Data Analysis and Related Applications 4
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This book is a collective work by a number of leading scientists, analysts, engineers, mathematicians and statisticians who have been working at the forefront of data analysis and related applications, arising from data science, operations research, engineering, machine learning or statistics. The chapters of this collaborative work represent a cross-section of current research interests in the above scientific areas. The collected material has been divided into appropriate sections to provide the reader with both theoretical and applied information on data analysis methods, models and techniques, along with appropriate applications.
Data Analysis and Related Applications 4 investigates a number of different topics in the areas mentioned above, touching on statistical analysis, stochastic processes, estimation methods, algorithms, distributions and networks, among others.
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Yiannis Dimotikalis is Assistant Professor of Quantitative Methods in the Department of Management Science and Technology at Hellenic Mediterranean University, Greece.
Christos H. Skiadas was the Founder and Director of Data Analysis and Forecasting and Former Vice-Rector at the Technical University of Crete, Greece. He is the Chair of the Applied Stochastic Models and Data Analysis conference series.
Content
Chapter 1. On the First-Passage Area of a One-Dimensional Diffusion Process with Stochastic Resetting 1
Mario ABUNDO
1.1. Formulation of the problem and general results 1
1.2. Brownian motion with resetting 5
1.3. Drifted Brownian motion with resetting 15
1.4. References 17
Chapter 2. Statistical Analysis of Groundwater Level in Slovakia 19
Dominika SONAK BALLOVA, Jana KALICKA and Michaela CERVENANSKA
2.1. Introduction 19
2.2. Data and methods 20
2.3. Results 25
2.4. Conclusion 27
2.5. Acknowledgment 28
2.6. References 28
Chapter 3. Stochastic Processes Associated with Fully Nonlinear Parabolic Equations Arising in Financial Mathematics 29
Yana BELOPOLSKAYA and Andrey CHUBATOV
3.1. Semilinear and fully nonlinear PDEs 29
3.2. BSDE, FBSDE and deep learning algorithms 34
3.3. Acknowledgments 42
3.4. References 42
Chapter 4. An Improved Shape Parameter Estimation Method for the Pareto Model 45
Frederico CAEIRO and Ayana MATEUS
4.1. Introduction 45
4.2. Estimators under study 47
4.3. An algorithm for selection of the control parameter of the LGPWM shape parameter estimator 50
4.4. Numerical results 51
4.5. Conclusion 54
4.6. Acknowledgments 55
4.7. References 55
Chapter 5. BSDE-¿ Scheme for the Heston Model: Valuation of American Options 57
Marko DIMITROV, Abigail BERTA, Achref BACHOUCH, Christian EWALD and Ying NI
5.1. Background 57
5.2. BSDE numerical schemes 59
5.3. Numerical experimental studies: valuation of American options 68
5.4. Conclusion and future work. 72
5.5. References 72
Chapter 6. Age-replacement Policy for Series Systems Under Parameter Uncertainty in Lifetime Distribution 75
Kentaro FUJIOKA, Ying NI and Lu JIN
6.1. Introduction 75
6.2.The model 78
6.3. Optimization 81
6.4. Numerical example 86
6.5. Conclusion 89
6.6. References 90
Chapter 7. New Bicluster Algorithm for Trading 93
Gloria GHENO
7.1. Introduction 93
7.2. Fuzzy logic and trading rules 98
7.3. Sentiment analysis, trading indicators and fuzzy rules 104
7.4. Conclusion 106
7.5. References 107
Chapter 8. A Flexible Generalization of the Latent Dirichlet Allocation 109
Alice GIAMPINO, Roberto ASCARI and Sonia MIGLIORATI
8.1. Introduction 109
8.2. Distributions on the simplex 111
8.3. Latent topic models 113
8.4. Collapsed Gibbs sampling 114
8.5. Simulation study 117
8.6. References 122
Chapter 9. Extreme Value Parameters Estimation: An Overview 123
Dora Prata GOMES and M. Manuela NEVES
9.1. Introduction and overview of extreme value theory 123
9.2. Some parameters of interest in EVT 126
9.3. EVI and EI estimation 127
9.4. Extreme quantile estimation 130
9.5. Application to the daily mean flow discharge in river Tejo 132
9.6. Conclusion and work in progress 135
9.7. Acknowledgments 135
9.8. References 135
Chapter 10. Some Properties on Optimal Maintenance Policies for k-out-of-n:G Systems Considering Imperfect Repair with Controllable Repair Levels 139
Sota IKENOYA and Lu JIN
10.1. Introduction 139
10.2. System description 141
10.2.1. Deterioration state 141
10.2.2. Maintenance actions 142
10.3. Total expected discounted cost 142
10.4. Optimization of maintenance policy 144
10.5. Numerical studies 149
10.6. Conclusion 154
10.7. References 155
Chapter 11. Stochastic Orders and Reliability Properties for the Deficit at Ruin and Bounds for the Laplace Transform of a Compound Geometric Distribution 157
Lazaros KANELLOPOULOS and Konstantinos POLITIS
11.1. Introduction 157
11.2. Model description and results 159
11.3. Bounds for the LT of the maximal aggregate loss 162
11.4. Examples 163
11.5. References 165
Chapter 12. A New Family of Continuous Univariate Distributions with Applications in Actuarial Science 167
Markos V. KOUTRAS and Spiros D. DAFNIS
12.1. Introduction 167
12.2. Definitions and notations 169
12.3. Probability bounds 173
12.4. Aging properties and unimodality 176
12.5. Tail behavior of Dg+(h) 180
12.6. Conclusion 181
12.7. Acknowledgment 181
12.8. References 181
Chapter 13. Simple Form of Probability Density Functions via Sampling 183
Maria LEDAKI and Myrto PAPAGEORGIOU
13.1. Introduction 183
13.2. The sense and the method 184
13.3. Using sampling data 187
13.4. Results and discussion 191
13.5. References 192
Chapter 14. Optimizing Financial Trading Strategies Using Dynamic Bayesian Networks 193
Karl LEWIS, Mark Anthony CARUANA and David Paul SUDA
14.1. Introduction 193
14.2. Theoretical framework 194arameters 201
14.3. Methodology of analysis and results 203
14.4. Conclusion 206
14.5. References 207
Chapter 15. Quantitative Methods for Analysing the Risk and Timing of Bankruptcy of Small and Medium Enterprises 209
Francesca PIERRI and Chrys CARONI
15.1.Introduction 209
15.2. Approaches to statistical modeling 210
15.3. Imbalanced data 211
15.4. Competing risks 214
15.5. Conclusion 218
15.6. References 219
Chapter 16. Network of Adaptive Frequency Oscillators in a Ballistic, Non-Gaussian, Noisy Environment 223
Julio RODRIGUEZ
16.1. Introduction 223
16.2. Dynamics of the network 224
16.3. Analyzing the dynamics 226
16.4. Numerical simulations 231
16.5. Discussion and perspectives 233
16.6. Appendices 234
16.7. References 249
Chapter 17. Penalised Regression Adaptations of the Longstaff-Schwartz Algorithm for Pricing American Options 251
David Paul SUDA, Monique BORG INGUANEZ and Lara CILIA
17.1. Introduction 252
17.2. The Longstaff-Schwartz algorithm and proposed extensions 253
17.3. Stochastic processes in finance and relevant theoretical considerations 255
17.4. Simulation design 257
17.5. Results 259
17.6. Conclusion 263
17.7. References 264
Chapter 18. International Auditing Standards and Their Contribution to the Limitation of Accounting Fraud 267
Efthalia TABOURATZI and Leonidas FANOURAKIS
18.1. Introduction 267
18.2. Literature review 269
18.3. Empirical analysis 277
18.4. Conclusion 292
18.5. References 292
Chapter 19. Equivariant Robust Estimators for Moment Condition Models 295
Aida TOMA, Amor KEZIOU, Luiza BADIN and Silvia DEDU
19.1. Introduction 295
19.2. Robust estimators for moment condition models 297
19.3. Equivariance of robust minimum empirical divergence estimators 301
19.4. Acknowledgments 305
19.5. References 305
Chapter 20. Continuous Increasing Probability Density Functions: An Approach Through Sampling 309
Maria TSIFOUTIDOU and Nikolaos FARMAKIS
20.1. Introduction 309
20.2. Theoretical approach 310
20.3. Examples for illustration 312
20.4. Results and discussion 315
20.5. References 315
Chapter 21. The Importance of the Initial Selection of Suppliers in the Food Service Divisions of Hotels, the Current Situation in the Supply Chain of Greece 317
Konstantinos VASILAKAKIS, Efthalia TABOURATZI and Despoina SDRALI
21.1. Introduction 317
21.2. Literature review 319
21.3. Benefits of supply chain management 320
21.4. Research methodology 321
21.5. Data analysis 322
21.6. Discussion. 331
21.7. Conclusion 332
21.8. References 332
Chapter 22. Compliance with IUU Fisheries of Manila Clams in the Tagus Estuary 337
Margarida XAVIER, Ana LORGA DA SILVA and Paula CHAINHO
22.1. Introduction 338
22.2. Research methodology 341
22.3. Analysis and interpretation of results 343
22.4. Conclusion 350
22.5. Acknowledgments 351
22.6. References 351
Chapter 23. The Expectation of a Mixed Moving Average Process Subject to Ambiguous Lévy Basis 355
Hidekazu YOSHIOKA and Yumi YOSHIOKA
23.1. Introduction 355
23.2. supOU process 356
23.3. Optimization problems 358
23.4. Application 363
23.5. Conclusion 367
23.6. Acknowledgments 368
23.7. References 368
List of Authors 371
Index 377
1
On the First-Passage Area of a One-Dimensional Diffusion Process with Stochastic Resetting
For a one-dimensional diffusion process with stochastic resetting , obtained from an underlying diffusion X(t), we study the statistical properties of its first-passage time through zero, when starting from x > 0, and its first-passage area, i.e. the random area swept out by until its first-passage time through zero. By making use of solutions of certain associated ODEs, we find explicit expressions for the Laplace transforms of the first-passage time and the first-passage area, and their single and joint moments.
1.1. Formulation of the problem and general results
This note deals with the first-passage area (FPA) of a diffusion process with stochastic resetting. It is a continuation of Abundo (2013, 2023b), Abundo and Del Vescovo (2017) and Abundo and Furia (2019), regarding the FPA of jump-diffusions, drifted Brownian motion, Le?vy process and Ornstein-Uhlenbeck process. In fact, here we considered a one-dimensional diffusion process in the presence of stochastic resetting , obtained from an underlying diffusion X(t); this kind of process is treated, for example, in den Hollander et al. (2019) and Evans et al. (2020) (see also the references in Abundo 2023a). We studied the statistical properties of the first-passage time (FPT) through zero of , starting from x > 0, and its FPA, namely the random area swept out by until its FPT through zero. In some cases, we explicitly obtained the Laplace transform of the FPT and FPA, and their single and joint moments. Moreover, we provided the distribution of the maximum displacement of .
In the case that the underlying diffusion X(t) is Brownian motion without drift, the FPA was already studied in Singh and Pal (2022), although the results found therein were obtained using special functions. In contrast, here we used nothing but elementary functions: our arguments were based on classical results for one-dimensional diffusions. In fact, the study of the distributions of the FPT and FPA was carried out via solutions of certain associated ODEs. We focused on the case when the underlying diffusion X(t) is a Wiener process, i.e. a Brownian motion with or without drift, but the results can be extended to other processes.
The FPT and FPA of a diffusion process with stochastic resetting have important applications in many areas, for example, in biology, in the ambit of stochastic models for the activity of a neuron subject to resetting (see, for example, Nobile et al. 1985 and the references contained therein). Other important applications are found in queuing theory, where the first hitting time to zero can be identified with the busy period, i.e. the first instant at which the queue is empty, and the FPA is the total waiting time of the "customers" during a busy period (see, for example, Kearney 2004).
Now, we precisely describe the process with stochastic resetting.
We consider a one-dimensional temporally homogeneous diffusion process X(t) driven by the SDE:
[1.1]and starting from the position X(0) = x > 0, where the drift µ(·) and diffusion coefficient s(·) are regular enough functions, such that there exists a unique strong solution of the SDE [1.1] for a given starting point, and Bt is the standard Brownian motion. We also assume that the FPT of the diffusion X(t) below zero is finite with probability one.
We assume that resetting events can occur according to a homogeneous Poisson process with rate r > 0. Until the first resetting event , the process coincides with X(t), and it evolves according to [1.1] with . When the reset occurs, is set instantly to a position xR > 0. After that, evolves again according to [1.1], starting afresh (independently of the past history) from xR, until the next resetting event occurs, etc. The inter-resetting times turn out to be independent and exponentially distributed random variables with parameter r. In other words, in any time interval (t, t + ?t), with ?t 0+, the process can pass from to the position xR with probability r?t + o(?t), or it can continue its evolution according to [1.1] with probability 1-r?t+o(?t). The process so obtained is called diffusion with stochastic resetting. For any C2 function f(x), its infinitesimal generator is given by (see, for example, Abundo 2013):
[1.2]where is the "diffusion part" of the generator, i.e. regarding the diffusion process X(t).
For an initial position x > 0, we are concerned with the FPT of through zero, namely:
[1.3]and the corresponding FPA
[1.4]which is the area enclosed between the time axis and the path of the process up to the FPT through zero. We assume that both t(x) and A(x) are finite with probability one, for any x > 0. Note that
[1.5]where tX(x) is the first-hitting time to zero of X(t) starting from x > 0 and s is an exponentially distributed random variable with parameter r > 0.
In fact, we limit ourselves to study the case when the underlying process X(t) is a Wiener process, i.e. Brownian motion with or without drift.
The main qualitative difference between the FPT of the process and the FPT of the underlying diffusion X(t) is that, for the process , the moments of the FPT are finite, while for the second one, they may be infinite. This is, for example, the case of Brownian motion starting from x > 0, where, as is well known, the first-hitting time to zero is finite with probability 1, but it has infinite expectation.
For ? > 0, let us consider the Laplace transform (LT) of , U(x) = ax + b (a, b = 0) being a polynomial of degree one, i.e.
[1.6]Taking U(x) = 1, we obtain the LT of the FPT t(x), while for U(x) = x, we get the LT of the FPA A(x). The following holds (see Abundo 2023a):
PROPOSITION 1.1.-
The LT M?(x) of satisfies the differential problem:
[1.7]where L denotes the infinitesimal generator of the underlying diffusion X(t), which is given, for any C2 function f, by
[1.8]and f' and f" denote the first and second derivatives of f.
REMARK 1.1.-
Proposition 1.1 was already proved in Singh and Pal (2022) in the case when X(t) is Brownian motion. Note that, for r = 0 (i.e. when no resetting is allowed), we obtain equation (2.12) of Abundo (2013), provided that the jump part in the infinitesimal generator is set to zero, while the second boundary condition is M?(+8) = 0.
If the n-th order moment of exists finite, it is provided by:
[1.9]By calculating the n-th derivative with respect to ?, at ? = 0, of both members of [1.7], we obtain that, setting T0(x) = 1, the n-th order moments Tn(x) satisfy the ODEs:
[1.10]with the constraint Tn(0) = 0 and the addition of an appropriate further condition (indeed, we need two conditions to obtain the unique solution of [1.10]). Note that for r = 0, [1.10] becomes equation (2.19) of Abundo (2013). In particular, for U(x) = 1, [1.10] is nothing but the celebrated Darling and Siegert's equation (1953) for the moments of the FPT of a diffusion without resetting.
As regards the joint moments of t(x) and A(x), we consider the joint LT of t(x) and A(x), i.e. :
[1.11]As easily seen, we get:
[1.12]and
[1.13]Applying the same reasoning as before and taking U(x) = ?1 + ?2x, we obtain that solves the problem
[1.14]Then, by applying and calculating it for ?1 = ?2 = 0, we obtain that V(x) ? E[t(x)A(x)] is the solution of the problem:
[1.15]with a suitable additional condition.
Note that for r = 0, [1.15] becomes the analogous equation, respectively, obtained in Abundo and Del...
