Random Noisy Data Research Articles

A process knowledge-based hybrid method for univariate time series prediction with uncertain inputs in process industry

In process industry, accurate and efficient univariate time series prediction (TSP) of key process parameters is crucial for optimizing production processes. However, noisy data and varying working conditions introduce uncertainties that hinder the modeling process. Data-driven methods have been proposed to address input-uncertain TSP models, but they often fail to obtain sufficient and high-quality industrial data for model establishment in complex and dynamic production process. The paper presents a process knowledge-based hybrid method for univariate TSP that handles noisy inputs in different working conditions. Primarily, a knowledge-embedded deep kernel network (KE-DKN) model is proposed that approximates the outputs to the knowledge space in noisy industrial environment, by embedding instantiated process knowledge as soft constraints into the conventional DKN network. Further, a two-stage training schema is designed for KE-DKN to reduce the interference superposition on the predicted variable from random noisy data and operational interferences. Moreover, the pre-trained parameters transferring is conducted from stable working conditions to mitigate the low modeling efficiency in unstable working conditions. In two cases from process industry, the proposed method delivers higher prediction accuracy and robustness than baseline methods in both working conditions, and the knowledge transfer significantly enhances the modeling efficiency in unstable working conditions.

Inverse learning in Hilbert scales

We study linear ill-posed inverse problems with noisy data in the framework of statistical learning. The corresponding linear operator equation is assumed to fit a given Hilbert scale, generated by some unbounded self-adjoint operator. Approximate reconstructions from random noisy data are obtained with general regularization schemes in such a way that these belong to the domain of the generator. The analysis has thus to distinguish two cases, the regular one, when the true solution also belongs to the domain of the generator, and the ‘oversmoothing’ one, when this is not the case. Rates of convergence for the regularized solutions will be expressed in terms of certain distance functions. For solutions with smoothness given in terms of source conditions with respect to the scale generating operator, then the error bounds can then be made explicit in terms of the sample size.

Identification of a harmonically varying external source in wave equation from Neumann-type boundary measurement

Abstract In this paper, the identification problem of recovering the spatial source F ∈ L 2 ⁢ ( 0 , l ) {F\in L^{2}(0,l)} in the wave equation u t ⁢ t = u x ⁢ x + F ⁢ ( x ) ⁢ cos ⁡ ( ω ⁢ t ) {u_{tt}=u_{xx}+F(x)\cos(\omega t)} , with harmonically varying external source F ⁢ ( x ) ⁢ cos ⁡ ( ω ⁢ t ) {F(x)\cos(\omega t)} and with the homogeneous boundary u ⁢ ( 0 , t ) = u ⁢ ( l , t ) = 0 {u(0,t)=u(l,t)=0} , t ∈ ( 0 , T ) {t\in(0,T)} , and initial u ⁢ ( x , 0 ) = u t ⁢ ( x , 0 ) = 0 {u(x,0)=u_{t}(x,0)=0} , x ∈ ( 0 , l ) {x\in(0,l)} , conditions, is studied. As a measurement output g ⁢ ( t ) {g(t)} , the Neumann-type boundary measurement g ⁢ ( t ) := u x ⁢ ( 0 , t ) {g(t):=u_{x}(0,t)} , t ∈ ( 0 , T ) {t\in(0,T)} , at the left boundary x = 0 {x=0} is used. It is assumed that the observation g ∈ L 2 ⁢ ( 0 , T ) {g\in L^{2}(0,T)} may has a random noise. We propose combination of the boundary control for PDEs, adjoint method and Tikhonov regularization, for identification of the unknown source F ∈ L 2 ⁢ ( 0 , l ) {F\in L^{2}(0,l)} . Our approach based on weak solution theory of PDEs and, as a result, allows use of nonsmooth input/output data. Introducing the input-output operator Φ ⁢ F := u x ⁢ ( 0 , t ; F ) {\Phi F:=u_{x}(0,t;F)} , Φ : L 2 ⁢ ( 0 , l ) ↦ L 2 ⁢ ( 0 , T ) {\Phi:L^{2}(0,l)\mapsto L^{2}(0,T)} , where u ⁢ ( x , t ; F ) {u(x,t;F)} is the solution of the wave equation with above homogeneous boundary and initial conditions, we first prove the compactness of this operator. This allows to obtain the uniqueness of regularized solution of the identification problem, i.e. the minimum of the regularized cost functional J α ⁢ ( F ) := J ⁢ ( F ) + 1 2 ⁢ α ⁢ ∥ F ∥ L 2 ⁢ ( 0 , l ) 2 {J_{\alpha}(F):=J(F)+\frac{1}{2}\alpha\|F\|_{L^{2}(0,l)}^{2}} , where J ⁢ ( F ) = 1 2 ⁢ ∥ u x ⁢ ( 0 , ⋅ ; F ) - g ∥ L 2 ⁢ ( 0 , T ) 2 {J(F)=\frac{1}{2}\|u_{x}(0,\cdot\,;F)-g\|_{L^{2}(0,T)}^{2}} . Then the adjoint problem approach is used to derive a formula for the Fréchet gradient of the cost functional J ⁢ ( F ) {J(F)} . Use of the gradient formula in the conjugate gradient algorithm (CGA) allows to construct a fast algorithm for recovering the unknown source F ⁢ ( x ) {F(x)} . A comprehensive set of benchmark numerical examples, with up to 10 noise level random noisy data, illustrate the usefulness and effectiveness of the proposed approach.

Tikhonov regularization with oversmoothing penalty for nonlinear statistical inverse problems

In this paper, we consider the nonlinear ill-posed inverse problem with noisy data in the statistical learning setting. The Tikhonov regularization scheme in Hilbert scales is considered to reconstruct the estimator from the random noisy data. In this statistical learning setting, we derive the rates of convergence for the regularized solution under certain assumptions on the nonlinear forward operator and the prior assumptions. We discuss estimates of the reconstruction error using the approach of reproducing kernel Hilbert spaces.

Point-wise gated restricted Boltzmann machines using clean data

The point-wise gated restricted Boltzmann machine (pgRBM), an RBM variant, finds the task-relevant patterns from data containing irrelevant patterns and thus achieves satisfied classification results. Given that the training data composes of noisy data and clean data, how the clean data is applied to promote the performance of the pgRBM is a problem. To address the problem, this paper proposes a method, named as pgRBM based on random noisy data and clean data (pgrncRBM). The pgrncRBM makes use of RBM and the clean data to obtain the initial values of the task-relevant weights. However, if the noise is an image, the pgrncRBM cannot learn the task-relevant patterns from the noisy data. Therefore, this paper combines the Spike-and-Slab RBM with the pgRBM and proposes a method, named as pgRBM based on image noisy data and clean data (pgincRBM). The pgincRBM uses the spike-and-slab RBM to model the noise, so it can learn the 'clean' data from the data containing image noisy. Finally, we discuss the validities of these methods on MNIST variation datasets.

Point-wise gated restricted Boltzmann machines using clean data

The point-wise gated restricted Boltzmann machine (pgRBM), an RBM variant, finds the task-relevant patterns from data containing irrelevant patterns and thus achieves satisfied classification results. Given that the training data composes of noisy data and clean data, how the clean data is applied to promote the performance of the pgRBM is a problem. To address the problem, this paper proposes a method, named as pgRBM based on random noisy data and clean data (pgrncRBM). The pgrncRBM makes use of RBM and the clean data to obtain the initial values of the task-relevant weights. However, if the noise is an image, the pgrncRBM cannot learn the task-relevant patterns from the noisy data. Therefore, this paper combines the Spike-and-Slab RBM with the pgRBM and proposes a method, named as pgRBM based on image noisy data and clean data (pgincRBM). The pgincRBM uses the spike-and-slab RBM to model the noise, so it can learn the 'clean' data from the data containing image noisy. Finally, we discuss the validities of these methods on MNIST variation datasets.

Numerical Inversion for the Multiple Fractional Orders in the Multiterm TFDE

The fractional order in a fractional diffusion model is a key parameter which characterizes the anomalous diffusion behaviors. This paper deals with an inverse problem of determining the multiple fractional orders in the multiterm time-fractional diffusion equation (TFDE for short) from numerics. The homotopy regularization algorithm is applied to solve the inversion problem using the finite data at one interior point in the space domain. The inversion fractional orders with random noisy data give good approximations to the exact order demonstrating the efficiency of the inversion algorithm and numerical stability of the inversion problem.

A conditional Lipschitz stability for determining a space-dependent source coefficient in the 2D/3D advection-dispersion equation

Abstract This paper deals with an inverse problem of determining a space-dependent source coefficient in the 2D/3D advection-dispersion equation with final observations using the variational adjoint method. Data compatibility for the inverse problem is analyzed by which an admissible set for the unknowns is induced. With the aid of an adjoint problem, a bilinear functional based on the variational identity is set forth with which a norm for the unknown is well-defined under suitable conditions, and then a conditional Lipschitz stability for the inverse problem is established. Furthermore, numerical inversions with random noisy data are performed using the optimal perturbation algorithm, and the inversion solutions give good approximations to the exact solution as the noise level goes to small.

Fourier collocation algorithm for identifying the spacewise-dependent source in the advection–diffusion equation from boundary data measurements

In this study, we investigate the inverse problem of identifying an unknown spacewise-dependent source F(x) in the one-dimensional advection–diffusion equation ut=Duxx−vux+F(x)H(t), (x,t)∈(0,1)×(0,T], based on boundary concentration measurements g(t):=u(ℓ,t). Most studies have attempted to reconstruct an unknown spacewise-dependent source F(x) from the final observation uT(x):=u(x,T), but from an engineering viewpoint, the above boundary data measurements are feasible. Thus, we propose a new algorithm for reconstructing the spacewise-dependent source F(x). This algorithm is based on Fourier expansion of the direct problem solution followed by minimization of the cost functional by taking a partial K-sum of the Fourier expansion. Tikhonov regularization is then applied to the ill-posed problem that is obtained. The proposed approach also allows us to estimate the degree of ill-posedness for the inverse problem considered in this study. We then establish the relationship between the noise level γ>0, the parameter of regularization α>0, and the truncation (or cut-off) parameter K. A new numerical filtering algorithm is proposed for smoothing the noisy output data. Our numerical results demonstrated that the results obtained for random noisy data up to noise levels of 7% had sufficiently high accuracy for all reconstructions.

A variational approach to reconstruction of an initial tsunami source perturbation

Tsunamis are gravitational, i.e. gravity-controlled waves generated by a given motion of the bottom. There are different natural phenomena, such as submarine slumps, slides, volcanic explosions, earthquakes, etc. that can lead to a tsunami. This paper deals with the case where the tsunami source is an earthquake. The mathematical model studied here is based on shallow water theory, which is used extensively in tsunami modeling. The inverse problem consists of determining an unknown initial tsunami source q(x,y) by using measurements fm(t) of the height of a passing tsunami wave at a finite number of given points (xm,ym), m=1,2,…,M, of the coastal area. The proposed approach is based on the weak solution theory for hyperbolic PDEs and adjoint problem method for minimization of the corresponding cost functional. The adjoint problem is defined to obtain an explicit gradient formula for the cost functional J(q)=‖Aq−F‖2, F=(f1,…,fM). Numerical algorithms are proposed for the direct as well as adjoint problems. Conjugate gradient algorithm based on explicit gradient formula is used for numerical solution of the inverse problem. Results of computational experiments presented for the synthetic noise free and random noisy data in real scale illustrate bounds of applicability of the proposed approach, also its efficiency and accuracy.

System Identification with Noisy Data

System identification from the field obtained data is not successful till date because of the noise in the sensor or measured. The unknown system parameter for any unknown damage is still a long way to move. In this paper finite element technique is applied on the simulated damaged structures. The damage detection for the noise free data is extended and tried for the random noisy sensor data. The proposed algorithm is used to predict the system parameter on static deflection data with introduction of some random noisy data using simulated structure. The developed algorithm is applied on bridge truss structure for the identification of the damaged system parameter prediction.

Inverse resistivity problem: Geoelectric uncertainty principle and numerical reconstruction method

Mathematical model of vertical electrical sounding by using resistivity method is studied. The model leads to an inverse problem of determination of the unknown leading coefficient (conductivity) of the elliptic equation in R 2 in a slab. The direct problem is obtained in the form of mixed BVP in axisymmetric cylindrical coordinates. The additional (available measured) data is given on the upper boundary of the slab, in the form of tangential derivative. Due to ill-conditionedness of the considered inverse problem the logarithmic transformation is applied to the unknown coefficient and the inverse problem is studied as a minimization problem for the cost functional, with respect to the reflection coefficient. The Conjugate Gradient method (CGM) is applied for the numerical solution of this problem. Computational experiments were performed with noise free and random noisy data.

A new gradient regularization algorithm for source term inversion in 1D solute transportation with final observations

In this paper, a new gradient regularization algorithm is introduced and applied to solve an inverse problem of determining source terms in one-dimensional advection–dispersion equation with final observations. By functional approximations, the algorithm is reduced to find an optimal perturbation for a given source parameter involving computations of gradient vectors. In the case of using accurate data, the optimal perturbation can be directly worked out by the least square method; in the case of with random noisy data, regularization strategy may be useful to the realization of the algorithm. As compared with ordinary gradient regularization algorithm, the new gradient regularization algorithm gives a possible approach to the precise choices of optimal regularization parameters. Several numerical simulations under different conditions are carried out showing that the algorithm is feasible and efficient. Furthermore, the algorithm is successfully applied to solve a real life example of determining an average magnitude of groundwater pollution sources.

Regularization of some linear ill-posed problems with discretized random noisy data

For linear statistical ill-posed problems in Hilbert spaces we introduce an adaptive procedure to recover the unknown solution from indirect discrete and noisy data. This procedure is shown to be order optimal for a large class of problems. Smoothness of the solution is measured in terms of general source conditions. The concept of operator monotone functions turns out to be an important tool for the analysis.