in estimating the parameters of the Weibull distribution, the experiment was repeated to (1000) times and the average criterion for MSE was calculated. Three wavelets (Db2), (Sym3), and (Coif4) were used with different methods in estimating the threshold level (SURE, Minimax, and Universal), with threshold rule (Soft), and for different samples (50, and 100) and percentage of contamination (10% and 20%). The results are summarized in tables (14) for the average of (MSE) criterion when at
Figure (2): Survival Function of the Weibull Distribution for the Contaminated Data
Figure (3): The Survival Function of the Weibull Distribution for the Treated Data
Table (1): Average of MSE Criterion when
Method 
Sample size 
Percentage of Contamination 
Wavelet 
Threshold Method 


Proposed 
50 
10% 
Db2 
SURE 
29.9383 
1.8361 
Minimax 
29.8261 
1.4718 

Universal 
29.6699 
1.3710 

Sym3 
SURE 
29.9890 
1.9961 

Minimax 
29.8394 
1.4203 

Universal 
29.6371 
1.3449 

Coif4 
SURE 
29.9009 
1.6753 

Minimax 
29.7546 
1.3025 

Universal 
29.6123 
1.2506 

Classical 
31.1941 
9.3698 

Proposed 
50 
20% 
Db2 
SURE 
120.1897 
2.1030 
Minimax 
119.7780 
1.7198 

Universal 
119.1890 
1.5535 

Sym3 
SURE 
120.3456 
2.2526 

Minimax 
119.8419 
1.6561 

Universal 
119.0830 
1.4870 

Coif4 
SURE 
120.0310 
1.8744 

Minimax 
119.5262 
1.4863 

Universal 
118.9887 
1.3718 

Classical 
122.8904 
10.2371 

Proposed 
100 
10% 
Db2 
SURE 
28.8237 
2.6367 
Minimax 
28.6013 
2.6987 

Universal 
28.4428 
3.5660 

Sym3 
SURE 
28.9553 
2.3568 

Minimax 
28.7226 
2.2413 

Universal 
28.5333 
3.0492 

Coif4 
SURE 
28.7310 
3.5742 

Minimax 
28.4770 
3.7580 

Universal 
28.3180 
2.2137 

Classical 
31.2688 
9.4165 

Proposed 
100 
20% 
Db2 
SURE 
115.9454 
2.2129 
Minimax 
115.1003 
2.1284 

Universal 
114.4698 
2.8339 

Sym3 
SURE 
116.4356 
2.0469 

Minimax 
115.5527 
1.8101 

Universal 
114.8186 
2.4264 

Coif4 
SURE 
115.5530 
2.8968 

Minimax 
114.6063 
2.9944 

Universal 
113.9868 
1.7810 

Classical 
123.1614 
10.2919 
Table (2): Average of MSE Criterion when
Method 
Sample size 
Percentage of Contamination 
Wavelet 
Threshold Method 


Proposed 
50 
10% 
Db2 
SURE 
29.1860 
1.5416 
Minimax 
29.0536 
1.2177 

Universal 
28.8879 
1.2386 

Sym3 
SURE 
29.2510 
1.7314 

Minimax 
29.0638 
1.1804 

Universal 
28.8460 
1.2830 

Coif4 
SURE 
29.1386 
1.4764 

Minimax 
28.9685 
1.1649 

Universal 
28.8163 
1.1589 

Classical 
31.1308 
8.0982 

Proposed 
50 
20% 
Db2 
SURE 
117.6636 
1.8820 
Minimax 
117.2345 
1.5141 

Universal 
116.6199 
1.4014 

Sym3 
SURE 
117.8677 
2.0424 

Minimax 
117.2856 
1.4626 

Universal 
116.4954 
1.3669 

Coif4 
SURE 
117.5177 
1.7053 

Minimax 
116.9569 
1.3329 

Universal 
116.3993 
1.2675 

Classical 
122.2504 
9.5304 

Proposed 
100 
10% 
Db2 
SURE 
27.9865 
3.7141 
Minimax 
27.7598 
3.9826 

Universal 
27.5976 
5.2004 

Sym3 
SURE 
28.1462 
3.1542 

Minimax 
27.8891 
3.2646 

Universal 
27.6915 
4.4804 

Coif4 
SURE 
27.8984 
4.9873 

Minimax 
27.6301 
5.4365 

Universal 
27.4674 
3.1415 

Classical 
31.2109 
8.1348 

Proposed 
100 
20% 
Db2 
SURE 
113.2828 
2.5646 
Minimax 
112.4170 
2.5755 

Universal 
111.7902 
3.4094 

Sym3 
SURE 
113.7934 
2.2832 

Minimax 
112.8932 
2.1179 

Universal 
112.1464 
2.9146 

Coif4 
SURE 
112.9306 
3.3937 

Minimax 
111.9261 
3.5960 

Universal 
111.2990 
2.1087 

Classical 
122.5386 
9.5784 
Table (3): Average of MSE Criterion when
Method 
Sample size 
Percentage of Contamination 
Wavelet 
Threshold Method 


Proposed 
50 
10% 
Db2 
SURE 
30.1136 
35.2721 
Minimax 
29.9956 
33.6752 

Universal 
29.8382 
31.8721 

Sym3 
SURE 
30.1549 
35.3359 

Minimax 
30.0078 
33.2520 

Universal 
29.8042 
30.9007 

Coif4 
SURE 
30.0588 
33.5041 

Minimax 
29.9209 
31.6553 

Universal 
29.7790 
30.0291 

Classical 
31.4130 
64.6975 

Proposed 
50 
20% 
Db2 
SURE 
120.5057 
37.4422 
Minimax 
120.0781 
35.8988 

Universal 
119.4895 
34.0720 

Sym3 
SURE 
120.6475 
37.4647 

Minimax 
120.1447 
35.4239 

Universal 
119.3857 
33.0576 

Coif4 
SURE 
120.3542 
35.7758 

Minimax 
119.8240 
33.8354 

Universal 
119.2866 
32.2075 

Classical 
123.3063 
67.0477 

Proposed 
100 
10% 
Db2 
SURE 
28.9857 
20.5670 
Minimax 
28.7648 
17.6639 

Universal 
28.6033 
15.7304 

Sym3 
SURE 
29.1198 
22.6352 

Minimax 
28.8843 
19.5389 

Universal 
28.6943 
17.1038 

Coif4 
SURE 
28.8839 
18.6996 

Minimax 
28.6369 
15.4699 

Universal 
28.4779 
13.6675 

Classical 
31.4843 
64.8822 

Proposed 
100 
20% 
Db2 
SURE 
116.2646 
22.6668 
Minimax 
115.3885 
19.6353 

Universal 
114.7571 
17.6281 

Sym3 
SURE 
116.7477 
24.6787 

Minimax 
115.8434 
21.5736 

Universal 
115.1075 
19.0601 

Coif4 
SURE 
115.8330 
20.6174 

Minimax 
114.8926 
17.3623 

Universal 
114.2733 
15.4556 

Classical 
123.5680 
67.2418 
Table (4): Average of MSE Criterion when
Method 
Sample size 
Percentage of Contamination 
Wavelet 
Threshold Method 


Proposed 
50 
10% 
Db2 
SURE 
29.5131 
31.5014 
Minimax 
29.3832 
29.9700 

Universal 
29.2157 
28.1807 

Sym3 
SURE 
29.5710 
31.7905 

Minimax 
29.3903 
29.5946 

Universal 
29.1729 
27.2522 

Coif4 
SURE 
29.4627 
30.0089 

Minimax 
29.2922 
28.0110 

Universal 
29.1405 
26.3864 

Classical 
31.5393 
61.0901 

Proposed 
50 
20% 
Db2 
SURE 
118.2867 
35.6962 
Minimax 
117.8332 
34.1064 

Universal 
117.2145 
32.3038 

Sym3 
SURE 
118.4436 
35.7407 

Minimax 
117.8802 
33.6804 

Universal 
117.0887 
31.3192 

Coif4 
SURE 
118.0879 
33.9611 

Minimax 
117.5441 
32.0812 

Universal 
116.9879 
30.4530 

Classical 
123.0293 
65.1388 

Proposed 
100 
10% 
Db2 
SURE 
28.3065 
17.3308 
Minimax 
28.0739 
14.4667 

Universal 
27.9102 
12.7196 

Sym3 
SURE 
28.4562 
19.3011 

Minimax 
28.2026 
16.2475 

Universal 
28.0054 
13.9791 

Coif4 
SURE 
28.2115 
15.7111 

Minimax 
27.9432 
12.4664 

Universal 
27.7793 
10.8975 

Classical 
31.6152 
61.2684 

Proposed 
100 
20% 
Db2 
SURE 
113.8650 
20.9711 
Minimax 
112.9944 
18.0499 

Universal 
112.3565 
16.0971 

Sym3 
SURE 
114.3819 
23.0055 

Minimax 
113.4643 
19.9312 

Universal 
112.7144 
17.4826 

Coif4 
SURE 
113.4534 
19.0606 

Minimax 
112.4899 
15.8356 

Universal 
111.8630 
14.0107 

Classical 
123.3046 
65.3248 
Tables (14) shows that all the proposed methods have better efficiency than the classical method in estimating scale and shape parameters for Weibull distribution depending on the average of criterion (MSE) for all cases. Also, (Coif4) wavelet with Universal threshold method is the best efficient compared with all other proposed methods and with the classical method because it has the lowest average of the criterion (MSE(α) and MSE(β)). The efficiency of the estimated parameters decreases with increasing contamination percentage for all simulations. Also, the efficiency of the estimated scale parameter α is not affected by an increase in its real value, and the efficiency of the estimated shape parameter β decreases as its real value increases for all simulations.
Real data represent the time of kidney failure. The distribution of the data was tested using KolmogorovSmirnov, and the test statistic (0.12074) is less than critical value (0.23059), that supports the hypothesis of the Weibull distribution for data (pvalue 0.45096 > 0.01). The statistic test (the goodness of fit, χ^{2}) for the classical and proposed methods is summarized in table (5).
Table (5): The Goodness of Fit for Real Data
Method 
Wavelet 
Threshold Method 
The Statistic 
Proposed 
Db2 
SURE 
3.8305 
Minimax 
3.4331 

Universal 
3.6593 

Sym3 
SURE 
3.8296 

Minimax 
1.9165 

Universal 
1.8655 

Coif4 
SURE 
3.8253 

Minimax 
2.1507 

Universal 
2.1214 

Classical 
3.8245 
The proposed method (Db2 with Universal Threshold Method) is the best because it has the statistic (1.8655) with scale parameter (8.3444) and shape parameter (18.922). On this basis, the probability density function, cumulative, and survival of kidney failure data are shown in the figures (46):
Figure (4): The Probability Density Function of the Weibull Distribution for Real Data
Figure (4) shows the Probability Function of the Weibull Distribution of Kidney Failure Data for the Time Period (1224).
Figure (5): The Cumulative Function of the Weibull Distribution for the Real Data
Figure (6): The Survival Function of the Weibull Distribution for Real Data