Estimation of Failure Function under Multiple Causes for Two-Parameter Weibull Distribution for Traffic Accident in Sulaimani
A Thesis Submitted to the Council of College of Administration & Economics– University of Sulaimani, as Partial Fulfillment for the Requirements of the Master Degree of Sciences in Statistics
By
Huda Mohammad Saeed
Supervised by: Assistant Professor
Dr. Talib Sharif Jalil
2015 A.D
2714 K
1435 H
تكدير دالة الفشن حتت تعدد اشباب الفشن لتوزيع ويبن ذي املعممتني لمحوادث املرورية يف الصميمانية
رشالة مكدمة إىل جممض كمية االدارة واالقتصاد -جامعة الصميمانية وهي جسء من متطمبات نين درجة ماجصتريعموم يف االحصاء
من قبن
هدى حممد شعيد
بإشراف األشتاذ املصاعد
د .طالب شريف جمين
5341هـ
7253ك
7051م
بِشِ ِم اللَّ ِى السَّحِمَ ِن السَّحَِمِ لَِتشِتًٌَُا َعلَى ظُوٌُزِيِ ثُمَّ تَ ِركُسًُا نِعِمَةَ زَبُِّلمِ ِإذَا اسِتَ ٌٍَُِتمِ َعلََِىِ ًََتقٌُلٌُا سُبِحانَ الَّرِي سَخَّسَ لَنَا هَرَا ًَمَا كُنَّا لَىُ ُمقِِسنِنيِ [ًَ ]31إِنَّا ِإلَى زَبِّنَا لَمُن َقلِبٌُن
[]31
(صدق اهلل العظَم) سٌزة الزخسف
Dedication To the spirit of my father… With faithfulness To my lovely mother… With grateful of favor To my brothers and sisters… With infinite love To my Advisor Asst. Prof. Dr. Talib Sh. Jalil… With respect and appreciation Thanks for all who supported me …
Researcher …
I
Acknowledgements First and foremost I would like to thank the Almighty God, our creator and guide, who gave me the strength and knowledge to go through this rigorous course and execute this work. I am deeply indebted to Asst. Prof. Dr. Talib Sh. Jalil for his undertaking task of supervising this thesis and offering many suggestions and corrections. I would like to express my thanks to the dean of the college of Administration and Economies, Dr. Kawa M. Jamal. I would like to express my gratitude to Asst. Lect. Aras Jalal and Mr. Handren Abdulla for offering many suggestions and corrections. My best thanks go to Asst. Lect. Alan Ghafur, Asst. Lect. Zainab Abdulla and Dr. Samira M. Salh for their help. Also I am thankful for those at the department of Statistics; especially Dr. Mohammad Faqi the head of the department for his assistance, with special thanks to Dr. Nawzad Mohammad, Dr. Shawnm Abdul-qadr, Dr. Munem Aziz and Dr. Abdul-Rahim K.Rahi for all their interesting lectures during the courses. My best thanks go to my friends Ms. Renas, Ms. Shaima, Ms. Awez, Ms. Shanaz and Mr. Farhad for all their useful assistance. I wish to thank all the members of college library for their help during field work. Also, I would like to thank all the members of college library in the university of Salahaddin-Erbil. I would like to express my gratitude to Mr. Rawand from traffic police records in Sulaimani. Finally, I would like to acknowledge my deep gratitude to my (mother, father, brothers and sisters) for their forbearance and support life and friends for their patience and help. Huda … II
Abstract A road accident is commonly defined as the collision of vehicles, pedestrian, or with an object that will result to death, disability and damage to property. Road crashes are a growing problem worldwide, resulting in around (1) million deaths and more than (23) million injuries annually. An increasing number of road accidents have been seen as an indicator of a poor performing economic activity. Most of the road accidents that caused death of the people who are themselves primarily involved like the drivers or those victims were attributed to the defective roads, non-standard road signs and drivers’ carelessness. The subject of reliability has become a domain in which doing research and studies has being increased due to the important role that it plays as a science in dealing with various typical applications whether component equipment or organisms both of Reliability Theory and Survival Theory. These two theories are different in that Reliability theory is suitable for equipment and theory of Survival is more suitable for a life organism. In real practice rarely happens that a failure is due to a single cause, in most cases there is more than one causes of failure. In this thesis, multiple causes of failure are considered which four are; (speed, neglecting the traffic rules, changing sides and drivers’ carelessness) under the following conditions; the first is the time-to-failure distribution of each cause is known and another is at the instant of a component failure, the failure is due to a single cause. In this study the shape and scale parameters of the Weibull distribution are estimated using Maximum Likelihood Estimation to find Failure Function, Survival Function and Hazard Function for each cause alone and for causes III
together respectively. Also using Maximum Likelihood Estimation to obtained failure rate of each cause as: the failure rate of speed cause equals to (44.5) percent, the failure rate of neglecting traffic rules cause equals to (23.3) percent, the failure rate of changing sides cause equals to (13) percent and the failure rate of carelessness cause equals to (19.2) percent. That is mean the speed cause has a great effect on the traffic accidents and changing sides cause has the least effect on the traffic accidents.
IV
Contents Title
Page
Al-Aya Karima Dedication
I
Acknowledgements Abstract
II III-IV
Contents
V-VII
List of Figures
VIII
1.1
List of Tables Introduction
IX 1
1.2
Literature Review
3
1.3 2.1
The Objectives of the thesis Some Basic Concepts
7 8
2.1.1
Failure Density Function
8
2.1.2
Failure Distribution Function
8
2.1.3 2.1.4
Survival Function Mean Time to Failure (MTTF)
9 10
2.1.5 2.1.6
Hazard Function (Instantaneous Failure Rate) Life-Time Periods
11 12
2.2 2.2.1
Some Failure Distributions Weibull Distribution
13 13
2.2.1.1 2.2.1.2
Two-parameter Weibull Distribution Effects of Scale and Shape Parameters of Weibull
14 16
2.2.1.3
Distribution Effect of the Shape Parameter on the Hazard Function and Some Properties
17
2.3
Relationship of two-parameter Weibull distribution with other distributions
18
2.3.1
Exponential Distribution
18
V
2.3.2
Raleigh distribution
18
2.4 2.5
Probability Failure Function under Multiple Causes Methods of Estimation
19 22
2.5.1 2.5.2
Maximum Likelihood Estimation Method (MLE) Maximum Likelihood Estimation of the Two-Parameter Weibull Distribution Failure Function of two-parameter Weibull distribution Under multiple causes
22 23
2.7 2.7.1
Multinomial Distributio Estimate the Parameters of the Multinomial Distribution
32 33
3.1
Traffic Accidents
36
3.2
Accident Causes
37
3.2.1
Personal Causes
37
3.2.2
The External Causes that Cause Accidents
39
3.3
Data Description
41
3.4 3.4.1.1
Statistical Analysis Goodness of fit test of speed cause
43 43
3.4.1.2
Failure Function Estimation of the Traffic Accidents of speed case Figures of Failure Density Function, Failure Distribution Function, Survival Function and Hazard Function for speed cause Goodness of fit test of Neglecting cause Failure Function Estimation of the Traffic Accidents of neglecting traffic rules case Figures of Failure Density Function, Failure Distribution Function, Survival Function and Hazard Function for neglecting traffic rules cause Goodness of fit test for Changing Sides cause
44
Failure Function Estimation of the Traffic Accidents for Changing Sides case Figures of Failure Density Function, Failure Distribution Function, Survival Function and Hazard Function for changing sides cause
57
2.6
3.4.1.3
3.4.2.1 3.4.2.2 3.4.2.3
3.4.3.1 3.4.3.2 3.4.3.3
VI
29
48
50 50 54
57
61
3.4.4.1 3.4.4.2
63 64
3.5
Goodness of fit test for Carelessness Cause Failure Function Estimation of the Traffic Accidents for Carelessness Cause Figures of Failure Density Function, Failure Distribution Function, Survival Function and Hazard Function for Carelessness cause Failure Function Estimation for Multiple Causes
3.6 3.7
Goodness of fit test of Multinomial Distribution Failure Rate of the traffic accidents of each cause
89 89
4.1
Results and Conclusions
92
4.2
Recommendation
96
3.4.4.3
Reference Appendixes
67
70
97-101 102-104
Arabic Abstract Kurdish Abstract
A B
VII
List of Figures Figure
Title
Figure (2-1)
Failure Distribution Function and Survival Function as the area under the curve of Failure Density Function
9
Figure (2-2)
Mean Time to Failure as the area under the curve of Survival Function Bathtub Hazard Rate Curve Two-parameter Weibull Distribution Function Hazard Function curves for different values of (β) Failure Density Function for speed cause Failure Distribution Function for speed cause Survival Function for speed cause Hazard Function for speed cause Failure Density Function for neglecting cause Failure Distribution Function for neglecting cause Survival Function for neglecting cause Hazard Function for neglecting traffic rules cause Failure Density Function for changing sides cause Failure Distribution Function for changing sides cause Survival Function for changing sides cause Hazard Function for changing sides cause Failure Density Function for carelessness cause Failure Distribution Function for carelessness cause Survival Function for carelessness cause Hazard Function for carelessness cause
10
Figure (2-3) Figure (2-4) Figure (2-5) Figure (3-1) Figure (3-2) Figure (3-3) Figure (3-4) Figure (3-5) Figure (3-6) Figure (3-7) Figure (3-8) Figure (3-9) Figure (3-10) Figure (3-11) Figure (3-12) Figure (3-13) Figure (3-14) Figure (3-15) Figure (3-16)
Page
VIII
12 16 17 48 48 49 49 55 55 56 56 61 62 62 63 68 68 69 69
List of Tables Table Title Table (3-1) Table (3-2) Table (3-3) Table (3-4) Table (3-5) Table (3-6) Table (3-7) Table (3-8) Table (3-9) Table (3-10) Table (3-11) Table (3-12) Table (3-13) Table (3-14) Table (3-15)
Page
Estimated value of Failure Function, Survival Function and Hazard Function for speed cause Estimated value of Failure Function, Survival Function and Hazard Function of neglecting traffic rules cause Estimated value of Failure Function, Survival Function and Hazard Function of changing sides cause Estimated value of Failure Function, Survival Function and Hazard Function of carelessness cause Estimated value of Failure Function of speed and neglecting cause together Estimated value of Failure Function of speed and changing sides cause together Estimated value of Failure Function of speed and carelessness cause together Estimated value of Failure Function of neglecting traffic rules and changing sides causes together Estimated value of Failure Function of neglecting traffic rules and carelessness cause together Estimated value of Failure Function of changing sides and carelessness cause together Estimated value of Failure Function of speed, neglecting traffic rules and changing sides cause together Estimated value of Failure Function of speed, changing sides and carelessness cause together Estimated value of Failure Distribution Function of neglecting, changing sides and carelessness cause together Estimated value of Failure Distribution Function of speed, neglecting, changing sides and carelessness cause together Failure rate of each cause
IX
46 52 59 66 70 72 74 76 78 79 81 83 85 87 91
Chapter One Introduction and Literature Review Literature Reiew
Chapter Two Theoretical Part
Chapter Three Application & Data Description
References
Chapter Four Conclusions &
Recommendations
Appendixes
Chapter One
1.1
Introduction and Literature Review
Introduction:[2][8][11]
The increasing interest for research and recent studies in the subject of reliability came as a result of the role played by this science in dealing with various typical applications whether component equipment or organisms both of Reliability Theory and Survival Theory, both share the performance measurement period efficiently whether for a machine or system or a living organism. The differences shall be governed by the optimization system reliability for multiple systems earners because such question is represented by numbers and location parts of the system and easily give alternative for these parts and fast processing for it which makes this system an optimal. However in the theory of survival, there are no such instances because the system here is a life organism and has been arranged very well, so changing the organs or organisms is difficult. Also, the organs deliver the best example of an optimal working system. There is a big probability for the machines to damage. As, a result, they stop working, or failure leading to increase costs and less production, consequently to the human losses, moral, material and loss of time and other damages. Therefore, reliability measurement of any device would be the basis for the development of most of the machines. The vast developments in the fields of industry and the rise of mechanical, electrical, and electronic complexities, which led to widespread attention in theories of reliability in the last century. Before 1940 when the researches were limited to quality control and maintenance machines and reliability was not diagnosed as an independent science but by the Second World War and the increase of complex military equipment, it had obtained bases and principles that led to the appearance and development of the rise of reliability. 1
Chapter One
Introduction and Literature Review
The most statistical research on reliability focused on the periods, whether the equipment or systems operate or fail and estimate the reliability of a certain niche is of the utmost importance in the vicinity of modern technology and future developments. Therefore, the estimate of reliability is the target of many of the statistical problems. In real practice it is seldom true that a component’s (or system’s) failure is happened due to a single cause, in most of the cases the failure is happened by more than one (or multiple) causes. Each cause has the effect of failure with different rate. In this thesis, a multiple causes of failure are considered under the following conditions: 1. The time-to- failure distribution of each cause is known. 2. At the instant of failure only one of the causes would cause the failure.
Due to the fact that the Weibull distribution is one of the important failure models, which is characterized by a high risk function of failure when you began operating, which decreases gradually as the time goes on.
The thesis includes four chapters; the first chapter provides an introduction, the literature review of some of researches which related to the topic and the objectives of the thesis. The second chapter deals with some basic concepts on the topic of failure and deals with the theoretical aspect of the Weibull distribution and using Maximum Likelihood Estimation to estimate the parameters. Chapter three includes an introduction to the traffic accidents and the major causes of their occurrence. Also, there is a data of traffic accidents in Sulaimani.
2
Chapter One
Introduction and Literature Review
The last chapter is devoted to the results, conclusions and recommendations obtained from this thesis application part for further studies.
1.2
Literature Review:
There are many papers that used different methods of estimation for the parameters and reliability functions of common failure time distributions, some of them are:
In (1973), (Canavos G. C. and Tsokos C. P.)
[8]
used Bayesian analysis to
estimate the scale parameter of Weibull distribution, when the shape parameter is known.
In (1981), (Diane I. G. and Lonnie C. V.)
[9]
compared between Maximum
Likelihood Estimator, Least Squares estimators and moment estimator for the scale and shape parameters of two-parameter Weibull distribution by using Mean Square Error. In (1988), (Sinha S. K. and Sloan J. A.) [14] obtained Bayes estimates of the parameters and reliability function of Three-parameter Weibull distribution and compare posterior standard-deviation estimates with the corresponding asymptotic standard-deviation estimates of their Maximum Likelihood counterparts. Numerical examples are given. In (2004), (Al – Helaly, Firas Saddam Abed)
[3]
compared Bayesian
approaches (Bayes method and Shrinkage method) with other methods of estimation for parameters and reliability function of the Three-parameter of 3
Chapter One
Introduction and Literature Review
Weibull distribution as a model of failure. He observed that the best method is the Shrinkage method and comes after the Maximum Likelihood method. In (2007), (Al- Yasseri, Tahani Mehdi Abbas) [4] estimated the approximate reliability function of two-parameter Weibull distribution. When the sample
data under study is contaminated, the previous information or both cases appear together in the case of not determining accurately. She made a comparison between Robust Bayes method with other methods of estimation as:
(Maximum Likelihood method and method of moments and White’s
method) by using the simulation procedure. Also, she suggested a proposed method to estimate the parameters of Weibull distribution.
In (2007), (Wahdi, Awat Sirdar )
[7]
compared between the
Maximum
Likelihood method and the Shrinkage method to estimate the parameters and the reliability function of the two-parameter Gamma distribution, in case of missing data, through simulation of the Monte Carlo method. Generally, through simulation by using the Monte Carlo method for the developed Maximum Likelihood method by using Thom method is found to be the best one for the reliability function estimation because it has the minimum (IMSE) and the minimum (IMAPE) in comparison with the other methods.
In (2007), (Alhad, Etaf Adwar Abed)
[1]
compared between different
methods of estimation for the reliability function of two-parameter Exponential distribution. He found that the two methods, Second Modification Maximum
4
Chapter One
Introduction and Literature Review
Likelihood (M.M.L.E-II) and First Modification Moments (M.M.E-I) are the best methods based on the following criteria:-
1- Integral Mean Squared Error (IMSE). 2- Integral Means Absolute Percentage error (IMAPE).
In (2008), (Al-Shamry, Najat Abed Al-Jabar Rajab)
[2]
used the Maximum
Likelihood and the mixture methods to estimate the shape parameter of twoparameter Weibull distribution (when the scale parameter is known) by taken different sizes of the censored samples. She had noted that the maximum likelihood method fairs better especially in large sample size and the mixture method is proper for small sample size.
In (2011), (Muhammad, Zainab Abdulla)
[6]
used Bayesian method to
estimate two-parameter Weibull distribution and the reliability function of the series system using mixture distribution.
In (2011), (Pandey B.N. , Dwivedi N. and Bandyopadhyay P.) [13] considered the estimation of the scale parameter of two-parameter Weibull distribution with known shape parameter. Maximum Likelihood Estimator is discussed. Bayes Estimator is applied using Jeffrey’s prior under linear loss function. They have calculated relative efficiency of the estimators in small and large samples for over-estimation and under-estimation using simulated data. They have observed that Bayes estimator is better especially in small sample size and when the over estimation is more critical than under estimation. 5
Chapter One
Introduction and Literature Review
In (2011), (Rahim, Alan Ghafur )
[5]
estimated both scale and shape
parameters of Weibull distribution and the failure function under multiple causes of failure, when the time-to-failure follows two-parameter Weibull distribution, using Maximum Likelihood Method taking the advantage of Taylor Expansion, with application to real medical data.
In (2012), (Prakash G.)
[11]
obtained Bayes estimators corresponding to the
informative and non-informative priors for the parameters of the mixture model of two-parameter Weibull distributions when failure data are available. Also he determined the Bayes predictive intervals.
In (2012), (Guure, Chris B., Ibrahim N. A. , and Ahmad, Al O. M.)
[10]
examined the performance of Maximum Likelihood Estimator and Bayesian estimator using extension of Jeffrey’s prior information with three loss functions, namely, the linear exponential loss, general entropy loss, and the square error loss function for estimating the two-parameter Weibull failure time distribution. These methods are compared using Mean Square Error through simulation study with varying sample sizes. The results show that Bayesian estimator using extension of Jeffrey’s prior under linear exponential loss function in most cases gave the smallest Mean Square Error and absolute bias for both the scale parameter
and the shape parameter
for the extension of
Jeffrey’s prior.
In (2013), (Yahgmaei F. , Babanezhad M. and Moghadam O.S.)
[16]
used
different methods of estimating the scale parameter in the Inverse Weibull 6
Chapter One
Introduction and Literature Review
Distribution (IWD). Specifically, the Maximum Likelihood Estimator of the scale parameter in IWD is introduced. Then they derived the Bayes estimators for this parameter by considering quasi, gamma, and uniform priors distributions under the square error, entropy, and precautionary loss functions. Finally, the different proposed estimators have been compared by the extensive simulation studies in corresponding the Mean Square Errors and the evolution of risk functions.
1.3
The Objectives of the thesis:
The purpose of this thesis is to estimate Failure Function under multiple causes of failure using the proposed model and also to estimate rate of failure of each cause using Maximum Likelihood Estimation method.
7
Chapter Two
Theoretical Part
2.1 Some Basic Concepts: Failure distributions are characterized as a set of common functions in the period
for the continuous random variable
that represents the time
to failure of a component or a system, these functions are:
2.1.1 Failure Density Function: [17] [28] Failure Density Function is denoted by
, that represents the probability
of failure of a component or a system during the period where
,
is a small value (absolute zero) that represents the change in the value
of random variable
. Also is called (Unconditional Failure Rate),
mathematically is defined as follows:
..….. (2.1)
2.1.2 Failure Distribution Function:[18][23] Failure Distribution Function is denoted by
, and it is defined as the
probability that a system or a component will fail before or at time t, and mathematically is defined as:
..….. (2.2)
∫
Also, it is called Unreliability Function and it is a Monotonic Increasing Function of t, with:
8
Chapter Two
Theoretical Part
2.1.3 Survival Function: [23] [28] Survival Function is the probability that a system or component will survive without failure during a specified time interval conditions. Survival function is denoted by
under given operating
, and mathematically is defined
as:
..….. (2.3)
∫
Survival Function is a Monotonic Decreasing Function of , with:
Also, it is linked with the Failure Distribution Function
and in fact it is the
complement of it, i.e.: ……. (2.4)
Figure (2-1): Failure Distribution Function and Survival Function as the area under the curve of Failure Density Function 9
Chapter Two
Theoretical Part
2.1.4 Mean Time to Failure (MTTF): [23] [26] Mean time to failure represents the expected value of time to the first failure. Mathematically
can be written as:
….... (2.5)
∫
By taken derivative of equation (2.4), we get: = =
By using integrating by parts of equation (2.5) we get: ∫
Time Figure (2-2): Mean Time to Failure as the area under the curve of Survival Function
10
Chapter Two
Theoretical Part
2.1.5 Hazard Function (Instantaneous Failure Rate): [25] Hazard Function is the limit of the conditional probability that a component fails (for the first time) in a small interval the beginning of the interval divided by usually by
or
given that it survived to . The Hazard Function is denoted
, and defined mathematically as:
[
]
[
]
[
]
[
]
where : is the Failure Density Function, and : is the Survival Function. 11
Chapter Two
Theoretical Part
2.1.6 Life-Time Periods: [19] Failure rates are often used as an index for reliability. A failure rate indicates how often a failure occurs per unit time, and failure rate values generally change over time and divided into three distinct periods: 1. Burn-in period: during this period, failures occur at a high rate following the initial operation and characterized by low efficiency due to errors in device design or manufacturing errors (bad quality). 2. Useful life period: this stage represents normal operating period and the failure rate is almost constant (time independent) because failure occur stochastically this means that failure distribution is the Exponential Distribution (see the theorem given above). 3. Wear-out period: during this stage, failures occur with increasing frequency over time and are caused by age-related wear and fatigue (or due to the age factor). It was observed that these stages are not limited to equipment, but beyond that to include human society. These periods illustrated in the following figure:
Figure (2-3): Bathtub Hazard Rate Curve 12
Chapter Two
Theoretical Part
2.2 Some Failure Distributions: [17] [26] There are many continuous distributions which are able to be probability density functions that have failure model depending on time, they are: 1. Exponential Distribution. 2. Weibull Distribution. 3. Gamma Distribution. 4. Normal Distribution. 5. Lognormal Distribution.
2.2.1 Weibull Distribution: [24] Weibull distribution is a continuous distribution. It is one of the most common and important failure models. In the thirteenth and fourteenth of the last century, the family of parametric distributions are emerged, after a period of time these models are used widely in the reliability or survival field and life time testing, in comparison with other distributions Weibull distribution has stature and importance in this area, because of the flexibility of its Probability Density Function and Risk Function. Also, this distribution is appropriate for the case that the failure rates are high relatively at the beginning of operation, then begin to decrease gradually with the increase of time. Weibull distribution is named after a Swedish scientist (Wallodi Weibull) who was the first to promote the usefulness of this to model data sets of widely differing character in his study in (1939) which delt with the strength of materials. The earliest known publication dealing with the Weibull distribution is a work by Fisher and Tippet (1928) where this distribution is obtained as the 13
Chapter Two
Theoretical Part
limiting distribution of the smallest extremes in a sample. Gumball (1958) refers to the Weibull distribution as the third asymptotic distribution of the smallest extremes. A subsequent study in English (Weibull, 1951) was a landmark work in which he modeled data sets from many different disciplines and promoted the versatility of the model in terms of its applications in different disciplines. A similar model was proposed earlier by Rosen and Rambler (1933) in the context of modeling the variability in the diameter of powder particles being greater than a specific size.
2.2.1.1 Two-parameter Weibull Distribution: [22] [27] As mentioned above, the Weibull distribution is one of the most commonly used distribution in reliability or survival analyses and life time testing, because of many shapes attain with different values of the shape parameter . The Twoparameter Weibull probability density function may be given by:
….… (2.7)
elsewhere
where: : represents the scale parameter; and : represents the shape parameter. 14
Chapter Two
Theoretical Part
Th mean and the variance of the Two-parameter Weibull distribution respectively are: (
(
)
) [
(
)
(
)
(
) ]
Failure Distribution Function of the two-parameter Weibull distribution is: ….… (2.8)
The Survival Function of the two-parameter Weibull distribution is: ….… (2.9)
Hazard Function of the two-parameter Weibull distribution is: ….… (2.10)
Which is the time dependent quantity, observe that for becomes,
, hazard function
, constant which is the hazard function of the Exponential
distribution.
15
Chapter Two
Theoretical Part
2.2.1.2 Effects of Scale and Shape Parameters of Weibull Distribution: [18] The two-parameter Weibull distribution can take different shape curves depending on the values of the parameter
, altering the parameter values
leads to a change in the Weibull curve. It is obvious why shape parameter. But the change of parameter values and reduction in the distribution curve that is why
is called the
leads to enlargement is called the scale
parameter of the Weibull distribution.
From equation (2.9), the curve of the Weibull distribution is close to the curve of Exponential distribution when when
, it tends to Raleigh distribution
and it tends to Normal distribution when
. As the figure
below shows:
Weibull Distribution Shape,Scale 1,1.5 2,1.5 3,1.5
1
f(t)
0.8 0.6 0.4 0.2 0 0
2
4
6
8
Time Figure (2-4): Two-parameter Weibull distribution function
16
Chapter Two
Theoretical Part
2.2.1.3 Effect of the Shape Parameter on the Hazard Function and Some Properties: [5] 1. When
the Hazard Function is decreasing with .
2. When
the Hazard Function is constant, and becomes
,
this shows that the Exponential distribution is a special case of the Weibull distribution. 1. When
the Hazard Function is increasing.
2. When
the Hazard Function is a linear function, which
. 3. When
the Hazard Function is a quadratic function, which .
The figure (2-5) summarized these properties as follows:
𝛽
𝛽
𝛽
0
Time Figure (2-5): Hazard Function curves for different values of (β)
17
Chapter Two
Theoretical Part
2.3 Relationship of two-parameter Weibull distribution with other distributions:
2.3.1 Exponential Distribution: The Exponential distribution is a special case of two-parameter Weibull distribution when the shape parameter
in equation (2-7) equals to 1, as
below: In equation (2-7) the two-parameter Weibull distribution is:
Let
, then
, which is the probability density function of the Exponential distribution.
2.3.2 Raleigh distribution: The Raleigh distribution is a special case of two-parameter Weibull distribution when the shape parameter
in equation (2-7) equals to 2, as
below: Also, in equation (2-7) the two-parameter Weibull distribution is:
18
Chapter Two
Let
Theoretical Part
, then
, which is the probability density function of the Raleigh distribution.
2.4 Probability Failure Function under Multiple Causes: [23] It is seldom true that a component’s failure is occurred by a single cause. In most of the cases the failure of a component is occurred due to more than one cause. The expression for failure density function can be derived in case of multiple causes (under the following conditions): 1. The time-to-failure distribution is known of each cause { }. 2. At the instant of failure only one of the causes would cause the failure.
Mathematically the failure density of a component under multiple causes is obtained as a mixture distribution of all causes failure densities: f
=
∏
∏
+
∏ 19
+…+
Chapter Two
Theoretical Part
∑
∏
[
]
….… (2.11)
where: : represents the failure density of the mixture distribution under multiple causes. f1
: represent failure density for failure cause 1,2,…,k
respectively. : represent failure distribution function for the causes respectively.
Since
is the Failure Distribution of the causes, we can obtain the
reliability (or Survival) function of each failure cause from equation (2.4) as:
Also,
=
by using the relation between [
], then the failure density (2.16) under multiple causes becomes: f
∑ ∑
∏ ∏
….... (2.12)
The Hazard Function of the multiple causes is: ∑ 20
Chapter Two
Theoretical Part
The Survival Function of the multiple causes is: ∏
And the Failure distribution under multiple causes is: F
=∫
, where
is given by (2.12).
Illustration for the special case: Now if
are the failure densities of the two-parameter
Weibull distribution, then
j= 1,2,…,k
}
21
……. (2.13)
Chapter Two
Theoretical Part
By substituting
(∑
and
in equation (2.12) we get:
)
∑
For identical components, we get:
This is time-to-failure of the K-causes.
2.5 Methods of Estimation: 2.5.1 Maximum Likelihood Estimation Method (MLE): [20] [5] The method of Maximum Likelihood is one of the most widely used in classical inference for estimating model parameters. Estimates value which obtained in this method is called Maximum Likelihood Estimates which is the value of the parameters that maximize the value of the likelihood function. It was suggested by (R.A. Fisher) in (1920). The Likelihood Function of (n) time-to-failure random variables when their failure density function be the joint density function.
22
is known is defined to
Chapter Two
Theoretical Part
2.5.2 Maximum Likelihood Estimation of the Two-Parameter Weibull Distribution: [23] Let
be a time-to-failure random sample of size
failure density function
, and the failure distribution function
two-parameter Weibull distribution, then the likelihood function of
with the of the under
multiple causes is: (
) ∏
Now the likelihood function of (2.14) is:
∑
∏∑
∏ *∑
+
∑
∑
The Log-Likelihood function is: ∑
*∑
+
∑∑
Differentiating with respect to ( ) and ( ) and equating to zero, we get: 23
Chapter Two
∑
Theoretical Part
∑
∑
∑*
+
∑
These equations can be solved only by using numerical methods for example Newton-Raphson method.
In a special case: Maximum likelihood function of two causes is: From (2.14) the failure function of two causes is:
Now the Likelihood function is:
∏(
)
24
∑
∑
Chapter Two
Theoretical Part
Since the two causes are independent the equation above can be written as:
*
(∏ ∑
)
(∏
)
+
∑
The Log-Likelihood function is:
*
∏
∏
+
∑
Differentiating with respect to
∑
and equating to zero, we get:
∏ *
∏
∏
∑
25
+
Chapter Two
Theoretical Part
∏ (∏
*
)
∏
[
(∏
∑
∏
∏
*
+
)
∑
∏
] +
∑
∏
[
∑
∏
∏
*
∏
] +
∑
These equations can be solved only by using numerical methods for example Newton-Raphson method.
26
Chapter Two
Theoretical Part
Maximum Likelihood function of one causes is: from (2-7) there is:
The likelihood function of
is:
∏
The log-likelihood function is: ∑
Differentiating (2.15) with respect to have
and
∑
∑
∑
∑
From (2.25): ∑ 27
and equating to zero, we
Chapter Two
Theoretical Part
When ̂ is obtained then ̂ can be determined. We propose to solve ̂ by using Newton-Raphson method as given below. By substituting (2.29) into (2.28), and let
be the same as (2.28), we have:
∑
∑
∑
and taking the first differential of
*
(∑
, we have:
∑
) (∑
(∑
)
+
)
Therefore, ̂ is obtained from the equation below by carefully choosing an initial value for
and iterating the process till it converges:
28
Chapter Two
Theoretical Part
2.6 Failure Function of two-parameter Weibull distribution Under multiple causes:[5] The Failure Function is: F
= ∫
From (2.12) the failure density function ∑
is:
∏
Where is the hazard function of j cause and
is the survival function of j
cause. And from (2.13) the hazard function and the survival function of multiple causes respectively are:
By substituting these equations in (2.42), we get: ∑
∏
29
Chapter Two
Theoretical Part
By substituting (2.43) in (2.41), get:
∫ (∑
∏
)
∫
∫
∫
By using integrating by partition of the first factor of (2-44), we get:
∫
∫
∫
By substituting the integrating by partition of the first factor in (2-36), can be get:
30
Chapter Two
Theoretical Part
[∫
( )]
[∫ ∫
]
Also, the Survival Function and the Hazard Function respectively are:
After finding the value of the parameters (
by
using the Maximum Likelihood Estimation and the Bayesian estimation and substituting in (2-37), (2-38) and (2-39), can be find the estimation of the Failure Function, the Survival Function and the Hazard Function under multiple causes of the two-parameter Weibull distribution.
31
Chapter Two
Theoretical Part
In the special case the Failure distribution of the two-parameter Weibull distribution of two, three and four cause together respectively are:
2.7 Multinomial Distribution: [17][21] The Multinomial Distribution is a common probability distribution for categorical data, if we want to model the number of occurrences of each category from a total of N random observations from categorical population. Also it is a generalization of the binomial distribution.
The multinomial probability mass function is: (
with
)∏
, ∑
32
and ∑
Chapter Two
Theoretical Part
The mean and the variance of the Multinomial distribution respectively are:
2.7.1 Estimate the Parameters of the Multinomial Distribution [21] By using Maximum Likelihood Estimation method of (2-43) can be find the estimate value of the parameters of the Multinomial distribution as below:
In the special case the probability mass function of the Multinomial distribution in (2-43) for four causes is being: (
)∏
Now the likelihood function of the equation above is:
( Let ( Since ∑
)
∑
∑
) , then
∑ 33
∑
∑
Chapter Two
Theoretical Part
After substituting
in (2-44), the log-likelihood function is:
∑ ∑
∑
∑
Differentiation with respect to ( ∑
and equating to zero can be get:
∑ ∑
∑
∑ ∑
∑
∑ ∑
∑
∑ ∑
and ∑
∑ ∑
∑
∑
∑ ∑ 34
Chapter Two
Theoretical Part
∑
Let
∑
From (2-45), (2-46) and (2-47): ∑
∑
∑
∑
and the estimated value of ̂ ̂ ̂ ̂
̂
̂
̂
∑
∑
∑
and
̂
∑
35
are:
Chapter Three
Application and Data description
3.1 Traffic Accidents:
[12] [15]
1. A road accident is commonly defined as the collision of vehicles, pedestrian, or with an object that will result to death, disability and damage to property. Road crashes are a growing problem worldwide, resulting in around 1 million deaths and more than 23 million injuries annually. An increasing number of road accidents have been seen as an indicator of a poor performing economic activity. Most of the road accidents that caused death of the people who are themselves primarily involved like the drivers or those victims were attributed to the defective roads and non-standard road signs.
2. While the Economic Commission for Europe, of the United Nations said the road traffic accident is the one that satisfy the following elements: Occurs in the open road or highway. That result death or injured a person or more.
The involving of at least one car.
3. Some researchers believe that the road traffic accident "means an accident is not deliberate result in death or injury or damage due to movement of vehicles or tonnage on the highway".
63
Chapter Three
3.2 Accident Causes:
Application and Data description [12]
It is important to understand what causes the accident. In general, there are personal causes and external causes behind the accidents occurrence.
3.2.1 Personal Causes: [12] [15] Personal causes remain the most common causes of road accidents and there are many personal causes or factors as:
1. Psychological construction: Scientists emphasize that the human factor is the main cause of the accident and that 10% of reasons accidents or slightly do not refer to human, such as cars, buildings and roads.
2. Experience: Experience related to the accident, and related to age by another hand, but experience alone is not a responsible of the number of accidents involving young people. Also, experience alone is not a mechanism of accidents involving older age. Many researchers emphasize that there is an inverse relationship between experience and accidents.
3. Age: Researches indicate that the age is one of the factors that have significance in the occurrence of accidents. Young age and senior age are more targeted for traffic accidents.
63
Chapter Three
Application and Data description
4. Health status: There is a relationship between accidents and health. Accidents of the drivers who suffer from high blood pressure are twice of those who do not suffer. There is also a relationship between accidents and the sight.
5. Level of education: The educational level has an important role in gaining experience, improving efficiency, and creating drivers.
6. Profession: There is little information about the profession and its relationship to involve the accidents. Studies found that the drivers employed group by the state were lower frequent in accident than other groups. The staffs of companies and businessmen are more frequent in the accidents.
7. Level of civilization: Level of civilization means residence in rural areas or the city. Studies indicate that 80% of those who were involved in accidents in the United States were from the urban population.
8. The social situation of the individuals: Social status and family stability are important, such as other variables, some studies indicate that the unmarried are more susceptible to accidents than the married which enjoy a stable family life. 68
Chapter Three
Application and Data description
9. Driver Characteristics: The major contributing cause for many accidents is the performance of the drivers in both single car and multi-car accidents. The changes in perceptionreaction time of drivers depend on age, whether the person is tired or under the influence of alcohol and/or drugs. An older driver wants longer perceptionreaction time. Human factor testing has shown that the average adult’s perception and reaction time can be ranged from 2 sec to as high as 10 sec depending on age, type and amount of information, weather conditions, car conditions, and other factors.
3.2.2 The External Causes that Cause Accidents: [12] Researchers agree that the non-human causes are not to exceed 20% of the total factors. These factors can be divided into six as the following:
1. Stress and Tiredness: The tiredness and stress factor has an impact on accidents; some researches indicate that tiredness makes the persons more targeted to accidents.
2. Car Characteristics: A small percentage of accidents are caused by mechanical failure of the cars. The design of a car can be a contributory factor in the severity of accident. Some of the features and recent improvements that affect the safety of a car are presented below: a- Airbags b- Antilock Braking System 63
Chapter Three
Application and Data description
c- Automatic locking retractor and emergency locking retractor seatbelts: This feature is especially important to parents installing children safety seats. d- Accident Resistant Door Pillars: Auto manufacturers have introduced this safety feature to deflect the force of a side-impact collision away from the head area and toward the legs.
3. Speed: Increase the rate of accidents by increasing speed and most of the researches emphasizes this result. This is because the speed works on the distribution of attention and thus increasing the chance of errors. This does not mean slow driving is safe driving because they cause fatigue for the driver and passenger.
4. Roadway Characteristics: The roadway’s conditions such as other causes can be a factor of the accidents. Highways must be designed as adequate sight distances of designed speed for the driver to have enough perception-reaction time. The traffic signals should provide enough time for decision sight distance when the signal changes from green to red. 5. Environment Characteristics: The climatic and environmental conditions can also be a factor in accidents. Many severe accidents have occurred during conditions of smoke or fog. Drivers traveling at high rate of speed are unable to see the slowing and/or stopped cars in front of them.
04
Chapter Three
Application and Data description
6. Compounds: Compounds will be considered a catalyst in a series of causal factors of accidents that cannot be separated between the cars and the personalities of the drivers.
3.3 Data Description: Data has a significant role in researches, but unfortunately in Kurdistan region there is no adequate data. This condition is due to two factors, first is related to the citizens and the second to the government. Traffic aspect is one of the most important fields in human life, but in our region until the present time the traffic has not been regarded but neglected.
There are also other factors for the lack of traffic data. Initially, there is not detailed information about the drivers particularly at the time of traffic accidents. Drivers like all other people of our region do not have intelligent smart (I.D. card). It is very important because by this card the investigator can get all information about the driver and that electro-card has a (chipper) cell. On this cell all the information about the driver like (age, gender, health,…etc.) are recorded. They are very useful in traffic accidents investigations.
Another factor of data shortage belongs to the people awareness and the routine of government institutions. Some people try to solve their problems outside the court. These cases have bad influence on traffic data, especially (injured and death numbers…etc.). At last these conditions make a countercurrent data, for example the victim censuses of traffic accident at emergency differ from the censuses at the traffic police which are recorded at the time of the accidents. 04
Chapter Three
Application and Data description
In this study, we deal with failure (or accidents) due to multiple causes; the data set used consists of a sample of 339 observations of four causes of failure which are obtained from traffic police records in Sulaimani. The data are selected from all filed records in 2013.
04
Chapter Three
Application and Data description
3.4 Statistical Analysis: Test each variable for suitable distribution by using XLSTST software, estimate the parameters of the two-parameter Weibull distribution by using Maximum Likelihood Estimation through Synthesis software and find Failure Distribution Function under multiple causes of traffic accidents of each cause can be find as:
3.4.1.1 Goodness of fit test of Speed Cause: By using Kolmogorov-Smirnov test can be find suitable distribution of the variables of speed cause as:
Kolmogorov-Smirnov test D p-value alpha
0.152 0.335 0.05
Test interpretation: H0 : The sample follows a Weibull (2) distribution H1 : The sample does not follow a Weibull (2) distribution As the computed p-value is greater than the significance level alpha=0.05, one cannot reject the null hypothesis H0.
06
Chapter Three
Application and Data description
3.4.1.2 Failure Function Estimation of the Traffic Accidents of Speed case: We can find the estimated value of the shape parameter parameter
and the scale
of the Weibull distribution by using Maximum Likelihood
Estimation method, through Synthesis software as below: ̂
̂
By substituting these estimated values of
̂ and
Failure Function estimation for speed cause as: ̂
̂
̂
̂
̂
̂
00
̂ in (2.8) we can find
Chapter Three
Application and Data description
The Survival Function estimation for speed cause is obtained by substituting these estimated values of ̂
̂
̂ and ̂ in (2.9) as:
̂
̂
̂
̂
Also, the Hazard Function estimation for speed cause is obtained by substituting these estimated values of ̂ and ̂ in (2.10) as: ̂
̂ ̂
̂
̂
̂
04
Chapter Three
Application and Data description
̂
These values are showed in table below: Table (3-1): Estimated value of Failure Function, Survival Function and Hazard Function for speed cause: t
1 4 1 3 3 3 6 5 5 1 5 3 3 2 5 4 2 5 2 3 5 1 4 5 6 6 7 8 6 5
̂
0.195272 0.61985 0.195272 0.472657 0.472657 0.472657 4708148. 47.00094 47.00094 0.195272 47.00094 0.472657 0.472657 0.321166 0.744452 0.61985 0.321166 47.00094 47849922 470.429. 0.43083 479194.4 0.362898 47.00094 4708148. 4708148. 47149410 4710.842 0.495671 47.00094
̂
̂
0.804728 0.38015 0.804728 0.527343 0.527343 0.527343 0.160763 0.255548 0.255548 0.804728 0.255548 0.527343 0.527343 0.678834 0.255548 0.38015 0.678834 0.255548 0.678834 0.527343 0.255548 0.804728 0.38015 0.255548 0.160763 0.160763 0.094906 0.052694 0.160763 0.255548
479401.0 47824010 479401.0 0.290878 474140.0 474140.0 470192.9 4708408 4708408 479401.0 4708408 474140.0 474140.0 47494124 4708408 47824010 47494124 4708408 47494124 474140.0 4708408 479401.0 47824010 4708408 470192.9 470192.9 47990499 47290810 470192.9 4708408 03
Chapter Three 12 7 1 2 5 1 1 3
4711.919 47149410 479194.4 47849922 47.00094 479194.4 479194.4 470.429.
Application and Data description 0.002805 0.094906 0.804728 0.678834 0.255548 0.804728 0.804728 0.527343
4700029 47990499 479401.0 47494124 4708408 479401.0 479401.0 474140.0
Illustration of the results of table (3-1): 1. In the (42) days the Failure Function of speed cause equals to (0.124974) when the Survival Function equals to (0.804728), and the Hazard Function in the same day is (0.195272) 2. In the (49) days the Failure Function of speed cause equals to (0.362898) when the Survival Function equals to (0.38015), and the Hazard Function in the same day is (0.61985). 3. In the (56) days the failure distribution of speed cause equals to (0.124974) when the Survival Function equals to (0.804728), and the Hazard Function in the same day is (0.195272).
38. In the (301) days the Failure Function of speed cause equals to (0.290878)
when the Survival Function equals to (0.527343), and the
Hazard Function in the same day is (0.472657).
03
Chapter Three
Application and Data description
3.4.1.3 Figures of Failure Density Function; Failure Distribution Function, Survival Function and Hazard Function for Speed Cause: Figures of the Failure Density Function, Failure Distribution Function, Survival Function and Hazard Function of the two-parameter Weibull distribution of speed cause:
Figure (3-1): Failure Density Function for speed cause
Figure (3-2): Failure Distribution Function for speed cause 08
Chapter Three
Application and Data description
Figure (3-3): Survival Function for speed cause
Figure (3-4): Hazard Function for speed cause
03
Chapter Three
Application and Data description
3.4.2.1 Goodness of fit test of Neglecting the Traffic Rules cause:
By using Kolmogorov-Smirnov test can be find suitable distribution of the variables of neglecting traffic rules cause as:
Kolmogorov-Smirnov test D p-value alpha
0.194 0.157 0.05
Test interpretation: H0 : The sample follows a Weibull (2) distribution H1 : The sample does not follow a Weibull (2) distribution As the computed p-value is greater than the significance level alpha= 0.05, one cannot reject the null hypothesis H0.
3.4.2.2 Failure Function Estimation of the Traffic Accidents for neglecting traffic rules case: We can find the estimated value of the shape parameter parameter
and the scale
of the Weibull distribution by using Maximum Likelihood
Estimation method of neglecting traffic rules cause through Synthesis software as below: ̂
̂
44
Chapter Three
Application and Data description
By substituting these estimated values of
̂ and
̂ in (2.8) we can find
Failure Function estimation for neglecting traffic rules cause as: ̂
̂
̂
̂
̂
̂
The Survival Function estimation of neglecting traffic rules cause is obtained by substituting these estimated values of ̂ and ̂ in (2.9) as: ̂
̂
̂
̂
̂
44
Chapter Three
Application and Data description
̂
Also, the Hazard Function estimation of neglecting traffic rules cause is obtained by substituting these estimated values of ̂ and ̂ in (2.10) as: ̂
̂ ̂
̂
̂
̂
̂
These values are showed in table below:
Table (3-2): Estimated value of Failure Function, Survival Function and Hazard Function of neglecting traffic rules cause as: t
1 3 3
̂
0.318447 0.789161 0.789161
̂
̂
0.681553 0.210839 0.210839
0.225157 0.995192 0.995192 44
Chapter Three 2 1 2 1 4 5 3 1 2 1 1 2 2 4 1 4 3 2 2 3 2 4 2 2 3 4 3 3 1 2
0.539992 0.318447 0.539992 0.318447 0.938246 0.988986 0.789161 0.318447 0.539992 0.318447 0.318447 0.539992 0.539992 0.938246 0.318447 0.938246 0.789161 0.539992 0.539992 0.789161 0.539992 0.938246 0.539992 0.539992 0.789161 47180402 0.789161 0.789161 4789000. 47981114
Application and Data description 0.460008 0.681553 0.460008 0.681553 0.061754 0.011014 0.210839 0.681553 0.460008 0.681553 0.681553 0.460008 0.460008 0.061754 0.681553 0.061754 0.210839 0.460008 0.460008 0.210839 0.460008 0.061754 0.460008 0.460008 0.210839 0.061754 0.210839 0.210839 0.681553 0.460008
0.575044 0.225157 0.575044 0.225157 1.468645 1.986144 0.995192 0.225157 0.575044 0.225157 0.225157 0.575044 0.575044 1.468645 0.225157 1.468645 0.995192 0.575044 0.57507844 0.995192 0.575044 1.468645 0.575044 0.575044 0.995192 1.468645 0.995192 0.995192 0.225157 0.575044
Illustration of the results of table (3-2): 1. In the (21) days the Failure Function of neglecting traffic rules cause equals to (0.318447) when the Survival Function equals to (0.681553), and the Hazard Function in the same day is (0.225157).
46
Chapter Three
Application and Data description
2. In the (28) day the Failure Function of neglecting traffic rules cause equals to (0.789161) when the Survival Function equals to equals to (0.210839), and the Hazard Function in the same day is (0.995192).
3. In the (35) day the Failure Function of neglecting traffic rules cause equals to (0.789161) when the Survival Function equals to (0.210839), and the Hazard Function in the same day is (0.995192).
33. In the (294) day the Failure Function of neglecting traffic rules cause equals to (0.539992) when the Survival Function equals to (0.460008), and the Hazard Function in the same day is (0.575044).
3.4.2.3 Figures of Failure Density Function, Failure Distribution Function, Survival Function and Hazard Function for Neglecting Traffic Rules Cause: Figures of the failure density function, failure distribution function, survival function and hazard function of the two-parameter Weibull distribution of neglecting traffic rules cause:
40
Chapter Three
Application and Data description
Figure (3-5): Failure Density Function for neglecting cause
Figure (3-6): Failure Distribution Function for neglecting cause
44
Chapter Three
Application and Data description
Figure (3-7): Survival Function for neglecting cause
Figure (3-8): Hazard Function for neglecting cause
43
Chapter Three
Application and Data description
3.4.3.1 Goodness of fit test for Changing Sides Cause: By using Kolmogorov-Smirnov test can be find suitable distribution of the variables of changing sides cause as:
Kolmogorov-Smirnov test D p-value alpha
0.280 0.070 0.05
Test interpretation: H0 : The sample follows a Weibull (2) distribution H1 : The sample does not follow a Weibull (2) distribution As the computed p-value is greater than the significance level alpha=0.05, one cannot reject the null hypothesis H0.
3.4.3.2 Failure Function Estimation of the Traffic Accidents for Changing Sides case: We can find the estimated value of the shape parameter parameter
and the scale
of the Weibull distribution by using Maximum Likelihood
Estimation method of changing sides cause through Synthesis software as below: ̂
̂
By substituting these estimated values of
̂ and
̂ in (2.8) we can find
Failure Function estimation for changing sides causes as:
43
Chapter Three
Application and Data description
̂
̂
̂
̂
̂
̂
The Survival Function estimation of changing sides cause is obtained by substituting these estimated values of ̂ and ̂ in (2.9) as: ̂
̂
̂
̂
̂
48
Chapter Three
Application and Data description
̂
Also, the Hazard Function estimation of changing sides cause is obtained by substituting these estimated values of ̂ and ̂ in (2.10) as: ̂
̂ ̂
̂
̂
̂
̂
These values are showed in the table below:
Table (3-3): Estimated value of Failure Function, Survival Function and Hazard Function of changing sides cause as: t 1 2 1 1 1 2
̂ 0.632822 0.779877 0.632822 0.632822 0.632822 0.779877
̂ 0.409426 0.60413 0.409426 0.409426 0.409426 0.60413
̂ 0.367178 0.220123 0.367178 0.367178 0.367178 0.220123 43
Chapter Three 1 1 1 1 3 5 1 1 1 1 4 2 3 4 6 2
0.632822 0.632822 0.632822 0.632822 0.888883 0.981228 0.632822 0.632822 0.632822 0.632822 0.951371 0.779877 0.888883 0.951371 0.993532 0.779877
Application and Data description 0.367178 0.367178 0.367178 0.367178 0.111117 0.018772 0.367178 0.367178 0.367178 0.367178 0.048629 0.220123 0.111117 0.048629 0.006468 0.220123
0.409426 0.409426 0.409426 0.409426 0.758513 1.010363 0.409426 0.409426 0.409426 0.409426 0.891428 0.60413 0.758513 0.891428 1.119228 0.60413
Illustration of the results of table (3-3): 1. In the (35) day the Failure Function of the changing sides cause equals to (0.632822) when the Survival Function equals to (0.367178), and the Hazard Function in the same day is (0.409426). 2. In the (42) day the Failure Function of the changing sides cause equals to (0.779877) when the Survival Function equals to (0.220123), and the Hazard Function in the same day is (0.60413).
3. In the (70) day the failure Function of the changing sides cause equals to (0.632822) when the Survival Function equals to (0.367178), and the Hazard Function in the same day is (0.409426).
34
Chapter Three
Application and Data description
21. In the (301) day the Failure Function of the changing sides cause equals to (0.779877) when the survival function equals to (0.220123), and the Hazard Function in the same day is (0.60413).
3.4.3.3 Figures of Failure Density Function, Failure Distribution Function, Survival Function and Hazard Function for Changing Sides Cause:
Figures of the Failure Density Function, Failure Distribution Function, Survival Function and Hazard Function of the two-parameter Weibull distribution of changing sides cause:
Figure (3-9): Failure Density Function for changing sides
34
Chapter Three
Application and Data description
Figure (3-10): Failure Distribution Function for changing sides
Figure (3-11): Survival Function for changing sides
34
Chapter Three
Application and Data description
Figure (3-12): Hazard Function for changing sides
3.4.4.1 Goodness of fit test for Carelessness cause: By using Kolmogorov-Smirnov test can be find suitable distribution of the variables of carelessness cause as:
Kolmogorov-Smirnov test D p-value alpha
0.233 0.102 0.05
Test interpretation: H0 : The sample follows a Weibull (2) distribution H1 : The sample does not follow a Weibull (2) distribution As the computed p-value is greater than the significance level alpha=0.05, one cannot reject the null hypothesis H0.
36
Chapter Three
Application and Data description
3.4.4.2 Failure Function Estimation of the Traffic Accidents for Carelessness case: We can find the estimated value of the shape parameter parameter
and the scale
of the Weibull distribution by using Maximum Likelihood
Estimation method of carelessness cause through Synthesis software as below: ̂
̂
By substituting these estimated values of
̂ and
̂ in (2.8) we can find
Failure Function estimation for carelessness causes as: ̂
̂
̂
̂
̂
̂
The Survival Function for carelessness cause is obtained by substituting these estimated values of ̂ and ̂ in (2.9) as: ̂
̂
̂
30
Chapter Three
Application and Data description
̂
̂
̂
Also, the Hazard Function for carelessness cause is obtained by substituting these estimated values of ̂ and ̂ in (2.10) as: ̂
̂ ̂
̂
̂
̂
̂
34
Chapter Three
Application and Data description
These values are showed in table below:
Table (3-4): Estimated value of Failure Function, Survival Function and Hazard Function of carelessness cause: t 1 3 2 1 4 1 1 2 5 1 3 3 4 1 1 4 3 4 3 1 4 1 2 3 1 2 4
̂ 0.463066 0.816699 0.64095 0.463066 0.928599 0.463066 0.463066 0.64095 0.978787 0.463066 0.816699 0.816699 0.928599 0.463066 0.463066 0.928599 0.816699 0.928599 0.816699 0.463066 0.928599 0.463066 0.64095 0.816699 0.463066 0.64095 0.928599
̂ 0.267714 0.8075 0.537259 0.267714 1.078194 0.267714 0.267714 0.537259 1.349224 0.267714 0.8075 0.8075 1.078194 0.267714 0.267714 1.078194 0.8075 1.078194 0.8075 0.267714 1.078194 0.267714 0.537259 0.8075 0.267714 0.537259 1.078194
̂ 0.536934 0.183301 0.35905 0.536934 0.071401 0.536934 0.536934 0.35905 0.071401 0.536934 0.183301 0.183301 0.071401 0.536934 0.536934 0.071401 0.183301 0.071401 0.183301 0.536934 0.071401 0.536934 0.35905 0.183301 0.536934 0.35905 0.071401
Illustration of the results of table (3-4): 1. In the (49) day the Failure Function of carelessness cause equals to (0.463066) when the Survival Function equals to (0.536934), and the Hazard Function in the same day is (0.267714). 33
Chapter Three
Application and Data description
2. In the (56) day the Failure Function of carelessness cause equals to (0.816699) when the survival function equals to (0.183301), and the hazard function in the same day is (0.8075).
3. In the (70) day the Failure Function of carelessness cause equals to (0.64095) when the survival function equals to (0.35905), and the Hazard Function in the same day is (0.537259).
27. In the (280) day the failure Function of carelessness cause equals to (0.928599) when the Survival Function equals to (0.071401), and the Hazard Function in the same day is (1.078194).
3.4.4.3 Figures of Failure Density Function, Failure Distribution Function,
Survival
Function
and
Hazard
Function
for
Carelessness cause:
Figures of the Failure Density Function, Failure Distribution Function, Survival Function and Hazard Function of the two-parameter Weibull distribution of carelessness cause:
33
Chapter Three
Application and Data description
Figure (3-13): Failure Density Function carelessness
Figure (3-14): Failure Distribution Function carelessness
38
Chapter Three
Application and Data description
Figure (3-15): Survival Function carelessness
Figure (3-16): Hazard Function carelessness
33
Chapter Three
Application and Data description
3.5 Failure Function Estimation for Multiple Causes: Estimate the Failure Distribution Function of two, three and four causes together respectively: By substituting the expected values of the parameters ̂ and ̂ of speed cause and neglecting traffic rules cause together in (2-40):
̂
and the results are shown in table (3-5) as:
Table (3-5): Estimated value of Failure Function of speed and neglecting traffic rules cause together as: None 1 None 3 None 3 1 None 4 2 1 1 3 2 3 1 3 4 6 5 5 3 5 None 1 1 5 2 3 1 3 None 2 1
̂ 0.083263 0.22863 0.22863 0.083263 0.615893 0.136828 0.483575 0.349161 0.732944 0.969617 0.826859 0.483575 0.136828 0.732944 0.349161 0.22863 0.22863 34
Chapter Three 5 4 2 5 2 3 5 1 4 5 6 6 7 8 6 5 12 7 1 2 5 1 1 3
2 2 4 1 4 3 2 2 3 2 None 4 None 2 None 2 None 3 4 3 3 1 2 None
Application and Data description 0.732944 0.615893 0.615893 0.615893 0.615893 0.615893 0.732944 0.22863 0.732944 0.732944 0.615893 0.941489 0.732944 0.941489 0.615893 0.732944 0.9854 0.941489 0.483575 0.483575 0.826859 0.136828 0.22863 0.22863
Illustration of the results of table (3-5): 1. In the (21) day the Failure Function of speed and neglecting traffic rules cause together equals to (0.083263). 2. In the (28) day the Failure Function of speed and neglecting traffic rules cause together equals to (0.22863). 3. In the (35) day the Failure Function of speed and neglecting traffic rules cause together equals to (0.22863).
34
Chapter Three
Application and Data description
41. In the (301) day the Failure Function of speed and neglecting traffic rules cause together equals to (0.22863).
By substituting the expected values of the parameters ̂ and ̂ of speed cause and changing sides cause together in (2-40) as:
̂
and the results are shown in table (3-6) as:
Table (3-6): Estimated value of Failure Function of speed and changing sides cause together: None 1 4 1 3 3 3 6 5 5 1 5 3 3 2 5 4
1 2 None None None 1 None None 1 None None None None 2 None 1 None
̂ 0.164888 0.381594 0.502037 0.164888 0.381594 0.502037 0.381594 0.71081 0.71081 0.613866 0.164888 0.613866 0.381594 0.613866 0.263575 0.71081 0.502037 34
Chapter Three 2 5 2 3 5 1 4 5 6 6 7 8 6 5 12 7 1 2 5 1 1 2
None None 1 1 1 None None 3 5 1 1 1 1 4 2 3 4 None 6 None None 3
Application and Data description 0.263575 0.613866 0.381594 0.502037 0.71081 0.164888 0.502037 0.852608 0.956835 0.790345 0.852608 0.89937 0.790345 0.89937 0.99001 0.933197 0.613866 0.263575 0.956835 0.164888 0.164888 0.613866
Illustration of the results of table (3-6): 1. In the (35) day the Failure Function of speed and changing sides cause together equals to (0.164888). 2. In the (42) day the Failure Function of speed and changing sides cause together equals to (0.381594). 3. In the (49) day the Failure Function of speed and changing sides cause together equals to (0.502037).
39. In the (301) day the Failure Function of speed and changing sides cause together equals to (0.613866).
36
Chapter Three
Application and Data description
By substituting the expected values of the parameters ̂ and ̂ of speed cause and carelessness cause together in (2-40) as:
̂
and the results are shown in table (3-7) as: Table (3-7): Estimated value of Failure Function of speed and carelessness cause together as: 1 4 1 3 3 3 6 5 5 1 5 3 3 2 5 4 2 5 2 3 5 1 4 5 6 6 7
None 1 3 None 2 1 None 4 1 None None None 1 2 5 None 1 3 3 4 1 1 4 3 4 3 1
̂ 0.16391 0.54193 0.43248 0.32522 0.54193 0.43248 0.64468 0.86802 0.64468 0.16391 0.54193 0.32522 0.43248 0.43248 0.91186 0.43248 0.32522 0.80943 0.54193 0.73483 0.64468 0.23147 0.80943 0.80943 0.91186 0.86802 0.80943 30
Chapter Three 8 6 5 12 7 1 2 5 1 1 3
4 None 1 2 3 1 2 4 None None None
Application and Data description 0.96467 0.64468 0.64468 0.98768 0.91186 0.23147 0.43248 0.86802 0.16391 0.16391 0.32522
Illustration of the results of table (3-7): 1. In the (42) day the Failure Function of speed and carelessness cause together equals to (0.16391). 2. In the (49) day the Failure Function of speed and carelessness cause together equals to (0.54193). 3. In the (56) day the Failure Function of speed and carelessness cause together equals to (0.43248).
38. In the (301) day the Failure Function of speed and carelessness cause together equals to (0.32522).
By substituting the expected values of the parameters
̂
and
̂
neglecting traffic rules cause and changing sides cause together in (2-40) as:
̂
34
of
Chapter Three
Application and Data description
and the results are shown in table (3-8) as: Table (3-8): Estimated value of Failure Function of neglecting traffic rules and changing sides causes together as: 1 3 3 None 2 1 2 1 4 5 3 1 2 1 None 1 2 2 4 1 4 3 2 2 3 2 None 4 None 2 None 2 None 3 4 3
None None 1 2 None None None 1 None None 1 None None None 2 None 1 None None None 1 1 1 None None 3 5 1 1 1 1 4 2 3 4 None
̂ 0.194917 0.584153 0.745624 0.386569 0.386569 0.194917 0.386569 0.386569 0.745624 0.858352 0.745624 0.194917 0.386569 0.194917 0.386569 0.194917 0.584153 0.386569 0.745624 0.194917 0.858352 0.745624 0.584153 0.386569 0.584153 0.858352 0.858352 0.858352 0.194917 0.584153 0.194917 0.927748 0.386569 0.927748 0.985285 0.584153 33
Chapter Three 3 6 1 None 2 None None 2
Application and Data description 0.994084 0.194917 0.386569 0.386569
Illustration of the results of table (3-8): 1. In the (21) day the Failure Function of neglecting traffic rules and changing sides cause together equals to (0.194917). 2. In the (28) day the Failure Function of neglecting traffic rules and changing sides cause together equals to (0.584153). 3. In the (35) day the Failure Function of neglecting traffic rules and changing sides cause together equals to (0.745624).
41. In the (301) day the Failure Function of neglecting traffic rules and changing sides cause together equals to (0.386569). By substituting the expected values of the parameters ̂ and ̂ of neglecting traffic rules cause and carelessness cause together in (2-40) as:
̂
and the results are shown in table (3-9) as:
33
Chapter Three
Application and Data description
Table (3-9): Estimated value of Failure Function of neglecting traffic rules and carelessness cause together as: 1 3 3 2 1 2 1 4 5 3 None 1 2 1 None 1 2 2 4 1 4 3 2 2 3 2 None 4 None 2 2 None 3 4 3 3 1 2
None None None 1 3 None 2 1 None 4 1 None None None 1 2 5 None 1 3 3 4 1 1 4 3 4 3 1 4 1 2 3 1 2 4 None None
̂ 0.189789 0.481859 0.481859 0.481859 0.642032 0.3187 0.481859 0.774674 0.774674 0.931903 0.189789 0.189789 0.3187 0.189789 0.189789 0.481859 0.931903 0.3187 0.774674 0.642032 0.931903 0.931903 0.481859 0.481859 0.931903 0.774674 0.642032 0.931903 0.189789 0.870486 0.481859 0.3187 0.870486 0.774674 0.774674 0.931903 0.189789 0.3187 38
Chapter Three
Application and Data description
Illustration of the results of table (3-9): 1. In the (21) day the Failure Function of neglecting traffic rules and carelessness cause together equals to (0.189789). 2. In the (28) day the Failure Function of neglecting traffic rules and carelessness cause together equals to (0.481859). 3. In the (35) day the Failure Function of neglecting traffic rules and carelessness cause together equals to (0.481859).
41. In the (294) day the Failure Function of neglecting traffic rules and carelessness cause together equals to (0.3187). By substituting the expected values of the parameters ̂ and ̂ of changing sides and carelessness cause together in (2-40) as:
̂
and the results are shown in table (3-10) as: Table (3-10): Estimated value of Failure Function of changing sides and carelessness cause together as: 1 None 2 None None 1 None 3 1 2 None 1
̂ 0.367238 0.494183 0.367238 0.628629 0.628629 0.367238 33
Chapter Three 1 4 None 1 2 1 None 2 1 5 None 1 None 3 1 3 1 4 1 1 None 1 None 4 3 3 5 4 1 3 1 1 1 4 1 None 4 1 2 2 3 3 4 1 None 2 6 4 2 None
Application and Data description 0.837754 0.367238 0.628629 0.494183 0.902207 0.367238 0.628629 0.746425 0.837754 0.494183 0.367238 0.746425 0.902207 0.984487 0.746425 0.494183 0.837754 0.367238 0.837754 0.746425 0.902207 0.837754 0.494183 0.992392 0.494183
Illustration of the results of table (3-10): 1. In the (35) day the Failure Function of changing sides and carelessness cause together equals to (0.367238). 2. In the (42) day the Failure Function of changing sides and carelessness cause together equals to (0.494183). 3. In the (49) day the Failure Function of changing sides and carelessness cause together equals to (0.367238).
84
Chapter Three
Application and Data description
31. In the (301) day the Failure Function of changing sides and carelessness cause together equals to (0.494183).
By substituting the expected value of the parameters ̂ and ̂ of speed cause, neglecting and changing sides cause together in (2-41):
̂
and the results are shown in table (3-11) as: Table (3-11): Estimated value of Failure Function of speed, neglecting traffic rules and changing sides cause together as: None None None 1 4 1 3 3 3 6 5 5 1 5 3 3
1 3 3 None 2 1 2 1 4 5 3 None 1 2 1 None
None None 1 2 None None None 1 None None 1 None None None None 2
̂ 0.081473 0.193112 0.283826 0.193112 0.496813 0.123756 0.387961 0.387961 0.60229 0.896487 0.779765 0.387961 0.123756 0.60229 0.283826 0.387961 84
Chapter Three 2 5 4 2 5 2 3 5 1 4 5 6 6 7 8 6 5 12 7 1 2 5 1 1 3
1 2 2 4 1 4 3 2 2 3 2 None 4 None 2 None 2 None 3 4 3 3 1 2 None
Application and Data description None 1 None None None 1 1 1 None None 3 5 1 1 1 1 4 2 3 4 None 6 None None 2
0.193112 0.697985 0.496813 0.496813 0.496813 0.60229 0.60229 0.697985 0.193112 0.60229 0.845852 0.896487 0.896487 0.697985 0.896487 0.60229 0.896487 0.975645 0.95884 0.779765 0.387961 0.975645 0.123756 0.193112 0.387961
Illustration of the results of table (3-11): 1. In the (21) days the Failure Function of speed, neglecting traffic rules and changing sides cause together equals to (0.081473). 2. In the (28) days the Failure Function of speed, neglecting traffic rules and changing sides cause together equals to (0.193112). 3. In the (35) days the Failure Function of speed, neglecting traffic rules and changing sides cause together equals to (0.283826).
84
Chapter Three
Application and Data description
31. In the (301) days the Failure Function of speed, neglecting traffic rules and changing sides cause together equals to (0.387961).
By substituting the expected value of the parameters ̂ and ̂ of speed, changing sides and carelessness cause together in (2-41):
̂
and the results are shown in table (3-12) as: Table (3-12): Estimated value of Failure Function of speed, changing sides and carelessness cause together as: None 1 4 1 3 3 3 6 5 5 1 5 3 3
1 2 None None None 1 None None 1 None None None None 2
None None 1 3 None 2 1 None 4 1 None None None 1
̂ 0.142787 0.284408 0.466326 0.374128 0.284408 0.555592 0.374128 0.555592 0.827478 0.555592 0.142787 0.466326 0.284408 0.555592 86
Chapter Three 2 5 4 2 5 2 3 5 1 4 5 6 6 7 8 6 5 12 7 1 2 5 1 1 3
None 1 None None None 1 1 1 None None 3 5 1 1 1 1 4 2 3 4 None 6 None None 2
Application and Data description 2 5 None 1 3 3 4 1 1 4 3 4 3 1 4 None 1 2 3 1 2 4 None None None
0.374128 0.870387 0.374128 0.284408 0.711527 0.555592 0.711527 0.638114 0.204275 0.711527 0.870387 0.965528 0.827478 0.774703 0.930763 0.638114 0.827478 0.976287 0.930763 0.555592 0.374128 0.965528 0.142787 0.142787 0.466326
Illustration of the results of table (3-12): 1. In the (35) day the Failure Function of speed, changing sides and carelessness cause together equals to (0.142787). 2. In the (42) day the Failure Function of speed, changing sides and carelessness cause together equals to (0.284408). 3. In the (49) day the Failure Function of speed, changing sides and carelessness cause together equals to (0.466326).
80
Chapter Three
Application and Data description
31. In the (301) day the Failure Function of speed, changing sides and carelessness cause together equals to (0.466326).
By substituting the expected value of the parameters ̂ and ̂ of neglecting traffic rules, changing sides and carelessness cause together in (2-41):
̂
and the results are shown in table (3-13) as: Table (3-13): Estimated value of Failure Distribution Function of neglecting traffic rules, changing sides and carelessness cause together as: 1 3 3 None 2 1 2 1 4 5 3 None 1 2 1 None
None None 1 2 None None None 1 None None 1 None None None None 2
None None None None 1 3 None 2 1 None 4 1 None None None 1
̂ 0.132423 0.387884 0.522384 0.251074 0.387884 0.522384 0.251074 0.522384 0.64245 0.64245 0.879353 0.132423 0.132423 0.251074 0.132423 0.387884 84
Chapter Three 1 2 2 4 1 4 3 2 2 3 2 None 4 None 2 None 2 None 3 4 3 3 1 2 None
None 1 None None None 1 1 1 None None 3 5 1 1 1 1 4 2 3 4 None 6 None None 2
Application and Data description 2 5 None 1 3 3 4 1 1 4 3 4 3 1 4 None 1 2 3 1 2 4 None None None
0.387884 0.879353 0.251074 0.64245 0.522384 0.879353 0.879353 0.522384 0.387884 0.820715 0.879353 0.921338 0.879353 0.251074 0.820715 0.132423 0.820715 0.522384 0.921338 0.921338 0.64245 0.989369 0.132423 0.251074 0.251074
Illustration of the results of table (3-13): 1. In the (21) day the Failure Function of neglecting traffic rules, changing sides and carelessness cause together equals to (0.132423). 2. In the (28) day the Failure Function of neglecting traffic rules, changing sides and carelessness cause together equals to (0.387884). 3. In the (35) day the Failure Function of neglecting traffic rules, changing sides and carelessness cause together equals to (0.522384).
83
Chapter Three
Application and Data description
41. In the (301) day the Failure Function of neglecting traffic rules, changing sides and carelessness cause together equals to (0.251074).
By substituting the expected value of the parameters ̂ and ̂ of speed cause, neglecting traffic rules, changing sides and carelessness cause together in (2-41):
̂
and the results are shown in table (3-14) as: Table (3-14): Estimated value of Failure Distribution Function of speed, neglecting traffic rules, changing sides and carelessness cause together as: None 1 None None None 3 None None None 3 1 None 1 None 2 None 4 2 None 1 1 1 None 3 3 2 None None 3 1 1 2 3 4 None 1 6 5 None None 5 3 1 4 5 None None 1 1 1 None None 5 2 None None 3 1 None None 3 None 2 1
̂ 0.079233 0.161316 0.22653 0.161316 0.470257 0.30266 0.30266 0.470257 0.55358 0.766374 0.864271 0.385331 0.111089 0.470257 0.22653 0.385331 83
Chapter Three 2 5 4 2 5 2 3 5 1 4 5 6 6 7 8 6 5 12 7 1 2 5 1 1 3
1 2 2 4 1 4 3 2 2 3 2 None 4 None 2 None 2 None 3 4 3 3 1 2 None
Application and Data description None 1 None None None 1 1 1 None None 3 5 1 1 1 1 4 2 3 4 None 6 None None 2
2 5 None 1 3 3 4 1 1 4 3 4 3 1 4 None 1 2 3 1 2 4 None None None
0.30266 0.864271 0.385331 0.470257 0.632126 0.703549 0.766374 0.632126 0.22653 0.766374 0.864271 0.927845 0.899935 0.632126 0.927845 0.470257 0.819937 0.949109 0.949109 0.703549 0.470257 0.976309 0.111089 0.161316 0.30266
Illustration of the results of table (3-14): 1. In the (21) day the Failure Function of speed, neglecting traffic rules, changing sides and carelessness cause together equals to (0.079233). 2. In the (28) day the Failure Function of speed, neglecting traffic rules, changing sides and carelessness cause together equals to (0.161316). 3. In the (35) day the Failure Function of speed, neglecting traffic rules, changing sides and carelessness cause together equals to (0.22653).
88
Chapter Three
Application and Data description
41. In the (301) day the Failure Function of speed, neglecting traffic rules, changing sides and carelessness cause together equals to (0.30266).
3.6 Goodness of fit test of Multinomial Distribution: By using Chi-Square test can be find the suitable distribution of the number of traffic accidents of four causes with hypothesis as below:
the variables follow the Multinomial distribution the variables does not follow the Multinomial distribution
Chi-Square test Deg. of freedom alpha Critical value Statistic
3 0.05 0.352 0.0019
Since the critical value (0.352) greater than statistic (0.0019), that means can be accept the hypothesis
.
3.7 Failure Rate of the traffic accidents of each cause: We have the number of traffic accidents of each cause as:
Let
represents the number of traffic accident for speed cause. represents the number of traffic accident for neglecting cause. represents the number of traffic accident for changing sides cause. 83
Chapter Three
Application and Data description
represents the number of traffic accident for carelessness cause.
By substituting these values of (
in (2-48), (2-49), (2-50) and (2-
51) We can get the failure rate of each cause:
̂
̂
̂
∑
∑
∑
and
34
Chapter Three
̂
Application and Data description
∑
These results are showed in the table below: Table (3-15): Failure rate of each cause Observation Failure rate
151 0.445
79 0.233
44 0.130
65 0.192
Illustration the results of the table (3-15): 1. The Failure rate of speed cause equals to (44.5) present. 2. The failure rate of neglecting traffic rules cause equals to (23.3) present. 3. The failure rate of changing sides cause equals to (13) present. 4. The failure rate of carelessness cause equals to (19.2) present.
34
Chapter four
Conclusions & Recommendations
4.1 Results and Conclusions: As the result of the practical work the following results and conclusions are: 1. The results of Failure Function, Survival Function and Hazard Function for Speed cause are; in the (42) days the Failure Function of speed cause equals to (0.124974) when the Survival Function equals to (0.804728), and the Hazard Function in the same day is (0.195272) and so on for the th
other days until (301) day the Failure Function of speed cause equals to (0.290878)
when the Survival Function equals to (0.527343) and
the Hazard Function in the same day is (0.472657).
2. Then for the neglecting traffic rules cause the Failure Function, Survival Function and Hazard Function are; in the (21) days the Failure Function for neglecting traffic rules cause equals to (0.318447) when the Survival Function equals to (0.681553), and the Hazard Function in the same day th
is (0.225157) and so on for the other days until (294) day the Failure Function of neglecting traffic rules cause equals to (0.539992) when the Survival Function equals to (0.460008) and the Hazard Function in the same day is (0.575044). 3. Also, the results of Failure Function, Survival Function and Hazard Function for changing sides cause are; in the (35) days the Failure Function of the changing sides cause equals to (0.632822) when the Survival Function equals to (0.367178) and the Hazard Function in the th
same day is (0.409426) and so on for the other days until (301) day the Failure Function for changing sides cause equals to (0.779877) when the 92
Chapter four
Conclusions & Recommendations
Survival Function equals to (0.220123) and the Hazard Function in the same day is (0.60413).
4. The results of Failure Function, Survival Function and Hazard Function for carelessness cause are; in the (49) days the Failure Function of carelessness cause equals to (0.463066) when the Survival Function equals to (0.536934) and the Hazard Function in the same day is (0.267714) and so on for the other days until (280)
th
day the Failure
Function for carelessness cause equals to (0.928599) when the Survival Function equals to (0.071401) and the Hazard Function in the same day is (1.078194).
5. Then for speed and neglecting traffic rules causes together the result of Failure Function in the (21) days equals to (0.083263) and so on for the other days until (301)
th
day the Failure Function for the two causes
together equals to (0.22863).
6. For speed and changing sides cause together the result of Failure Function in the (35) days equals to (0.164888) and so on for the other days until (301)
th
day the Failure Function for the two causes together
equals to (0.613866).
7. Also, for speed and carelessness cause together the result of the Failure Function in the (42) days equals to (0.16391) and so on for the other 93
Chapter four
days until (301)
Conclusions & Recommendations th
day the Failure Function of the two causes together
equals to (0.32522).
8. Then for neglecting traffic rules and changing sides cause together the result of Failure Function in the (21) days equals to (0.194917) and so on th
for the other days until (301) day the Failure Function for these causes together equals to (0.386569).
9. For neglecting traffic rules and carelessness cause together the result of Failure Function in (21) days equals to (0.189789) and so on for the other days until (294)
th
day the Failure Function for these causes
together equals to (0.3187).
10. Also, for changing sides and carelessness cause together the results of Failure Function in the (35) days equals to (0.367238) and so on for the th
other days until (301)
day the Failure Function for these causes
together equals to (0.494183).
11. For speed, neglecting traffic rules and changing sides cause together the results of Failure Function in the (21) days equals to (0.081473) and so on for the other days until (301)
th
causes together equals to (0.387961).
94
day the Failure Function for these
Chapter four
Conclusions & Recommendations
12. Also, for speed, changing sides and carelessness cause together the results of Failure Function in the (35) days equals to (0.142787) and so th
on for the other days until (301) day the Failure Distribution Function for these causes together equals to (0.466326).
13. Then for neglecting traffic rules, changing sides and carelessness causes together the results of Failure Function in the (21) days equals to (0.132423), and so on for the other days until (301)
th
day the Failure
Distribution Function for these causes together equals to (0.251074).
14. For speed, neglecting traffic rules, changing sides and carelessness cause together the results of Failure Function in the (21) days equals to (0.079233) and so on for the other days until (301)
th
the Failure
Distribution Function for these causes together equals to (0.30266).
15. The failure rate of speed cause equals to (44.5) present, the failure rate of neglecting cause equals to (23.3) present, the failure rate of changing sides cause equals to (13) present and the failure rate of carelessness cause equals to (19.2) present.
16. Speed cause has a great effect on the traffic accident and changing sides cause has the least effect on the traffic accidents.
95
Chapter four
Conclusions & Recommendations
4.2 Recommendations: Based on the results of the study the following points are recommended:
1. In consequence of the thesis we come to the conclusion that the speed cause has a great effect on traffic accidents, thus government must put more traffic control cameras in the highways and put traffic signs and traffic lights in all the roads of the city.
2. The government should provide the population with detailed guidance and awareness of the traffic rules which can be studied in education programs.
3. The similar study of failure due to multiple causes to be done for heart attaches, cancer disease, pollution, psychological problems and bank failures.
96
References
References First: Arabic References
.1االحد ,عطاف ادوارعبد " ,)2002( ,تمذَراث الوعىلُت للتىزَع األسٍ بوعلوتُن– دراست همارنت–" رسالت هاجستيرـ كليت اإلدارة واالقتصاد /جاهعت بغداد. .2الشوريً ,جاة عبدالجبار رجب " ,)2002( ,استخذام الوحاكاة فٍ همارنت همذراث التملص لوعلوت الشكل لتىزَع وَبل لبُاناث الورالبت" ,اطروحت دكتىراٍ -كليت اإلدارة واالقتصاد /جاهعت بغداد. .3الهاللي ,فراس صدام عبد" ,)2002( ,همارنت طرائك تمذَر هعالن أنوىرج وَبل للفشل بثالثت هعالن" ,رسالت هاجستير ـ كليت اإلدارة واالقتصاد /جاهعت بغداد. .4الياسري ,تهاًي ههدي عباس" ,)2002( ,همارنت همذراث بُس الحصُن هع همذراث أخري لتمذَر دالت الوعىلُت التمرَبُت لتىزَع وَبل" ,اطروحت دكتىراٍ -كليت اإلدارة واالقتصاد /جاهعت بغداد.
.5رحين ,ئاالى غفىر " ,)2011( ,تمذَر دالت الفشل لتىزَع وَبل فٍ حالت تعذد أسباب الفشل الكلىٌ لورضً الوستشفً الجوهىرٌ فٍ أربُل /العراق" ,رسالت هاجستيرـ كليت اإلدارة واالقتصاد /جاهعت صالح الديي/اربيل.
.6هحود ,زيٌب عبدهللا" ,)2011( ,هعىلُت نظام التىالٍ باستخذام التىزَع الوختلط وتمذَر دالت الوعىلُت للنظام هع اجراء الوحاكاة لتىزَع وَبل رٌ الوعلوتُن" ,رسالت هاجستيرـ كليت اإلدارة واالقتصاد /جاهعت صالح الديي/اربيل.
97
References
ٌ "همارنت طرائك تمذَر هعلواث ودالت هعىلُت تىزَع كاها ر,)2002( , اواث سردار, وادي.7 رسالت هاجستيرـ كليت اإلدارة,"الوعلوتُن فٍ حالت البُاناث الوفمىدة باستخذام الوحاكاة . جاهعت بغداد/واالقتصاد
Second: Foreign References a. Papers:
8. Canavos G.C. & Tsokos C.P., (1973), (Bayesian Estimation of Life Parameters in the Weibull Distribution), Operation Research Society, Vol.(21), No. (2), PP. (755-763).
9. Diane I.G. & Lonnie C.V., (1981), (A simulation Study of Estimators for the 2 – Parameter Weibull Distribution), IEEE Transaction on Reliability, Vol. (R-30), No.(I).
10. Guure, Chris B., Ibrahim N. A., and Ahmad, Al O. M.), (2012), “Bayesian Estimation of Two-Parameter Weibull Distribution Using Extension of Jeffrey’s Prior Information with Three Loss Functions”, Hindawi Publishing Corporation Mathematical Problems in Engineering doi:10.1155/2012/589640. 11. Prakash G., (2012), “Bayes Estimation for a Mixture of the Weibull Distributions”, International Journal of Mathematics and Scientific Computing, VOL. 2, NO. 1
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References
12. Mhamad, Aras J., (2011), “Using Logistic Regression Model in Traffic Problem in Sulaimani”, Master thesis/ College of Administration & Economics- Sulaimani.
13. Pandey B.N., Dwivedi N. & Bandyopadhyay P., (2011), “Comparison between Bayesian and Maximum Likelihood Estimation of Scale Parameter In Weibull Distribution with known Shape Under Linex Loss Function”,
Journal of Scientific Research\Banaras Hindu
University, Varanasi, Vol. 55, 2: 163-172.
14. Sinha S.K. & Sloan J.A., (1988), (Bayes Estimation of the Parameter & Reliability Function of the 3-Parameters Weibull Distribution), IEEE Transaction on Reliability, Vol.(37), No.(4), PP. (364 - 368). 15. Tamayo, Adrian M., (2007), “Occurrence of Traffic Accidents in the Philippines: An Application of Poisson Regression Analysis”, http://ssrn.com/abstract=1438478. 16. Yahgmaei F. , Babanezhad M. & Moghadam O.S., (2013), “Bayesian Estimation of the Scale Parameter of Inverse Weibull Distribution under the Asymmetric Loss Functions”, Journal of Probability and Statistics Volume 2013 (2013), Article ID 890914.
99
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b. Books
17. Dhillon, B. C., (1999), “Design Reliability: Fundamentals and Application”, CRC Press LLC, London, UK. 18. Grosh, Doris Lloyd, (1989), “A Primer of Reliability Theory”, John Wiley & Sons, Inc. Canada.
19. Hashimoto, Koichi, (2003),"Review of Quality and Reliability Handbook”, NEC Electronics Corporation, Japan.
20. Hamada M. S., Wilson, A.G., Reese C. Sh. & Martz H.F., (2008), “Bayesian Reliability”, Springer, Science & Business Media, LLC, New York, USA. 21. Hogg R. V. & Craig A. T., (1978), “Introduction to Mathematical Statistics”, Fourth Edition, Macmillan Publishing Co., Inc. NEW YORK 22. Kececioglu D.B., (2002), “Reliability Engineering handbook”, Vol. (1), DEStech publications, Inc., USA. 23. Kumar, U D., Crocker, J., Chitra T. & Saranga, H., (2006), “Reliability and Six Sigma”, Springer Science & Business Media, Inc. USA. 24. Murthy D.N.P., (2004), “Weibull Models”, John Wiley & Sons.INC. Canada.
100
References
25. Ridon, Steven E. & Basu, Asit P., (2000), “Statistical Methods for the Reliability of Repairable System”, John Wiley & Sons, Inc. Canada. 26. Smith, Charls O., (1976), "Introduction to reliability in Design", McGraw –Hill Kogakusha, Ltd. Tokyo, Japan.
27. Sinha, S.K. (1986), "Reliability and Life Time", Wiley Estern Ltd. 28.Todinov, M.T., (2005), “Reliability and Risk Models”, John Wiley & Sons
Ltd, England.
101
Appendix
Appendix (A) Theoretical part 1. To find the Survival function through the Hazard function:
h(t )
f (t ) S (t )
from equation (2-6) in theoretical part
Rt f ( s)ds
;t 0
t
d d Rt f ( s)ds dt dt t
f t
d Rt dt
h(t )
R(t ) R(t )
by taking the integration of both sides, we get: t
t
0
0
hs ds
Rt ds Rt
ln Rs |
t 0
ln Rt ln R0 ln Rt By using the above relationship we can write the Survival function in general as:
102
Appendix
t R(t ) exp h( s)ds 0 2. Some properties of the following distribution:[23][26][30]
a. Weibull distribution: Table (A-1): properties of two-parameter Weibull distribution
Properties
Formula 1
1 Mean
1 1 E T 1
2 Variance
1 Var T
3 Median
ln 2 Me
4 Mode
1 x
2
2 2 1 1 1
1
5 Moment Generating Function
M x t n 0
103
1
tn n!
n 1 n
Appendix
b. Multinomial Distribution: Table (A-3): Properties of the Multinomial Distribution:
Properties
Formula
1 Mean
E Y nj
2 Variance
Var Y nj (1 j )
3 Covariance
Cov(Yj , Yk ) njk
4 Moment Generating Function
k
M y t ( je tj ) n j 1
104
املمـــخص
ٖتي تعسٖف احلٕادث املسٔزٖ ٛعىٕوا باضي تصادً املسنبات ٔاملصا ،ٚأٔ وع نأ ،َٟوَ شأٌّا أُ تؤدٙ إىل الٕفأ ٚالعحص األضساز يف املىتمهات .حٕادث الطسم ِ٘ املصهم ٛاملتٍاوٗ ٛيف مجٗع أحنا ٞالعامل ،مما أدٝ إىل ٔفا ٚحٕال٘ ( )1ومُٕٗ حالٔ ٛفأ ٚأنجس وَ ( )32ومُٕٗ إصاب ٛضٍٕٖأ .قد شّد عدد وتصاٖد وَ حٕادث الطسم نىؤشس لضػف الٍصاط االقتصادٌٔ .ٙطب ٛوعظي حٕادث الطسم اليت تطببت يف وٕت الٍاس ِي أٌفطّي ٖصازنُٕ يف املكاً األٔه ،وجن :الطاٟكني أٔ ِؤال ٞالضحاٖا إىل الطسم املعٗبٔ ،ٛعالوات الطسٖل غري الكٗاضٗٔ ٛإِىاه الطاٟكني. لكد أصبح وٕضٕع املعٕلٗ ٛأنجس اضتخداوا يف االحباث ٔالدزاضات ٌظسا لمدٔز املّي الرٖ ٙمعبْ نعمي يف التعاون وع خمتمف التطبٗكات الٍىٕذجٗ ٛضٕا ٞعٍصس املعدات أٔ الهاٍٟات .اُ نال وَ ٌظسٖ ٛاملعٕلٗٛ ٌٔظسٖ ٛالبكا ٞعم ٜقٗد احلٗاِ .ٚرٓ الٍظسٖتاُ ختتمفاُ يف أُ ٌظسٖ ٛاملعٕلٗ ٛوٍاضب ٛلمىعدات ٔالٍظسٖ ٛالبكا ِٕ ٞأنجس والٟىا لمها َٟاحل٘. يف املىازض ٛاحلكٗكٌٗ ٛادزا وا حيدث أُ الفصن ِٕ ٌتٗح ٛلطبب ٔاحد ،يف وعظي احلاالت ٍِاك أضباب نجس ٚلمفصن .يف ِرٓ األطسٔح ،ٛتعترب أضباب وتعدد ٚوَ الفصن اليت ِ٘ األزبعٛاضباب (الطسع،ٛ جتاِن قٕاعد املسٔز ،تػري املطاز ٔإِىاه الطاٟكني) ٔفكا لمصسٔط التالٗ ِ٘ ،ٛجيب اُ تهُٕ ٔقت الفصن لهن ضبب وعسٔف تٕشٖعْ ،عٍدوا حيدث الفصن أ ٙحلظ ٛوَ الصوَ ٖهُٕ لطبب ٔاحد فكط. يف ِرٓ الدزاض ٛوعمى ٛالصهن ٔالكٗاس لتٕشٖع ٖٔبن مت تكدٖسِىا بأضتخداً طسٖك ٛاالوهاُ االعظي ٔوَ خالهلىا مت تكدٖس نن وَ دال ٛالفصن البكا ٔ ٞاخلطٕز ٚلهن ضبب لٕحدٓ ٔوعا عم ٜالتٕال٘ .أٖضا ٖطتخدً طسٖك ٛاالوهاُ االعظي لمحصٕه عم ٜوعده الفصن وَ نن ضبب عم ٜالٍحٕ التال٘ :وعده الفصن يف الطسعٖ ٛطأ )..44( ٙو ،ٖٕٛ٠فإُ وعده الفصن بطبب جتاِن قٕاعد املسٔزٖٖ ٛطأٙ ( )3242و ،ٖٕٛ٠فإُ وعده فصن بطبب تػري املطاز ٖطأ )12( ٙؤ ٖٕٛ٠وعده الفصن بطبب اإلِىاه الطاٟكني ٖطأ )1.43( ٙؤِ .ٖٕٛ٠را ٖعين أُ الطسع ٛلْ تأثري نبري عم ٜاحلٕادث املسٔزٖٔ ٛبطبب تػري املطاز لدّٖا أقن تأثري عم ٜاحلٕادث املسٔزٖ.ٛ A
ثوخـتة زِوداوى ياتوضؤ بةشيَوةيةكى باو ثيَهاضة ئةكسيَت بة ثيَكدادانى ئؤتؤوبين و ثيادة زِةو ياى بةزكةوتو بة يةز شتيَكدا كة ئةجناوةكةى وسدى ,كةم ئةندام بووى ياى لةدةضتدانى تايبةمتةندى ية وسؤيى يةكاى بيََت .زِوداوةكانى َنة ئةبيَتة يؤى وسدنى نصيكةى ( )1وميؤى ياتووضؤ لة زِيَطاو بانةكاندا كيَصةيةكى جيًانى طةشةضةندووة ,كة ضاال كةس و بسيهداز بوونى شياتس لة ( )32وميؤى كةس .بيَطوواى شيادبوونى زِيَرةى ثيَكداداى بة ئاواذةيةك دائةنسىَ بؤ جيَبةجىَ كسدنيَكى الواشى ضاالكى ية ئابووزي يةكاى .شؤزيهةى زِوداوةكاى لة زِيَطاو بانةكاى كة بونةتة يؤى وسدى ,ئةو كةضانةى كة بةشيَوةيةكى ضةزةتايى خؤياى بةشدازى لة ثسِؤضةكةدا وةك شؤفيَسةكاى وة قوزبانى يةكاى ئةدزيَهة ثاهَ كةوتةزخةوى شؤفيَساى و خساثى و ويَسانى زِيَطاو بانةكاى و يةزوةيا ضتاندازد نةبوونى ييَىاكانى ياتووضؤ. َيهةوة تيايدا شؤز بابةتى تواناى وانةوة ( )Reliabilityئيًطتا بووةتة بوازيَك كة ئةجناودانى باس و ليَكؤل َيهةوةى تايبةتى جياواش َة طسنطةى كة ئةيطيَسِيَت وةك شانطتيَك بؤ ضةنديو ليَكؤل شيادى كسدووة ئةوةش بةيؤى ئةو زِؤل بؤ يةزيةك لة ثيَكًاتةى ئاويَسةكاى و شيهدةوةزاى ,يةزيةك لة بريدؤشى ( )Reliabilityو ( )Survivalجياواشى َيهةوة لة تواناى وانةوةى ثيَكًاتةى ئاويَسةكاى لة كاتيَكدا لةوةدا كة بريدؤشى ( )Reliabilityطوجناوة بؤ ليَكؤل َيهةوة لة تواناى وانةوةى شيهدةوةزاى. بريدؤشى ( )Survivalطوجناوة بؤ ليَكؤل َكو لة شؤزبةى لة ذيانى كسدازيدا كةم زِووئةدات كة شكطت ييَهاى تةنًا بةيؤى يةك يؤكازةوة بيَت ,بةل َةتةكاندا شياتس لة يةك يؤكاز يةية بؤ شكطت ييَهاى .لةم تويَريهةوةيةدا كاز كساوة لةضةز فسة يؤكازى شكطت حال ييَهاى كة بسيتني لة ضواز يؤكاز وةك( :خيَسايى ,طويَهةداى بة ياضاكانى ياتووضؤ ,كؤزِيهى ضايدةكاى و كةوتةزخةوى شؤفيَساى) بة ثيَى ئةم وةزجانة يةكةم دابةشكساوةى كات-بؤ-شكطت ييَهانى يةز يؤكازيَك شانساو بيَت ,دووةم لة ضاتى شكطت ييَهاندا ,ئةو شكطتة تةنًا بة يؤى يةك يؤكازةوة بيَت.
A
َيهةوةيةدا ثازاويتةزةكانى ( )shapeو ( )scaleى دابةشكساوةى ويبن ( )Weibullخةوَميَهساوى لةم ليَكؤل َندنى تواناى بةزش ( )Maximum Likelihood Estimationبؤ دؤشيهةوةى بة بةكاز ييَهانى زِيَطاى خةوال نةخصةى شكطت ييَهاى ( ,)Failure Functionنةخصةى وانةوة ( )Survival Functionو نةخصةى وةتسضى ( )Hazard Functionبؤ يةزيةك لة يؤكازةكاى بة جياو بؤ تةواوى يؤكازةكاى ثيَكةوة يةك لةدواى َندنى تواناى بةزش ئةطةزى زِوودانى زِووداوى ياتوضؤ بؤ يةز يؤكازيَك يةك .يةزوةيا بة بةكاز ييَهانى زِيَطاى خةوال َيََهساوة بةم شيَوةية :ئةطةزى زِوودانى زِووداوى ياتوضؤ بة يؤى خيَسايى يةوة ئةكاتة ( ,)..44%ئةطةزى زِوودانى خةوم زِووداوى ياتوضؤ بةيؤى طويَهةداى بة ياضاكانى ياتووضؤوة بسيتى ية لة ( ,)%3242ئةطةزى زِوودانى زِووداوى ياتوضؤ بة يؤى طؤزِيهى ضايدةكاى كةم ئةبيَتةوة بؤ ( )12%لةكاتيَكدا ئةطةزى زِوودانى زِووداوى ياتوضؤ بة يؤى كةوتةزخةوى شؤفيَسانةوة بةزش ئةبيَتةوة بؤ ( .)1.43%ئةوةش ئةوة ئةطةيةنيَت كة يؤكازى خيَسا ليَخوزيهى شؤفيَساى طةوزةتسيو كازيطةزى يةية لة ضةز زِوودانى زِووداوى ياتووضؤ لةكاتيَكدا يؤكازى طؤزِيهى ضايدةكاى كةورتيو كازيطةزى يةية.
B
