Yunxin Xie, Chenyang Zhu, Wei Jiang, Jia Bi, Zhengwei Zhu, "Analyzing Machine Learning Models with Gaussian Process for the Indoor Positioning System", Mathematical Problems in Engineering, vol. The hyperparameter \(\ell\) is a locality parameter, i.e. Their approach reaches the mean error of 1.6 meters. Thus, these parameters are tuned to with cross-validation to get the best XGBoost model. Gaussian Processes (GP) are a generic supervised learning method designed to solve regression and probabilistic classification problems. A model is built with supervised learning for the given input and the predicted value is . We are committed to sharing findings related to COVID-19 as quickly as possible. Besides, the GPR is trained with … The gaussian process fit automatically selects the best hyperparameters which maximize the log-marginal likelihood. basis functions number of basis function.” (Gaussian Processes for Machine Learning, Ch 2.2). prior distribution to contain only those functions which agree with the observed The radiofrequency-based system utilizes signal strength information at multiple base stations to provide user location services [2]. \begin{array}{cc} The number of boosting iterations and other parameters concerning the tree structure do not affect the prediction accuracy a lot. Thus, we select this as the kernel of the GPR model to compare with other machine learning models. Machine learning approaches can avoid the complexity of determining an appropriate propagation model with traditional geometric approaches and adapt well to local variations of indoor environment [6]. p. cm. Equation (10) shows the Rational Quadratic kernel, which can be seen as a mixture of RBF kernels with different length scales. When the maximum depth of the individual tree reaches 10, the model comes to the best performance. Indoor position estimation is usually challenging for robots with only built-in sensors. [1989] Indoor positioning modeling procedure with offline phase and online phase. Gaussian Process Regression Gaussian Processes: Deﬁnition A Gaussian process is a collection of random variables, any ﬁnite number of which have a joint Gaussian distribution. Recently, there has been growing interest in improving the efficiency and accuracy of the Indoor Positioning System (IPS). During the training process, the model is trained with the four folds of data and test with the left fold of data. Updated Version: 2019/09/21 (Extension + Minor Corrections). Distance error with confidence interval for different Gaussian progress regression kernels. Gaussian process regression (GPR) models are nonparametric kernel-based probabilistic models. Then the current model is updated with the previous model with the shrunk base model . The model can determine the indoor position based on the RSS information in that position. Learning the hyperparameters Automatic Relevance Determination 7. From the consistency requirement of gaussian processes we know that the prior distribution for \(f_*\) is \(N(0, K(X_*, X_*))\). Let us plot the resulting fit: In contrast, we see that for these set of hyper parameters the higher values of the posterior covariance matrix are concentrated along the diagonal. function corresponds to a Bayesian linear regression model with an infinite Then, we got the final model that maps the RSS to its corresponding position in the building. The training set’s size could be adjusted accordingly based on the model performance, which would be discussed in the following section. Besides, the GPR is trained with three kernels, namely, Radial-Basis Function (RBF) kernel, Matérn kernel, and Rational Quadratic (RQ) kernel, and evaluated with the average error and standard deviation. A machine-learning algorithm that involves a Gaussian pro In this paper, we use the validation curve with 5-fold cross-validation to show the balanced trade-off between the bias and variance of the model. We will be providing unlimited waivers of publication charges for accepted research articles as well as case reports and case series related to COVID-19. Updated Version: 2019/09/21 (Extension + Minor Corrections). The Gaussian process model is mainly divided into Gaussian process classification and Gaussian process regression (GPR), … Gaussian processes (GPs) provide a principled, practical, probabilistic approach to learning in kernel machines. Results reveal that there has been a gradual decrease in distance error with the increasing of the training size for all machine learning models. In the validation curve, the training score is higher than the validation score as the model will be a better fit to the training data than test data. When I was reading the textbook and watching tutorial videos online, I can follow the majority without too many difficulties. Given a set of data points associated with set of labels , supervised learning could build a regressor or classifier to predict or classify the unseen from . More recently, there has been extensive research on supervised learning to predict or classify some unseen outcomes from some existing patterns. Hyperparameter tuning is used to select the optimum parameter set for each model. The graph also shows that there has been a sharp drop in the distance error in the first three APs for XGBoost, RF, and GPR models. Later in the online phase, we can use the generated model for indoor positioning. We consider de model \(y = f(x) + \varepsilon\), where \(\varepsilon \sim N(0, \sigma_n)\). GP Deﬁnition and Intuition 4. Gaussian processes (GPs) provide a principled, practical, probabilistic approach to learning in kernel machines. This is actually the implementation used by Scikit-Learn. In the first step, cross-validation (CV) is used to test whether the model is suitable for the given machine learning model. (d) Learning rate. Results show that the distance error decreases gradually for the SVR model. (d) Min samples leaf. How to apply these techniques to classification problems. Thus, linear models cannot describe the model correctly. The data are available from the corresponding author upon request. During the procedure, trees are built to generate the forest. Here, is the covariance matrix based on training data points , is the covariance matrix between the test data points and training points, and is the covariance matrix between test points. The infrared-based system uses sensor networks to collect infrared signals and deduce the infrared client’s location by checking the location information of different sensors [3]. We reshape the variables into matrix form. The size of the APs determines the size of the features. Here, defines the stochastic map for each data point and its label and defines the measurement noise assumed to satisfy the Gaussian noise with standard deviation: Given the training data with its corresponding labels as well as the test data with its corresponding labels with the same distribution, then equation (6) is satisfied. Please refer to the docomentation example to get more detailed information. Hyperparameter tuning for different machine learning models. Section 2 summarizes the related work that constructs models for indoor positioning. First, they areextremely common when modeling “noise” in statistical algorithms. Machine Learning Summer School 2012: Gaussian Processes for Machine Learning (Part 1) - John Cunningham (University of Cambridge) http://mlss2012.tsc.uc3m.es/ XGBoost also outperforms the SVR with RBF kernel. In this paper, we evaluate different machine learning approaches for indoor positioning with RSS data. Gaussian processes for machine learning / Carl Edward Rasmussen, Christopher K. I. Williams. This means that we expect points far away to have no effect on each other, i.e. During the field test, we collect 799 RSS data as the training set. Random Forest (RF) algorithm is one of the ensemble methods that build several regression trees and average the result of the final prediction of each regression tree [19]. Figure 4 shows the tuning process that calculates the optimum value for the number of trees in the random forest as well as the tree structure of the individual tree in the forest. To overcome these challenges, Yoshihiro Tawada and Toru Sugimura propose a new method to obtain a hedge strategy for options by applying Gaussian process regression to the policy function in reinforcement learning. [39] proposed methods for preference-based Bayesian optimization and GP regression, re-spectively, but they were not active. \text{cov}(f_*) = K(X_*, X_*) - K(X_*, X)(K(X, X) + \sigma^2_n I)^{-1} K(X, X_*) \in M_{n_*}(\mathbb{R}) Gaussian process regression offers a more flexible alternative to typical parametric regression approaches. where \(\sigma_f , \ell >0\) are hyperparameters. Connection to … The model is then trained with the RSS training samples. Gaussian process regression (GPR). The CV can be used for feature selection and hyperparameter tuning. Accumulated errors could be introduced into the localization process when the robot moves around. ISBN 0-262-18253-X 1. The model-based positioning system involves offline and online phases. N(\bar{f}_*, \text{cov}(f_*)) Wireless indoor positioning is attracting considerable critical attention due to the increasing demands on indoor location-based services. (b) Learning rate. Overall, XGBoost still has the best performance among RF and GPR models. Recall that a gaussian process is completely specified by its mean function and covariance (we usually take the mean equal to zero, although it is not necessary). The hyperparameter \(\sigma_f\) enoces the amplitude of the fit. (a) Impact of the number of RSS samples. 2020, Article ID 4696198, 10 pages, 2020. https://doi.org/10.1155/2020/4696198, 1School of Petroleum Engineering, Changzhou University, Changzhou 213100, China, 2School of Information Science and Engineering, Changzhou University, Changzhou 213100, China, 3Electronics and Computer Science, University of Southampton, University Road, Southampton SO17 1BJ, UK. No guidelines of the size of training samples and the number of AP are provided to train the models. The training procedure is repeated five times to calculate the average accuracy of the model with the specific parameter. The implementation is based on Algorithm 2.1 of Gaussian Processes for Machine Learning (GPML) by Rasmussen and Williams. The task is then to learn a regression model that can predict the price index or range. This trend indicates that only three APs are required to determine the indoor position. I… Trained with a few samples, it can obtain the prediction results of the whole region and the variance information of the prediction that is used to measure confidence. Moreover, there is no state-of-the-art work that evaluates the model performance of different algorithms. We propose a new robust GP regression algorithm that iteratively trims a portion of the data points with the largest deviation from the predicted mean. There are my kernel functions implemented in Scikit-Learn. f_*|X, y, X_* Gaussian processes—Data processing. f_* Acknowledgments: Thank you to Fox Weng for pointing out a typo in one of the formulas presented in a previous version of the post. every finite linear combination of them is normally distributed. Hyperparameter tuning for Random Forest model. Their results show that the SVR models have better positioning performance compared with NN models. Thus, ensemble methods are proposed to construct a set of tree-based classifiers and combine these classifiers’ decision with different weighting algorithms [18]. We demonstrate … However, using one single tree to classify or predict data might cause high variance. Series. This means that we expect points far away can still have some interaction, i.e. Sign up here as a reviewer to help fast-track new submissions. Figure 5 shows the tuning process that calculates the optimum value for the number of boosting iterations and the learning rate for the AdaBoost model. Hsieh, K.-W. Chang, M. Ringgaard, and C.-J. Bekkali et al. Gaussian processes for classiﬁcation Laplace approximation 8. But they are also used in a large variety of applications … Section 6 concludes the paper and outlines some future work. III. time or space. Examples of use of GP 2. In GPR, covariance functions are also essential for the performance of GPR models. (a) Number of estimators. A relatively rare technique for regression is called Gaussian Process Model. Results show that GP with a rational quadratic kernel and eXtreme gradient tree boosting model has the best positioning accuracy compared to other models. The RSS data of seven APs are taken as seven features. The training process of supervised learning is to minimize the difference between predicted value and the actual value with a loss function . The prediction results are evaluated with different sizes of training samples and numbers of AP. However, in some cases, the distribution of data is nonlinear. \right) Drucker et al. The output is the coordinates of the location on the two-dimensional floor. Next, we generate some training sample observations: We now consider test data points on which we want to generate predictions. Brunato evaluated the k-nearest-neighbor approach for indoor positioning with wireless signals from several access points [8], which has an average uncertainty of two meters. With the increase of the training size, GPR gets the better performance, while its performance is still slightly weaker compared with the XGBoost model. \]. Thus, given the training data points with label , the estimated of target can be calculated by maximizing the joint likelihood in equation (7). As is shown in Section 2, the machine learning models require hyperparameter tuning to get the best model that fits the data. The RSS readings from different AP are collected during the offline phase with the machine learning approach, which captures the indoor environment’s complex radiofrequency profile [7]. The method is tested using typical option schemes with … The advantages of Gaussian processes are: The prediction interpolates the observations (at least for regular kernels). This paper mainly evaluates three covariance functions, namely, Radial Basis Function (RBF) kernel, Matérn kernel, and Rational Quadratic kernel. data points, that is, we are interested in computing \(f_*|X, y, X_*\). The marginal likelihood is the integral of the likelihood times the prior. However, based on our proposed XGBoost model with RSS signals, the robot can predict the exact position without the accumulated error. Hyperparameter tuning for SVR with linear and RBF kernel. \text{cov}(f(x_p), f(x_q)) = k_{\sigma_f, \ell}(x_p, x_q) = \sigma_f \exp\left(-\frac{1}{2\ell^2} ||x_p - x_q||^2\right) We now compute the matrix \(C\). Figure 3 shows the tuning process that determines the optimum value for the penalty parameter and kernel coefficient parameter for the SVR with RBF and linear kernels. Tables 1 and 2 show the distance error of different machine learning models. how far the points interact. In probability theory and statistics, a Gaussian process is a stochastic process, such that every finite collection of those random variables has a multivariate normal distribution, i.e. As the coverage range of infrared-based clients is up to 10 meters while the coverage range of radiofrequency-based clients is up to 50 meters, radiofrequency has become the most commonly used technique for indoor positioning. A better approach is to use the Cholesky decomposition of \(K(X,X) + \sigma_n^2 I\) as described in Gaussian Processes for Machine Learning, Ch 2 Algorithm 2.1. Table 2 shows the distance error with a confidence interval for different kernels with length scale bounds. proposed a support vector regression (SVR) algorithm that applies a soft margin of tolerance in SVM to approximate and predict values [15]. As SVR has the best prediction performance in the current work, we select SVR as a baseline model to evaluate the performance of the other three machine learning approaches and the GPR approach with different kernels. \(K(X_*, X) \in M_{n_* \times n}(\mathbb{R})\), Sampling from a Multivariate Normal Distribution, Regularized Bayesian Regression as a Gaussian Process, Gaussian Processes for Machine Learning, Ch 2, Gaussian Processes for Timeseries Modeling, Gaussian Processes for Machine Learning, Ch 2.2, Gaussian Processes for Machine Learning, Appendinx A.2, Gaussian Processes for Machine Learning, Ch 2 Algorithm 2.1, Gaussian Processes for Machine Learning, Ch 5, Gaussian Processes for Machine Learning, Ch 4, Gaussian Processes for Machine Learning, Ch 4.2.4, Gaussian Processes for Machine Learning, Ch 3. Let’s assume a linear function: y=wx+ϵ. —(Adaptive computation and machine learning) Includes bibliographical references and indexes. In recent years, there has been a greater focus placed upon eXtreme Gradient Tree Boosting (XGBoost) models [21]. GPs have received increased attention in the machine-learning community over the past decade, and this book provides a long-needed systematic and unified treatment of theoretical and practical aspects of GPs in machine learning. Using the results of Gaussian Processes for Machine Learning, Appendinx A.2, one can show that, \[ A GP is usually parameterized by a mean function and a covariance function , formalized in equations (3) and (4). The RBF and Matérn kernel have the 4.4 m and 8.74 m confidence interval with 95% accuracy while the Rational Quadratic kernel has the 0.72 m confidence interval with 95% accuracy. This paper is organized as follows. compared different kernel functions of the support vector regression to estimate locations with GSM signals [6]. Figure 7(a) shows the impact of the training sample size on different machine learning models. \], \[ In the building, we place 7 APs represented as red pentagram on the floor with an area of 21.6 M 15.6 m. The RSS measurements are taken at each point in a grid of 0.6 m spacing between each other. During the online phase, the client’s position is determined by the signal strength and the trained model. There are many questions which are still open: I hope to keep exploring these and more questions in future posts. Results show that the XGBoost model outperforms all the other models and related work in positioning accuracy. At last, the weak models are combined to generate the strong model . Gaussian Processes in Reinforcement Learning Carl Edward Rasmussen and Malte Kuss Max Planck Institute for Biological Cybernetics Spemannstraße 38, 72076 Tubingen,¨ Germany carl,malte.kuss @tuebingen.mpg.de Abstract We exploit some useful properties of Gaussian process (GP) regression models for reinforcement learning in continuous state spaces and dis-crete time. Hyperparameter tuning for XGBoost model. On the machine learning side, Gonzalez´ et al. In the training process, we use the RSS collected from different APs as features to train the model. When the validation score decreases, the model is overfitting. This course covers the fundamental mathematical concepts needed by the modern data scientist to … However, the global positioning system (GPS) has been used for outdoor positioning in the last few decades, while its positioning accuracy is limited in the indoor environment. In contrast, the eXtreme gradient tree boosting model could achieve higher positioning accuracy with smaller training size and fewer access points. During the training process, the number of trees and the trees’ parameter are required to be determined to get the best parameter set for the RF model. In all stages, XGBoost has the lowest distance error compared with all the other models. The hyperparameter tuning technique is used to select the optimum parameter set for each model. built Gaussian process models with the Matérn kernel function to solve the localization problem in cellular networks [5]. Wu et al. Unlike many popular supervised machine learning algorithms that learn exact values for every parameter in a function, the Bayesian approach infers a probability distribution over all possible values. Results also reveal that 3 APs are enough for indoor positioning as the distance error does not decrease with more APs. Let us denote by \(K(X, X) \in M_{n}(\mathbb{R})\), \(K(X_*, X) \in M_{n_* \times n}(\mathbb{R})\) and \(K(X_*, X_*) \in M_{n_*}(\mathbb{R})\) the covariance matrices applies to \(x\) and \(x_*\). \], \[ Abstract We give a basic introduction to Gaussian Process regression models. Battiti et al. Thus, kernel functions map the nonlinear separable feature space to linear separable feature space with kernel functions [16]. In the past decade, machine learning played a fundamental role in artificial intelligence areas such as lithology classification, signal processing, and medical image analysis [11–13]. More APs are not helpful as the indoor positioning accuracy is not improving with more APs. However, the confidence interval has a huge difference between the three kernels. Besides machine learning approaches, Gaussian process regression has also been applied to improve the indoor positioning accuracy. K(X, X) + \sigma^2_n I & K(X, X_*) \\ Let us now sample from the posterior distribution: We now study the effect of the hyperparameters \(\sigma_f\) and \(\ell\) of the kernel function defined above. Moreover, the traditional geometric approach that deduces the location based on the angle and distance estimates from different signal transmitters is problematic as the transmitted signal might be distorted due to reflections and refraction and the indoor environment [5]. The validation curve shows that the maximum depth of the tree might affect the performance of the RF model. Thus, validation curves can be used to select the best parameter of a model from a range of values. In machine learning they are mainly used for modelling expensive functions. Overall, the GPR with Rational Quadratic kernel has the lowest distance error among all the GP models, and XGBoost has the lowest distance error compared with other machine learning models. We write Android applications to collect RSS data at reference points within the test area marked by the seven APs, whereas the RSS comes from the Nighthawk R7000P commercial router. Results show that the NN model performs better than the k-nearest-neighbor model and can achieve a standard average of 1.8 meters. Can we combine kernels to get new ones? The validation curve shows that when is 0.01, the SVR has the best performance in predicting the position. Thus, we use machine learning approaches to construct an empirical model that models the distribution of Received Signal Strength (RSS) in an indoor environment. Machine learning—Mathematical models. As a concrete example, let us consider (1-dim problem). Lin, “Training and testing low-degree polynomial data mappings via linear svm,”, T. G. Dietterich, “Ensemble methods in machine learning,” in, R. E. Schapire, “The boosting approach to machine learning: an overview,” in, T. Chen and C. Guestrin, “Xgboost: a scalable tree boosting system,” in, J. H. Friedman, “Stochastic gradient boosting,”. After a sequence of preliminary posts (Sampling from a Multivariate Normal Distribution and Regularized Bayesian Regression as a Gaussian Process), I want to explore a concrete example of a gaussian process regression. Our work assesses the positioning performance of different models and experiments on the size of training samples and the number of APs for the optimum model. Section 3 introduces the background of machine learning approaches as well as the kernel functions for GPR. Each model is trained with the optimum parameter set obtained from the hyperparameter tuning procedure. \left( We focus on understanding the role of the stochastic process and how it is used to define a distribution over functions. Observe that we need to add the term \(\sigma^2_n I\) to the upper left component to account for noise (assuming additive independent identically distributed Gaussian noise). In this section, we evaluate the result by evaluating the performance of the models with 200 collected RSS samples with location coordinates. I. Williams, Christopher K. I. II. Probabilistic modelling, which falls under the Bayesian paradigm, is gaining popularity world-wide. Park C and Apley D (2018) Patchwork Kriging for large-scale Gaussian process regression, The Journal of Machine Learning Research, 19:1, (269-311), Online publication date: 1-Jan-2018. Here, is the penalty parameter of the error term : SVR uses a linear hyperplane to separate the data and predict the values. compared the neural network- (NN-) based model and k-nearest-neighbor model to determine the mobile terminal under the wireless LAN environment [9]. The Matérn kernel adds parameter that controls the resulting function’s smoothness, which is given in equation (9). Given the feature space and its corresponding labels, the RF algorithm takes a random sample from the features and constructs the CART tree with randomly selected features. K(X_*, X) & K(X_*, X_*) The Housing data set is a popular regression benchmarking data set hosted on the UCI Machine Learning Repository. We calculate the confidence interval by multiplying the standard deviation with 1.96. The 200 RSS data are collected during the day with people moving or environment changes, which are used to evaluate the model performance. \]. Tuning is a process that uses a performance matrix to rank the regressors with different parameters to optimize a parameter for each specific model [11]. Consistency: If the GP speciﬁes y(1),y(2) ∼ N(µ,Σ), then it must also specify y(1) ∼ N(µ 1,Σ 11): A GP is completely speciﬁed by a mean function and a The weights of the model are calculated given that model function is at most from the target ; formally, . To avoid overfitting, we also tune the subsample parameter that controls the ratio of training data before growing trees. \left( Here \(f\) does not need to be a linear function of \(x\). Alfakih et al. Machine Learning Srihari Topics in Gaussian Processes 1. Results show that XGBoost has the best performance compared with all the other machine learning models. \], \[ Maximum likelihood estimation (MLE) has been used in statistical models, given the prior knowledge of the data distribution [25]. Analyzing Machine Learning Models with Gaussian Process for the Indoor Positioning System, School of Petroleum Engineering, Changzhou University, Changzhou 213100, China, School of Information Science and Engineering, Changzhou University, Changzhou 213100, China, Electronics and Computer Science, University of Southampton, University Road, Southampton SO17 1BJ, UK, Determine the leaf weight for the learnt structure with, A. Serra, D. Carboni, and V. Marotto, “Indoor pedestrian navigation system using a modern smartphone,” in, P. Bahl, V. N. Padmanabhan, V. Bahl, and V. Padmanabhan, “Radar: an in-building rf-based user location and tracking system,” in, A. Harter and A. Hopper, “A distributed location system for the active office,”, H. Hashemi, “The indoor radio propagation channel,”, A. Schwaighofer, M. Grigoras, V. Tresp, and C. Hoffmann, “Gpps: a Gaussian process positioning system for cellular networks,”, Z. L. Wu, C. H. Li, J. K. Y. Ng, and K. R. Leung, “Location estimation via support vector regression,”, A. Bekkali, T. Masuo, T. Tominaga, N. Nakamoto, and H. Ban, “Gaussian processes for learning-based indoor localization,” in, M. Brunato and C. Kiss Kallo, “Transparent location fingerprinting for wireless services,”, R. Battiti, A. Villani, and T. Le Nhat, “Neural network models for intelligent networks: deriving the location from signal patterns,” in, M. Alfakih, M. Keche, and H. Benoudnine, “Gaussian mixture modeling for indoor positioning wifi systems,” in, Y. Xie, C. Zhu, W. Zhou, Z. Li, X. Liu, and M. Tu, “Evaluation of machine learning methods for formation lithology identification: a comparison of tuning processes and model performances,”, Y.

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