Besides, we further interpreted the model and identify potential biomarkers for the aging process. Among these identified biomarkers, we found that Finegoldia magna, Bifidobacterium dentium, and Clostridium clostridioforme had increased abundance in the elderly. Moreover, the utilization of amino acids by the gut microbiome undergoes substantial changes with increasing age which have been reported as the risk factors for age-associated malnutrition and inflammation. This model will be helpful for the comprehensive utilization of multiple omics data, and will allow greater understanding of the interaction between microorganisms and age to realize the targeted intervention of aging.The human gut microbiome is a complex ecosystem that is closely related to the aging process. However, there is currently no reliable method to make full use of the metagenomics data of the gut microbiome to determine the age of the host. In this study, we considered the influence of geographical factors on the gut microbiome, and a total of 2604 filtered metagenomics data from the gut microbiome were used to construct an age prediction model. Then, we developed an ensemble model with multiple heterogeneous algorithms and combined species and pathway profiles for multi-view learning. By integrating gut microbiome metagenomics data and adjusting host confounding factors, the model showed high accuracy (R In line with this, the resolution of gut microbiome signatures will be invaluable for the development of noninvasive microbiome-based tests to determine signs of accelerated or delayed aging in the elderly, and to evaluate gut microbiota-based interventions to alleviate the aging-related diseases.In humans, aging is a continual and progressive process that results in decreased physiologic function across all organ systems. The gut microbiota is considered the most important symbiotic microecosystem, and has multiple functions in human health, including in digestion, immunity, metabolite production, and even neural function.1 Recent studies have shown that the alpha-diversity of the gut microbiome continues to increase from infancy to adulthood, where it remains relatively stable before increasing again in old age.3 Moreover, the proportion of beneficial bacteria, such as short-chain fatty acid-producing species, reduces with age, whereas that of opportunistic species and pathobionts increases.4 In addition to the taxonomic component of gut bacteria, functional genes, such as those associated with xenobiotic biodegradation and metabolism, have also been shown to correlate with age, with a more prominent functional rearrangement observed in the elderly.5Metagenome-wide association studies have begun to explore gut microbiome alterations in the aging process to investigate the associations of the gut microbiome with age. Microflora evolution does not stop in childhood but extends into adulthood and even old age.However, most prior studies have focused on a single cohort, and very few have integrated data from multiple populations or other studies. These limitations hinder our ability to clarify the robustness of microbiome–age associations and obscure our understanding of the potential mechanisms by which the microbiome contributes to aging.6 The results demonstrated that the models based on skin microbiome 16s rRNA data provided the best prediction of age . However, although these models achieved very good results in age prediction, amplicon sequencing is restricted in its ability to provide an in-depth analysis of the association between microbiome and aging. Besides, in another study, a model based on deep neural network and metagenomics data was proposed , but the accuracy of its prediction was still lacking and the accuracy was related to the reliability of the final association interpretation.7The robustness of microbiome–age associations can be assessed by a meta-analysis of data integrated from all relevant investigations. The investigation of microbiome alterations across a wide age range requires a comprehensive dataset of consistently collated microbiome profiles. Although a deluge of metagenomic data on the associations between the human gut microbiome and age has been generated, obtaining molecular footprints of aging in molecular-level features from these data remains a major challenge. Such findings are disorganized and need to be unified into a theory of gut community dynamics. Machine learning offers next-level analyses that allow the development of new perspectives and novel hypotheses relating to the human gut microbiome. Human microbiome aging clocks are considered a reliable way to measure the passage of time in a gut community and to distinguish two temporally different states. Until now, this pursuit has produced limited results in the gut metagenomics field, despite there being a mass of reports on the association between specific microbes and aging. The ability of the oral, gut, and skin microbiomes to predict age has been previously tested in adults using random forest regression on data combined from multiple publicly available studies.9 Second, because metagenomic data are usually sparse and presented as a high-dimensional matrix with high noise,10 there is an urgent need for an optimal machine learning method to decipher the links between aging and microflora composition and functions. Furthermore, while previous studies have proven insightful, their focus on taxonomy may limit our understanding of the association between microbiome functions and health.11 The ability to resolve the association between microbiome functions and health may prove critical to determining the mechanisms through which the microbiome accelerates or delays aging. Thus, these deficiencies limit the ability to establish a robust age regression prediction model based on gut microbiome metagenomic data, and prevent further understanding of the potential interaction between the gut microbiome and human age. These challenges have led to the introduction of an emerging machine learning field – multi-view learning and envision its potentially powerful applications to gut microflora composition and functions.12 In particular, multi-view learning is more effective than previous integrative methods that have been used to learn the heterogeneity of metagenomics data and reveal cross-talk patterns.12Although the power of machine-learning algorithms is attracting increasing attention, some limitations remain. First, the complexity of the microbiome and its susceptibility to numerous covariates and confounding variates apart from age complicate the basic task of aggregating the available information into an intestinal aging clock.The aim of this study was to deliver a universal and reliable aging clock based on the functions and composition of the gut microflora. To this end, we used meta-analysis to rectify the problem of data heterogeneity between multiple metagenomic cohorts based on a large-scale dataset to effectively avoid the influence of host or technical factors. Moreover, we built multi-view learning models to integrate the taxonomic and metabolic pathways of the gut microbiota with the aim to establish the functional and mechanistic interpretations between the gut microbiota and age. Finally, we describe how the abundance and function of microbes affect age prediction, and how this knowledge can be used to identify potential biomarkers of intestinal aging. The results of this study will improve the understanding of the relationship between the gut microbiome and age, provide insights into a new aging-diagnosis strategy, and suggest potential intervention targets for future research.p < .001).A total of 4478 fecal samples were collected from 31 study cohorts covering 13 subregions . Collect2 > 0.1; We next investigated whether the distribution of geographical factors was age-oriented. In general, the nature of the age distribution trend between subregions leads to a decrease in the performance of the age prediction model, and it is difficult to demonstrate that the age-related markers are not driven by geographical factors. Moreover, given the differences in distribution characteristics between countries and ages, addition of the geographical factor as an extra feature to the prediction model leads to a false positive situation owing to the data tendency of the feature itself as opposed to a learned association between the feature and intestinal flora. For these reasons, we utilized two different algorithms (Linear Regression [LR] and Random Forest [RF]) to evaluate the association between country and age. Both algorithms showed that there was indeed a specific trend of sample enrichment in a certain age range in some national datasets .We combined two methods to reduce the correlation between geographic location and age to avoid age–regional distribution problems. We first grouped countries into different subregion-level bins based on geographic location; this ensured that, after decrease regions with special distributions, still maintain most samples. After clustering, the subregional factor was still an important factor affecting the species and pathways of the gut microbiome . And the13 Thus, we performed a systematic evaluation of different machine learning algorithms on the adjusted dataset .The diversity of the base regressor was necessary to achieve an effective model ensemble as heterogeneous algorithms were expected to obtain more diversity, so as to reduce deviation through integration.2 = 0.5, MAE = 9.48).First, we aimed to determine which models were able to achieve age prediction for the gut microbiome species-level profiles. A total of 11 models were considered, including LR, Lasso, Elastic Net (EN), Bayesian Ridge (BR), Support Vector Machine (SVM), Decision Tree (DT), RF, Gradient Boosted Regression Trees (GBRT), eXtreme Gradient Boosting (XGB and XGBRF) and LightGBM (LGB). Except for the LR and DT models, several other models can achieve age prediction ; indeed Second, we adopted a series feature selection method with the aim to reduce feature dimensions and decrease computational cost. Among them, the GBRT-based selection method decreased the data feature dimension with no reduction in performance .As mentioned previously, we first judged whether this series of algorithms could predict age based on pathway data. After 10 times 5-fold cross-validation, the algorithms, with the exception of LR and DT, showed varying degrees of age regression capabilities . InteresWe next implemented the same feature selection strategy and the GBRT-based selection method has the least feature dimension without affecting the predictive performance of each model . SimilarWe next examined whether the fusion of multiple models can improve the prediction accuracy compared to a single model. To this end, nine pre-verified regression methods  were applied to construct the ensemble model and LR was used as the weight learning algorithm of the generalizer. First, we compared the predictive accuracy of the single and ensemble algorithm with multiple data types . To evaluate the performance, the LGB that showed the highest accuracy on both datasets was decided as the benchmark for the comparison. The result has shown that the ensemble method can significantly improve prediction accuracy in all data types, which emphasizes that the method was feasible . MoreoveIt should be noted that, as the ensemble algorithm was simultaneous modeling of two datasets independently and then weighting results as the prediction outcome, and most of the current models were not designed to model multiple data sets simultaneously, meaning that the corresponding multiple datasets were not suitable for the LGB model. Therefore, due to the restriction of the data structure, it is hard to directly compare the performance difference between the ensemble model and LGB. To solve this problem, we adopted another data fusion way to compare the predictive ability between the LGB model and the ensemble model . To realize the comprehensive utilization of the species and pathway data, we directly concatenated the two datasets. The metabolic pathway data of each sample were aggregated with the species data as additional features using the same method as the previous one that used the subregion feature. We used the concatenated dataset to construct the LGB and ensemble models (abbreviated as EM1) and compared them with the ensemble model that used the aforementioned integration strategy . As a result, we found that the expansion of features can also improve regression accuracy. Under this integration strategy, the performance of our ensemble model was still significantly higher than that of the LGB, and was not significantly different from that of the original ensemble method .We also tested dissimilar weights learning methods and judged the changes to the ensemble effect. The simple average (SA) result was used as the baseline for different weighting methods and the non-ensemble tree model was considered as the potential generalizer, including LR, Lasso, BR, SVM with linear kernel (LSVM), and SVM with non-linear kernel (SVM). The rationale behind this was to avoid overfitting of complex models in the generalization process and resulting in performance loss. We observed that the basic linear models  were able to obtain the ideal integration effect compared with the SA, and there was no significant difference between these methods . It was Finegoldia magna. The remaining factors included taxadiene biosynthesis, Streptococcus thermophilus, Prevotella copri, hexitol fermentation, Bifidobacterium dentium and Streptococcus infants.We next sought to interpret the ensemble model to identify the age-related features included species and pathway two aspects. The permutation feature importance (PFI) method was used to investigate the relationship between individual features and aging. We obtained a set of biomarkers that were significantly related to the aging process . Among tF. magna, B. dentium, and Clostridium clostridioforme had increased abundance in the elderly, while P. copri and Burkholderialse bacterium 1_1_47 decreased in abundance with age  to extend the finding across a broader landscape of human aging; this may lead to the discovery of more general patterns of age-associated microbiome shifts.17 Simultaneously, we applied a two-step screening method to realize the decoupling of sample age and country. Our results showed that the above method can decouple geographical and age factors , and show that the multi-model ensemble has higher accuracy than any of the individual regressors acting solo on all of the datasets  were also identified as biomarkers.30 In line with a previous study, a significant proportion of the species and pathways we identified were found to be related to the frailty of the elderly population, including C. clostridioforme, Clostridium hathewayi, Clostridium bolteae, Clostridium leptum, Clostridiales bacterium 1_7_47FAA and the pathway of pyruvate fermentation to acetone.8 In addition, some indicators may clarify the collective characteristics of the gut environment. Among the previously obtained biomarkers, we observed that the utilization of amino acids by the gut microbiome undergoes substantial changes with increasing age. This metabolic characteristic that leads to a reduction in amino acids may aggravate malnutrition in the aging process, which is not conducive to maintaining immune system function and prevent frailty.32 It should be noted that the consumption of branched-chain amino acids (BCAAs) is increased with age, especially the enrichment of the leucine metabolism and loss of the isoleucine synthesis pathway (Supplementary Table S4). BCAAs are believed to improve muscle protein synthesis, and related research has shown that the elderly population needs to consume higher levels of leucine;34 these findings are consistent with the key metabolic characteristics revealed by our analysis. In addition, pathways related to tryptophan biosynthesis are also believed to be related to aging. The kynurenine pathway is considered to be the main tryptophan metabolism pathway in humans. The kynurenine is neurotoxic and can directly compromise mitochondria, which in turn leads to aging-related inflammation.35 The synthesis of tryptophan in the intestinal flora may lead to the accumulation of kynurenine in the body. These age-related change eventually may lead to disease. These age-associated biomarkers clarify the potential role of the gut microbiome in the aging process, and the integration method allowed us to rank different types of data to achieve more refined directional control for aging.By relying on a more accurate age prediction model, we revealed potential biomarkers of aging by using comprehensive metagenomics annotation data with a wide age range. A total of 102 species and 41 pathways were considered to be closely related to the aging process. It is worth noting that many of the biomarkers are correlated with aging-related diseases or those more prevalent in the elderly . For exa36 Through improvements in methods, it will be possible to extend data to ensure the reliability of the conclusion and maximize its utility.The current study consists of a multi-view data-adapted ensemble machine learning framework. Through the integration of heterogeneous models and data, we observed broad patterns of gut microbiome changes in aging and clarified the impact of the species and functions in this process. Future studies should focus on further improving the collection of big data on the gut microbiome. Although we have removed some host confounding factors in this study, more high-quality and diverse data are necessary to explore the in-depth associations between gut bacteria and age. The machine learning framework should also be improved. Indeed, new algorithms are constantly being proposed, especially neural network models, which have shown considerable predictive potential in other fields.37 We first selected stool samples and subsequently removed the samples that contained incorrectly annotated information. From these datasets, we deleted samples having age less than 18 years of age, a total of 4478 stool samples from 31 cohorts are included in this study .The data used in this analysis comes from a pre-processed public metagenome data from curatedMetagenomicData.In order to explore the impact of metadata between different datasets, we considered the reliable and non-redundant metadata which is defined as missing values is less than 40% and not associated with existing metadata (E.g. Age and Age-category), and finally 7 metadata remained, divided into 5 covariates  and 2 confounding factors . And for subsequent data analysis, we regrouped country into subregion level .Permutational multivariate analysis of variance using distance matrices  was used to quantitatively evaluate the impact of metadata. The PERMANOVA analysis for each covariate was using the following formula: adonis. Bray-Curtis dissimilarity was used in PERMANOVA to judge the beta diversity of the species and pathway samples, and the diversity distances are calculated by R packages (vegan::vegdist). In addition, Bonferroni was applied for p-value correction.To filter a subset which was no correlation between geographical and age, we first constructed a random forest age prediction model based on full subregion labels (encode by sklearn OneHotEncoder). Next, we removed the corresponding subregion data in descending order of feature importance score until one area remained. In each epoch, a random forest model was constructed with screened data, and 10 times 5-fold cross-validation was used to judge the association.http://scikit-bio.org/). Next, use the centered log-ratio  to transform the preprocessed profiles.To achieve an effective model ensemble, heterogeneous machine regression models were constructed, including Linear Regression (LR), Lasso, Elastic Net (EN), Bayesian Ridge (BR), Support Vector Machine (SVM), Decision Tree (DT), Random Forest (RF), Gradient Boosted Regression Trees (GBRT), eXtreme Gradient Boosting (XGB) and LightGBM (LGB). Before construct machine learning models, the zeros in species and pathway profiles were replaced with a very small number to suit logarithmic conversions  were used as the performance evaluation metrics. Among them, LR, Lasso, EN, BR, SVM, DT, RF and GBRT models are constructed using packages of the scikit-learn,38 XGB and XGBRF are constructed using xgboost package39 and LGB model is constructed using lightgbm package.40For each algorithm, built regression model based on species and metabolic pathway data. 10 times 5-fold cross-validation was performed to avoid random errors, R2 score was used as an evaluation indicator. We next used paired Wilcoxon rank-sum test to compare the performance changes between the original features and the filtered features, and the p-value was corrected by Bonferroni.Feature selection was performed to improve prediction accuracy and decrease feature dimension which helped to reduce the computational cost, contain Univariate linear regression tests , Mutual information estimation and model-based selection method . For each feature selection method, the filtered data was applied to each regressor with 10 times 5-fold cross-validation to avoid random errors, and the average RThe feature selection method was evaluated by the number of models that performance was significantly different after selection, the method with the least number of models with the lowest performance degradation is the optimal algorithm. In the case of multiple methods with the same counts, the number of models with the performance increase and the feature dimension as the secondary screening basis.A stacking ensemble strategy was utilized to construct a powerful meta-learner and the stacking process includes two stages. In the first stage , the traIn the second stage , The preFor multi-view learning, the integration of different data types was performed in the second stage. After performed the training of each model in the first stage on different types of datasets and obtained their predicted training set. Next concatenated each set to obtain the merged set for weights learning and perform the same process as mentioned above.2 score reduction as the metric and the feature which has a positive effect of R2 score was not considered). Based on multiple repeated calculation results, paired Wilcoxon rank-sum test was used to compare the prediction performance with the original model which without feature replacement. Features that are significantly different from baseline data are considered as potential biomarkers and ranked by the extent of influence on the prediction performance. Furthermore, Spearman’s rho test was applied to judge the association between the biomarkers and age. All p-values were Bonferroni-corrected.After constructed the ensemble model, in order to determine the importance of each feature in the prediction process, the permutation feature importance (PFI) was used to measure individual feature importance which evaluates by eliminating a single feature (randomly shuffle the feature 50 times) to calculate the extent of change in the prediction effect (RClick here for additional data file.


Damage classification is an important topic in the development of structural health monitoring systems. When applied to wind-turbine foundations, it provides information about the state of the structure, helps in maintenance, and prevents catastrophic failures. A data-driven pattern-recognition methodology for structural damage classification was developed in this study. The proposed methodology involves several stages: (1) data acquisition, (2) data arrangement, (3) data normalization through the mean-centered unitary group-scaling method, (4) linear feature extraction, (5) classification using the extreme gradient boosting machine learning classifier, and (6) validation applying a 5-fold cross-validation technique. The linear feature extraction capabilities of principal component analysis are employed; the original data of 58,008 features is reduced to only 21 features. The methodology is validated with an experimental test performed in a small-scale wind-turbine foundation structure that simulates the perturbation effects caused by wind and marine waves by applying an unknown white noise signal excitation to the structure. A vibration-response methodology is selected for collecting accelerometer data from both the healthy structure and the structure subjected to four different damage scenarios. The datasets are satisfactorily classified, with performance measures over  Structural health monitoring (SHM) allows identifying the states of structures to prevent damage that can occur because of operational and/or environmental conditions. It is possible to detect the beginning of a possible damage/failure in a structure and its components using methods associated with SHM systems . Damage Wind power plants are a good example of systems exposed to environmental influences. Offshore wind power is clean energy that exploits high and uniform wind speed conditions with even larger offshore wind turbines . HoweverRecently, sensors and signal processing techniques have been successfully applied for the analysis and evaluation of structures for obtaining significant and reliable results when the state of structures is evaluated . The EOCk nearest neighbors (kNN) and quadratic-kernel support vector machine (SVM) [Damage diagnosis for offshore wind-turbine foundations remains an open field of research. Common SHM approaches are based on guided waves with a known input excitation; however, in this type of structure, the applicability of guided waves is not functional because external perturbation effects caused by wind and marine waves are ignored. An approach based only on vibration-response accelerometer signals needs to be considered to address the challenge of online and in-service SHM for wind turbines. Some variants of the vibration-response-only SHM strategy for wind-turbine foundations have been reported. Vidal et al.  developene (SVM)  are testkNN, quadratic SVM, and Gaussian SVM were used as classifiers. The best algorithm was found to be the Gaussian SVM, which achieved a classification accuracy of In contrast to the conventional data-driven SHM techniques, the deep learning approach has been demonstrated its successfully application to solve SHM problems ,12. An aa et al.  solved tThis research seeks to solve the damage classification problem of in situ real structures exposed to strong changes in EOCs  using ML algorithms that use signals from sensor networks as inputs. The main goal of data-driven algorithms is to analyze large or complex sensor networks that provide multivariate information using ML approaches. These complex sensor networks can be found in some SHM solutions ,16, clasDifferent strategies have been used to solve structural damage-detection problems. There are traditional, machine learning methods with their parametric and non-parametric variants and deep learning methods . In 2020kNN [t-distributed stochastic neighbor embedding (t-SNE) [Previous studies focused on structural damage classification using multisensor systems; they used several ML algorithms and data reduction for pattern recognition. For instance, PCA , self-orkNN , artifickNN , SVMs [3kNN , and t-d (t-SNE) ,34. ThisThe remainder of this paper is organized as follows. This study aims to provide an accurate structural damage classification methodology that can specify whether a wind-turbine foundation is damaged; if it is damaged, the methodology can detect the nature of the damage. The classification methodology is derived by first obtaining a training set of data from both damaged and non-damaged structures (data acquisition). These data are properly pre-processed , and data transformation and dimensionality reduction techniques are applied to discard features that are not relevant to the classification problem (linear feature extraction). These new small-dimensional data are used as inputs to train the supervised machine learning classifier. Once the classification methodology is employed, given a new experimental sample of the wind-turbine foundation, the classification algorithm can predict the structural state of the foundation. k-fold cross-validation error estimation method.Before using the classification algorithm to classify new data, it is crucial to evaluate the performance of the algorithm. This evaluation is a challenging problem because the available data samples must be used to define the classifier and estimate its performance when making predictions of new samples. The training dataset and test set must be sufficiently large and representative of the underlying problem so that the resulting performance of the classifier is not too optimistic or pessimistic. In fact, if the collected dataset is very large and representative, one can split the dataset into two parts and use the first part to train the model and the second part to test it. However, this is rarely the case, and it is standard to use a k folds (or subsets). This allows the generation of k models, each of which takes the data from The basic idea of this procedure is to split the full dataset into runs see .This section details and provides a theoretical background for each step of the training data preparation developed in this work, as shown in The structure of a small-scale wind-turbine foundation is used to validate the damage detection and localization methodology developed in this study. This benchmark structure, based on the one designed by Zugasti , was contructure  and therData used to validate the structural damage classification methodology were obtained using the following experimental process. First, an arbitrary function generator (GW INSTEK AFG-2005) was used to apply a white noise signal to the structure. Four different wind speeds were simulated by applying different amplitudes of the white noise signal with factors of Data were obtained by performing several experiments applying different white noise signals to the small-scale wind-turbine foundation, both to the undamaged structure and the four different damaged structures. Each test had a duration of These collected data are arranged into a three-dimensional matrix of size mean-centered data. This requires the columns of matrix A prerequisite for most linear and nonlinear extraction techniques is obtaining d (MCGS) ,37,38,39k at time instant j; jth column of matrix n-dimensional column vector.The MCGS group-scaling technique is devised as a two-step procedure. First, the columns of all matrices are modified by computing the mean of each column and subtracting this mean from its corresponding column values. This procedure is also called column-wise scaling ,36. For k measurements. Thus, the elements of k measurements, i.e., Then, the matrix In this study, a novel alternative to the MCGS scaling technique is considered to yield normalized data with both sensor-block and global unitary standard deviation. The key idea is to scale the mean-centered data using the standard deviation, i.e.,For simplicity, the MCUGS matrices experiments × time instants × sensors is rearranged into a two-dimensional matrix experiments × (time instants × sensors). The sensor matrices After normalizing the data, the three-dimensional matrix ith row of matrix ith test isEach row of matrix These rows  are the input samples for the feature extraction technique described in d-dimensional manifold described by the transformed matrix Let Equation . Featured eigenvectors of the decomposition yieldsd columns of matrix d eigenvectors with the largest eigenvalues. Because the total variance is given by the sum of all eigenvalues multiplied by The PCA is one of the most effective techniques for-dimensionality reduction. This technique discovers linear manifolds that characterize the data by diagonalizing the covariance matrix In general, linear and nonlinear manifold learning algorithms are based on the idea that the high dimensionality of the dataset is artificial, and that the samples are characterized by a low-dimensional manifold (or subspace) embedded in a high-dimensional space . Manifolf from ith row of matrix f to be the linear transformation given by Manifold learning techniques introduce a transformation or map d components of the vector d features associated with the ith sample arejth vector of the canonical basis.The This section explains the main characteristics of the XGBoost classifier. First, a detailed explanation of the XGBoost classifier as an ensemble method (providing a forest of regression trees) is provided, followed by a description of the main XGBoost parameters see . FinallyCurrently, one of the most popular and accurate machine learning classifiers is the extreme gradient boosting technique (XGBoost) . The graforest of regression trees, where the prediction of the forest is a weighted average of the predictions of its trees.The XGBoost method is an ensemble method that involves sequentially creating and adding regression trees. New trees are created to correct the errors in the predictions from the existing set of trees, which boosts the attributes that lead to the misclassification from the previous tree. Thus, multiple trees are built on top of each other to correct the errors in the previous tree. The XGBoost classifier thus provides a XGBoost exploits the limit of computational resources in the gradient boosting decision tree algorithm . The booSome important characteristics that make the XGBoost classifier one of the best classifier algorithms are the use of inbuilt cross-validation, which handles missing values; the use of regularization to avoid overfitting in the model; a novel tree learning algorithm for handling sparse data; save resources and time with a cache access pattern mechanism; and incorporate sharding and data compression.ith leaf.The use of the XGBoost technique for multiclass classification purposes is shown here for a particular example of a sample composed of 3 classes, unds see . In the Given a new sample to be classified, the decision rules in each tree are used to produce a set of three raw scores (one per class) Predictions obtained in this first round are used to train the second round of trees in the forest, thereby obtaining a new regression tree per class R denotes the number of boosting rounds and l denotes the total number of classes, given a sample, the row scores per class/tree are computed The same technique applies if more boosting rounds are added. If the XGBoost technique provides a forest on cp as :(6)rsi=∑Consider the specific example given in In addition to the great classification power of the XGboost method, the final classifier provides two relevant additional benefits. The first benefit is that for a given sample, the XGBoost classifier not only provides a prediction for the class of the sample, but also provides a probability-like measure of the sample belonging to each class category. From Equation , given aClass prediction Probability associated with the predicted class Probability associated with each class This additional information can be used to assess the reliability of the prediction .The performance, complexity, and overfitting properties of the XGBoost method depend on the proper tuning of the hyperparameters. The best model should balance the model complexity with its predictive power. This tuning can be performed via experimentation or by using specific routines .The first category of learning task parameters, we find the parameters that specify the learning task and corresponding learning objective. In particular, we set the value of objective to be multi:softmax to predict the class of each data sample or multi:softprob to predict the probabilities of each data sample belonging to each class category, and the number of classes is defined by the parameter num_class. Once the probabilities are computed, the results of multi:softmax can be directly obtained by selecting the class with a higher probability, as shown in seed and random_state parameters. The output of the XGBoost method is a random forest, and as the name indicates, randomness is introduced in the process to avoid overfitting . Setting specific values for random seeds allow for reproducibility of the results. However, picking a convenient seed may result in over-optimistic results. Therefore, the seed should only be fixed for reproducibility and not to increase performance.In the second category, booster parameters that guide the individual booster trees at each step. The three main parameters controlling the complexity of the final random forest are n_estimators, max_depth, and learning_rate. The n_estimators parameter determines the number of trees to grow per class (number of rounds); therefore, the final number of trees in the forest is max_depth parameter controls the maximum depth of each decision tree. The maximum number of nodes in the forest is learning_rate parameter or shrinkage parameter is analogous to the learning rate in gradient-boosted models. The learning rate corresponds to how quickly the error is corrected from each tree to the next, and it is a simple constant learning_rate to add smaller corrections in each round; therefore, small learning rates slows down the learning and makes the boosting process more conservative. Another set of interesting parameters in this category is reg_alpha and reg_lambda, which introduce subsample and colsample_bytree control the fraction of items to be subsampled to train a particular tree. Every time a new tree in the random forest is trained, the algorithm selects a random sample from the complete training dataset and a random subset of the features to train the tree. The subsample parameter is the ratio of training instances to be selected; that is, if subsamplecolsample_bytree parameter is the fraction of features to be sampled randomly for each tree. The default value of 1 indicates no subsampling. Lower values make the algorithm more conservative and prevent overfitting.The final category corresponds to the ing (see ). IncreaIt is crucial to evaluate the performance of the machine learning model on unseen data. A test dataset with new instances must be available to check the correctness of the predicted classes for evaluating the performance of a model. In multiclass classification problems , each sample from the test dataset has a class label that is compared to the predicted class label. A measure of correctly or incorrectly recognized classes must be defined.(1)how to define the test and training datasets so that no overfitting occurs ; and,(2)how to define the performance/accuracy measure from the correctness/incorrectness of the predicted classes.Therefore, model validation has two key points:k-fold cross-validation is one of the most used techniques to determine the training and test data sets when a limited amount of data is available. This is because it avoids overfitting and results in a less biased or less optimistic estimate of the model skill compared to a simple train/test split [st split . In thisEach fold of the cross-validation procedure provides a measure of the performance of the classification algorithms, i.e., The standard deviation of these performance measures can also be computed asA large standard deviation In general, the classification performance for a given fold is defined by specific measures of the confusion matrices (see details below). Although this is not the approach used in this work, it is common practice to obtain the total performance by first adding the five confusion matrices associated with each fold directly and then computing the performance of the total matrix, despite the performance measures being nonlinear.Specific performance measures used in the present work are described as follows: Given the classification results associated with a fold, the correctness of the classification method associated with this fold is evaluated by first computing the number of correctly recognized class samples , the number of correctly recognized samples that do not belong to the class , and samples that either were incorrectly assigned to a class  or that were not recognized as class samples . This information is summarized in a multiclass confusion matrix ,47. Indel denotes the total number of classes or labels, i.e., Then, for a given class ollowing , we intrThe quality of the classification strategy for the fold is assessed in this case because of data imbalance, which uses macro-averaging global performance measures that treat all classes equally instead of favoring the larger ones. Global measures are computed by averaging the measures obtained in each class, namelyThe final overall performance measure of the classifier is obtained by computing the average and standard deviation of the five-fold performance measures.baseline data  was used to determine the pre-trained XGBoost classifier. The offline stage stores the MCUGS normalization parameters, the PCA projection matrix selects the relevant features, and the classifier is given in the form of a forest regression tree. This information is then used in the online stage for real-time classification of a new observation. A flowchart of the proposed approach to illustrate how the SHM strategy is applied when a new wind turbine (WT) should be classified is depicted in In this context, an offline strategy is adopted. In the offline stage, the Given a new single observation of the wind turbine to be diagnosed that contains signals measured by the The collected data were first normalized using the pre-stored MCUGS parameters (the mean quations  and 2),,μjk and d principal components stored in the matrix The normalized data were then projected using the pre-stored PCA projection matrix to select the relevant features to be used in the XGBoost classifier. d-dimensional vector that is the projection of d features Equation .forest classifier, which returns a set of probabilities l denotes the total number of different structural states and i. Thus, the real-time classification strategy provides the following: (1) the structural state/class prediction As shown in The problem considered in the present work is related to real-time structural damage diagnosis and classification. A closely related problem is the early detection of incipient damage (prognosis). Prognosis methodologies contribute a predictive maintenance option that provide the decision-maker the flexibility to determine whether and when to act before the structure is severely damaged . To thisThe results of damage detection and classification for small-scale jacket-type wind turbines are presented in this section. More precisely, Equation ) before Raw unpreprocessed data containing the values of the acceleration over time for each sensor  are shown along with its associated normalized MCUGS signal. The MCGS normalized signal is not seen because it is nearly equal to the MCUGS signal. The raw data are shown along with the time evolution of the mean value of all samples per time and sensor d most relevant features . d, thereby characterizing the retained linear manifold.The PCA is applied to the data matrix Equation , where tThe number of principal components needed to retain a relevant amount of variance is considerably high  yields more accurate classification results; however, it generates overfitted models. The optimal XGBoost parameters used to train all models are summarized in Once the relevant features are extracted using PCA, the transformed datad) used as input feature matrix for the XGBoost. The effectivity index associated with each measure fier see . In addiSpecific performance measures Very good performance metrics are obtained even when only 5 features are used to classify the samples. For values larger than or equal to 21 principal components, a nearly perfect classification is obtained. For the parameters described in damage 3 and damage 4) are easily separated from the other samples. These samples are perfectly classified in three cases: The confusion matrices confirm the observations of most relevant XGBoost features of the data in The small-scale jacket-type wind turbine is a benchmark structure for testing, validating, and comparing different damage-detection approaches. Vidal et al.  used a cVidal et al.  considerEquation . In thisIn this study, when state in  is damagThe confusion matrices presented in As a complement to The sample in the first column (undamaged) is wrongly classified as damage 1 with a The sample in the second column (damage 1) is wrongly classified as undamaged with a The sample in the third column (damage 1) is wrongly classified as undamaged with a Nine columns  taking into account the differences in magnitude obtained by the 8 triaxial accelerometers attached to the jacket-type wind-turbine foundation, and data unfolding.Data transformation: a dimensionality reduction stage is applied to perform feature extraction and reduce the bid data obtained with the sensors using PCA. The selection of the number of principal components is done according to the average classification accuracy obtained with the methodology. The best results were obtained when considering 21 principal components as relevant features, which only account for Damage Detection and Classification: the machine learning model is designed to define XGBoost hyperparameters . These parameters play a fundamental role in improving the model performance while avoiding overfitting. The XGBoost classifier manages to classify nearly all samples appropriately (only three out of 5470 samples are misclassified).The estimation of class membership probabilities allows the assignment of a high level of confidence to most results (only 13 samples are classified with a probability smaller than Online Monitoring: the proposed methodology can process new data using the MCUGS normalization parameters, the PCA projection matrix and the XGBoost classifier for real-time classification of a new observation.The main conclusions of this work are:Future work is expected to focus on two main areas. Nonlinear manifold learning techniques will be explored to increase the separability capacity of the dimensionality reduction stage before entering the classifier; further, the possibility of combining both linear and nonlinear feature extraction techniques will be studied. This research line is addressed to better identify low-dimensional manifolds present when the data include nonlinear behaviors. The developed damage classification methodology will be tested in more complex and realistic structures.


In this study, we investigated cancer cellular networks in the context of gene interactions and their associated patterns in order to recognize the structural features underlying this disease. We aim to propose that the quest of understanding cancer takes us beyond pairwise interactions between genes to a higher-order construction. We characterize the most prominent network deviations in the gene interaction patterns between cancer and normal samples that contribute to the complexity of this disease. What we hope is that through understanding these interaction patterns we will notice a deeper structure in the cancer network. This study uncovers the significant deviations that topological features in cancerous cells show from the healthy one, where the last stage of filtration confirms the importance of one-dimensional holes  in cancerous cells and two-dimensional holes  in healthy cells. In the small threshold region, the drop in the number of connected components of the cancer network, along with the rise in the number of loops and voids, all occurring at some smaller weight values compared to the normal case, reveals the cancerous network tendency to certain pathways. Through the gene expression process, genetic code modulates biological functions and associated molecular pathways. The subsequent cellular phenotype is modulated by a dynamic network of interactions among genes. Perturbations in these interactions affect the overall manifestation of genetically driven diseases such as cancer. Genes and their dynamic interactions can be modeled by complex networks represented by nodes and links4. In network systems, each node is considered as a dynamic entity, evolving under the influence of others9. Systems of interacting units consist of links having positive, negative, or zero weight and they together develop a weighted signed network, called Gene Regulatory Network (GRN)14. GRNs can be constructed by maximum entropy models, analyzed by balance theory approaches15 and topological methods16. Moreover, responses to driving forces on the structure formation of these networks cause the development of new features and subsequently lead to the identification of unique patterns in the observational data. These patterns can arise from non-trivial connections that go beyond classical pairwise interactions, leading to a higher-order construction16. These constructions can be described by simplices of different dimensions and hence, can be studied in the framework of Balance Theory and Topological Data Analysis (TDA). From TDA, we employ the Persistent Homology (PH) analysis tool, which is based on algebraic topology and has been applied to problems in a variety of fields such as network science, physics, chemistry, biology, and medicine31. PH has been previously used to study protein-protein interaction networks to inform cancer therapy by determining the correlation between Betti numbers and the survival of cancer patients32.Cancer is one of the most common human genetic diseases characterized by cellular over-proliferation15. The resulting signed weighted network analysis in the context of Balance Theory showed significant differences between cancer and healthy cases of GRNs in the number of characteristics such as energy, number, and distribution of imbalanced triangles. This paper aims to study the higher-order representation of gene regularity interaction networks derived from cancer and normal samples. Using PH, we address theoretical concepts using empirical data and report network features of cancer samples compared to normal counterparts. Finally, we propose PH as unsupervised network analysis to study human diseases such as cancer.In order to study states of balanced and imbalanced cancer networks, we previously modeled GRNs by groups of three interacting genes, forming triangles (triads) of interactions33 which its nodes represent the genes, and its links between them represent the interactive couplings between genes which can be used to predict the global properties of the cells.Gene expression is the process by which information from a gene is used in the synthesis of a functional gene product which leads to the production of protein as the final functional product. Cells go through a wide range of mechanisms known as Gene regulation to increase or decrease the production of specific gene products. Gene expression data is large-scale measurements of the degrees of freedom of a biological system such as a cell. In the language of statistical physics, these describe the micro-states of a cell. A gene regulatory network is a complex network35. Since RPKM (Reads Per Kilobase transcript per Million reads) puts together the ideas of normalizing by sample and by gene, we used the RPKM normalized data to find the correlation between the expression levels of the genes. The Reads Per Kilobase transcript per Million reads (RPKM) normalized data was used in order to put together the ideas of normalizing by sample and by the gene. When we calculate RPKM, we are normalizing for both the library size (the sum of each column) and the gene length. Due to computational purposes, we only kept the top 483 most variable genes for all analyses by calculating for each gene the variance of its expression level over its samples. For each gene, we have calculated the variance of its expression level over its samples, and accordingly stored the first 483 genes with the highest variance, which is due to more different activity patterns these genes show. This cohort consisted of two sets of 114 healthy and 764 cancer samples.We used mRNA (expression level) data of 20532 genes in the case of Breast Cancer (BRCA: Breast invasive carcinoma) from The Cancer Genome Atlas (TCGA)37 from our data-set based on pairwise gene expressions in the obtained data-set. To find the regulatory connections between genes, we needed a statistical description of the data in terms of suitable observables and infer39 the underlying regulatory connections. Therefore, we restricted ourselves to an undirected pairwise maximum-entropy probability model with terms up to second order42, which we derive for continuous, real-valued variables. This can be considered as a problem in inverse Statistical Physics44 where we want to infer parameters of a model based on observations, instead of calculating observables on the basis of model parameters. We applied the following model with pairwise couplingsi as a continuous real-valued variable, and interaction matrix Z is the so-called partition function, for normalizing the model. The corresponding Hamiltonian (energy function) for this Boltzmann distribution function is then We constructed a pairwise correlation matrixL is the number of genes in the distribution and the couplings can be inferred simply by inverting the matrix of variances and covariances of expression levels 46 such that the inverse of the covariance matrix, Model parameters can be found by satisfying these conditions through the use of Lagrange multipliers; (i) Our model should give the same first and second moments as we measure from the data and (ii) it must maximize the Gibbs-Shannon entropy function defined as J, represent pairwise interaction between genes in the proposed model, where the weight of the link i and gene j. Furthermore, genetic interaction (GI) between two genes can be inferred from the sign of their interactions, indicating the way they may affect each other’s functions. Positive and negative interactions on the foundation of the constructed network imply gene expression modulation within the network. Therefore, we expect J to be a sparse matrix since each gene interacts only with a couple of other genes. Inverting a large covariance matrix computationally, however, yields to a matrix which almost none of its elements are zero. To keep this at bay, the inverse of the covariance matrix has been obtained by means of the Graphical Lasso (GLasso) algorithm47. GLasso is generally a sparse penalized maximum likelihood estimator for the concentration or inverse of covariance matrix of a multivariate elliptical distribution. When dealing with a multivariate Gaussian distribution with limited observations (lack of enough samples)49, GLasso yields a sparse network , is an essential parameter for the specific graph configuration and according to structural balance, the corresponding energies can be compared. This formalism examines the probability distribution of the jammed states’ levels of energy, assuming that for the triads, the shift from balanced to imbalanced can be determined based on all triads that share a common link61. As discussed, constructing 3rd and 4th order interaction networks and examining their corresponding energy for normal and cancer networks provide us with practical insights and can be used to compare stability, energy, and the tendency toward changing their states in cancer and normal samples. This concept motivated us to move forward to study higher-order interaction methods, so as to gain a thorough perspective of these interactions, the patterns of these interactions, by which we address further unsolved questions in this matter. In this paper, as an alternative to studying higher dimensional simplices, we employed a topological scheme to examine the interaction patterns of two networks. This method involves studying cancer and normal gene networks using behaviours of defined k-dimensional holes as a general approach to study their higher-order interactions.Examining the distribution of the triangles suggested the correlations between such triangles were also different between the two networks65. A simplicial complex is represented by a set of a finite collection of k-dimensional simplices (k-simplices) k-dimensional chain (k-chain) as a linear combination of k-simplices of k-simplices k-simplices forms an abstract vector space k-dimensional chain group (k-chain group), where its dimension is the number of k-simplices of the complex. For any simplices in any dimension k, in order to measure the topological features and study the homology of the complex a k-dimensional boundary operator has to be defined as:k-dimensional chain group By analogy, studying and comparing patterns of interactions in the networks as an alternative to transcending triads or quartic order can be considered as describing a building by its floors and bedrooms, and hallways rather than its building blocks. To study higher-order interactions in cancer networks, formed not only by nodes and links but also by triangles and cliques of higher dimensions, we employed algebraic topology strategy toward analyses that require the encapsulation of higher dimensionality as a substitute for simple pairwise interactions. We suggest that these representations are implemented to complement our previously employed network techniques to distinguish the features of cancerous and normal networks. Here we preview some fundamentals of algebraic topology, and homology theory that is utilized in topological data analysismentclass2pt{minimmentclass2pt{minimk-dimensional cycle (k-cycle) k-chain k-dimensional cycle group (k-cycle group), of vector space k-chain k-dimensional boundary (k-boundary) k-dimensional boundary group (k-boundary group) k-cycles that are also k-boundary. To this end, we put an equivalence relation on k-cycles k-homology group of complex One can define a kth Betti number of complex k-homology group of the complex k-dimensional topological holes (k-holes) of complex 66. Due to studying complex networks in terms of homology theory, we use the persistent homology (PH) technique which is the main part of topological data analysis (TDA) as a modern mathematical tool in data science. Following persistent homology strategy, rather than working with the set of nodes (1-simplices) and links (2-simplices), and the statistical properties of the network defined in network science67, we consider higher-order connections as high-dimensional simplices to map the network. In fact, a clique simplicial complex of a network is a simplicial complex in which any k-simplex w, a topological feature 68. Persistence barcode (PB) or equivalently persistence diagram (PD) for each dimension, are representations of PH that summarize topological information of the data-set. For instance, in PD plot of kth dimension for weighted complex k-dimensional topological holes and their persistence upon increasing the threshold by the mean of persistence diagram and barcode for the filtration.By this approach, one can capture global features of the network at any threshold (weight) and monitor the persistence and the robustness of the topological features. Hence, adopting a simplicial modeling, a gene is defined as a 0-simplex The mentclass2pt{minimmentclass2pt{minimmentclass2pt{minimk-homology group of the topological space  between the nodes (agents) of the network, such that by increasing the number of 0-holes k, plotting their persistence barcode and persistence diagrams and their evolution as a function of weight, is our key point to analyze the topological features of these two data sets. Figure By analyzing the interaction networks from the topological point of view, we aim to uncover prominent insights into cellular gene interaction patterns. To this end, applying the PH technique on the weighted complex networks of the normal and cancerous data sets, we analyze the evolution of the dimension of the We then asked whether this apparent separation was due to the variation in the strength and distribution of links in those two networks, where the range of weight function seems to be shortened in the cancerous data and more scattered in the normal one, Fig. Figure 69. It is plausible that the persistent topological spaces in cancer samples are sets of tightly related genes that modulate a specific oncogenic pathway, critical for cellular survival and proliferation. Referring back to our building analogy, with its floors and considering its building plan, our question now is if there exist some established patterns for the genes in cancerous networks upon which genes interact, or how these patterns, deviating significantly from the healthy one develop within the networks.Adopting a novel computational approach, we propose that topological data analysis methods, such as Persistent Homology can be used to study cancer sample data to gain a better perspective on the complexity of this disease at the network level. Cancer is the most common human genetic disease, generated by a number of certain modifications into genes that control the way our cells function. Genes interact with each other, which their highly correlated expressions, and their interactions within a regulatory frame and leading to the emergence of complex structures in the cells, led researchers to investigate the Gene Regulatory Network (GRN) of cells in the framework of graph theory. In this study, we found that network structures are distinctive for normal and cancer samples in both the number and persistence of topological features. Biologically, it is possible that patterns of Betti curves in cancer samples are a manifestation of oncogene addiction at the network level. This phenomenon is defined based on experimental observations that cancer cells appear to be highly dependent on a specific oncogenic pathway


Many activities require accurate wind and wave forecasts in the coastal ocean. The assimilation of fixed buoy observations into spectral wave models such as SWAN (Simulating Waves Nearshore) can provide improved estimates of wave forecasts fields. High-frequency (HF) radar observations provide a spatially expansive dataset in the coastal ocean for assimilation into wave models. A forward model for the HF Doppler spectrum based on first- and second-order Bragg scattering was developed to assimilate the HF radar wave observations into SWAN. This model uses the spatially varying wave spectra computed using the SWAN model, forecast currents from the Navy Coastal Ocean Model (NCOM), and system parameters from the HF radar sites to predict time-varying range-Doppler maps. Using an adjoint of the HF radar model, the error between these predictions and the corresponding HF Doppler spectrum observations can be translated into effective wave-spectrum errors for assimilation in the SWAN model for use in correcting the wind forcing in SWAN. The initial testing and validation of this system have been conducted using data from ten HF radar sites along the Southern California Bight during the CASPER-West experiment in October 2017. The improved winds compare positively to independent observation data, demonstrating that this algorithm can be utilized to fill an observational gap in the coastal ocean for winds and waves. Regional coastal modeling of the ocean and the atmosphere have come a long way in the last 20 years but most models still suffer from errors due to parameterization or inaccurate model inputs . These model errors can be addressed with the assimilation of local observations. In this study, we examine coastal wind estimation by assimilating observations of Doppler spectra from coastal high-frequency (HF) radar sites into a regional wave model with a variational assimilation approach. The wave model used here is SWAN  whichHF radar data assimilation studies commonly utilize the surface current measurements traditionally associated with this data source. A few of these studies are ,4,5. TheThe methodology for the variational assimilation framework in the SWAN model was first laid out by Walker (2006)  and thenShore-based HF radars are routinely used to map surface currents by measuring the Doppler shifted backscatter from ocean waves. The backscattered energy from the ocean surface is due to Bragg scattering of the transmitted electromagnetic wave by ocean surface waves. Coherent reflections in the backscattered spectrum are generated at ocean wavelengths exactly aks. See  for a brThe objectives of this work were to develop a system for assimilation of HF radar data for general nearshore domains, to produce improved wave nowcasts/forecasts, and provide wind forcing corrections. Initial implementation of our HF radar assimilation algorithm in SRI’s swanX 5DVAR system (based on SWAN 41.20) is complete. It includes a ground-wave HF radar model, based on first- and second-order Bragg scattering. The model uses the wave spectrum output from SWAN and the winds and ocean current field from COAMPS-OS along with the HF system parameters. The adjoint of the HF radar model translates errors in the HF radar spectrum into errors in the wave spectrum. The adjoint of SWAN then produces an effective wind error, consistent with the ST6 wind growth model. The whole system is converged, and wind field updates are made to the COAMPS wind inputs. Here, we present an initial demonstration of the SWAN HF radar assimilation system for CASPER-West experiment. Comparisons between the wind estimates from the SWAN HF assimilation and independent buoy data have shown the algorithm capable of correcting winds inputs for this coastal domain.The rest of the paper has the following structure: The algorithm used here is based on the open-source SWAN wave spectrum model and makes use of HFR Doppler spectra data. We briefly describe the model, and then discuss the observed HFR Doppler spectra data. Then, we discuss the HFR forward model and the assimilation framework. Then, we examine the wind model and finish with a brief discussion about implementation.E is the variance density spectrum, t is time. The action balance equation is given byA, the so-called “linear” wind-growth term, is independent of E, and all other terms are proportional to E, but with a collective coefficient E.SWAN is a third-generation wave model for coastal regions ,14. The This set of equations can be solved for the action spectrum for spatial region SWAN version 41.20 (used here) has been updated from 40.85 to include dissipation of swell energy, non-breaking dissipation, optional observation-consistent wind input, and white-capping physics; the complete suite of improvements is known as ST6. There are two options for the implementation of non-breaking dissipation. The first uses the formulation by Young et al. (2013) , updatedWe start with the following expression for the observed energy in the Doppler spectrum x,s) at a fixed time), The first-order scattering is given byAt each spatial location, there is a near-surface ocean current follows,σ^=∫k–plane. It should be noted that the last term in For second-order scattering, the contribution to the Doppler spectrum, equivalent to Equation , is(7)σAccounting for the Doppler shift associated with near-surface currents yieldsAssimilation of HFR data is carried out in a variational framework based on the models described above. This requires definition of a cost function and one or more control variables that are adjusted to minimize the cost function. In this case, the cost function is a measure of the error between the predicted and observed HFR Doppler spectrum, and the control variable is the surface wind field used in the SWAN model. The adjustments to the wind field are guided by the solution of an adjoint version of the SWAN model with input from the adjoint HFR model which, in turn, has as its input the error in the present HFR prediction. The gradient of the cost function with respect to the wind, used to adjust the wind field, is computed from the adjoint wave action spectrum, using an expression derived from the wind-growth term in the SWAN model. In the sections below, we first describe the cost function, followed by the adjoint forms of the models, and then the expression for the gradient. We then describe the sequence of operations for the assimilation algorithm.s, on the other hand, are uncalibrated and contain noise, and can be represented asn) calibration constant and e is a fixed noise level. Rearranging yieldsThe Doppler spectrum at second order has two components that are summed to produce the complete spectrum. Since they are summed, we can treat them separately and then combine the results. First, we must examine our data. The model for the first- or second-order returns produces the noise-free Doppler spectral density in units of normalized radar cross section (square meters per square meter) per radian  with respect to the wind that can be used to guide adjustments to the wind. The source function Equation  containsailed in ). The deJ. In practice, the following sequence of steps is followed:The adjoint solution is calculated using the error in the most recent prediction as input;Using the adjoint solution, the gradient is determined;The conjugate-gradient descent algorithm is used to calculate a new estimate of the wind field and The SWAN model is run with corrected inputs and a new wave-spectrum prediction for the region is generated;The forward HF radar model is run with the new spectrum as input and a new prediction of the data is generated.The initial guess for the wave spectrum over the entire region is a previously corrected SWAN model background that has corrected the swell due to assimilation of National Data Buoy Center (NDBC) buoy data. The adjoint Equation  is solveA graphical description of the complete algorithm structure is shown in The HFR inverse modeling framework developed here was applied for estimation of the 10-m winds in the Southern California Bight during the CASPER-West field experiment in October 2017. The aim here was to utilize HF radar Doppler spectra observations to estimate the winds near the coast. The SWAN model (version 41.20) was run with all the relevant model physics enabled  and optional observation-consistent wind input and white-capping physics). The complete suite of improvements are known as ST6 with all input parameters set to their default values. The details of the ST6 physics can be found in Rogers et al. . The iniHere, we utilize the Doppler spectra returns of sea state from 10 SeaSonde type Coastal Ocean Dynamics Applications Radar (CODAR). The locations of HFR sites can be seen in In this section, we present comparisons between wave parameters from NDBC buoy 46217 and the predicted wave spectra from the same location. Comparison at buoys 46025 and 46221 are similar, so they are not shown in detail. We limit our comparisons to the significant wave height Here, we compare the 10-m winds from the SWAN HF assimilation algorithm against winds from NDBC buoy 46053. The buoy winds have not been incorporated in any way into our assimilation framework, so, they can be treated as independent observations. We restrict our comparisons to the U and V components of the wind.In this work, we have described the development, implementation, and testing of an HF radar assimilation algorithm for the SWAN model. This framework utilizes a cost function to determine the error between the observed and predicted Doppler spectra from an HF radar first- and second-order forward model. Adjoint models of both the HF radar forward model and the SWAN model are then solved. The first- and second-order gradients of the cost functions are calculated from the adjoint spectrum solutions. Lastly, a conjugate-gradient algorithm is used to determine the appropriate Doppler spectrum and wind conditions that minimizes the cost function.The algorithm was applied and validated using real-world observations from coastal wave buoys and HFR sites during the CASPER-West field experiment in October of 2017. The background wave field was taken from a buoy assimilation experiment where the open-boundaries were corrected. Predicted and observed significant wave height In conclusion, a variational HF radar Doppler spectrum assimilation algorithm for the SWAN model has been developed. Corrections in the near surface winds during the CASPER-West experiment were generated. A comparison of these winds to independent buoy observations shows good agreement and the skill of the algorithm in adjusting errors in coastal wind forecasts. Additional testing of the algorithm in other regions of interest is necessary before this tool can be utilized for operational forecasting. Further expansion of the algorithm to potentially account for bistatic or skywave HF radar configurations is ongoing and should allow for more general utility and relocatablity.


Model-based force estimation is an emerging methodology in the mechatronic community given the possibility to exploit physically inspired high-fidelity models in tandem with ready-to-use cheap sensors. In this work, an inverse input load identification methodology is presented combining high-fidelity multibody models with a Kalman filter-based estimator and providing the means for an accurate and computationally efficient state-input estimation strategy. A particular challenge addressed in this work is the handling of the redundant state-description encountered in common multibody model descriptions. A novel linearization framework is proposed on the time-discretized equations in order to extract the required system model matrices for the Kalman filter. The presented framework is experimentally validated on a slider-crank mechanism. The nonlinear kinematics and dynamics are well represented through a rigid multibody model with lumped flexibilities to account for localized interaction phenomena among bodies. The proposed methodology is validated estimating the input torque delivered by a driver electro-motor together with the system states and comparing the experimental data with the estimated quantities. The results show the stability and accuracy of the estimation framework by only employing the angular motor velocity, measured by the motor encoder sensor and available in most of the commercial electro-motors. In mechatronic systems, operational forces and moments are essential quantities in the different stages of the development cycle and strongly impact the design, durability, diagnostic, prognostic, maintenance, and advanced control strategies . HoweverIn past decades, different test-driven and model-based inverse force methods have been presented in the literature to overcome these limitations. Initially, the challenge of inverse load identification was tackled in offline test-based strategies. One of the most commonly used technique for mechanical applications relates to the field of Transfer Path Analysis (TPA) .TPA summarizes the family of test-based methodologies to study the vibration transmission in mechanical systems where the Matrix Inversion and Mount Stiffness approaches are the most commonly used to estimate inputs and transmitted forces respectively. Despite the wide variety of methods and extensive use in the industrial world, TPA still remains quite expensive from an experimental point of view in terms of preparation and execution time.The growing computational power of modern computers opened new opportunities to exploit numerical model-based methods. These models can be exploited in virtual sensing approaches  which enState estimation techniques such as the Kalman Filter (KF) methods allow the joint estimation of unknown inputs and model states  in an efMultibody (MB) modeling approaches are regularly used in the literature and industry ,6,7 for However, the link between MB models and estimation algorithms is nontrivial since most estimators require an ordinary differential state-space representation of the system dynamics. Instead, the MB model dynamics, depicted by the Equations of Motion (EOMs), are generally described by a set of Implicit-Differential Algebraic Equations (I-DAEs) that makes the state-space representation difficult to be met. On the other hand, Explicit-Ordinary Differential Equations (E-ODEs) are well suited for a state-space representation but specific MB formulations should be employed to obtain this structure , otherwIn  the MatrSimilarly in ,11,12 thAlternatively, in  a kinemaIn ,15 an inIn  an AugmeIn this work, a generalization of the methodology described in  is preseThe proposed methodology starts from a novel linearization approach of the EOMs that includes the algebraic variables (Lagrange multipliers) to the system states. Consequently, the resulting unconstrained discrete-time state-space model is employed in a constraint KF scheme where the kinematic constraints are enforced trough the augmented measurement equations, therefore eliminating the effort of selecting effective penalty factors.The scientific contribution is structured as follows: in This section summarizes the derivation of the EOMs of flexible multibody systems, as they will be employed in this work.The mathematical description of the system dynamics can be derived by means of the Lagrange’s equations for constrained mechanical systems :(1)ddt∂LEquation  can be wA set of natural coordinates posed in , where rposed in  allows dEquation  can be wHere, Here, n referred to the natural coordinate formulation for the remainder of this manuscript.Due to the structure of the EOM, Equation , for theDespite the computational advantages of the above mentioned MB approach, the methodologies that will be introduced in the next sections can be easily extended to alternative MB formulation, such as the floating-frame of reference component mode synthesis approach or the generalized component mode synthesis ,20.The I-DAEs form of Equation  are geneIn the next section, we present a new methodology to directly linearize the I-DAEs starting from its discrete form but without employing any explicit constraint elimination technique.To transform the second order differential equation into first-order differential equations, it is common to introduce the velocity variable Equation  in a firThese equations represent a system of constrained-DAEs of index 3 . NumericIn this work, a first order, constant time-step Backward Differentiation Formula (BDF-1), also known as Backward Euler, is employed. This choice does not imply that other differentiation schemes cannot be applied to discretize the EOMs in time. However, the time-discretization must be consistent with the defined estimation sampling and particular attention should be paid to choosing the differentiation schemes  because it influences the achievement of the discrete-time E-ODE form of the EOMs, as will be discussed in The single step method BDF-1 can be written asv˙k+1=1h tailored for MB models is discussed in this section. In this section we discuss all the required components to set up a Kalman filter for assimilating the different multibody states and unknown inputs. More in details, in The system of EOMs in Equation  describeHere, h are necessary which are affected by measurement noise y, the concept of virtual sensor (VS) The estimation framework relies on combining the prior model information with measurements A virtual sensor represents model-based information; more specifically, it is a physical quantity that can be expressed as a function of the system variables, such as joint forces, body motion trajectories  and body strains/stresses. A VS equation can be treated similarly to a measurement equation but evaluated after the a posteriori Kalman filter step and hence to exploited in the estimation process itself.In  differeny of Equation 45):(46)gd.The linearized system matrix Equation  at the tEquation , is obtaEquation , while BEquation  but withd.Subsequently, the linearized augmented measurement matrix Equation  but withStarting from these model, measurement and tangent matrices, an extended Kalman filter (EKF) procedure can be set up in order to estimate both the states and the unknown input forces. The application of this estimation approach is discussed in the following section.In this work, we employ an extended Kalman filter (EKF)  to perfoIn general, we can group the EKF scheme in two main steps: the a-priori and a posteriori steps.A-priori step: Assuming that the augmented states Equation :(49)gd(Knowing the estimated state covariance matrix Equation  as(50)PThe predicted measurement Equation  as:(51)The Kalman filter gain Equation  and R˜k A-posteriori step: When the real measurement The inclusion of the actual measurements also affects the posterior covariance matrix In With this scheme the entire state-input estimation can be performed for an arbitrary (flexible) multibody model. In the following section we will validate it on an experimental setup.In practical applications the performance of the estimation scheme will strongly depend on the tuning of the model covariance In this section, the presented joint state-input estimation methodology is experimentally validated on a slider-crank mechanism.The slider-crank system in C), crank shaft (S), rod-slider bearing . It includes a motor encoder sensor allowing the measurement of the rotation angle and velocity of the motor shaft, indicated by www.pcb.com) to measure its translational acceleration along the sliding direction indicated by T.The system motion is driven by a brushless servomotor “MAC3000”’ with integrated controller MAC00-B4 from JVL  and on the angular velocity transitions (zoom-in figures on the right) thanks to the proposed methodology.In The choice of performing the experiments for a relatively long timespan was made to demonstrate the filter stability in terms of both mean value and covariance prediction. For these kind of applications, where the uncertainty is dominated by difficult to model load effects , the choice of focussing the model covariance on the input load allows effectively propagating the uncertainties. The dashed blue curves in These results for the estimated torques show a significantly larger error than those obtained for the (derivative) states. For multibody problems in general, the state-estimates can be expected to be dominated by the kinematics of the system, which are generally well known. For the load estimates however, the dynamics of the system will play a crucial role. Besides key dynamic quantities such as the system inertia which can be modelled very accurately, other effects such as friction forces also influence this outcome. In this work we employed the friction model from Equation , but anyerved in  where thAn important aspect in estimation problems is the choice of the Kalman filter parameters such as the process and measurement noise covarince matrices, To attain the best accuracy from the presented estimation scheme, the model and measurement covariances need to be judiciously selected. In this work we have started with the selection of the measurement covariance matrix Similarly to the augmented measurement covariance As we assume the model to be practically perfect compared to the high uncertainty on the unknown inputs, the process noise matrix Error norm and Smoothing error. These norms are defined ask. On the left side of The remaining parameter posed by  where itOn the right side of In Although this study does not address the full scope of noise covariance tuning, the author deemed it sufficient to explain the methodological developments considered in this contribution. Further research on covariance tuning might be necessary to obtain a more holistic approach.This work presents a new estimation methodology tailored for MB models to enable the definition of virtual sensors for various system states and inputs.Through the choice of a general MB modeling approach various key physical effects can be accurately accounted for, ranging from nonlinear kinematics to complex dynamic effects. The developed framework allows using these multibody models in an estimation framework without particular additional modeling assumptions or reformulations. More specifically, no constraint elimination methods are required to employ the defined MB model into the estimation framework, reducing the preparation time and the user effort to setup the estimation problem while ensuring the correct physical and mathematical interpretation of the system under investigation. As the proposed methodology has no particular assumptions with respect to the multibody model formulation it can be easily extended to any of the commonly available (flexible) MB approaches, e.g., FNCF, FFR-CMS, or GCMS. However, to fully benefit from the proposed approach the equations of motion and tangent stiffness matrices of the system should be analytically available to efficiently assemble the linearized system and measurement matrices.In the present work, we exploited the FNCF MB approach, as this methodology inherently enables an easy and efficient evaluation of the different tangent model matrices required for the estimation framework.Finally, the developed methodology has been experimentally validated on a slider-crank mechanism. Very high accuracy is obtained for the estimated states with respect to the available measurements. Good accuracy is also obtained for the estimated input torque, but due to the larger dynamic model errors in e.g., friction effects the resulting errors are higher than those obtained for the states. The validation has been performed over a relatively large time-span which also demonstrates the capability of the presented framework to obtain long-term stable estimates with a bounded uncertainty, in the form of a bounded covariance.The presented methodology has some drawbacks since a large number of states (including the Lagrange multipliers) and measurements (including the constraints equations) are employed. These lead to a computationally less efficient approach as compared to other state of the art techniques . A possFuture work will focus on how these methodologies can be employed to obtain more accurate descriptions for key dynamic effects such as the friction present in these multibody systems.