A BRIEF LITERATURE ON PROBLEMS AND PERSPECTIVES OF MATHEMATICAL AND STOCHASTIC MODELLING

This research article mainly explores on problems and perspectives of mathematical and stochastic modeling. There is a large element of compromise in mathematical modelling. The majority of interacting systems in the real world are far too complicated to model in their entirely.In this research paper an extensive discussion has been made on linear models,nonlinear models,static models,dynamic models.A comparative study is done between the pairs explicit and implicit model,discrete and continuous model,deterministic and probabilistic model. In this talk a brief discussion on different types of models has been proposed and the concept of stages of model building is extensively discussed.Problems of stochastic model building are presented in a lucid manner and this literature is highly helpful for young researchers in stochastic modeling.

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Nested Nonlinear Regression Models

10.1093/oso/9780198505044.003.0015 ◽

2017 ◽

Author(s):

Russell Cheng

Keyword(s):

Regression Models ◽

Free Choice ◽

Model Building ◽

Linear Models ◽

Nonlinear Models ◽

Real Data ◽

Nonlinear Regression Models ◽

Stepwise Method ◽

Forward Stepwise ◽

Nested Regression

Stepwise fitting of nonlinear nested regression models is considered in this chapter. The forward stepwise method of linear model building is used as far as possible. With linear models this is straightforward as there is in principle a free choice of the order that individual terms or factors are selected for inclusion. The only real issue is that sufficient submodels are examined to ensure that those finally selected really are amongst the best. The nonlinear case is not so straightforward, as embeddedness and parameter indeterminacy issues impose restrictions on the order in which steps can be taken to build a valid model, as certain parameters can only be meaningfully included if other specific parameters are definitely present. A systematic way of building valid nonlinear models of increasing complexity is described and illustrated by two examples using real data. A brief review of non-nested model building is also given.

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Turbofan Engine Robust H-∞ Dynamical Output Feedback Control Design Based on LMI Method

Volume 1: Turbo Expo 2005 ◽

10.1115/gt2005-68124 ◽

2005 ◽

Author(s):

Xi Wang ◽

Daoliang Tan ◽

Tiejun Zheng

Keyword(s):

Output Feedback ◽

Linear Models ◽

Nonlinear Models ◽

Design Method ◽

State Space Model ◽

Output Feedback Control ◽

Linear Matrix ◽

Turbofan Engine ◽

Engine Simulation ◽

Linear State

This paper presents an approach to turbofan engine dynamical output feedback controller (DOFC) design in the framework of LMI (Linear Matrix Inequality)-based H∞ control. In combination with loop shaping and internal model principle, the linear state space model of a turbofan engine is converted into that of some augmented plant, which is used to establish the LMI formulations of the standard H∞ control problem with respect to this augmented plant. Furthermore, by solving optimal H∞ controller for the augmented plant, we indirectly obtain the H∞ DOFC of turbofan engine which successfully achieves the tracking of reference instructions and effective constraints on control inputs. This design method is applied to the H∞ DOFC design for the linear models of an advanced multivariate turbofan engine. The obtained H∞ DOFC is only in control of the steady state of this turbofan engine. Simulation results from the linear and nonlinear models of this turbofan engine show that the resulting controller has such properties as good tracking performance, strong disturbance rejection, and satisfying robustness.

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On the Wave Spectrum Generated by Tropical Heating

Journal of the Atmospheric Sciences ◽

10.1175/2011jas3718.1 ◽

2011 ◽

Vol 68 (9) ◽

pp. 2042-2060 ◽

Cited By ~ 9

Author(s):

David A. Ortland ◽

M. Joan Alexander ◽

Alison W. Grimsdell

Keyword(s):

Linear Models ◽

Nonlinear Models ◽

Wave Spectrum ◽

Convective Heating ◽

Temperature Structure ◽

Mean Flow ◽

Quasi Biennial Oscillation ◽

Wave Response ◽

Zonal Wavenumber ◽

The Mean

Abstract Convective heating profiles are computed from one month of rainfall rate and cloud-top height measurements using global Tropical Rainfall Measuring Mission and infrared cloud-top products. Estimates of the tropical wave response to this heating and the mean flow forcing by the waves are calculated using linear and nonlinear models. With a spectral resolution up to zonal wavenumber 80 and frequency up to 4 cpd, the model produces 50%–70% of the zonal wind acceleration required to drive a quasi-biennial oscillation (QBO). The sensitivity of the wave spectrum to the assumed shape of the heating profile, to the mean wind and temperature structure of the tropical troposphere, and to the type of model used is also examined. The redness of the heating spectrum implies that the heating strongly projects onto Hough modes with small equivalent depth. Nonlinear models produce wave flux significantly smaller than linear models due to what appear to be dynamical processes that limit the wave amplitude. Both nonlinearity and mean winds in the lower stratosphere are effective in reducing the Rossby wave response to heating relative to the response in a linear model for a mean state at rest.

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Optimal Attack against Autoregressive Models by Manipulating the Environment

Proceedings of the AAAI Conference on Artificial Intelligence ◽

10.1609/aaai.v34i04.5760 ◽

2020 ◽

Vol 34 (04) ◽

pp. 3545-3552

Author(s):

Yiding Chen ◽

Xiaojin Zhu

Keyword(s):

Predictive Control ◽

Linear Models ◽

Nonlinear Models ◽

Linear Quadratic Regulator ◽

Black Box ◽

Combine System ◽

Linear Quadratic ◽

Time Series Forecast ◽

Forecast Models ◽

Adversarial Attack

We describe an optimal adversarial attack formulation against autoregressive time series forecast using Linear Quadratic Regulator (LQR). In this threat model, the environment evolves according to a dynamical system; an autoregressive model observes the current environment state and predicts its future values; an attacker has the ability to modify the environment state in order to manipulate future autoregressive forecasts. The attacker's goal is to force autoregressive forecasts into tracking a target trajectory while minimizing its attack expenditure. In the white-box setting where the attacker knows the environment and forecast models, we present the optimal attack using LQR for linear models, and Model Predictive Control (MPC) for nonlinear models. In the black-box setting, we combine system identification and MPC. Experiments demonstrate the effectiveness of our attacks.

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modelBuildR: an R package for model building and feature selection with erroneous classifications

PeerJ ◽

10.7717/peerj.10849 ◽

2021 ◽

Vol 9 ◽

pp. e10849

Author(s):

Maximilian Knoll ◽

Jennifer Furkel ◽

Juergen Debus ◽

Amir Abdollahi

Keyword(s):

Feature Selection ◽

Cross Validation ◽

Model Building ◽

Linear Models ◽

Binary Classification ◽

Ground Truth ◽

R Package ◽

Methylation Array ◽

Survival Difference ◽

Error Probabilities

Background Model building is a crucial part of omics based biomedical research to transfer classifications and obtain insights into underlying mechanisms. Feature selection is often based on minimizing error between model predictions and given classification (maximizing accuracy). Human ratings/classifications, however, might be error prone, with discordance rates between experts of 5–15%. We therefore evaluate if a feature pre-filtering step might improve identification of features associated with true underlying groups. Methods Data was simulated for up to 100 samples and up to 10,000 features, 10% of which were associated with the ground truth comprising 2–10 normally distributed populations. Binary and semi-quantitative ratings with varying error probabilities were used as classification. For feature preselection standard cross-validation (V2) was compared to a novel heuristic (V1) applying univariate testing, multiplicity adjustment and cross-validation on switched dependent (classification) and independent (features) variables. Preselected features were used to train logistic regression/linear models (backward selection, AIC). Predictions were compared against the ground truth (ROC, multiclass-ROC). As use case, multiple feature selection/classification methods were benchmarked against the novel heuristic to identify prognostically different G-CIMP negative glioblastoma tumors from the TCGA-GBM 450 k methylation array data cohort, starting from a fuzzy umap based rough and erroneous separation. Results V1 yielded higher median AUC ranks for two true groups (ground truth), with smaller differences for true graduated differences (3–10 groups). Lower fractions of models were successfully fit with V1. Median AUCs for binary classification and two true groups were 0.91 (range: 0.54–1.00) for V1 (Benjamini-Hochberg) and 0.70 (0.28–1.00) for V2, 13% (n = 616) of V2 models showed AUCs < = 50% for 25 samples and 100 features. For larger numbers of features and samples, median AUCs were 0.75 (range 0.59–1.00) for V1 and 0.54 (range 0.32–0.75) for V2. In the TCGA-GBM data, modelBuildR allowed best prognostic separation of patients with highest median overall survival difference (7.51 months) followed a difference of 6.04 months for a random forest based method. Conclusions The proposed heuristic is beneficial for the retrieval of features associated with two true groups classified with errors. We provide the R package modelBuildR to simplify (comparative) evaluation/application of the proposed heuristic (http://github.com/mknoll/modelBuildR).

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Sustainability engagement’s impact on tourism sector performance: linear and nonlinear models

Journal of Organizational Change Management ◽

10.1108/jocm-10-2020-0308 ◽

2021 ◽

Vol ahead-of-print (ahead-of-print) ◽

Author(s):

Amina Buallay ◽

Jasim Al-Ajmi ◽

Elisabetta Barone

Keyword(s):

Significant Relationship ◽

Linear Models ◽

Performance Indicator ◽

Nonlinear Models ◽

Sustainability Reporting ◽

Analytical Framework ◽

Content Type ◽

Return On Equity ◽

Using Data ◽

The Relationship

PurposeThis study aims to investigate the relationship between the level of sustainability reporting and tourism sector’s performance (operational, financial and market).Design/methodology/approachUsing data culled from 1,375 observations from 37 different countries for ten years (2008–2017), an independent variable derived from the environmental, social and governance (ESG score) is regressed against dependent performance indicator variables (return on assets (ROA), return on equity (ROE) and Tobin's Q (TQ)). Two types of control variables complete the regression analysis in this study: firm-specific and macroeconomic.FindingsThe findings elicited from the empirical results of the linear models demonstrate that there is a significant relationship between ESG and operational performance (ROA) and market performance (TQ). However, there is no significant relationship between ESG and financial performance (ROE). Furthermore, the results of the nonlinear models suggest that the relationship between sustainability performance and firm's profitability and valuation is nonlinear (inverted U-shape).Originality/valueThe models in this study presents a valuable analytical framework for exploring sustainability reporting as a driver of performance in the tourism sector's economies. In addition, this study highlights the tourism sector's management lacunae manifesting in terms of the weak nexus between each component of ESG and tourism sector's performance.

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Simple population models

Population Dynamics for Conservation ◽

10.1093/oso/9780198758365.003.0002 ◽

2019 ◽

pp. 21-53

Author(s):

Louis W. Botsford ◽

J. Wilson White ◽

Alan Hastings

Keyword(s):

Continuous Time ◽

Population Model ◽

Linear Models ◽

Nonlinear Models ◽

Single Variable ◽

Fibonacci Series ◽

Basic Concepts ◽

Dependent Growth ◽

Geometric Growth ◽

Simple Models

This chapter introduces basic concepts in population modeling that will be applied throughout the book. It begins with the oldest example of a population model, the rabbit problem, which was described by Leonardo of Pisa (“Fibonacci”) and whose solution is the Fibonacci series. The chapter then explores what is known about simple models of populations (i.e. those with a single variable such as abundance or biomass). The two major classes are: (1) linear models of exponential (or geometric) growth and (2) models of logistic, density-dependent growth. It covers both discrete time and continuous time versions of each of these. These simple models are then used to illustrate several different population dynamic concepts: dynamic stability, linearizing nonlinear models, calculation of probabilities of extinction, and management of sustainable fisheries. Each of these concepts is discussed further in later chapters, with more complete models.

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Oscillatory behavior of two nonlinear microbial models of soil carbon decomposition

Biogeosciences ◽

10.5194/bg-11-1817-2014 ◽

2014 ◽

Vol 11 (7) ◽

pp. 1817-1831 ◽

Cited By ~ 39

Author(s):

Y. P. Wang ◽

B. C. Chen ◽

W. R. Wieder ◽

M. Leite ◽

B. E. Medlyn ◽

...

Keyword(s):

Stability Analysis ◽

Microbial Biomass ◽

Soil Carbon ◽

Nonlinear Model ◽

Linear Models ◽

Nonlinear Models ◽

Oscillatory Behavior ◽

Carbon Pools ◽

Carbon Decomposition ◽

Pool Model

Abstract. A number of nonlinear models have recently been proposed for simulating soil carbon decomposition. Their predictions of soil carbon responses to fresh litter input and warming differ significantly from conventional linear models. Using both stability analysis and numerical simulations, we showed that two of those nonlinear models (a two-pool model and a three-pool model) exhibit damped oscillatory responses to small perturbations. Stability analysis showed the frequency of oscillation is proportional to √(&varepsilon;−1−1) Ks/Vs in the two-pool model, and to √(&varepsilon;−1−1) Kl/Vl in the three-pool model, where &varepsilon; is microbial growth efficiency, Ks and Kl are the half saturation constants of soil and litter carbon, respectively, and /Vs and /Vl are the maximal rates of carbon decomposition per unit of microbial biomass for soil and litter carbon, respectively. For both models, the oscillation has a period of between 5 and 15 years depending on other parameter values, and has smaller amplitude at soil temperatures between 0 and 15 °C. In addition, the equilibrium pool sizes of litter or soil carbon are insensitive to carbon inputs in the nonlinear model, but are proportional to carbon input in the conventional linear model. Under warming, the microbial biomass and litter carbon pools simulated by the nonlinear models can increase or decrease, depending whether &varepsilon; varies with temperature. In contrast, the conventional linear models always simulate a decrease in both microbial and litter carbon pools with warming. Based on the evidence available, we concluded that the oscillatory behavior and insensitivity of soil carbon to carbon input are notable features in these nonlinear models that are somewhat unrealistic. We recommend that a better model for capturing the soil carbon dynamics over decadal to centennial timescales would combine the sensitivity of the conventional models to carbon influx with the flexible response to warming of the nonlinear model.

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Predicting the Electrical Energy Consumption of Electric Arc Furnaces Using Statistical Modeling

Metals ◽

10.3390/met9090959 ◽

2019 ◽

Vol 9 (9) ◽

pp. 959 ◽

Cited By ~ 8

Author(s):

Leo S. Carlsson ◽

Peter B. Samuelsson ◽

Pär G. Jönsson

Keyword(s):

Statistical Modeling ◽

Statistical Models ◽

Linear Models ◽

Nonlinear Models ◽

Electrical Energy ◽

Electric Arc ◽

Future Research ◽

Published Data ◽

Industrial Setting ◽

Input Variables

Statistical modeling, also known as machine learning, has gained increased attention in part due to the Industry 4.0 development. However, a review of the statistical models within the scope of steel processes has not previously been conducted. This paper reviews available statistical models in the literature predicting the Electrical Energy (EE) consumption of the Electric Arc Furnace (EAF). The aim was to structure published data and to bring clarity to the subject in light of challenges and considerations that are imposed by statistical models. These include data complexity and data treatment, model validation and error reporting, choice of input variables, and model transparency with respect to process metallurgy. A majority of the models are never tested on future heats, which essentially renders the models useless in a practical industrial setting. In addition, nonlinear models outperform linear models but lack transparency with regards to which input variables are influencing the EE consumption prediction. Some input variables that heavily influence the EE consumption are rarely used in the models. The scrap composition and additive materials are two such examples. These observed shortcomings have to be correctly addressed in future research applying statistical modeling on steel processes. Lastly, the paper provides three key recommendations for future research applying statistical modeling on steel processes.

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Invariant Causal Prediction for Nonlinear Models

Journal of Causal Inference ◽

10.1515/jci-2017-0016 ◽

2018 ◽

Vol 6 (2) ◽

Cited By ~ 7

Author(s):

Christina Heinze-Deml ◽

Jonas Peters ◽

Nicolai Meinshausen

Keyword(s):

Linear Models ◽

Causal Model ◽

Nonlinear Models ◽

Causal Structure ◽

Nonparametric Tests ◽

Population Projections ◽

World Population ◽

Residual Distribution ◽

Using Data

AbstractAn important problem in many domains is to predict how a system will respond to interventions. This task is inherently linked to estimating the system’s underlying causal structure. To this end, Invariant Causal Prediction (ICP) [1] has been proposed which learns a causal model exploiting the invariance of causal relations using data from different environments. When considering linear models, the implementation of ICP is relatively straightforward. However, the nonlinear case is more challenging due to the difficulty of performing nonparametric tests for conditional independence.In this work, we present and evaluate an array of methods for nonlinear and nonparametric versions of ICP for learning the causal parents of given target variables. We find that an approach which first fits a nonlinear model with data pooled over all environments and then tests for differences between the residual distributions across environments is quite robust across a large variety of simulation settings. We call this procedure “invariant residual distribution test”. In general, we observe that the performance of all approaches is critically dependent on the true (unknown) causal structure and it becomes challenging to achieve high power if the parental set includes more than two variables.As a real-world example, we consider fertility rate modeling which is central to world population projections. We explore predicting the effect of hypothetical interventions using the accepted models from nonlinear ICP. The results reaffirm the previously observed central causal role of child mortality rates.

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