Maximum Likelihood Estimation Algorithm Research Articles

A New Algorithm to Estimate the Parameters of Log-Logistic Distribution Based on the Survival functions

Estimation of the parameters is quite important in the numerous fields for the development of mathematical models. Maximum likelihood estimation is a good method, which is usually used to elaborate on the parameter estimation. The likelihood function formed for the parameter estimation of Log-Logistic is very hard to maximize. Therefore, this paper proposes a new hybrid of Maximum Likelihood Estimator (MLE) and Simplex Downhill Algorithm (SDA) called (MLESDA) to estimate parameters of Log-Logistic distribution based on Survival functions. To compare the suggested method (MLESDA) and classical Maximum Likelihood (MLE) method, simulation is used. The results demonstrate that MLESDA is more efficient than the MLE method.

Hidden mover-stayer model for disease progression accounting for misclassified and partially observed diagnostic tests: Application to the natural history of human papillomavirus and cervical precancer.

Hidden Markov models (HMMs) have been proposed to model the natural history of diseases while accounting for misclassification in state identification. We introduce a discrete time HMM for human papillomavirus (HPV) and cervical precancer/cancer where the hidden and observed state spaces are defined by all possible combinations of HPV, cytology, and colposcopy results. Because the population of women undergoing cervical cancer screening is heterogeneous with respect to sexual behavior, and therefore risk of HPV acquisition and subsequent precancers, we use a mover-stayer mixture model that assumes a proportion of the population will stay in the healthy state and are not subject to disease progression. As each state is a combination of three distinct tests that characterize the cervix, partially observed data arise when at least one but not every test is observed. The standard forward-backward algorithm, used for evaluating the E-step within the E-M algorithm for maximum-likelihood estimation of HMMs, cannot incorporate time points with partially observed data. We propose a new forward-backward algorithm that considers all possible fully observed states that could have occurred across a participant's follow-up visits. We apply our method to data from a large management trial for women with low-grade cervical abnormalities. Our simulation study found that our method has relatively little bias and out preforms simpler methods that resulted in larger bias.

Modelling of Fuel- and Energy-Switching Prices by Mean-Reverting Processes and Their Applications to Alberta Energy Markets

This paper introduces a fuel-switching price to the Alberta market, which is designed for encouraging power plant companies to switch from coal to natural gas when they produce electricity; this has been successfully applied to the European market. Moreover, we consider an energy-switching price which considers power switch from natural gas to wind. We modeled these two prices using five mean reverting processes including a Regime-switching processes, Lévy-driven Ornstein–Uhlenbeck process and Inhomogeneous Geometric Brownian Motion, and estimate them based on multiple procedures such as Maximum likelihood estimation and Expectation-Maximization algorithm. Finally, this paper proves previous results applied to the Albertan Market where the jump modeling technique is needed when modeling fuel-switching data. In addition, it not only gives promising conclusions on the necessity of introducing Regime-switching models to the fuel-switching data, but also shows that the Regime-switching model is better fitted to the data.

Probability of signal demodulation jump errors in the maximum-likelihood-estimation algorithm for a low SNR interference spectrum.

When the signal-to-noise ratio (SNR) of an optical fiber sensor is lower than a specific value, jump errors will appear in the maximum-likelihood-estimation (MLE) algorithm. This research proposes a quantitative calculation method based on the MLE algorithm. This method can calculate the probability of jump errors by different SNRs. The SNR threshold can also be obtained in the MLE algorithm for the low-finesse interference spectrum. From the simulation, we find that when the SNR varies from 0 dB to 25 dB, the possibility of jump errors decreases exponentially. Subsequent experiments also verify the correctness of the simulation results.

Modeling of Lake Malombe Annual Fish Landings and Catch per Unit Effort (CPUE)

Forecasting, using time series data, has become the most relevant and effective tool for fisheries stock assessment. Autoregressive integrated moving average (ARIMA) modeling has been commonly used to predict the general trend for fish landings with increased reliability and precision. In this paper, ARIMA models were applied to predict Lake Malombe annual fish landings and catch per unit effort (CPUE). The annual fish landings and CPUE trends were first observed and both were non-stationary. The first-order differencing was applied to transform the non-stationary data into stationary. Autocorrelation functions (AC), partial autocorrelation function (PAC), Akaike information criterion (AIC), Bayesian information criterion (BIC), square root of the mean square error (RMSE), the mean absolute error (MAE), percentage standard error of prediction (SEP), average relative variance (ARV), Gaussian maximum likelihood estimation (GMLE) algorithm, efficiency coefficient (E2), coefficient of determination (R2), and persistent index (PI) were estimated, which led to the identification and construction of ARIMA models, suitable in explaining the time series and forecasting. According to the measures of forecasting accuracy, the best forecasting models for fish landings and CPUE were ARIMA (0,1,1) and ARIMA (0,1,0). These models had the lowest values AIC, BIC, RMSE, MAE, SEP, ARV. The models further displayed the highest values of GMLE, PI, R2, and E2. The “auto. arima ()” command in R version 3.6.3 further displayed ARIMA (0,1,1) and ARIMA (0,1,0) as the best. The selected models satisfactorily forecasted the fish landings of 2725.243 metric tons and CPUE of 0.097 kg/h by 2024.

Manipulations and quantum tomography of bright squeezed states

Generation and manipulation of continuous variable quantum states are the building blocks of quantum communication, quantum key distribution and quantum networks. According to the second-order nonlinear process of the periodically-poled potassium titanyl phosphate (PPKTP) crystal, we design a semi-monolithic optical parametric amplifier (OPA) cavity to generate the bright squeezed light at a wavelength of 1064 nm. With the injection of a seed beam, the squeezed state generated by the OPA has a coherent amplitude, so called bright squeezed state. The squeezing level is directly observed to be –11.6 dB when the pump power is 310 mW at an analysis frequency of 3 MHz. However, with the increase of the pump power, the purity of the squeezed state gets lower and lower due to the increased influence of the anti-squeezing quadrature component on the squeezed quadrature component in the detection process. To obtain a higher purity of the squeezed state for achieving linear optical manipulation and quantum tomography, we choose the pump power of 50 mW, the squeezing level decreases to –6 dB, and the purity of the squeezed state is 98.5% in this case. An electro-optic modulator is adopted to realize the liner manipulation of the squeezed light in the phase space. During the measurement of the bright squeezed state, all the data are taken on condition that the length of the OPA cavity and relative phase between the seed beam and the pump beam are locked by a locking loop. The direct current (DC) signal of the balanced homodyne detection (BHD) is used to accurately determine the phase corresponding to the time domain signal of the squeezed state, while the alternate current (AC) signal of the BHD is mixed with the signal generated by the function generator, after passing through a low-pass filter and a high-pass filter, the signal is then amplified by using a low-noise amplifier. A high-performance oscilloscope is finally used to simultaneously collect the signals, thus obtaining the quantum noise signal of the bright squeezed light after linear manipulation. Together with the maximum likelihood estimation algorithm, the quantum tomography, the density matrix and the Wigner function of the bright squeezed light are obtained, that is, all the information such as the photon number distribution of the quantum state is determined. Multiple iterations are taken in the maximum likelihood estimation algorithm process to eliminate the influence of the low quantum efficiency on the detection system, so that the density matrix is fitted well with the theoretical results.

Nonuniform Clustering of Wireless Sensor Network Node Positioning Anomaly Detection and Calibration

In order to detect and correct node localization anomalies in wireless sensor networks, a hierarchical nonuniform clustering algorithm is proposed. This paper designs a centroid iterative maximum likelihood estimation location algorithm based on nonuniformity analysis, selects the nonuniformity analysis algorithm, gives the flowchart of node location algorithm, and simulates the distribution of nodes with MATLAB. Firstly, the algorithm divides the nodes in the network into different network levels according to the number of hops required to reach the sink node. According to the average residual energy of nodes in each layer, the sink node selects the nodes with higher residual energy in each layer of the network as candidate cluster heads and selects a certain number of nodes with lower residual energy as additional candidate cluster heads. Then, at each level, the candidate cluster heads are elected to produce the final cluster heads. Finally, by controlling the communication range between cluster head and cluster members, clusters of different sizes are formed, and clusters at the level closer to the sink node have a smaller scale. By simulating the improved centroid iterative algorithm, the values of the optimal iteration parameters α and η are obtained. Based on the analysis of the positioning errors of the improved centroid iterative algorithm and the maximum likelihood estimation algorithm, the value of the algorithm conversion factor is selected. Aiming at the problem of abnormal nodes that may occur in the process of ranging, a hybrid node location algorithm is further proposed. The algorithm uses the ℓ2, 1 norm to smooth the structured anomalies in the ranging information and realizes accurate positioning while detecting node anomalies. Experimental results show that the algorithm can accurately determine the uniformity of distribution, achieve good positioning effect in complex environment, and detect abnormal nodes well. In this paper, the hybrid node location algorithm is extended to the node location problem in large‐scale scenes, and a good location effect is achieved.

Simulation of generalized Gamma distribution with maximum likelihood estimation and expectation-maximization algorithm on right censored data type 1

The Generalized Gamma distribution is very suitable for modeling data with various forms of hazard (risk) functions, which makes the Generalized Gamma distribution useful in survival analysis. Survival analysis aims are to predict chances of survival, disease recurrence, death, and other events over a period of time. One characteristic of survival data is the possibility of sensors. Let X be the life span of the person being studied and the right censorship time of Cr, X is assumed to be independent with the probability density function f(x), the survival function S(x), and the hazard function h(x). A person's X life span will be known if X is less than or equal to Cr. If X is greater than Cr, the individual X survives or is censored right now. Statistical inference, especially parameter estimation is needed in analyzing empirical data. Obviously the estimation results obtained are expected to be a good estimator, namely to meet the nature of unbiased and minimum variance. This paper will discuss the results of the estimation of Generalized Gamma distribution parameters with type 1 right censored data through simulations using the Expectation Maximization method and the Maximum Likelihood Estimation method. The simulation is conducted by generating data with the sample size: 25, 50, 100, 200, 500, 1000, 1500 and 2000 as well as determining censored data of 10%, 20% and 30% by first setting the parameters used which are obtained from the data of patients with gastric cancer namely α = 1.0649, β = 1,072, θ = 59.766. Based on the results obtained from the simulations on the two estimation methods that the parameter estimation using the Maximum Likelihood Estimation method is better than the Expectation Maximization method because it provides a smaller bias and MSE value where the larger the sample size used, the estimated parameter value will get closer to the parameter in fact. In addition, the Expectation Maximization method can also be used as an alternative estimation of generalized gamma distribution parameters with type 1 right censored data because it has a bias value and MSE approaching the MLE method.

A Comparative Study of DEM Reconstruction Using the Single-Baseline and Multibaseline InSAR Techniques

Interferometric SAR (InSAR) is a practical technique to derive 3-D structure of the earth's surface. This article conducts a comparative study of digital elevation model (DEM) reconstruction using the single-baseline and multibaseline InSAR techniques. To improve the accuracy of the DEM generated from the single-baseline InSAR, we improve the weight definition approach according to the coherence magnitudes for the fusion of single-baseline InSAR DEMs. For multibaseline InSAR, we adopt a method based on the maximum-likelihood estimation (MLE) algorithm, which uses external DEM as a constraint. A TerraSAR-X dataset comprising four SAR images acquired over Barcelona is used for experimental investigation. The accuracies of four types of DEM are comparatively studied, including DEM by single-baseline InSAR, DEM by the fusion of single-baseline InSAR DEMs, DEM by unconstrained multibaseline InSAR, and DEM by constrained multibaseline InSAR. The results show that the proposed weighted fusion of single-baseline InSAR performed slightly better than the constrained multibaseline InSAR method, and both of them performed better than the single-baseline results, demonstrating that multiple observations can effectively improve the accuracy of the reconstructed DEM.

Height Measurement With Meter Wave MIMO Radar Based on Precise Signal Model Under Complex Terrain

Meter wave radar height measurement is one of the most difficult problems in meter wave radar system. At present, the most popular height measurement methods of meter wave radar include beam splitting method, MUSIC algorithm and maximum likelihood estimation method, which can achieve good performance under flat ground. However, the real ground is not ideal condition, and the situation of undulating ground is also very complicated. Under this condition, the performance of the above algorithm dropped sharply. Many researchers solve this problem by improving the robustness of the algorithm. However, the authors think that the key to solve the problem is to establish a refined height measurement signal model. In this paper, the accurate model of reflection coefficient, reflection height and reflection point under the condition of undulating ground are established to provide accurate signal model for meter wave radar height measurement. Then we deduce the application of generalized MUSIC algorithm and maximum likelihood estimation algorithm for MIMO radar in the accurate terrain signal model. In fact, the method proposed in this paper is a terrain matching MIMO radar target height measurement, and the traditional algorithm can be regarded as terrain mismatch. The simulation results show that the accuracy of terrain matching estimation of the precise signal model established in this paper is much better than that of the traditional simplified signal model.

Mathematical modeling and impact analysis of the use of COVID Alert SA app.

The human life-threatening novel Severe Acute Respiratory Syndrome Corona-virus-2 (SARS-CoV-2) has lasted for over a year escalating and posing simultaneous anxiety day-by-day globally since its first report in the late December 2019. The scientific arena has been kept animated via continuous investigations in an effort to understand the spread dynamics and the impact of various mitigation measures to keep this pandemic diminished. Despite a lot of research works having been accomplished this far, the pandemic is still deep-rooted in many regions worldwide signaling for more scientific investigations. This study joins the field by developing a modified SEIR (Susceptible-Exposed-Infectious-Removed) compartmental deterministic model whose key distinct feature is the incorporation of the COVID Alert SA app use by the general public in prolific intention to control the spread of the epidemic. Validation of the model is performed by fitting the model to the Republic of South Africa's COVID-19 cases reported data using the Maximum Likelihood Estimation algorithm implemented in fitR package. The model's sensitivity analysis and simulations stipulate that gradual to complete use of the app would be perfect in contact tracing and substantially reduce the plateau number of COVID-19 infections. This would consequentially contribute remarkably to the eradication of the SARS-CoV-2 over time. Proportional amalgamation of the app use and test for COVID-19 on individuals not using the app would also reduce the peak number of infections apart from the 50 – 50% ratio which spikes the plateau number beyond any other proportion. The study establishes that at least 30% implementation of the app use with gradual increase in tests conducted for individuals not using the app would suffice to stabilize the disease free equilibrium resulting to gradual eradication of the pandemic.

On the Uncertainty Modelling for Linear Continuous-Time Systems Utilising Sampled Data and Gaussian Mixture Models

Abstract In this paper a Maximum Likelihood estimation algorithm for model error modelling in a continuous-time system is developed utilising sampled data and a Stochastic Embedding approach. Orthonormal basis functions are used to model both the continuous-time nominal model and the error-model. The stochastic properties of the error-model distribution are defined by using a Gaussian mixture model. For the estimation of the nominal model and the error-model distribution we develop a technique based on the Expectation-Maximization algorithm using sampled data from independent experiments. The benefits of our proposal are illustrated via numerical simulations.

Estimation of Path Loss Parameters of a Sub-Terahertz Wireless Channel Using Monostatic Radar

This paper presents a novel technique for estimating the path loss parameters of radio channels using monostatic radar. To this end, a two-way path loss model for a scenario using monostatic radar for a sub-terahertz (sub-THz) band channel measurement is expressed as a function of a parameter of a one-way path loss model, and then the path loss parameters are estimated using the maximum-likelihood estimation algorithm based on linear regression models, which is based on single reference distance and dual reference distances. To verify the effectiveness of the proposed method, a path loss measurement experiment is conducted in an indoor environment using a frequency modulated continuous wave (FMCW) radar system with the base frequency of 119.5 GHz and the bandwidth of 5 GHz, and then the estimation results based on the proposed method are analyzed.

PRNU Extraction from Stabilized Video: a Patch Maybe Better than a Bunch

This paper presents an algorithm to solve the problem of Photo-Response Non-Uniformity (PRNU) noise facing stabilized video. The stabilized video undergoes in-camera processing like rolling shutter correction. Thus, misalignment exists between the PRNU noises in the adjacent frames owing to the global and local frame registration performed by the in-camera processing. The misalignment makes the reference PRNU noise and the test PRNU noise unable to extract and match accurately. We design a computing method of maximum likelihood estimation algorithm for extracting the PRNU noise from stabilized video frames. Besides, unlike most prior arts tending to match the PRNU noise in whole frame, we propose a new patch-based matching strategy, aiming at reducing the influence from misalignment of frame the PRNU noise. After extracting the reference PRNU noise and the test PRNU noise, this paper adopts the reference and the test PRNU overlapping patch-based matching. It is different from the traditional matching method. This paper conducts different experiments on 224 stabilized videos taken by 13 smartphones in the VISION database. The area under curve of the algorithm proposed in this paper is 0.841, which is significantly higher than 0.805 of the whole frame matching in the traditional algorithm. Experimental results show good performance and effectiveness the proposed strategy by comparing with the prior arts.

Algebraic analysis of rotation data

We develop algebraic tools for statistical inference from samples of rotation matrices. This rests on the theory of D-modules in algebraic analysis. Noncommutative Gr\"obner bases are used to design numerical algorithms for maximum likelihood estimation, building on the holonomic gradient method of Sei, Shibata, Takemura, Ohara, and Takayama. We study the Fisher model for sampling from rotation matrices, and we apply our algorithms for data from the applied sciences. On the theoretical side, we generalize the underlying equivariant D-modules from SO(3) to arbitrary Lie groups. For compact groups, our D-ideals encode the normalizing constant of the Fisher model.

A Fast Sequential MLE Algorithm for Jet Engine Classification With Radar

Radar signals scattered by an operating jet engine contain a unique signature that can be used to classify the type of engine, and therefore the aircraft. We present a new algorithm based on the maximum likelihood estimation (MLE) principle that can identify the number of blades of the rotors in the jet engine from the returned radar signal. Unlike the conventional algorithms, the proposed algorithm avoids the estimation/inspection of the autocorrelation of radar signal and utilizes the property of jet engine modulation (JEM) harmonics together with the successive interference cancellation (SIC) technique in the multi-dimensional search, which saves about 66.7% computing time compared to the autocorrelation based algorithm (specifically, the empirical mode decomposition (EMD) based algorithm). The proposed algorithm is assessed with synthetic data as well as real radar data using a mock-up jet engine. Specifically, by evaluating the mean-squared-errors and Cramer-Rao lower bound, we verify that the proposed SIC algorithm performs better than previously known jet engine-classification algorithms.

18408 A combined clinicopathologic and gene expression model (CP-GEP) identifies primary cutaneous melanoma patients who can safely forgo sentinel lymph node biopsy

More than 80% of patients undergoing sentinel lymph node (SLN) biopsy have no nodal metastasis, and are unnecessarily exposed to procedure-induced morbidity such as lymphedema. Our objective was to develop a model combining clinicopathologic and gene expression (CP-GEP) to discriminate high-risk patients from patients who can safely forego SLN biopsy, thus reducing procedure-associated morbidity, and prioritizing care for high risk patients. A panel of 108 candidate biomarkers was identified, and the expression of these genes was quantified in FFPE diagnostic biopsy tissue across a cohort of 754 patients; 128/754 (17%) SLN positive patients. All patients underwent SLN biopsy at Mayo Clinic within 90 days of diagnosis between 2004 and 2018. We trained logistic regression models, using a penalized maximum likelihood estimation algorithm, in a repeated cross-validation scheme. The CP-GEP model, combining age and Breslow depth with genes involved in extracellular matrix remodeling (glia-derived nexin, growth differentiation factor 15, integrin β3, interleukin 8, lysyl oxidase homolog 4, TGFβ receptor type 1 and tissue-type plasminogen activator), and melanosome function (antigen recognized by T-cells), outperformed models based on only clinicopathologic variables, or only on gene expression, in discriminating SLN positive and negative patients (AUC, 0.82, 95% CI 0.78-0.86). The CP-GEP model achieved a SLN biopsy reduction rate of 42% at a negative predictive value of 96%. The 5-year relapse-free survival for CP-GEP negative patients was 88% compared with 50% for CP-GEP positive and SLN positive patients, confirming the value of the CP-GEP model as a tool to inform SLN biopsy decisions.

Genetic Modulators of Niclosamide Sensitivity and Resistance in Acute Myeloid Leukemia

Introduction Acute myeloid leukemia (AML) is a malignancy of myeloid progenitor cells that leads to the accumulation of immature blasts in the blood and bone marrow. While the 5-year relative survival rate has increased from 6.4% to 28.7% in the last 40 years, death rates have remained steady at 3-4%. Less toxic, more effective targeted treatments are needed to improve outcomes for patients with AML. Previous studies identified cAMP response element-binding protein (CREB) as a potential therapeutic target for AML therapy. CREB is overexpressed in AML and is associated with poor prognosis. We identified a small molecule, XX-650-23, that inhibits interaction between CREB and its co-activator, CREB Binding Protein. However, XX-650-23 does not have optimal physical chemical properties for clinical application in humans. Niclosamide, an FDA approved anthelmintic drug, shares structural similarity with XX-650-23, and suppresses the proliferation of AML cells in vitro and in vivo by inhibiting CREB-dependent signaling. The aim of this study is to define the molecular pathways that mediate the effects of niclosamide in AML cells. This will allow for more precise selection of patient populations for treatment, identify potential sources of niclosamide resistance and reveal combination therapies that make use of the distinct pathways targeted. Methods To identify genetic factors modulating cellular sensitivity to niclosamide treatment, we performed a CRISPR/Cas9 library screen in the presence or absence of niclosamide. HL60 cells expressing Cas9 were infected with a custom sgRNA lentivirus library of 220 genes from a previous genome-wide shRNA screen. After puromycin selection, HL60 cells expressing pooled sgRNAs were grown in either niclosamide or 0.1% DMSO for 20 days. Genomic DNA was extracted from control and niclosamide-treated cells. Frequency of sgRNA were measured by next generation sequencing and analyzed using the Cas9 high-throughput maximum likelihood estimator (casTLE) algorithm. Results Sixty-five gene hits were found to be significant (p≤0.05). Of the top 220 hits from the previous genome-wide shRNA screen, 26 were identified as positive hits (p ≤0.05) in the CRISPR/Cas9 knock out screen. The CRISPR screen identified genes enriched in a number of pathways including programmed cell death, nucleotide biosynthesis, mitochondrial processes and canonical glycolysis, as measured by gene ontology term analysis. Numerous gene deletions conferred resistance to niclosamide treatment. Genes DHODH (score= 123; effect= 2.9) and HSPA9 (score= 144; effect = 2.5), involved in nucleotide biosynthesis and mitochondrial processes, were significantly enriched in the surviving population. Perturbation of known metabolic pathways sensitized AML cells to niclosamide, including ATP synthase F1 subunit beta (ATP5B) (score= 316; effect= -5.1), hexokinase 2 (HK2) (score= 312; effect= -5.4), phosphofructokinase (PFKP) (score= 195; effect= -1.9) and mitochondrial phosphate carrier protein (SLC25A3) (score= 271; effect= -7.8). SLC25A3 and ATP5B play a role in oxidative phosphorylation and mitochondrial proton transmembrane transport, whereas HK2 and PFKP are key to glycolysis, suggesting that niclosamide's mechanism of action is dependent on energy metabolism. Furthermore, we found that disruption of Bcl-2 (BCL2) was sensitizing to niclosamide treatment (score= 129; effect= -4.7). To confirm the sensitizing effect of Bcl-2 inhibition on niclosamide treatment, HL60 cells were treated with niclosamide and Bc1-2 inhibitor venetoclax. Pretreatment of HL60 cells with niclosamide led to significantly decreased cell viability upon venetoclax treatment (p<0.0001). Simultaneous treatment of HL60 cells with niclosamide and venetoclax had a synergistic effect on cellular viability, with combination index values <1 at ED50 to ED90 (Chou-Talalay method). Overall, our study identified genes with strong signatures for cell death, nucleotide biosynthesis, mitochondrial function and metabolic processes. Compounds inhibiting genes identified in this screen could be combined with niclosamide for synergistic effect. Targets overrepresented in the surviving population represent sources of resistance to niclosamide therapy. This study provides insight into mechanisms of action and resistance in AML cells treated with niclosamide and novel targets for combination therapy. Disclosures No relevant conflicts of interest to declare. OffLabel Disclosure: Niclosamide is an anthelmintic, FDA approved to treat tapeworm infections.

Soft Sensing of Nonlinear and Multimode Processes Based on Semi-Supervised Weighted Gaussian Regression

This paper presents a new semi-supervised probabilistic density-based regression approach, called Semi-supervised Weighted Gaussian Regression (SWGR), for the soft sensing of nonlinear and multimode industrial processes given a limited number of labeled data samples. In SWGR, different weights are assigned to each training sample based on their similarities to a query sample. Then a local weighted Gaussian density is built for capturing the joint probability of historical samples around the query sample. The training process of parameters in SWGR incorporates both labeled and unlabeled data samples via a maximum likelihood estimation algorithm. In this way, the soft sensor model is able to approximate the nonlinear mechanics of input and output variables and remedy the insufficiency of labeled samples. At last, the output prediction as well as the uncertainty of prediction can be obtained by the conditional distribution. Two case studies validate that the proposed semi-supervised soft sensing method outperforms some recent methods.

Identification of Two-Dimensional Causal Systems With Missing Output Data via Expectation–Maximization Algorithm

For 2-D causal systems, the variables depend both on time, and on spatial coordinates. This article develops two identification algorithms for two-dimensional causal systems. First, a maximum likelihood estimation algorithm is developed for two-dimensional causal systems when there is no missing data. Second, an expectation-maximization based auxiliary model algorithm, and an expectation-maximization based modified Kalman filtering and smoothing algorithm are derived for 2-D causal systems with missing outputs. It is demonstrated that the modified Kalman filtering, and smoothing algorithm is more effective for systems with missing outputs. The effectiveness of these two algorithms is verified by a simulation example.