In this paper we study probabilistic and neural network approximations for solutions to Poisson e... more In this paper we study probabilistic and neural network approximations for solutions to Poisson equation subject to Hölder or C 2 data in general bounded domains of R d. We aim at two fundamental goals. The first, and the most important, we show that the solution to Poisson equation can be numerically approximated in the sup-norm by Monte Carlo methods based on a slight change of the walk on spheres algorithm. This provides estimates which are efficient with respect to the prescribed approximation error and without the curse of dimensionality. In addition, the overall number of samples does not not depend on the point at which the approximation is performed. As a second goal, we show that the obtained Monte Carlo solver renders ReLU deep neural network (DNN) solutions to Poisson problem, whose sizes depend at most polynomially in the dimension d and in the desired error. In fact we show that the random DNN provides with high probability a small approximation error and low polynomial complexity in the dimension.
2019 18th IEEE International Conference On Machine Learning And Applications (ICMLA)
Naive Bayes estimator is widely used in text classification problems. However, it doesn't perform... more Naive Bayes estimator is widely used in text classification problems. However, it doesn't perform well with smallsize training dataset. We propose a new method based on Naive Bayes estimator to solve this problem. A correlation factor is introduced to incorporate the correlation among different classes. Experimental results show that our estimator achieves a better accuracy compared with traditional Naive Bayes in real world data.
We study the spectrum reconstruction technique. As is known to all, eigenvalues play an important... more We study the spectrum reconstruction technique. As is known to all, eigenvalues play an important role in many research fields and are foundation to many practical techniques such like PCA(Principal Component Analysis). We believe that related algorithms should perform better with more accurate spectrum estimation. There was an approximation formula proposed, however, they didn't give any proof. In our research, we show why the formula works. And when both number of features and dimension of space go to infinity, we find the order of error for the approximation formula, which is related to a constant $c$-the ratio of dimension of space and number of features.
In this paper we propose two schemes for the recovery of the spectrum of a covariance matrix from... more In this paper we propose two schemes for the recovery of the spectrum of a covariance matrix from the empirical covariance matrix, in the case where the dimension of the matrix is a subunitary multiple of the number of observations. We test, compare and analyze these on simulated data and also on some data coming from the stock market.
Given {Xk} is a martingale difference sequence. And given another {Yk} which has dependency withi... more Given {Xk} is a martingale difference sequence. And given another {Yk} which has dependency within the sequence. Assume {Xk} is independent with {Yk}, we study the properties of the sums of product of two sequences ∑n k=1XkYk. We obtain product-CLT, a modification of classical central limit theorem, which can be useful in the study of random projections. We also obtain the rate of convergence which is similar to the Berry-Essen theorem in the classical CLT.
Naive Bayes (NB) estimator is widely-used in text classification problems. However, it does not p... more Naive Bayes (NB) estimator is widely-used in text classification problems. However, it does not perform well with smallsize training datasets. Most previous literature focuses on either creating and modifying features or combing clustering to improve the performance of NB. We directly tackle the problem by constructing a new estimator, called Naive Bayes with correlation factor. We introduce a correlation factor to NB estimator that incorporates overall correlation among the different classes. This effectively exploits the idea of bootstrapping, which reuses data for all classes even if they only belong to one class. Moreover, we obtain a formula for the optimal correlation factor by balancing bias and variance of the estimator. Experimental results on real-world data show that our estimator achieves better accuracy compared with traditional Naive Bayes, yet at the same time maintaining the simplicity of NB.
We define a semidirect product groupoid of a system of partially defined local homeomorphisms $T=... more We define a semidirect product groupoid of a system of partially defined local homeomorphisms $T=(T_{1},..., T_{r})$. We prove that this construction gives rise to amenable groupoids. The associated algebra is a Cuntz-like algebra. We use this construction for higher rank graph algebras in order to give a topological interpretation for the duality in $E$-theory between $C^{*}(\Lambda)$ and $C^{*}(\Lambda^{op})$.
ABSTRACT. In this short note we give a refinement of Brascamp-Lieb [1] in the style of Houdré-Ka... more ABSTRACT. In this short note we give a refinement of Brascamp-Lieb [1] in the style of Houdré-Kagan [3] extension for Poincare ́ inequality in one dimension. This is inspired by works of Helffer [2] and Ledoux [4]. 1. THE BRASCAMP-LIEB INEQUALITY We take a convex potential V: R → R which is Ck with k ≥ 2 and the measure µ(dx) = e−V (x)dx which we assume it is a probability measure on R. Theorem 1 (Brascamp-Lieb [1]). If V ′ ′> 0 then for any C2 compactly supported function f on the real line
We develop original models to study interacting agents in financial markets and in social network... more We develop original models to study interacting agents in financial markets and in social networks. Within these models randomness is vital as a form of shock or news that decays with time. Agents learn from their observations and learning ability to interpret news or private information in time-varying networks. Under general assumption on the noise, a limit theorem is developed for the generalized DeGroot framework for certain type of conditions governing the learning. In this context, the agents' beliefs (properly scaled) converge in distribution that is not necessarily normal. Fresh insights are gained not only from proposing a new setting for social learning models but also from using different techniques to study discrete time random linear dynamical systems.
In this paper we propose a three stages analysis of the evolution of Covid19 in Romania. There ar... more In this paper we propose a three stages analysis of the evolution of Covid19 in Romania. There are two main issues when it comes to pandemic prediction. The first one is the fact that the numbers reported of infected and recovered are unreliable, however the number of deaths is more accurate. The second issue is that there were many factors which affected the evolution of the pandemic. In this paper we propose an analysis in three stages. The first stage is based on the classical SIR model which we do using a neural network. This provides a first set of daily parameters. In the second stage we propose a refinement of the SIR model in which we separate the deceased into a distinct category. By using the first estimate and a grid search, we give a daily estimation of the parameters. The third stage is used to define a notion of turning points (local extremes) for the parameters. We call a regime the time between these points. We outline a general way based on time varying parameters of SIRD to make predictions.
Analysing and understanding the transmission and evolution of the COVID-19 pandemic is mandatory,... more Analysing and understanding the transmission and evolution of the COVID-19 pandemic is mandatory, to be able to design the best social and medical policies, foresee their outcomes and deal with all the subsequent socioeconomic effects. We address this important problem from a computational and machine learning perspective. We are, to the best of our knowledge, the first to tackle the task for the case of Romania. More specifically, we want to statistically estimate all the relevant parameters of the spreading of the new coronavirus COVID-19, such as the reproduction number, fatality rate or length of infectiousness period, based on Romanian patients, as well as be able to predict future outcomes. This endeavor is important, since it is well known that these factors vary across the globe, and might be dependent on many causes, including social, medical, age and genetic factors. At the core of our computational approach lies the recently published, state-of-the-art work Chowdhury et al. [2020] that proposes an improved version of SEIR, which is the classic, established model for infectious diseases. We want to infer all the parameters of the model, which govern the evolution of the pandemic in Romania, based on the only reliable, true measurement, which is the number of deaths. The true number of infected people is in reality impossible to know precisely. Once the model parameters are estimated, we are able to predict, all the other relevant measures, such as the number of exposed and infectious people and many other factors, as shown in this paper. To this end, we propose a self-supervised approach to train a deep convolutional network to guess the correct set of Modified-SEIR model parameters, given the observed number of daily fatalities. Then, starting from these initial parameters, we refine the solution with a stochastic coordinate descent approach. We compare our deep learning optimization scheme with the classic grid search approach and show great improvement in both computational time and prediction accuracy. We find an optimistic result in the case fatality rate for Romania which may be around 0.3% and we also demonstrate that our model 1
In this paper we propose a three stages analysis of the evolution of Covid19 in Romania. There ar... more In this paper we propose a three stages analysis of the evolution of Covid19 in Romania. There are two main issues when it comes to pandemic prediction. The first one is the fact that the numbers reported of infected and recovered are unreliable, however the number of deaths is more accurate. The second issue is that there were many factors which affected the evolution of the pandemic. In this paper we propose an analysis in three stages. The first stage is based on the classical SIR model which we do using a neural network. This provides a first set of daily parameters. In the second stage we propose a refinement of the SIR model in which we separate the deceased into a distinct category. By using the first estimate and a grid search, we give a daily estimation of the parameters. The third stage is used to define a notion of turning points (local extremes) for the parameters. We call a regime the time between these points. We outline a general way based on time varying parameters o...
We consider the model space M n K of constant curvature K and dimension n ≥ 1 (Euclidean space fo... more We consider the model space M n K of constant curvature K and dimension n ≥ 1 (Euclidean space for K = 0, sphere for K > 0 and hyperbolic space for K < 0), and we show that given a function ρ : [0, ∞) → [0, ∞), ρ(0) = d(x, y) there exists a co-adapted coupling (X(t), Y (t)) of Brownian motions on M n K such that ρ(t) = d(X(t), Y (t)) for every t ≥ 0 if and only if ρ is continuous and satisfies for almost every t ≥ 0 the differential inequality −(n − 1) √ K tan √ Kρ(t) 2 ≤ ρ (t) ≤ −(n − 1) √ K tan √ Kρ(t) 2 + 2(n−1) √ K sin(√ Kρ(t)). In other words, we characterize all co-adapted couplings of Brownian motions on the model space M n K for which the distance between the processes is deterministic. In addition, the construction of the coupling is explicit for every choice of ρ satisfying the above hypotheses.
2π dx, this Poincaré inequality states that for any smooth function φ on [−2, 2], 2 −2 2 −2 φ(x) ... more 2π dx, this Poincaré inequality states that for any smooth function φ on [−2, 2], 2 −2 2 −2 φ(x) − φ(y) x − y 2 (4 − xy) 4π 2 (4 − x 2)(4 − y 2) dx dy ≤ (φ ′) 2 dα. (1.4) Note that left-hand side of (1.4), which replaces the classical variance term, is essentially the fluctuation quantity of random matrices. Further, note that (1.4) has a different flavor from its classical counterpart. For example, in case of the standard Gaussian measure, (1.1) is the expression of the spectral gap of the Ornstein-Uhlenbeck operator. In the free
This article appeared in a journal published by Elsevier. The attached copy is furnished to the a... more This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier's archiving and manuscript policies are encouraged to visit: http://www.elsevier.com/authorsrights
In this paper, we propose an analysis of Covid19 evolution and prediction on Romania combined wit... more In this paper, we propose an analysis of Covid19 evolution and prediction on Romania combined with the mathematical model of SIRD, an extension of the classical model SIR, which includes the deceased as a separate category. The reason is that, because we can not fully trust the reported numbers of infected or recovered people, we base our analysis on the more reliable number of deceased people. In addition, one of the parameters of our model includes the proportion of infected and tested versus infected. Since there are many factors which have an impact on the evolution of the pandemic, we decide to treat the estimation and the prediction based on the previous 7 days of data, particularly important here being the number of deceased. We perform the estimation and prediction using neural networks in two steps. Firstly, by simulating data with our model, we train several neural networks which learn the parameters of the model. Secondly, we use an ensemble of ten of these neural networks to forecast the parameters from the real data of Covid19 in Romania. Many of these results are backed up by a theorem which guarantees that we can recover the parameters from the reported data.
Johnson-Lindenstrauss lemma states random projections can be used as a topology preserving embedd... more Johnson-Lindenstrauss lemma states random projections can be used as a topology preserving embedding technique for fixed vectors. In this paper, we try to understand how random projections affect probabilistic properties of random vectors. In particular we prove the distribution of inner product of two independent random vectors X, Z ∈ R n with i.i.d. entries is preserved by random projection S : R n → R m. More precisely, sup t
This paper develops original models to study interacting agents in financial markets. The key fea... more This paper develops original models to study interacting agents in financial markets. The key feature of these models is how interactions are formulated and analysed. Agents learn from their observations and learning ability to interpret news or private information. Central limit theorems are developed but they arise rather unexpectedly. Under certain type of conditions governing the learning, agents beliefs converge in distribution that can be even fractal. The underlying randomness in the systems is not restricted to be of a certain class. Fresh insights are gained not only from developing new non-linear social learning models but also from using different techniques to study discrete time random linear dynamical systems.
In this paper we deal with free functional inequalities on the circle. There are some interesting... more In this paper we deal with free functional inequalities on the circle. There are some interesting changes from their classical counterparts. For example, the free Poincaré inequality has a slight change which seems to account for the lack of invariance under rotations of the base measure. Another instance is the modified Wasserstein distance on the circle which provides the tools for analyzing transportation, Log-Sobolev, and HWI inequalities. These new phenomena also indicate that they have classical counterparts, which does not seem to have been investigated thus far.
We consider two independent binary i.i.d. random strings X and Y of equal length n and the optima... more We consider two independent binary i.i.d. random strings X and Y of equal length n and the optimal alignments according to a symmetric scoring functions only. We decompose the space of scoring functions into five components. Two of these components add a part to the optimal score which does not depend on the alignment and which is asymptotically normal. We show that when we restrict the number of gaps sufficiently and add them only into one sequence, then the alignment score can be decomposed into a part which is normal and has order O(√ n) and a part which is on a smaller order and tends to a Tracy-Widom distribution. Adding gaps only into one sequence is equivalent to aligning a string with its descendants in case of mutations and deletes. For testing relatedness of strings, the normal part is irrelevant, since it does not depend on the alignment hence it can be safely removed from the test statistic.
In this paper we study probabilistic and neural network approximations for solutions to Poisson e... more In this paper we study probabilistic and neural network approximations for solutions to Poisson equation subject to Hölder or C 2 data in general bounded domains of R d. We aim at two fundamental goals. The first, and the most important, we show that the solution to Poisson equation can be numerically approximated in the sup-norm by Monte Carlo methods based on a slight change of the walk on spheres algorithm. This provides estimates which are efficient with respect to the prescribed approximation error and without the curse of dimensionality. In addition, the overall number of samples does not not depend on the point at which the approximation is performed. As a second goal, we show that the obtained Monte Carlo solver renders ReLU deep neural network (DNN) solutions to Poisson problem, whose sizes depend at most polynomially in the dimension d and in the desired error. In fact we show that the random DNN provides with high probability a small approximation error and low polynomial complexity in the dimension.
2019 18th IEEE International Conference On Machine Learning And Applications (ICMLA)
Naive Bayes estimator is widely used in text classification problems. However, it doesn't perform... more Naive Bayes estimator is widely used in text classification problems. However, it doesn't perform well with smallsize training dataset. We propose a new method based on Naive Bayes estimator to solve this problem. A correlation factor is introduced to incorporate the correlation among different classes. Experimental results show that our estimator achieves a better accuracy compared with traditional Naive Bayes in real world data.
We study the spectrum reconstruction technique. As is known to all, eigenvalues play an important... more We study the spectrum reconstruction technique. As is known to all, eigenvalues play an important role in many research fields and are foundation to many practical techniques such like PCA(Principal Component Analysis). We believe that related algorithms should perform better with more accurate spectrum estimation. There was an approximation formula proposed, however, they didn't give any proof. In our research, we show why the formula works. And when both number of features and dimension of space go to infinity, we find the order of error for the approximation formula, which is related to a constant $c$-the ratio of dimension of space and number of features.
In this paper we propose two schemes for the recovery of the spectrum of a covariance matrix from... more In this paper we propose two schemes for the recovery of the spectrum of a covariance matrix from the empirical covariance matrix, in the case where the dimension of the matrix is a subunitary multiple of the number of observations. We test, compare and analyze these on simulated data and also on some data coming from the stock market.
Given {Xk} is a martingale difference sequence. And given another {Yk} which has dependency withi... more Given {Xk} is a martingale difference sequence. And given another {Yk} which has dependency within the sequence. Assume {Xk} is independent with {Yk}, we study the properties of the sums of product of two sequences ∑n k=1XkYk. We obtain product-CLT, a modification of classical central limit theorem, which can be useful in the study of random projections. We also obtain the rate of convergence which is similar to the Berry-Essen theorem in the classical CLT.
Naive Bayes (NB) estimator is widely-used in text classification problems. However, it does not p... more Naive Bayes (NB) estimator is widely-used in text classification problems. However, it does not perform well with smallsize training datasets. Most previous literature focuses on either creating and modifying features or combing clustering to improve the performance of NB. We directly tackle the problem by constructing a new estimator, called Naive Bayes with correlation factor. We introduce a correlation factor to NB estimator that incorporates overall correlation among the different classes. This effectively exploits the idea of bootstrapping, which reuses data for all classes even if they only belong to one class. Moreover, we obtain a formula for the optimal correlation factor by balancing bias and variance of the estimator. Experimental results on real-world data show that our estimator achieves better accuracy compared with traditional Naive Bayes, yet at the same time maintaining the simplicity of NB.
We define a semidirect product groupoid of a system of partially defined local homeomorphisms $T=... more We define a semidirect product groupoid of a system of partially defined local homeomorphisms $T=(T_{1},..., T_{r})$. We prove that this construction gives rise to amenable groupoids. The associated algebra is a Cuntz-like algebra. We use this construction for higher rank graph algebras in order to give a topological interpretation for the duality in $E$-theory between $C^{*}(\Lambda)$ and $C^{*}(\Lambda^{op})$.
ABSTRACT. In this short note we give a refinement of Brascamp-Lieb [1] in the style of Houdré-Ka... more ABSTRACT. In this short note we give a refinement of Brascamp-Lieb [1] in the style of Houdré-Kagan [3] extension for Poincare ́ inequality in one dimension. This is inspired by works of Helffer [2] and Ledoux [4]. 1. THE BRASCAMP-LIEB INEQUALITY We take a convex potential V: R → R which is Ck with k ≥ 2 and the measure µ(dx) = e−V (x)dx which we assume it is a probability measure on R. Theorem 1 (Brascamp-Lieb [1]). If V ′ ′> 0 then for any C2 compactly supported function f on the real line
We develop original models to study interacting agents in financial markets and in social network... more We develop original models to study interacting agents in financial markets and in social networks. Within these models randomness is vital as a form of shock or news that decays with time. Agents learn from their observations and learning ability to interpret news or private information in time-varying networks. Under general assumption on the noise, a limit theorem is developed for the generalized DeGroot framework for certain type of conditions governing the learning. In this context, the agents' beliefs (properly scaled) converge in distribution that is not necessarily normal. Fresh insights are gained not only from proposing a new setting for social learning models but also from using different techniques to study discrete time random linear dynamical systems.
In this paper we propose a three stages analysis of the evolution of Covid19 in Romania. There ar... more In this paper we propose a three stages analysis of the evolution of Covid19 in Romania. There are two main issues when it comes to pandemic prediction. The first one is the fact that the numbers reported of infected and recovered are unreliable, however the number of deaths is more accurate. The second issue is that there were many factors which affected the evolution of the pandemic. In this paper we propose an analysis in three stages. The first stage is based on the classical SIR model which we do using a neural network. This provides a first set of daily parameters. In the second stage we propose a refinement of the SIR model in which we separate the deceased into a distinct category. By using the first estimate and a grid search, we give a daily estimation of the parameters. The third stage is used to define a notion of turning points (local extremes) for the parameters. We call a regime the time between these points. We outline a general way based on time varying parameters of SIRD to make predictions.
Analysing and understanding the transmission and evolution of the COVID-19 pandemic is mandatory,... more Analysing and understanding the transmission and evolution of the COVID-19 pandemic is mandatory, to be able to design the best social and medical policies, foresee their outcomes and deal with all the subsequent socioeconomic effects. We address this important problem from a computational and machine learning perspective. We are, to the best of our knowledge, the first to tackle the task for the case of Romania. More specifically, we want to statistically estimate all the relevant parameters of the spreading of the new coronavirus COVID-19, such as the reproduction number, fatality rate or length of infectiousness period, based on Romanian patients, as well as be able to predict future outcomes. This endeavor is important, since it is well known that these factors vary across the globe, and might be dependent on many causes, including social, medical, age and genetic factors. At the core of our computational approach lies the recently published, state-of-the-art work Chowdhury et al. [2020] that proposes an improved version of SEIR, which is the classic, established model for infectious diseases. We want to infer all the parameters of the model, which govern the evolution of the pandemic in Romania, based on the only reliable, true measurement, which is the number of deaths. The true number of infected people is in reality impossible to know precisely. Once the model parameters are estimated, we are able to predict, all the other relevant measures, such as the number of exposed and infectious people and many other factors, as shown in this paper. To this end, we propose a self-supervised approach to train a deep convolutional network to guess the correct set of Modified-SEIR model parameters, given the observed number of daily fatalities. Then, starting from these initial parameters, we refine the solution with a stochastic coordinate descent approach. We compare our deep learning optimization scheme with the classic grid search approach and show great improvement in both computational time and prediction accuracy. We find an optimistic result in the case fatality rate for Romania which may be around 0.3% and we also demonstrate that our model 1
In this paper we propose a three stages analysis of the evolution of Covid19 in Romania. There ar... more In this paper we propose a three stages analysis of the evolution of Covid19 in Romania. There are two main issues when it comes to pandemic prediction. The first one is the fact that the numbers reported of infected and recovered are unreliable, however the number of deaths is more accurate. The second issue is that there were many factors which affected the evolution of the pandemic. In this paper we propose an analysis in three stages. The first stage is based on the classical SIR model which we do using a neural network. This provides a first set of daily parameters. In the second stage we propose a refinement of the SIR model in which we separate the deceased into a distinct category. By using the first estimate and a grid search, we give a daily estimation of the parameters. The third stage is used to define a notion of turning points (local extremes) for the parameters. We call a regime the time between these points. We outline a general way based on time varying parameters o...
We consider the model space M n K of constant curvature K and dimension n ≥ 1 (Euclidean space fo... more We consider the model space M n K of constant curvature K and dimension n ≥ 1 (Euclidean space for K = 0, sphere for K > 0 and hyperbolic space for K < 0), and we show that given a function ρ : [0, ∞) → [0, ∞), ρ(0) = d(x, y) there exists a co-adapted coupling (X(t), Y (t)) of Brownian motions on M n K such that ρ(t) = d(X(t), Y (t)) for every t ≥ 0 if and only if ρ is continuous and satisfies for almost every t ≥ 0 the differential inequality −(n − 1) √ K tan √ Kρ(t) 2 ≤ ρ (t) ≤ −(n − 1) √ K tan √ Kρ(t) 2 + 2(n−1) √ K sin(√ Kρ(t)). In other words, we characterize all co-adapted couplings of Brownian motions on the model space M n K for which the distance between the processes is deterministic. In addition, the construction of the coupling is explicit for every choice of ρ satisfying the above hypotheses.
2π dx, this Poincaré inequality states that for any smooth function φ on [−2, 2], 2 −2 2 −2 φ(x) ... more 2π dx, this Poincaré inequality states that for any smooth function φ on [−2, 2], 2 −2 2 −2 φ(x) − φ(y) x − y 2 (4 − xy) 4π 2 (4 − x 2)(4 − y 2) dx dy ≤ (φ ′) 2 dα. (1.4) Note that left-hand side of (1.4), which replaces the classical variance term, is essentially the fluctuation quantity of random matrices. Further, note that (1.4) has a different flavor from its classical counterpart. For example, in case of the standard Gaussian measure, (1.1) is the expression of the spectral gap of the Ornstein-Uhlenbeck operator. In the free
This article appeared in a journal published by Elsevier. The attached copy is furnished to the a... more This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier's archiving and manuscript policies are encouraged to visit: http://www.elsevier.com/authorsrights
In this paper, we propose an analysis of Covid19 evolution and prediction on Romania combined wit... more In this paper, we propose an analysis of Covid19 evolution and prediction on Romania combined with the mathematical model of SIRD, an extension of the classical model SIR, which includes the deceased as a separate category. The reason is that, because we can not fully trust the reported numbers of infected or recovered people, we base our analysis on the more reliable number of deceased people. In addition, one of the parameters of our model includes the proportion of infected and tested versus infected. Since there are many factors which have an impact on the evolution of the pandemic, we decide to treat the estimation and the prediction based on the previous 7 days of data, particularly important here being the number of deceased. We perform the estimation and prediction using neural networks in two steps. Firstly, by simulating data with our model, we train several neural networks which learn the parameters of the model. Secondly, we use an ensemble of ten of these neural networks to forecast the parameters from the real data of Covid19 in Romania. Many of these results are backed up by a theorem which guarantees that we can recover the parameters from the reported data.
Johnson-Lindenstrauss lemma states random projections can be used as a topology preserving embedd... more Johnson-Lindenstrauss lemma states random projections can be used as a topology preserving embedding technique for fixed vectors. In this paper, we try to understand how random projections affect probabilistic properties of random vectors. In particular we prove the distribution of inner product of two independent random vectors X, Z ∈ R n with i.i.d. entries is preserved by random projection S : R n → R m. More precisely, sup t
This paper develops original models to study interacting agents in financial markets. The key fea... more This paper develops original models to study interacting agents in financial markets. The key feature of these models is how interactions are formulated and analysed. Agents learn from their observations and learning ability to interpret news or private information. Central limit theorems are developed but they arise rather unexpectedly. Under certain type of conditions governing the learning, agents beliefs converge in distribution that can be even fractal. The underlying randomness in the systems is not restricted to be of a certain class. Fresh insights are gained not only from developing new non-linear social learning models but also from using different techniques to study discrete time random linear dynamical systems.
In this paper we deal with free functional inequalities on the circle. There are some interesting... more In this paper we deal with free functional inequalities on the circle. There are some interesting changes from their classical counterparts. For example, the free Poincaré inequality has a slight change which seems to account for the lack of invariance under rotations of the base measure. Another instance is the modified Wasserstein distance on the circle which provides the tools for analyzing transportation, Log-Sobolev, and HWI inequalities. These new phenomena also indicate that they have classical counterparts, which does not seem to have been investigated thus far.
We consider two independent binary i.i.d. random strings X and Y of equal length n and the optima... more We consider two independent binary i.i.d. random strings X and Y of equal length n and the optimal alignments according to a symmetric scoring functions only. We decompose the space of scoring functions into five components. Two of these components add a part to the optimal score which does not depend on the alignment and which is asymptotically normal. We show that when we restrict the number of gaps sufficiently and add them only into one sequence, then the alignment score can be decomposed into a part which is normal and has order O(√ n) and a part which is on a smaller order and tends to a Tracy-Widom distribution. Adding gaps only into one sequence is equivalent to aligning a string with its descendants in case of mutations and deletes. For testing relatedness of strings, the normal part is irrelevant, since it does not depend on the alignment hence it can be safely removed from the test statistic.
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