Assessing the failure time of systems with interconnected components is a significant problem in reliability theory and leads to important questions in mathematical statistics and probability modelling. This PhD thesis aims to develop research in this field by proposing a stochastic model for evaluating the expected time of failure of a system from a rational expectations perspective. In particular, our aim is to address a reliability problem by exploiting the characteristics of k-out-of-n systems (with homogeneous and heterogeneous components) through rational expectations. Accordingly, the proposed approach consists of two indispensable frameworks: the framework for reliability theory with special focus on k-out-of-n systems and the framework for rational expectations. There are two different approaches to dealing with the study of reliability systems: a probabilistic approach by analyzing the probability distribution of a system's failure time and a Bayesian computational approach by estimating the average failure time of a system, conditioned by the description of a scenario in which the evolution of the given reliability system is observed. There are many studies that approach this problem through the probability distribution, as well as many researchers have addressed reliability problems using the Bayesian approach. Our research belongs to the second group of publications. In fact, we will estimate the average failure time of systems, where the failures depend on the number and importance of the components. We deal with both kinds of system, composed of homogeneous or heterogeneous components, considering different initial distributions of the components relevance in order to find out which distribution is better in this context and if we are consistent with the existing literature. The reliability of the systems is assessed by conditioning the results to the available information recorded over time. We want to explore how the efficient use of information collected over time on the dynamic evolution of component weights can affect the improvement of failure time predictions of our stochastic systems. This is an innovative study with original results as there are no similar contributions in the existing literature. The method is inspired by the one proposed by Andersen and Sornette (2005, 2006). However, unlike them, we insert an interaction into our systems in such a way that the failure of a component affects the rest of the still active system. The systems are compared on the basis of some synthetic measures (variance, kurtosis, asymmetry, Gini coefficient and Shannon entropy) calculated on the realizations of the components weights at each time (configurations). The weights are extracted from five different initial distributions: uniform in (0,1), beta with α = 1 and β = 3, beta with α = β = 0.5, beta with α = β = 2, beta with α = 1 and β = 0.5. These are dynamic values as the relevance of the components changes every time one of the components fails. The “reallocation rule” is applied. The reallocation rule works according to preference relations among the components. Once one of the components fails, its relevance is reallocated to the remaining active components. We seek to provide the most reliable estimate of the failure time of a stochastic system, whose components are interconnected through rational expectations. We present systems whose information on statistical indicators of configurations and failure times are contained in a set (information set). To make the predictions, we compare the information obtained on real systems (in-vivo systems), conditioning them to that of the information set. The result is the implementation of a scenario analysis of the errors obtained from this comparison which takes into account two fundamental aspects: the statistical measurements of the components weights and the time. We first illustrate a theoretical framework followed by two different computational models based on numerical simulations with different focuses and different results. The two models provide theoretical validation to the theoretical model and are strictly dependent on time and statistical indicators. In the first model we make a transversal analysis over time: we emphasize the contribution of the statistical indicator, showing how the error varies as a function of the measure considered aggregated over time. This is an individual analysis of the various indicators and how the prediction effectiveness changes depending on the initial weight distribution identified. Instead, in the second model, we emphasize the importance of the time factor on different scenarios of the statistical indicators based on the percentiles we condition them on. Here, we first measure the error trend for each indicator, and then we proceed with a comparative analysis to investigate which statistical indicator performs better also in relation to the initial weight distribution chosen. These models are two complementary studies that show two different approaches, focusing first on statistical indicators and then on time. The general goal is the implementation of a procedure that can be useful for any system with interactive components. In this thesis we propose a theoretical setting followed by two different frameworks of application that both exploit rational expectations in the context of the reliability of systems with interconnected components. All the results presented encourage the application of rational expectations to our simulated data. In both models we compare our research field with the existing literature and can thus evaluate the robustness of our results, the proposed methods, and the computational efficiency of our algorithms. We examine how the paths of the errors are influenced by several indicators, the different levels of these indicators, and initial weight distributions used in the study. In both models we successfully demonstrate that it is possible to obtain a prediction of failure times of stochastic systems which improves for certain combinations of indicators and initial distributions and with the passage of time. In future research further implementations of the model will have to be developed by increasing the complexity of the simulation procedure. The problem can be addressed in other rational expectations models and should certainly be attempted. It is our intention to change the rules and assumptions proposed in this thesis, and to compare new results with those already collected. We would like to extend the analysis by adding other statistical measures (Frosini index, Pearson index or other entropies). We can also consider a distance measure to check which combination of parameters will allow us to get the lowest error. One intention there is to study specifically how the evolution of the distribution of the components can affect our results. Again, with regard to this field, a network could be built on the connection existing between the components by considering of the probability with which they subsequently fail. We can also integrate cluster analysis by clustering our predictors over different time windows using a dendrogram to show in which periods the prediction was most effective. This thesis could have real practical relevance in economics and finance. It is also our intention to apply the simulation procedure to real data to model systems with interconnected components. Numerical experiments in the economic and financial field will follow. There are therefore many future research ideas that we hope to be able to deepen soon.
Attraverso la predizione del tempo di fallimento di sistemi stocastici con componenti interconnesse possiamo trarre interessanti conclusioni nell’ambito della teoria dell’affidabilità. In questa tesi di dottorato vogliamo approfondire questo filone di ricerca proponendo un modello stocastico per valutare il tempo di fallimento atteso di sistemi stocastici sotto una prospettiva di aspettative razionali. Il nostro obiettivo è quello di esplorare l’affidabilità dei sistemi k-out-of-n con componenti eterogenee e omogenee attraverso le aspettative razionali. Sono quindi due i framework sui quali si baserà questo lavoro di ricerca: la teoria dell’affidabilità con focus sui sistemi k-out-of-n e le aspettative razionali. I due principali approcci alla teoria dell’affidabilità possono essere così distinti: un approccio probabilistico che si concentra sulla distribuzione di probabilità dei tempi di fallimento dei sistemi e un approccio di tipo Bayesiano computazionale che stima il tempo medio di fallimento di un sistema condizionandolo a diverse caratteristiche. Sono moltissimi gli studi che si approcciano a questo problema attraverso la distribuzione di probabilità, così come moltissimi ricercatori hanno affrontato problemi di affidabilità sfruttando l’approccio Bayesiano. La nostra ricerca si colloca in quest’ultima area di studio. Infatti andremo a stimare il tempo medio di fallimento di sistemi il cui fallimento è direttamente dipendente dal numero e dalla rilevanza delle componenti. Ci occupiamo sia di sistemi con componenti omogenee che eterogenee, considerando diverse distribuzioni inziali della rilevanza delle componenti stesse con lo scopo di scoprire quale distribuzione funziona meglio in questo contesto. L’affidabilità dei sistemi oggetto di studio è valutata condizionando i risultati alle informazioni disponibili registrate con il passare del tempo e quindi nel contesto delle aspettative razionali. Vogliamo esplorare come l’utilizzo efficiente delle informazioni collezionate nel tempo sull’evoluzione dinamica dei pesi delle componenti dei sistemi, possa influenzare il miglioramento delle previsioni dei tempi di fallimento dei nostri sistemi stocastici. Si tratta di uno studio innovativo con risultati originali in quanto non esistono contributi simili nella letteratura esistente. La nostra nuova metodologia di previsione è ispirata a quella proposta da Andersen e Sornette (2005, 2006). A differenza loro però proponiamo un’interazione nel tempo tra le componenti che influenza la composizione e il funzionamento dell’intero sistema. I sistemi sono confrontati sulla base di alcune misure sintetiche (la varianza, la curtosi, l’asimmetria, il coefficiente di Gini e l’entropia di Shannon) calcolate sulle realizzazioni dei pesi delle componenti ad ogni tempo (configurazioni). I pesi sono estratti da cinque diverse distribuzioni iniziali: uniforme in (0,1), beta con α=1 e β=3, beta con α=β=0.5, beta con α=β=2, beta con α=1 e β=0.5. Si tratta di valori dinamici in quanto la rilevanza delle componenti cambia ogni volta che ne fallisce una. Si applica infatti la “regola di riallocazione” che ci permette di avere sistemi dinamici con componenti interattive: il peso della componente che fallisce viene riallocato in maniera proporzionale sui pesi delle componenti ancora attive all’interno del sistema. Le aspettative razionali ci permetteranno quindi di calcolare il valore atteso dei tempi di fallimento sotto il vincolo degli indicatori statistici delle configurazioni che variano nel tempo. Presentiamo dei sistemi le cui le informazioni sulle misure statistiche delle configurazioni e sui tempi di fallimento sono catalogate in un set (set informativo). Per realizzare le previsioni mettiamo a confronto le informazioni ottenute sui sistemi reali (sistemi in-vivo) condizionandole a quelle del set informativo. Il risultato è l’implementazione di un’analisi di scenario degli errori ottenuti da questo confronto che tiene conto di due aspetti fondamentali: le misure statistiche dei pesi delle componenti e il tempo. Illustriamo prima di tutto un framework teorico seguito da due diversi modelli computazionali basati su simulazioni numeriche con diversi focus e risultati differenti. I due modelli forniscono validazione teorica al modello teorico e dipendono strettamente dal tempo e dagli indicatori statistici. Nel primo modello viene enfatizzato il ruolo dei vari indicatori statistici con un’analisi trasversale nel tempo. La nostra intenzione è quella di predire i tempi residui di fallimento dei sistemi stocastici andando a studiare gli errori che commettiamo in corrispondenza dei diversi livelli delle misure statistiche oggetto di analisi. Si tratta di un’analisi individuale dei vari indicatori e come cambia l’efficacia di predizione a seconda della distribuzione iniziale dei pesi individuata. Nel secondo modello ci focalizziamo sul ruolo del tempo attraverso tre diversi condizionamenti. Esploriamo in questo caso le aspettative razionali per studiare il comportamento rispetto al tempo degli errori di predizione condizionati a percentili differenti delle distribuzioni degli indicatori statistici. Misuriamo in questo contesto in una prima analisi l’andamento degli errori per ogni indicatore, e poi procediamo con una comparazione tra le varie analisi per investigare quale indicatore statistico performa meglio anche in relazione alla distribuzione iniziale dei pesi scelta. Si tratta di due studi complementari mostrati con due distinti approcci computazionali. L’obiettivo è quello di implementare delle procedure che posso essere sfruttate per verificare l’affidabilità di qualsiasi sistema con componenti interconnesse. In questa tesi quindi proponiamo un modello teorico per descrivere le aspettative razionali in un contesto di teoria dell’affidabilità, validandolo attraverso due diversi approcci. I risultati presentati incoraggiano l’uso delle aspettative razionali nei modelli di predizione. In entrambi i modelli proposti riusciamo a fornire predizioni molto accurate e a stabilire quali indicatori statistici sono più adatti a seconda delle circostanze oggetto di analisi. Esaminiamo nei due approcci presentati come i percorsi degli errori sono influenzati dall’indicatore utilizzato, dai valori assunti dall’indicatore stesso e dalle distribuzioni iniziali dei pesi delle componenti. Dimostriamo quindi che è possibile ottenere la predizione dei tempi di fallimento di sistemi stocastici. Sono molte le ricerche future che abbiamo intenzione di proseguire. È nostra intenzione cambiare le regole e le assunzioni proposte in questa tesi, e confrontare nuovi risultati con quelli già collezionati. Vorremo estendere l’analisi aggiungendo altre misure statistiche (indice di Frosini, indice di Pearson oppure altre entropie). Vorremo approfondire il modo in cui l’evoluzione delle distribuzioni dei pesi delle componenti influenza i risultati ottenuti oppure sarebbe interessante costruire un network sulle connessioni esistenti tra le componenti in considerazione della probabilità con la quale falliscono. Questa tesi di dottorato può avere una reale rilevanza in un contesto economico-finanziario nel caso di modelli per previsioni basate sulle informazioni disponibili o nell’analisi del rischio sistemico. La nostra intenzione è quella di applicare i nostri modelli a dati reali fornendo esperimenti numerici nel campo economico-finanziario. In generale è adattabile ad ogni sistema con componenti interconnesse. Sono quindi molti i futuri spunti di ricerca che speriamo di riuscire presto ad approfondire.
Reliability stochastic systems and rational expectations / Riccioni, Jessica. - ELETTRONICO. - (2021).
Reliability stochastic systems and rational expectations
RICCIONI JESSICA
2021-01-01
Abstract
Assessing the failure time of systems with interconnected components is a significant problem in reliability theory and leads to important questions in mathematical statistics and probability modelling. This PhD thesis aims to develop research in this field by proposing a stochastic model for evaluating the expected time of failure of a system from a rational expectations perspective. In particular, our aim is to address a reliability problem by exploiting the characteristics of k-out-of-n systems (with homogeneous and heterogeneous components) through rational expectations. Accordingly, the proposed approach consists of two indispensable frameworks: the framework for reliability theory with special focus on k-out-of-n systems and the framework for rational expectations. There are two different approaches to dealing with the study of reliability systems: a probabilistic approach by analyzing the probability distribution of a system's failure time and a Bayesian computational approach by estimating the average failure time of a system, conditioned by the description of a scenario in which the evolution of the given reliability system is observed. There are many studies that approach this problem through the probability distribution, as well as many researchers have addressed reliability problems using the Bayesian approach. Our research belongs to the second group of publications. In fact, we will estimate the average failure time of systems, where the failures depend on the number and importance of the components. We deal with both kinds of system, composed of homogeneous or heterogeneous components, considering different initial distributions of the components relevance in order to find out which distribution is better in this context and if we are consistent with the existing literature. The reliability of the systems is assessed by conditioning the results to the available information recorded over time. We want to explore how the efficient use of information collected over time on the dynamic evolution of component weights can affect the improvement of failure time predictions of our stochastic systems. This is an innovative study with original results as there are no similar contributions in the existing literature. The method is inspired by the one proposed by Andersen and Sornette (2005, 2006). However, unlike them, we insert an interaction into our systems in such a way that the failure of a component affects the rest of the still active system. The systems are compared on the basis of some synthetic measures (variance, kurtosis, asymmetry, Gini coefficient and Shannon entropy) calculated on the realizations of the components weights at each time (configurations). The weights are extracted from five different initial distributions: uniform in (0,1), beta with α = 1 and β = 3, beta with α = β = 0.5, beta with α = β = 2, beta with α = 1 and β = 0.5. These are dynamic values as the relevance of the components changes every time one of the components fails. The “reallocation rule” is applied. The reallocation rule works according to preference relations among the components. Once one of the components fails, its relevance is reallocated to the remaining active components. We seek to provide the most reliable estimate of the failure time of a stochastic system, whose components are interconnected through rational expectations. We present systems whose information on statistical indicators of configurations and failure times are contained in a set (information set). To make the predictions, we compare the information obtained on real systems (in-vivo systems), conditioning them to that of the information set. The result is the implementation of a scenario analysis of the errors obtained from this comparison which takes into account two fundamental aspects: the statistical measurements of the components weights and the time. We first illustrate a theoretical framework followed by two different computational models based on numerical simulations with different focuses and different results. The two models provide theoretical validation to the theoretical model and are strictly dependent on time and statistical indicators. In the first model we make a transversal analysis over time: we emphasize the contribution of the statistical indicator, showing how the error varies as a function of the measure considered aggregated over time. This is an individual analysis of the various indicators and how the prediction effectiveness changes depending on the initial weight distribution identified. Instead, in the second model, we emphasize the importance of the time factor on different scenarios of the statistical indicators based on the percentiles we condition them on. Here, we first measure the error trend for each indicator, and then we proceed with a comparative analysis to investigate which statistical indicator performs better also in relation to the initial weight distribution chosen. These models are two complementary studies that show two different approaches, focusing first on statistical indicators and then on time. The general goal is the implementation of a procedure that can be useful for any system with interactive components. In this thesis we propose a theoretical setting followed by two different frameworks of application that both exploit rational expectations in the context of the reliability of systems with interconnected components. All the results presented encourage the application of rational expectations to our simulated data. In both models we compare our research field with the existing literature and can thus evaluate the robustness of our results, the proposed methods, and the computational efficiency of our algorithms. We examine how the paths of the errors are influenced by several indicators, the different levels of these indicators, and initial weight distributions used in the study. In both models we successfully demonstrate that it is possible to obtain a prediction of failure times of stochastic systems which improves for certain combinations of indicators and initial distributions and with the passage of time. In future research further implementations of the model will have to be developed by increasing the complexity of the simulation procedure. The problem can be addressed in other rational expectations models and should certainly be attempted. It is our intention to change the rules and assumptions proposed in this thesis, and to compare new results with those already collected. We would like to extend the analysis by adding other statistical measures (Frosini index, Pearson index or other entropies). We can also consider a distance measure to check which combination of parameters will allow us to get the lowest error. One intention there is to study specifically how the evolution of the distribution of the components can affect our results. Again, with regard to this field, a network could be built on the connection existing between the components by considering of the probability with which they subsequently fail. We can also integrate cluster analysis by clustering our predictors over different time windows using a dendrogram to show in which periods the prediction was most effective. This thesis could have real practical relevance in economics and finance. It is also our intention to apply the simulation procedure to real data to model systems with interconnected components. Numerical experiments in the economic and financial field will follow. There are therefore many future research ideas that we hope to be able to deepen soon.File | Dimensione | Formato | |
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