Likelihood Research Articles

Expected First Occurrence Time of Uncertain Future Events in One-Dimensional Linear Systems

The rapid advancement of machine learning algorithms has significantly enhanced tools for monitoring system health, making data-driven approaches predominant in Prognostics and Health Management (PHM). In contrast, model-based approaches have seen modest progress, as they are often constrained by the need for prior knowledge of specific governing equations, limiting their applicability to a wide range of problems. Recently, rigorous theoretical foundations have been established to extend dynamical systems theory by incorporating prognosis of uncertain events. This article leverages this formal framework to introduce and demonstrate a fundamental mathematical result for one-dimensional linear dynamical systems. The presented theorem offers an exact expression for calculating the expected time at which an event will first occur in the future. Unlike typical thresholds, this event is triggered by a hazard zone, defined as an uncertain event likelihood function over the system's state space. Applications of this theorem can be found in implementing real-time prognostic frameworks, where it is crucial to quickly estimate the magnitude of impending failures. Emphasis is placed on minimizing computational burden to facilitate prognostic decision-making.

Single Station Seismic Event Location With a Neural‐Guided Mixture Model: Model‐Based Weighting for Improved Accuracy

AbstractIn many regions of the world with complex seismic activity, the availability of seismic stations is limited. This motivated us to develop a probabilistic method for single‐station earthquake location. Single station location methods may provide an efficient and cost‐effective way to monitor small‐magnitude seismic events over large areas of the world. This method relies on the outputs of two neural networks, which predict the initial spatial location of specific areas (convolutional) used in the prior probability and seismic phase recognition (combination of convolutional and Long Short Term Memory) used in the likelihood function. These estimates and a 3D travel time likelihood function are used in a Bayesian framework. In this model, the initial spatial location is expressed as a Multivariate Gaussian Mixture Model, and the likelihood function refines the localization using the arrival time difference between S and P waves in a 3D space. The likelihood function uses a Fast Marching Eikonal method with a 3D velocity structure. We tested this method on the Ridgecrest 2019 seismic sequence (Mw = 7.1) and on West Texas and found promising results for locating with a single station. Our approach demonstrates that using a Bayesian model with strong priors obtained through deep learning algorithms is effective. Our model can predict epicenter and depth with mean errors of 7.55 km in longitude, 13.54 km in latitude and 0.630 km in depth at the two different locations studied. This method demonstrates the effectiveness of combining deep learning for initial estimations with probabilistic 3D location refinement.

Stochastic Nonlinear Model Updating in Structural Dynamics Using a Novel Likelihood Function within the Bayesian-MCMC Framework

The study presents a novel approach for stochastic nonlinear model updating in structural dynamics, employing a Bayesian framework integrated with Markov Chain Monte Carlo (MCMC) sampling for parameter estimation by using an approximated likelihood function. The proposed methodology is applied to both numerical and experimental cases. The paper commences by introducing Bayesian inference and its constituents: the likelihood function, prior distribution, and posterior distribution. The resonant decay method is employed to extract backbone curves, which capture the non-linear behaviour of the system. A mathematical model based on a single degree of freedom (SDOF) system is formulated, and backbone curves are obtained from time response data. Subsequently, MCMC sampling is employed to estimate the parameters using both numerical and experimental data. The obtained results demonstrate the convergence of the Markov chain, present parameter trace plots, and provide estimates of posterior distributions of updated parameters along with their uncertainties. Experimental validation is performed on a cantilever beam system equipped with permanent magnets and electromagnets. The proposed methodology demonstrates promising results in estimating parameters of stochastic non-linear dynamical systems. Through the use of the proposed likelihood functions using backbone curves, the probability distributions of both linear and non-linear parameters are simultaneously identified. Based on this view, the necessity to segregate stochastic linear and non-linear model updating is eliminated.

Real-time modeling of transient crustal deformation through the quantification of uncertainty deduced from GNSS data

We propose a new method for real-time uncertainty monitoring of earthquake and volcano source models using data from the global navigation satellite system (GNSS) and explore its application concerning observation station placement. The Geospatial Information Authority of Japan operates two main types of GNSS earth observation network system (GEONET) coordinates for crustal deformation monitoring on different time scales: post-processing analysis values and real-time GEONET analysis system for rapid deformation monitoring (REGARD). REGARD uses the Markov Chain Monte Carlo (MCMC) method termed real-time automatic uncertainty estimation of a coseismic single rectangular model using GNSS data (RUNE) for single rectangular fault model estimation to handle uncertainty. Thus far, no GNSS monitoring system can automatically detect transient crustal deformation events, such as volcanic activity and earthquake swarms, on timescales of a day or less. We extended RUNE and developed a core program for a new monitoring system for earthquake and volcanic source models and their uncertainties. Our program achieved automatic and stable MCMC utilization for rectangular fault, dike, Mogi, and spheroid models by increasing the computational speed, improving search efficiency, and adjusting hyperparameters. The program automatically determines the standard deviation of the likelihood function assuming a normal distribution with weights for each observation station. The calculation time was within 15 s for 1 × 106 samples on a standard 1U server. We assessed the reliability of the developed method using synthetic and observed GNSS data from the 2015 Sakurajima volcanic event. The results were consistent with the assumed model and previous studies and indicated an advantage in automatically quantifying uncertainty in a short computation time. Based on MCMC samples, we developed a new visualization algorithm to indicate areas on a map in which the number of observation stations should be expanded. We assessed the reliability using data from the 2023 Noto Peninsula earthquake [Mj 6.5]. The results indicate that the algorithm is helpful in studying the placement of stations. The above model extensions and their application are essential to achieve a rapid quantitative understanding of disaster events near urban areas and for utilizing this information in emergency response activities.Graphical

Benchmarking and building DNA binding affinity models using allele-specific and allele-agnostic transcription factor binding data

BackgroundTranscription factors (TFs) bind to DNA in a highly sequence-specific manner. This specificity manifests itself in vivo as differences in TF occupancy between the two alleles at heterozygous loci. Genome-scale assays such as ChIP-seq currently are limited in their power to detect allele-specific binding (ASB) both in terms of read coverage and representation of individual variants in the cell lines used. This makes prediction of allelic differences in TF binding from sequence alone desirable, provided that the reliability of such predictions can be quantitatively assessed.ResultsWe here propose methods for benchmarking sequence-to-affinity models for TF binding in terms of their ability to predict allelic imbalances in ChIP-seq counts. We use a likelihood function based on an over-dispersed binomial distribution to aggregate evidence for allelic preference across the genome without requiring statistical significance for individual variants. This allows us to systematically compare predictive performance when multiple binding models for the same TF are available. To facilitate the de novo inference of high-quality models from paired-end in vivo binding data such as ChIP-seq, ChIP-exo, and CUT&Tag without read mapping or peak calling, we introduce an extensible reimplementation of our biophysically interpretable machine learning framework named PyProBound. Explicitly accounting for assay-specific bias in DNA fragmentation rate when training on ChIP-seq yields improved TF binding models. Moreover, we show how PyProBound can leverage our threshold-free ASB likelihood function to perform de novo motif discovery using allele-specific ChIP-seq counts.ConclusionOur work provides new strategies for predicting the functional impact of non-coding variants.

Joint inference for telemetry and spatial survey data.

Data integration, the joint statistical analysis of data from different observation platforms, is pivotal for data-hungry disciplines such as spatial ecology. Pooled data types obtained from the same underlying process, analyzed jointly, can improve both precision and accuracy in models of species distributions and species-habitat associations. However, the integration of telemetry and spatial survey data has proved elusive because of the fundamentally different analytical approaches required by these two data types. Here, "spatial survey" denotes a survey that records spatial locations and has no temporal structure, for example, line or point transects but not capture-recapture or telemetry. Step selection functions (SSFs-the canonical framework for telemetry) and habitat selection functions (HSFs-the default approach to spatial surveys) differ in not only their specifications but also their results. By imposing the constraint that microscopic mechanisms (animal movement) must correctly scale up to macroscopic emergence (population distributions), a relationship can be written between SSFs and HSFs, leading to a joint likelihood using both datasets. We implement this approach using maximum likelihood, explore its estimation precision by systematic simulation, and seek to derive broad guidelines for effort allocation in the field. We find that complementarities in spatial coverage and resolution between telemetry and survey data often lead to marked inferential improvements in joint analyses over those using either data type alone. The optimal allocation of effort between the two methods (or the choice between them, if only one can be selected) depends on the overall effort expended and the pattern of environmental heterogeneity. Examining inferential performance over a broad range of scenarios for the relative cost between the two methods, we find that integrated analysis usually offers higher precision. Our methodological work shows how to integrate the analysis of telemetry and spatial survey data under a novel joint likelihood function, using traditional computational methods. Our simulation experiments suggest that even when the relative costs of the two methods favor the deployment of one field approach over another, their joint use is broadly preferable. Therefore, joint analysis of the two key methods used in spatial ecology is not only possible but also computationally efficient and statistically more powerful.

Machine learning-optimized non-invasive brain stimulation and treatment response classification for major depression

Background/ObjectivesTranscranial direct current stimulation (tDCS) is a non-invasive brain stimulation intervention that shows promise as a potential treatment for depression. However, the clinical efficacy of tDCS varies, possibly due to individual differences in head anatomy affecting tDCS dosage. While functional changes in brain activity are more commonly reported in major depressive disorder (MDD), some studies suggest that subtle macroscopic structural differences, such as cortical thickness or brain volume reductions, may occur in MDD and could influence tDCS electric field (E-field) distributions. Therefore, accounting for individual anatomical differences may provide a pathway to optimize functional gains in MDD by formulating personalized tDCS dosage.MethodsTo address the dosing variability of tDCS, we examined a subsample of sixteen active-tDCS participants’ data from the larger ELECT clinical trial (NCT01894815). With this dataset, individualized neuroimaging-derived computational models of tDCS current were generated for (1) classifying treatment response, (2) elucidating essential stimulation features associated with treatment response, and (3) computing a personalized dose of tDCS to maximize the likelihood of treatment response in MDD.ResultsIn the ELECT trial, tDCS was superior to placebo (3.2 points [95% CI, 0.7 to 5.5; P = 0.01]). Our algorithm achieved over 90% overall accuracy in classifying treatment responders from the active-tDCS group (AUC = 0.90, F1 = 0.92, MCC = 0.79). Computed precision doses also achieved an average response likelihood of 99.981% and decreased dosing variability by 91.9%.ConclusionThese findings support our previously developed precision-dosing method for a new application in psychiatry by optimizing the statistical likelihood of tDCS treatment response in MDD.

A convexity-constrained parameterization of the random effects generalized partial credit model.

An alternative closed-form expression for the marginal joint probability distribution of item scores under the random effects generalized partial credit model is presented. The closed-form expression involves a cumulant generating function and is therefore subjected to convexity constraints. As a consequence, complicated moment inequalities are taken into account in maximum likelihood estimation of the parameters of the model, so that the estimation solution is always proper. Another important favorable consequence is that the likelihood function has a single local extreme point, the global maximum. Furthermore, attention is paid to expected a posteriori person parameter estimation, generalizations of the model, and testing the goodness-of-fit of the model. Procedures proposed are demonstrated in an illustrative example.

Bayesian intelligence for medical diagnosis: a pilot study on patient disposition for emergency medicine chest pain.

Clinicians can rapidly and accurately diagnose disease, learn from experience, and explain their reasoning. Computational Bayesian medical decision-making might replicate this expertise. This paper assesses a computer system for diagnosing cardiac chest pain in the emergency department (ED) that decides whether to admit or discharge a patient. The system can learn likelihood functions by counting data frequency. The computer compares patient and disease data profiles using likelihood. It calculates a Bayesian probabilistic diagnosis and explains its reasoning. A utility function applies the probabilistic diagnosis to produce a numerical BAYES score for making a medical decision. We conducted a pilot study to assess BAYES efficacy in ED chest pain patient disposition. Binary BAYES decisions eliminated patient observation. We compared BAYES to the HEART score. On 100 patients, BAYES reduced HEART's false positive rate 18-fold from 58.7 to 3.3 %, and improved ROC AUC accuracy from 0.928 to 1.0. The pilot study results were encouraging. The data-driven BAYES score approach could learn from frequency counting, make fast and accurate decisions, and explain its reasoning. The computer replicated these aspects of diagnostic expertise. More research is needed to reproduce and extend these finding to larger diverse patient populations.

Efficient algorithm for thermal nondestructive testing and evaluation by considering the heteroscedastic nature of noise sources in infrared thermography

Abstract Thermal Imaging is a promising Non Destructive Testing & Evaluation (NDT & E) approach to monitor the health
of composite materials. Among various post processing approaches adopted in thermal imaging for NDT & E, statistical
analysis schemes gained importance due to their reliability and data reduction capabilities. This paper provides an insight
to a factor analysis-based statistical approach to detect the hidden defects in the Glass Fiber Reinforced Polymer (GFRP)
sample. The proposed approach models the observed data covariance into combination of temporal signal covariance
and noise covariance matrices. The modeling of the diagonal covariance matrix (with different elements) is motivated by
the presence of heterogeneity in the experimental data obtained from GFRP sample. This novel method is based on the
coordinate descent technique, which estimates the covariance matrix of the noise variances iteratively by minimizing the
negative log likelihood function. The obtained results from the chosen GFRP samples compared with the widely used
statistical Principal Component Thermography (PCT) technique illustrate the improved performance in terms of defect
detection with the proposed technique.

Bayesian inference: more than Bayes’s theorem

Bayesian inference gets its name from Bayes’s theorem, expressing posterior probabilities for hypotheses about a data generating process as the (normalized) product of prior probabilities and a likelihood function. But Bayesian inference uses all of probability theory, not just Bayes’s theorem. Many hypotheses of scientific interest are composite hypotheses, with the strength of evidence for the hypothesis dependent on knowledge about auxiliary factors, such as the values of nuisance parameters (e.g., uncertain background rates or calibration factors). Many important capabilities of Bayesian methods arise from use of the law of total probability, which instructs analysts to compute probabilities for composite hypotheses by marginalization over auxiliary factors. This tutorial targets relative newcomers to Bayesian inference, aiming to complement tutorials that focus on Bayes’s theorem and how priors modulate likelihoods. The emphasis here is on marginalization over parameter spaces—both how it is the foundation for important capabilities, and how it may motivate caution when parameter spaces are large. Topics covered include the difference between likelihood and probability, understanding the impact of priors beyond merely shifting the maximum likelihood estimate, and the role of marginalization in accounting for uncertainty in nuisance parameters, systematic error, and model misspecification.

Noise aware parameter estimation in bioprocesses: Using neural network surrogate models with nonuniform data sampling

AbstractThis article demonstrates a parameter estimation technique for bioprocesses that utilizes measurement noise in experimental data to determine credible intervals on parameter estimates, with this information of potential use in prediction, robust control, and optimization. To determine these estimates, the work implements Bayesian inference using nested sampling, presenting an approach to develop neural network‐ (NN) based surrogate models. To address challenges associated with nonuniform sampling of experimental measurements, an NN structure is proposed. The resultant surrogate model is utilized within a Nested Sampling Algorithm that samples possible parameter values from the parameter space and uses the NN to calculate model output for use in the likelihood function based on the joint probability distribution of the noise of output variables. This method is illustrated against simulated data, then with experimental data from a Sartorius fed‐batch bioprocess. Results demonstrate the feasibility of the proposed technique to enable rapid parameter estimation for bioprocesses.

Ship detection method based on attention guidance and multi-sample decision making

Single-stage target detection methods have the characteristics of fast training speed and short detection time. However, its feature pyramid network is difficult to suppress the background and noise information of SAR ship images, and the detection head has prediction errors. To address this problem, this paper proposes a detection model based on attention guidance and multi-sample decision for synthetic aperture radar ship detection. Firstly, an attention guidance network is proposed and added to the highest level of the feature pyramid to suppress background and noise interference, thereby improving the representation ability of features. Secondly, a multi-sample decision network is proposed to participate in the prediction of target position. This network alleviates the impact of prediction errors on detection results by increasing the number of samples output in the regression branch. Finally, a novel maximum likelihood loss function is designed. This loss function constructs a maximum likelihood function using the samples output from the multi-sample decision network, which is used to standardize the training of the decision network and further improve the accuracy of target positioning. Taking the RetinaNet network model as the baseline method, compared with the baseline method and the current advanced target detection methods, this method shows the highest detection accuracy on the ship detection dataset SSDD, with AP reaching 52.8%. Compared with the baseline method, the proposed method improves the AP evaluation index 3.4% ∼ 5.7%, and the training parameter Params only increases by 2.03 M, and the frame rate FPS only decreases 0.5Iter/s.

Multivariate lifetime data in presence of censoring and covariates: Use of semiparametric models under a Bayesian approach

A generalization of the popular proportional hazards model introduced by Cox (1972) is given by the class of semiparametric or transformation models to be used in the analysis of lifetime data in presence of censored data and covariates. In this study, we consider the use of semiparametric (transformation models) for the situation where there are two or more responses associated to the same individual or unit. We assume a hierarchical Bayesian analysis for semiparametric models considering the complete likelihood function obtained from the transformation models considering the unknown hazard functions as unknown latent variables and Markov Chain Monte Carlo (MCMC) methods to get the posterior summaries of interest. The dependence between multivariate responses for the same individual is captured by the introduction of another latent variable or frailty. Illustrations of the proposed methodology are presented considering two medical multivariate lifetime data sets.

Using the Conditional Maximum Likelihood function with the genetic algorithm to estimate STARIMA models (pλ, d, qm)

Strategic planning for a specific phenomenon depends mainly on accurate prediction by developing a model to represent that phenomenon. Therefore, this research dealt with spatiotemporal series models such as the STARIMA model (pλ, d, qm) model when there is a spatial correlation for neighbouring sites and a spatial correlation for the same geographical location. The methods for estimating the parameters are the Conditional Maximum Likelihood method and the method of estimating the parameters using the genetic algorithm (MLEGA). These models were compared using the statistical comparison metrics (RMSE). The data represents the daily infections with the COVID-19 epidemic for (10) sectors on the Rusafa side of the city of Baghdad from 29/2/2020 until 25/4/2023; it was found that spatiotemporal series data consider the analysis of spatial relationships between geographic points, which allows for a better understanding of the development of phenomena across time and space. The applied results reached the superiority of the spatiotemporal model STARIMA (11,1,31) with the presence of spatial correlation of neighbouring sites using the genetic algorithm because it has a lower (RMSE), so it was used to predict the daily infections of the Covid-19 epidemic for (10) sectors on the Rusafa side of the city of Baghdad For the period from 26/2/2020 until 5/5/2023. Paper type :Research paper.

Deep regression learning with optimal loss function

In this paper, we develop a novel efficient and robust nonparametric regression estimator under a framework of a feedforward neural network (FNN). There are several interesting characteristics for the proposed estimator. First, the loss function is built upon an estimated maximum likelihood function, which integrates the information from observed data as well as the information from the data distribution. Consequently, the resulting estimator has desirable optimal properties, such as efficiency. Second, different from the traditional maximum likelihood estimation (MLE), the proposed method avoids the specification of the distribution, making it adaptable to various distributions such as heavy tails and multimodal or heterogeneous distributions. Third, the proposed loss function relies on probabilities rather than direct observations as in least square loss, contributing to the robustness of the proposed estimator. Finally, the proposed loss function involves a nonparametric regression function only. This enables the direct application of the existing packages, simplifying the computational and programming requirements. We establish the large sample property of the proposed estimator in terms of its excess risk and minimax near-optimal rate. The theoretical results demonstrate that the proposed estimator is equivalent to the true MLE where the density function is known in terms of excess risk. Our simulation studies show that the proposed estimator outperforms the existing methods based on prediction accuracy, efficiency and robustness. Particularly, it is comparable to the MLE with the known density and even gets slightly better as the sample size increases. This implies that the adaptive and data-driven loss function from the estimated density may offer an additional avenue for capturing valuable information. We further apply the proposed method to four real data examples, resulting in significantly reduced out-of-sample prediction errors compared to existing methods.

Reliability Analysis for Degradation-Shock Processes with State-Varying Degradation Patterns Using Approximate Bayesian Computation (ABC) for Parameter Estimation

The degradation of products is an integral part of their life-cycle, often following predictable trajectories. However, sudden, unexpected events, termed ’shocks’, can substantially alter these degradation paths. Shocks can significantly influence the pace of degradation, leading to accelerated system failure. Moreover, they may initiate changes in degradation patterns, transitioning from linear to non-linear or random trajectories. To address this challenge, we present a novel multi-state reliability model for competing failure processes that account for degradation-shock dependencies by considering the state-varying degradation pattern. The degradation process is divided into s-states, with each state treated according to its pattern based on the time-transform Wiener process. The reliability function is derived based on soft failure caused by continuous degradation involving the s-states, the sudden increase in degradation caused by random shocks, and hard failure due to some shock processes. Additionally, we performed a sensitivity analysis to determine which parameters have the most significant impact on product reliability. Due to the complexity of the likelihood function, we adopted the ABC method to estimate the model parameters. A simulation study and a practical application with micro-electro-mechanical systems (MEMS) degradation results are used to demonstrate the efficiency and effectiveness of the proposed approach.

Identification of an additive interaction using parameter regularization and model selection in epidemiology

Background In epidemiology, indicators such as the relative excess risk due to interaction (RERI), attributable proportion (AP), and synergy index (S) are commonly used to assess additive interactions between two variables. However, the results of these indicators are sometimes inconsistent in real world applications and it may be difficult to draw conclusions from them. Method Based on the relationship between the RERI, AP, and S, we propose a method with consistent results, which are achieved by constraining $e^{\theta_3}-e^{\theta_1}-e^{\theta_2}+1=0$, and the interpretation of the results is simple and clear. We present two pathways to achieve this end: one is to complete the constraint by adding a regular penalty term to the model likelihood function; the other is to use model selection. Result Using simulated and real data, our proposed methods effectively identified additive interactions and proved to be applicable to real-world data. Simulations were used to evaluate the performance of the methods in scenarios with and without additive interactions. The penalty term converged to 0 with increasing λ, and the final models matched the expected interaction status, demonstrating that regularized estimation could effectively identify additive interactions. Model selection was compared with classical methods (delta and bootstrap) across various scenarios with different interaction strengths, and the additive interactions were closely observed and the results aligned closely with bootstrap results. The coefficients in the model without interaction adhered to a simplifying equation, reinforcing that there was no significant interaction between smoking and alcohol use on oral cancer risk. Conclusion In summary, the model selection method based on the Hannan-Quinn criterion (HQ) appears to be a competitive alternative to the bootstrap method for identifying additive interactions. Furthermore, when using RERI, AP, and S to assess the additive interaction, the results are more consistent and the results are simple and easy to understand.

Estimating and Modeling Pinus contorta Transpiration in a Montane Meadow Using Sap-Flow Measurements

This study quantifies the transpiration of encroached lodgepole pine (Pinus contorta var. murryana (Grev. & Balf.) Engelm.) in a montane meadow using pre-restoration sap-flow measurements. Lodgepole pine transpiration and its response to environmental variables were examined in Rock Creek Meadow (RCM), Southern Cascade Range, CA, USA. Sap-flow data from lodgepole pines were scaled to the meadow using tree survey data and then validated with MODIS evapotranspiration estimates for the 2019 and 2020 growing seasons. A modified Jarvis–Stewart model calibrated to 2020 sap-flow data analyzed lodgepole pine transpiration’s correlation with solar radiation, air temperature, vapor pressure deficit, and soil volumetric water content. Model validation utilized 2021 growing season sap-flow data. Calibration and validation employed a Markov Chain Monte Carlo (MCMC) approach through the DREAM(ZS) algorithm with a generalized likelihood (GL) function, enabling parameter and total uncertainty assessment. The model’s scaling was compared with simple scaling estimates. Average lodgepole pine transpiration at RCM ranged between 220.6 ± 25.3 and 393.4 ± 45.7 mm for the campaign (mid-July 2019 to mid-August 2020) and 100.2 ± 11.5 to 178.8 ± 20.7 mm for the 2020 partial growing season (April to mid-August), akin to MODIS ET. The model aligned well with observed normalized sap-velocity during the 2020 growing season (RMSE = 0.087). However, sap-velocity, on average, was underpredicted by the model (PBIAS = −6.579%). Model validation mirrored calibration in performance metrics (RMSE = 0.1233; PBIAS = −2.873%). The 95% total predictive uncertainty confidence intervals generated by GL-DREAM(ZS) enveloped close to the theoretically expected 95% of total observations for the calibration (94.5%) and validation (81.8%) periods. The performance of the GL-DREAM(ZS) approach and uncertainty assessment in this study shows promise for future MJS model applications, and the model-derived 2020 transpiration estimates highlight the MJS model utility for scaling sap-flow measurements from individual trees to stands of trees.

Effect of the Likelihood Function on the Calibration of the Effective Manning Roughness Factor

Effect of the Likelihood Function on the Calibration of the Effective Manning Roughness Factor