BL Model Research Articles

Anisotropic pressure effect on central EOS of PSR J0740+6620 in the light of dimensionless TOV equation

Abstract It is generally agreed upon that the pressure inside a neutron star is isotropic. However, a strong magnetic field or superfluidity suggests that the pressure anisotropy may be a more realistic model. We derived the dimensionless TOV equation for anisotropic neutron stars based on two popular models, namely the BL model and the H model, to investigate the effect of anisotropy. Similar to the isotropic case, the maximum mass $M_{max}$ and its corresponding radius $R_{Mmax}$ can also be expressed linearly by a combination of radial central pressure $p_{rc}$ and central energy density $\varepsilon_{c}$, which is insensitive to the equation of state (EOS). We also found that the obtained central EOS would change with different values of $\lambda_{BL}$ ($\lambda_{H}$), which controls the magnitude of the difference between the transverse pressure and the radial pressure. Combining with observational data of PSR J0740+6620 and comparing to the extracted EOS based on isotropic neutron star, it is shown that in the BL model, for $\lambda_{BL}$ = 0.4, the extracted central energy density $\varepsilon_{c}$ changed from 546 -- 1056 MeV/fm$^{3}$ to 510 -- 1005 MeV/fm$^{3}$, and the extracted radial central pressure $p_{rc}$ changed from 87 -- 310 MeV/fm$^{3}$ to 76 -- 271 MeV/fm$^{3}$. For $\lambda_{BL}$ = 2, the extracted $\varepsilon_{c}$ and $p_{rc}$ changed to 412 -- 822 MeV/fm$^{3}$ and 50 -- 165 MeV/fm$^{3}$, respectively. In the H model, for $\lambda_{H}$ = 0.4, the extracted $\varepsilon_{c}$ changed to 626 -- 1164 MeV/fm$^{3}$, and the extracted $p_{rc}$ changed to 104 -- 409 MeV/fm$^{3}$. For $\lambda_{H}$ = 2, the extracted $\varepsilon_{c}$ decreased to 894 -- 995 MeV/fm$^{3}$, and the extracted $p_{rc}$ changed to 220 -- 301 MeV/fm$^{3}$.

Impacts of dynamic stall on engineering model predictions of wind turbines loads under design load cases

The present work is aimed specifically at providing an order of estimate of the load changes due to the inclusion of unsteady aerodynamic effects. Two dynamic stall models are assessed: (1) the incompressible version of the Beddoes-Leishman model and (2) the newly developed IAG dynamic stall model. Two sets of attached flow constants are considered, based on the flat plate approximations according to Jones and the original constants adopted by the IAG model. On top of that, the influence of the impulsive contributions is also highlighted. This work is aimed at filling the gap of the state-of-the-art since at present there is no available study which compares the impacts of using the IAG model against the BL model in real design load case (DLC) scenarios required for wind turbine designs, considering it is very important to estimate loads changes depending on the engineering model being used. It will significantly influence the design of wind turbine structures, especially blades and the strength of the materials for the manufacturing process. Toward the end, providing a proper estimate into the load changes will greatly influence the cost required in blade design and manufacturing. The present study investigates the influence of dynamic stall modeling on the loads acting on wind turbines with various sizes, ranging from 2.5 MW to 8 MW of the rated power. In the studies, both onshore and offshore conditions are considered to obtain the full picture into the engineering modeling effects.

Blade aerodynamic model for thick airfoils of the series NACA00xx in pre-stall conditions

Blade aerodynamic modeling is needed for design, control, and aeroelastic studies of wind turbines. The ultimate aim of this study is to establish a blade aerodynamic model with well-quantified accuracy for thick airfoils, predominant in wind turbine blades. The study is limited to pre-stall conditions, involving only attached and trailing-edge separated flows. The account of dynamic stall will be considered in further studies. The analysis of the Glasgow University database on the family of symmetrical airfoils NACA 00xx (xx = 12,…,30) has been made by considering particularly the 2D spatiotemporal contours of the surface pressure coefficients on the suction side. Such contours provide a clear visualization of flow regime type (attached, separated or stalled) and, therefore, allow the selection of oscillatory test cases in attached or trailing-edge separated flows. The aerodynamic model of the normal force coefficient is established by improving the Beddoes–Leishman BL model. An important modification is carried on the calculation of the delayed angle of attack using the Goman–Khrabrov model, instead of the complex original procedure. There is a new aerodynamic component for simulating the trailing-edge separation. The present model, although limited to pre-stall conditions, involves ten parameters for the unsteady aerodynamic behavior. They can be obtained with the global optimization of the deviations between experimental results and model predictions. Previous optimization studies of the parameters of the BL model involve all flow regimes for test cases and do not lead to conclusive results. The parameter values obtained in the present study show a coherent and physics-expected variation with airfoil thickness that is not negligible.

B/PS bulges and barlenses from a kinematic viewpoint – II

ABSTRACT Internal dynamics and kinematics of galaxies have imprints on the line-of-sight velocity distribution (LOSVD). Gauss–Hermite parametrization allows one to identify the kinematics features of the system in terms of skewness (h3) and broadness (h4) deviations of an LOSVD. Such a method provides information about the type of orbits since a $h_3\!-\!\overline{V}$ correlation is a sign of elongated orbits, and the anticorrelation is a sign of circular or near-circular orbits. In previous works, analysis of the $h_3\!-\!\overline{V}$ relation provided a tool to identify a hidden bar or B/PS bulge (edge-on, PA = 90°) and to probe their strength. We prepared two N-body galaxy models with clear B/PS bulges: one has an ordinary bar (the X model), and the second one has a barlens embedded into a bar (the BL model) to investigate the mechanism of formation of h3 features at any position of an observer. We show that the $h_3\!-\!\overline{V}$ correlation appears in the regions where bar and disc particles are mixing. We also reveal that the model with a barlens has an $h_3\!-\!\overline{V}$ anticorrelation in the centre, and we show that barlens-specific orbits are responsible for this signal. Moreover, this feature can be observed only for galaxies with compact bulges and barlenses. The results of this work are applicable for the interpretation of future integral-field unit (IFU) data for real galaxies with B/PS bulges, especially for objects with barlenses.

Analysis Of Vocational Students' Critical Thinking Skills Using The OER-Assisted Blended Learning

One of the competencies in 21st-century learning is critical thinking (CT). Many lessons carried out by teachers do not touch this ability because of the lack of learning innovation and the use of IT-based learning resources. This study aimed to determine and analyze the effect of using the OER (Open Educational Resources)-assisted Blended learning model on vocational students' critical thinking skills. The type of research used mixed methods with embedded design. The samples were 54 students of the Multimedia program at VHS (Vocational High School) in Surakarta, Indonesia, and the instrument used a test based on critical thinking skills, according to Facione. Quantitative data analysis used an independent t-test and the qualitative data used criteria for achieving students' critical thinking skills. The results of the study support the expectation that the BL model assisted by OER has a significant effect on student learning outcomes and that students have excellent critical thinking skills on the indicators of interpretation, analysis, evaluation, explanation, and inference. On the other hand, students are weak in self-regulation.

A Novel Black-Litterman Model with Time-Varying Covariance for Optimal Asset Allocation of Pension Funds

The allocation of pension funds has important theoretical value and practical significance, which improves the level of pension investment income, achieves the maintenance and appreciation of pension funds, and resolves the pension payment risk caused by population aging. The asset allocation of pension funds is a long-term asset allocation problem. Thus, the long-term risk and return of the assets need to be estimated. The covariance matrix is usually adopted to measure the risk of the assets, while calculating the long-term covariance matrix is extremely difficult. Direct calculations suffer from the insufficiency of historical data, and indirect calculations accumulate short-term covariance, which suffers from the dynamic changes of the covariance matrix. Since the returns of main assets are highly autocorrelated, the covariance matrix of main asset returns is time-varying with dramatic dynamic changes, and the errors of indirect calculation cannot be ignored. In this paper, we propose a novel Black–Litterman model with time-varying covariance (TVC-BL) for the optimal asset allocation of pension funds to address the time-varying nature of asset returns and risks. Firstly, the return on assets (ROA) and the covariance of ROA are modeled by VARMA and GARCH, respectively. Secondly, the time-varying covariance estimation of ROA is obtained by introducing an effective transformation of the covariance matrix from short-term to long-term. Finally, the asset allocation decision of pension funds is achieved by the TVC-BL model. The results indicate that the proposed TVC-BL pension asset allocation model outperforms the traditional BL model. When the risk aversion coefficient is 1, 1.5, and 3, the Sharp ratio of pension asset allocation through the TVC-BL pension asset allocation model is 13.0%, 10.5%, and 12.8% higher than that of the traditional BL model. It helps to improve the long-term investment returns of pension funds, realize the preservation and appreciation of pension funds, and resolve the pension payment risks caused by the aging of the population.

Comparative evaluation of classical and SARIMA-BL time series hybrid models in predicting monthly qualitative parameters of Maroon river

In the present study, seasonal autoregressive integrated moving average (SARIMA) time series models, nonlinear BL model, multi-layer perceptron artificial neural network and SARIMA-bilinear hybrid models were employed to predict the quality parameters of total dissolved solids (TDS), sodium adsorption ratio (SAR), electrical conductivity (EC) in the Maroon basin in Khuzestan Province. For fitting the mentioned models, the monthly data were used for the calibration (1970–2000), confirmation (2001–2011) and prediction of the model (2012–2018). The appropriate models of SARIMA, the bilinear BL and SARIMA-BL hybrid models of the rationalized parameters of the above-mentioned quality parameters were selected based on the adequacy tests, such as the Akaike criterion and the independence test of the model residuals (Ljung–Box). To determine the effective input parameters of the network, the partial mutual information (PMI) algorithm was used to model the three monthly EC, TDS, and SAR parameters. Also, for input and output layers linear transfer function and for hidden layer various active functions with the back-propagation learning algorithm were used for modeling and predicting this water quality parameters. Comparison of the models showed the tangible superiority of SARIMA-bilinear hybrid models than the artificial neural network with effective input parameters based on PMI algorithm, SARIMA and bl models in performed prediction of the all three monthly qualitative parameters in all three stages, training-validation-test in Maroon basin.

Flood Forecasting Method Based on Improved VMD-FOS-QR-RBL

To improve the accuracy, reliability and validity of flood prediction models, this study proposes a regularized broad learning (RBL) model based on an improved variational mode decomposition (VMD). Firstly, grey correlation analysis was used to improve the endpoint effect of the VMD and the particle swarm optimisation (PSO) algorithm was used to optimise the VMD parameters. Then, using orthogonal triangular decomposition (QR), redefining the hidden layer output of BL model and adding forgettable online sequence learning mechanism (FOS) to construct online sequence BL (FOS-QR-RBL), which can significantly improve the computational efficiency of BL model. Finally, a flood forecasting method based on improved VMD-FOS-QR-RBL was constructed by combining the FOS-QR-RBL with the improved VMD and applying it to regional flood forecasting. The experimental results show that the computational efficiency of FOS-QR-RBL is improved by 35% and 23.68% compared with RBL and QR-RBL, respectively. The mean absolute error (MAE) of the coupling model of VMD and FOS-QR-RBL is reduced by 80.30% and 84.10% respectively, and the nash efficiency coefficient ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\text{E}_{\mathrm {ns}}$ </tex-math></inline-formula> ) is increased by 15.51% and 28.16% respectively, compared with that of the coupling model of FOS-QR-RBL with ensemble empirical mode decomposition (EEMD) and adaptive noise complete ensemble empirical mode decomposition (CEEMDAN). The results of the optimal operation based on VMD-FOS-OR-RBL show that the model can effectively reduce the economic losses caused by regional flooding.

I−Love−C relation for an anisotropic neutron star

One of the most common assumptions has been made that the pressure inside the star is isotropic in nature. However, the pressure is locally anisotropic in nature which is a more realistic case. In this study, we investigate certain properties of anisotropic neutron stars with the scalar pressure anisotropy model. Different perfect fluid conditions are tested within the star with the relativistic mean-field model equation of states (EOSs). The anisotropic neutron star properties such as mass ($M$), radius ($R$), compactness ($C$), Love number ($k_2$), dimensionless tidal deformability ($\Lambda$), and the moment of inertia ($I$) are calculated. The magnitude of the quantities as mentioned above increases (decreases) with the positive (negative) value of anisotropy except $k_2$ and $\Lambda$. The Universal relation $I-$Love$-C$ is calculated with almost 58 EOSs spans from relativistic to non-relativistic cases. We observed that the relations between them get weaker when we include anisotropicity. With the help of the GW170817 tidal deformability limit and radii constraints from different approaches, we find that the anisotropic parameter is less than 1.0 if one uses the BL model. Using the universal relation and the tidal deformability bound given by the GW170817, we put a theoretical limit for the canonical radius, $R_{1.4}=10.74_{-1.36}^{+1.84}$ km, and the moment of inertia, $I_{1.4} = 1.77_{-0.09}^{+0.17}\times10^{45}$ g cm$^2$ with 90\% confidence limit for isotropic stars. Similarly, for anisotropic stars with $\lambda_{\rm BL}=1.0$, the values are $R_{1.4}=11.74_{-1.54}^{+2.11}$ km, $I_{1.4} = 2.40_{-0.08}^{+0.17} \times10^{45}$ g cm$^2$ respectively.

The evaluation of stress on bone level and tissue level short implants: A Finite Element Analysis (FEA) study

PurposeThis study aimed to evaluate the difference between the stress level and distribution around the BL and TL short implants, and their surrounding structures, using finite element analysis. MethodsTwo different study models were constructed: BL model and TL model. Two dental implant systems (ITI (Straumann, Waldenburg, Switzerland) and NTA Short) with a diameter of 4.1 mm and 4 mm and with a length of 6 mm were used in this study. In each model, implants were placed in the mandibular 1st molar region. The von Mises stress and maximum principal (tensile) and minimum principal (compressive) stresses were evaluated. ResultsThe highest stress values recorded in the BL implants (von Mises: 342.77 MPa), in the peri‑implant bone around the BL implants (maximum principal stress: 114.1 MPa), as a result of oblique loading, and overall stress values were found to be higher in the BL model. However, these measured values appeared to be low to cause a fracture, when considering the yield strengths of the materials and bone. ConclusionsThe stress values were higher in the BL model, but not high enough to cause failure. Short implants could be an effective method of treatment for patients unsuitable for advanced surgical techniques.

Research on Black-Litterman Index Enhancement Strategy——Based on the Ledoit-Wolf Compression Estimation Method to Optimize the CSI 500 Index Enhancement Strategy

Financial risks may often lead to significant losses. A reasonable capital management model can prevent financial risks and enhance financial services to the real economy. The Black-Litterman model can reduce risks through asset allocation. This paper uses the Black-Litterman model to construct an enhanced strategy applied to the CSI 500 Index, and selects the backtest from December 1, 2019 to December 1, 2021. Through the strategy backtest, it can be found that: whether it is considered or not Transaction costs, using analysts&amp;rsquo; consensus target price as the input point of view of the BL model, can provide excess returns for the index enhancement strategy under relatively stable conditions within the sample interval, and improve the sharpness ratio, information ratio, maximum drawdown, etc. Within the risk-return parameters. &#x0D; &#x0D; In order to solve the problem of model instability and extreme values of configuration weights in the first step, this paper adjusts the covariance based on the Leodit-wolf compression estimation, thereby optimizing the exponential enhancement model. The backtest results showed that although the volatility and maximum drawdown of the optimized enhanced index model increased slightly, it showed a higher excess return rate and information ratio. Therefore, the BL model optimized based on the compression estimation method can make the model applicable to a wider range, and can be extended to large-scale assets and multi-asset allocation, so that investors have more choices in quantitative investment strategies.

Retrievals of Arctic sea ice melt pond depth and underlying ice thickness using optical data

Melt pond is a distinctive characteristic of the summer Arctic, which affects energy balance in the Arctic system. The Delta-Eddington model (BL) and Two-strEam rAdiative transfer model (TEA) are employed to retrieving pond depth Hp and underlying ice thickness H i according to the ratio X of the melt-pond albedo in two bands. Results showed that when λ1 = 359 nm and λ2 = 605 nm, the Pearson’s correlation coefficient r between X and Hp is 0.99 for the BL model. The result of TEA model was similar to the BL model. The retrievals of Hp for the two models agreed well with field observations. For H i, the highest r (0.99) was obtained when λ 1 = 447 nm and λ2 = 470 nm for the BL model, λ 1 = 447 nm and λ 2 = 451 nm for the TEA model. Furthermore, the BL model was more suitable for the retrieval of thick ice (0 Citation :Zhang H, Yu M, Lu P, et al. Retrievals of Arctic sea ice melt pond depth and underlying ice thickness using optical data. Adv Polar Sci, 2021, 32(2): 105-117, doi: 10.13679/j.advps.2021.0021

CB-BL model (challenge based on blended learning) for mathematical creativity

To improve students’ mathematical creativity, which is the competency in the industrial revolution 4.0 and is in 4-C and 5-C, it is necessary to have learning innovation for it. Challenge-based on blended learning (CB-BL) is a framework that integrates challenge-based learning and blended learning so that the need for face-to-face learning and learning using IT is satisfied. This research is a literature study research with the research aims to describe the CB–BL model steps that consist of The Big Idea, Essential Question, The Challenges, Solution – Action, and Assessment. The Big Idea and Essential Question are carried out face-to-face such that students know the learning objectives through solving challenges. The Challenges and Solution – Action are carried out online by providing the instructional video, textbook, and discussion forum integrated with the learning management system to solve the challenges. The last step is Assessment, which consists of Publishing Student Sample and Reflection, carried out face-to-face. In this stage, students publish the results of solving challenges and through self-reflection based on learning experiences they have carried out. By integrating challenge-based learning and blended learning, mathematics creativity can be developed. It is also a learning model that can be applied during New Normal.

A simplified model for transition prediction applicable to wind-turbine rotors

Abstract. This work aims to develop a simple framework for transition prediction over wind-turbine blades, including effects of the blade rotation and spanwise velocity without requiring fully three-dimensional simulations. The framework is based on a set of boundary-layer equations (BLEs) and parabolized stability equations (PSEs), including rotation effects. An important element of the developed BL method is the modeling of the spanwise velocity at the boundary-layer edge. The two analyzed wind-turbine geometries correspond to a constant airfoil and the DTU 10-MW Reference Wind Turbine blades. The BL model allows an accurate prediction of the chordwise velocity profiles. Further, for regions not too close to the stagnation point and root of the blade, profiles of the spanwise velocity agree with those from Reynolds-averaged Navier–Stokes (RANS) simulations. The model also allows predicting inflectional velocity profiles for lower radial positions, which may allow crossflow transition. Transition prediction is performed at several radial positions through an “envelope-of-envelopes” methodology. The results are compared with the eN method of Drela and Giles, implemented in the EllipSys3D RANS code. The RANS transition locations closely agree with those from the PSE analysis of a 2D mean flow without rotation. These results also agree with those from the developed model for cases with low 3D and rotation effects, such as at higher radial positions and geometries with strong adverse pressure gradients where 2D Tollmien–Schlichting (TS) waves are dominant. However, the RANS and PSE 2D models predict a later transition in the regions where 3D and rotation effects are non-negligible. The developed method, which accounts for these effects, predicted earlier transition onsets in this region (e.g., 19 % earlier than RANS at 26 % of the radius for the constant-airfoil geometry) and shows that transition may occur via highly oblique modes. These modes differ from 2D TS waves and appear in locations with inflectional spanwise velocity. However, except close to the root of the blade, crossflow transition is unlikely since the crossflow velocity is too low. At higher radial positions, where 3D and rotation effects are weaker and the adverse pressure gradient is more significant, modes with small wave angles (close to 2D) are found to be dominant. Finally, it is observed that an increase in the rotation speed modifies the spanwise velocity and increases the Coriolis and centrifugal forces, shifting the transition location closer to the leading edge. This work highlights the importance of considering the blade rotation and the three-dimensional flow generated by that in transition prediction, especially in the inner part of the blade.

Portfolio Selection Based on BP Neural Network and Black-Litterman Model

With the revolution of technology, the rapid development of machine learning has brought new impetus to the development of finance. Combining machine learning technology with traditional investment portfolio theoretical models can effectively reduce investment risks and increase returns. The Mean- Variance model proposed by the pioneer of contemporary portfolio theory, Markowitz, defines risk as the volatility of the rate of return. For the first time, the method of mathematical statistics is applied to the study of portfolio selection. This method makes the multi-objective optimization of returns and risks achieve the best balance effect. However, because this model is too sensitive to the input value, this article will use the Black-Litterman model and use the BP neural network to make predictions for the BL model's view matrix. The empirical results show that the improved Black-Litterman model obtains excess returns compared to the Mean-Variance model and other models, which optimizes the investment portfolio.

Genomic selection of forage agronomic traits in winter wheat

AbstractGenomic selection (GS) can improve genetic gain of complex traits in plant breeding. Phenotyping agronomic traits of winter wheat (Triticum aestivum L.) for dual‐purpose use is expensive and time‐consuming. In this study, we compared the prediction accuracies of four GS models (RR‐BLUP, GBLUP, GAUSS, and BL) for forage yield (FY), plant height (PH) and heading date (HD) of the hard winter wheat diversity panel (n = 298) using random and stratified sampling methods. In addition, we determined the appropriate training population (TP) size and marker density for GS of the traits. Moderate to high prediction accuracies ranging from 0.66 to 0.69 for FY, 0.46 to 0.49 for PH, and 0.71 to 0.74 for HD were observed for the GS models. However, the sampling method had little or no impact on prediction accuracy. The RR‐BLUP, GBLUP, and GAUSS models produced slightly greater prediction accuracies than BL for all traits studied. Prediction accuracies increased with increasing TP size and marker density in all the GS models tested. However, increase of prediction accuracy started to plateau at nTP = 180 lines and 1,000; 1,500; or 3,000 SNPs suggesting that the minimum TP size and marker density were about 180 lines and 1,000 or more SNPs, depending on the model and trait. The impact of TP size on prediction accuracy was greater for RR‐BLUP, GAUSS, and GBLUP than for BL model. This study suggests that RR‐BLUP, GBLUP, and GAUSS are viable models for selecting the forage agronomic traits during dual‐purpose wheat breeding.

EDUCATORS’ PERCEPTION OF BLENDED LEARNING MODELS ON MATHEMATICS LEARNING

The main objective of this research is to analyze and compare educators' perceptions of the application of the BL model in mathematics learning. The stages of research carried out in this study began with the manufacture of questionnaires both for educators and students. Next the researchers asked one of the lecturers and mathematics teachers to validate it. Based on the results of the validation conducted, the questionnaire was declared to be appropriate to be used by correcting a selection of words so it would not have a double meaning. In the next stage, researchers distributed the questionnaire to educators in schools and universities in several provincies in Indonesia After obtaining the data, researchers analyzed the data using WinStep. The results of the questionnaire showed that there was one question on the questionnaire that was difficult to be agreed upon by lecturers and teachers, namely the 16th statement that revealed students felt they needed more time to complete online assignments when dealing with graphics / diagrams / tables / other on mathematical material

A change in oil viscosity during field development influences the Muskat — Leverett model solutions

It has been repeatedly noted in the literature that the physic-chemical properties of oil extracted at different stages of exploitation can vary significantly. We have investigated a modification of the Buckley — Leverett model (BL model) in our previous works to study the influence of oil viscosity change on the solutions of twophase filtration problems. In this model the change in oil viscosity was modeled using the dependence of the viscosity ratio on dynamic water saturation. The solutions to the linear displacement problem at a given total filtration rate were analyzed. This article considers a modification of the two-phase isothermal Muskat — Leverett model (ML model), which takes into account capillary forces. The influence of oil viscosity change in the process of oil production on the structure of the solution is studied numerically a case study of the problem of radial displacement with a given pressure drop.

Economic Cycle and the Large-Scale Asset Allocation Strategy of Chinese National Social Security Fund

Based on the quarterly data of GDP from 1993 to 2018, this paper forecasts that China is in a recession period at present, and the trend of economic cycle in the next ten years is divided into five stages, which can be approximately regarded as a sequential evolution of Juglar-cycle. Combined with the prediction results, the BL model is used to allocate the assets of the three parts of the social security fund: stock, debt and cash. The results show that: 1) when only considering cash, stock and bond, the impact of economic cycle on the allocation of large categories of social security fund assets is not severe, but it has a significant impact on the style and industry rotation of large categories of assets; 2) according to the calculation of the model, the cash assets of social security fund are deposited. In case of over matching, it should be adjusted to the level near the minimum limit of 10%.

Parameter estimation of Markov switching bilinear model using the (EM) algorithm

Markov Switching models have known a strong growth since their introduction by James Hamilton in the late 1980’s. These models are used as an essential tool for the analysis of the economic cycles. In this paper, we are interested in a class of bilinear models with markov-switching regime MS−BL. These models first appeared in Bibi and Aknouche (2010). Parameter estimation via maximum likelihood (ML) of the MS−BL model has been considered in Bibi and Ghazel (2015). However, construction and numerical maximization in the approach proposed by Bibi and Ghazel (2015) are computationally intractable. Hence, we propose an expectation–maximization EM procedure that provides an alternative method for maximizing the likelihood function in such situations. Convergence and consistency of the EM algorithm are discussed in this context. Finally, a Monte Carlo study is presented and two real data examples are proposed.