Parameter estimation results are reported in Table 1. Subspace identification methods have the potential to provide extremely useful information in the two critical selections mentioned above. Aquifer hydraulics models coupled with geostatistical estimations techniques can adequately guide studies of hydrogeological characterisation. This section presents an overview of the available methods used in life data analysis. The software ensures P(t) is a positive-definite matrix by using a square-root algorithm to update it .The software computes P assuming that the residuals (difference between estimated and measured outputs) are white noise, and the variance of these residuals is 1.R 2 * P is the covariance matrix of the estimated parameters, and R 1 /R 2 is the covariance matrix of the parameter changes. Arun Pankajakshan, ... Federico Galvanin, in Computer Aided Chemical Engineering, 2018. The work presented in this contribution provides a methodology for finding the optimal experiment design for nonlinear dynamic systems in the context of guaranteed parameter estimation. Results are discussed in terms of i) estimated profiles; ii) parameter estimation, including estimated values and a-posteriori statistics (t-values); iii) information profiles (trace of FIM). x�c```b``������#� � `620�3�YΕ+����7M&��*4AH�YP'7��, � 2ll?�r�����]�Bl��y](qy�Q� ��� Almost all modern machine learning algorithms work like this: (1) specify a probabilistic model that has parameters. In this case, the parameter estimation algorithm (optim_methodargument) and the criterion function (crit_function argument) must be set in input of estim_param function.The list of available criteria for Bayesian methods is given by ? A statistical procedure or learning algorithm is used to estimate the parameters of the probability distributions to best fit the density of a given training dataset. For an example of parameter estimates, suppose you work for a spark plug manufacturer that is studying a problem in their spark plug gap. stream Latest endeavours have made use of geostatistical tools in hydrology to guide parameter derivations for unsampled locations. Confidence intervals are a range of values likely to contain the population parameter. 4 shows the interface in UML that is being proposed within the GLOBAL-CAPE-OPEN project. %���� stream There are many te… 19 0 obj machine learning algorithms to generate and generalize the parameter estimates, Kunce and Chatterjee build a bridge between the traditional and machine learning approaches. The dynamics shown in the dissolved oxygen profile in Figure 2 are due to the link between the oxygen uptake rate and the feed rate. The response variable is linear with the parameters. On the other hand, providing the user with reliable information on both selection items has long remained an open and challenging research topic. s0_�q�,�"Q�F1'"�Q�m8��w�~�;#[�vN��6]�S�s]?T������+]غ�W���Q�UZ�s�����ggfKg�{%�R�k6a���ʢ=��C�͆��߷��_P[��l�sY�@� �2��V:#�C�vI�}7 Step responses are often used in industrial applications in order to acquire initial information to design dedicated identification experiments. endstream Optimization algorithms work by identifying hyper-parameter assignments that could have been drawn, and that appear promising on the basis of the loss function’s value at other ... We keep the Estimation of Distribution (EDA, Costs incurred during field data collection, poor access to appropriate sampling location are additional constraints limiting guaranteed randomness during sampling. Parameter estimation in modelling reaction kinetics is affected by the prior knowledge on the domain of variability of model parameters which can be very limited at the beginning of model building activities. The term parameter estimation refers to the process of using sample data (in reliability engineering, usually times-to-failure or success data) to estimate the parameters of the selected distribution. The set of guaranteed parameter estimates is firstly over-approximated by a box using nonlinear programming (NLP). ) is a function of the Fisher informatics matrix F, defined as c=M/2log(λa/λg), with λa, the arithmetic mean of the eigenvalues (easy computable as trace(F)/M), and λg, the geometric mean of the eigenvalues (easy computable as det(F)1/M). Note that for diabetic subjects the global information profile exhibits two peaks: one at the very beginning of the test (maximum of the information obtained from GEXO readings) and one around 110 min (maximum of the information obtained from c-peptide and insulin readings); the level of information obtained from endogenous glucose concentration readings is very low. likelihoods. Product concentration is shown. stream A parameter estimation algorithm for the thermodynamically consistent reptation model (Öttinger, 1999; Fang et al., 2000), which is based on stochastic differential equations, is proposed. endobj The step response experiment is taken for generating the measured data. On the basis of the stochastic gradient algorithm (i.e., the gradient based search estimation algorithm), this work extends the scalar innovation into an innovation vector and presents a multi-innovation gradient parameter estimation algorithm for a state-space system with d-step state-delay … 1995. This paper considers the state and parameter estimation problem of a state-delay system. First of all, a PEDR Client can choose to perform either a DR or a PE task. For healthy subjects, a significant amount of information can be obtained from c-peptide readings, while GEXO measurements provide a limited amount of information. In this study, the authors consider the parameter estimation problem of the response signal from a highly non-linear dynamical system. The optimization problem solution are the estimated parameter values. ?�.� 2�;�U��=�\��]{ql��1&�D���I|@8�O�� ��pF��F܊�'d��K��`����nM�{?���D�3�N\�d�K)#v v�C ��H Ft������\B��3Q�g�� The proposed approach is illustrated in a case study of consecutive reactions in a plug flow reactor. The software formulates parameter estimation as an optimization problem. Figure 2 shows the results of the dynamic model for one batch of data. Results show a very good fitting capability of the model in spite of the significant difference in the insulin behaviour observed for the two subjects. The subject's response is indicated by diamonds. Thus, A Machine-Learning Approach to Parameter Estimation is the first monograph published by the CAS that shows how to use machine learning to enhance traditional ratemaking. Objective. This section is concerned with estimation procedures for the unknown parameter vector \[\beta=(\mu,\phi_1,\ldots,\phi_p,\theta_1,\ldots,\theta_q,\sigma^2)^T. << /Filter /FlateDecode /S 90 /Length 113 >> M. Kigobe, M. Kizza, in Proceedings from the International Conference on Advances in Engineering and Technology, 2006. (2) Learn the value of those parameters from data. In this work, we propose the use of binary classification techniques to define a feasible parametric region of parameter variability satisfying a set of user-defined model-based constraints. endstream This is known as a plug-in estimator. << /Pages 36 0 R /Type /Catalog >> %PDF-1.5 Estimation theory is a branch of statistics that deals with estimating the values of parameters based on measured empirical data that has a random component. For the purpose of improving the accuracy, a multi-innovation stochastic gradient parameter estimation algorithm is presented using the moving window data. In addition to the identification of dynamic systems operating in open-loop, extensions to address the identification in closed-loop is given as well. The Bayesian approach attempts to expend * P(w | D) w w Figure 8: Optimisers find the mode of … 16 0 obj The product prediction for all 11 batches is shown in Figure 3. Many parameter estimation algorithms used in system identification are based on numerical schemes to solve parametric optimization problems. In addition to that, the a-posteriori statistics for parameters τd (M1), MAXEGO, p3 and sL (M4) cannot be evaluated because the curvature of the likelihood function related to these model parameters becomes null. Parameters related to M3 are still very correlated and hard to be identified in a precise way. Federico Galvanin, ... Fabrizio Bezzo, in Computer Aided Chemical Engineering, 2013. Let X t {\displaystyle X_{t}} be a discrete hidden random variable with N {\displaystyle N} possible values (i.e. << /Linearized 1 /L 97144 /H [ 922 192 ] /O 20 /E 61819 /N 6 /T 96780 >> The coupled parameter estimation and dynamic model are applied offline to an eleven batch pilot scale data set, as described in the Materials and Methods section. Availability of sparsely sampled data as point data or spatially lumped data further complicates the estimation procedures. Along with the LSE, techniques for the design of dynamic experiments were developed determining the conditions for an experiment under which the most-informative data can be obtained. Although not shown here, parameters kGD, kID, k54, and k45 of M3 show a very limited impact on the measured responses (low sensitivities) and a very high correlation (always close to unity). Then, it selects the measured data to be reconciled or used for, ODE METHOD VERSUS MARTINGALE CONVERGENCE THEORY, Adaptive Systems in Control and Signal Processing 1983, Subspace Model Identification of MIMO Processes, Multivariable System Identification For Process Control, [0.482 0.721 0.894 4.193 2.328 0.687 1.965], [0.808 5.748 0.348 1.437 0.662 0.017 0.031]. where θ_(k) is an estimate of process parameter vector θ_oφ_(k) and x_(k) are vectors of process input-output and filtered-input-output respectively. Our proposed algorithm is aiming at the condition of existing synchronous and asynchronous frequency-hopping (FH) signals, and meanwhile considering the frequency switching time. This is especially true for the biomass and product concentrations which are modeled very well utilizing the updated parameters. This result is quite common for models affected by structural identifiability issues [9]. Parameter estimation results from an IVGTT for a healthy subject and a subject affected by T2DM. Many parameter estimation algorithms used in system identification are based on numerical schemes to solve parametric optimization problems. A crucial step in the analysis and solution of subspace identification methods is to relate input and output data to the system matrices in a structured manner so both data and model information are represented as matrices and not just as vectors and matrices as is the case in the classical definition of state space models. Apart from the fact that the user has to make a selection on a particular model parametrization, the iterative nature of many of these optimization schemes requires accurate initial estimates. �ɅT�?���?��, ��V����෸68L�E*RG�H5S8HɊHD���J֌���4�-�>��V�'�Iu6ܷ/�ȸ�R��"aY.5�"�� ���3\�,�����!�a�� 3���� V 8:��%���Z�+�4o��ڰ۸�MQ����� ���j��sR��B)�_-�T���J���#|L���X�J��]Lds�j;���a|Y��M^2#��̶��( endobj Coupled parameter estimator and dynamic model applied to pilot scale batch data. Information profiles (in terms of trace of the information matrix) obtained from IVGTT after parameter estimation for (a) a healthy subject and (b) a subject affected by T2DM. Batch data obtained from Novozymes A/S. endobj We use cookies to help provide and enhance our service and tailor content and ads. As a result, models that cannot be linearized have enjoyed far less recognition because it is necessary to use a search algorithm for parameter estimation. Your choices are to either use one of several 'standard' parameter settings or to calculate your own settings for your specific problem. The measured online data for carbon evolution rate (qc), oxygen uptake rate (qo) and ammonia addition rate (qn) are used as input to the parameter estimation block in order to simulate the system as would be done online. The tests performed suggest that given sufficient data, use of semivariograms and kriging tools can sufficiently provide estimates for aquifer parameters. Parameters Before we dive into parameter estimation, first let’s revisit the concept of parameters. Figure 2. eO is the apostiori error, 0≤Γ(k) <2 represents the weight of actual data and 0≤A(k) ≤ 1 is the supression factor for all past data. Furthermore, a vast amount of practical evidence has shown that the results obtained by the non-iterative subspace identification schemes do not need further improvement in iterative parametric optimization methods. The generalization to different and more general input sequences is analyzed in Section 8.5.1. Convergence on a solution does not necessarily guarantee that the model fit is optimal or that the sum of squared errors (SSE) are minimized. You can generate MATLAB ® code from the app, and accelerate parameter estimation using parallel computing and Simulink fast restart. Figure 3. Recursive parameters and state estimation algorithms are presented using the least squares technique, the adjustable model, and the Kalman filter theory. Michel Verhaegen, in Multivariable System Identification For Process Control, 2001. HAL Id: inria-00074015 Y = A+BX. For the sake of conciseness, only results for a single healthy subject (male, aged 22, BMI = 19.5, “1”) and a subject affected by T2DM (male, aged 44, BMI = 29.7, “S2”) are shown. Fig. Table 1. For details about the algorithms, see Recursive Algorithms for Online Parameter Estimation. Mature parameter estimation techniques exist that find the best fit between a (nonlinear, dynamic) model and data gathered in dynamic experiments that are performed at, for example, processing plants. we plug in the value for the maximum-likelihood parameter set, w∗. You can also estimate models using a recursive least squares (RLS) algorithm. Figure 3. The proposed parameter estimation algorithm is an off-line Bayesian parameter estimation algorithm, and it is an updated version of the marginalization based algorithms. Across the 11 batches, the root mean sum of squared errors between the model prediction and the data for product concentration ranges from 4% to 26%. In conventional parameter estimation approaches a reasonably wide domain of variability for kinetic parameters is initially assumed, but this uncertainty on domain definition might deeply affect the efficiency of model-based experimental design techniques for model validation. This paper presented a computationally efficient coherent detection and parameter estimation algorithm (i.e., SAF-SFT) for radar maneuvering target. Grey Wolf Optimization [21] and Bio – Inspired Optimization Algorithm PARAMETER ESTIMATION IN STOCHASTIC VOLATILITY MODELS WITH MISSING DATA USING PARTICLE METHODS AND THE EM ALGORITHM by Jeongeun Kim BS, Seoul National University, 1998 For subject S2 (Figure 2b) the glucose regulation is slower than the one realised in S1 (Figure 2a), as a result of a deficit in the insulin release. In statistics, maximum likelihood estimation (MLE) is a method of estimating the parameters of a probability distribution by maximizing a likelihood function, so that under the assumed statistical model the observed data is most probable. Lisa Mears, ... Krist V. Gernaey, in Computer Aided Chemical Engineering, 2016. In this paper, a parameter estimation algorithm for wideband multiple FH (multi-FH) signals based on compressed sensing (CS) is proposed. The problem is formulated using the maximum likelihood (MLE) objective function, and a modified Levenberg-Marquardt algorithm is developed for its solution. Information analysis (Figure 3) underlines some important aspects of the identification of the BM from IVGTT data. For subject S2 the estimation of model parameters is even more critical. endobj Finally, despite its internal modularity, PEDR manager had to expose a common interface to be invoked by any external client. For subject S1, a statistically sound estimation can be achieved only for the M1 and partially for the M2 submodel (although, as underlined by the low t-value, parameter ε is estimated with a large uncertainty). Chouaib Benqlilou, ... Luis Puigjaner, in Computer Aided Chemical Engineering, 2002. << /Contents 21 0 R /MediaBox [ 0 0 612 792 ] /Parent 36 0 R /Resources 29 0 R /Type /Page >> Coupled parameter estimator and dynamic model applied to 11 historical pilot scale batches. The problem of design of experiments, which determines the OED-optimal sequence of control inputs is then formulated as a dynamic optimization problem over the NLP which over-approximates the GPE solution set. The objective of the method is to estimate the parameters of the model, based on the observed pairs of values and applying a certain criterium function (the observed pairs of values are constituted by selected values of the auxiliary variable and by the corresponding observed values of the response variable), that is: Furthermore, the PEDR Manager provides a graphical and user-friendly interface (Fig. Parameter Estimation Techniques: A Tutorial with Application to Conic Fitting. x��]�ܶ��~���E-�_���n�Ɓ��M�A��=�֊I����b8�VZ��(�>�����p������͸��*��g�*���BRQd7��7�9��3�f�Ru�� ���`�y?�C5��n~���qj�B 6Ψ0*˥����֝����5�v��׮��o��:x@��ڒg�0�X��^W'�yKm)J��s�iaU�+N��x�ÈÃu��| ��J㪮u��C��V�����7� {׹v@�����n#'�A������U�.p��:_�6�_�I�4���0ԡw��QW��c4H�IJ�����7���5��iO�[���PW. A parameter estimation session has been carried out on the available clinical data from IVGTT comprising c-peptide measurements (available with a standard deviation σy1 = 0.1 nM), insulin measurements (σy2 = 10 pM), and glucose measurements (σy3 = σy4 = 0.15 mM) for 6 subjects (3 healthy subjects and 3 diabetics) of different age, sex, weight and body mass index (BMI). The pop-up window which permits to follow the progress of the task is shown below. << /Filter /FlateDecode /Length 2300 >> This explains the dynamics which are exhibited in the dissolved oxygen profile. PSO is used for parameter estimation of a Nonlinear Auto-Regressive with Exogenous (NARX) model for dc motor [20]. Since the latter are based on elementary linear algebra results, a summary of the relevant matrix analysis tools is given in Appendix A. 21 0 obj Several parameter estimation methods are available. �0���. Case Study: Hydrological Parameter Estimation in Mpigi-Wakiso, Proceedings from the International Conference on Advances in Engineering and Technology, 23rd European Symposium on Computer Aided Process Engineering, Federico Galvanin, ... Fabrizio Bezzo, in, European Symposium on Computer Aided Process Engineering-12, Chouaib Benqlilou, ... Luis Puigjaner, in, ) designed according to the methods that the Manager exposes. Finally, the Client could ask the system to solve the problem. ScienceDirect ® is a registered trademark of Elsevier B.V. ScienceDirect ® is a registered trademark of Elsevier B.V. URL: https://www.sciencedirect.com/science/article/pii/B9780124115576000057, URL: https://www.sciencedirect.com/science/article/pii/B9780444634283501314, URL: https://www.sciencedirect.com/science/article/pii/B9780444642356500656, URL: https://www.sciencedirect.com/science/article/pii/B9780080453125500248, URL: https://www.sciencedirect.com/science/article/pii/B9780444632340500233, URL: https://www.sciencedirect.com/science/article/pii/S1570794602801705, URL: https://www.sciencedirect.com/science/article/pii/B9780080305653500320, URL: https://www.sciencedirect.com/science/article/pii/B978044463428350223X, URL: https://www.sciencedirect.com/science/article/pii/B9780080439853500107, Computer Aided Chemical Engineering, 2018, Modelling Methodology for Physiology and Medicine (Second Edition), 26th European Symposium on Computer Aided Process Engineering, Anwesh Reddy Gottu Mukkula, Radoslav Paulen, in, 28th European Symposium on Computer Aided Process Engineering, Arun Pankajakshan, ... Federico Galvanin, in, Dealing With Spatial Variability Under Limited Hydrogeological Data. D. Matko, J. Tasič, in Adaptive Systems in Control and Signal Processing 1983, 1984, All parameter estimation methods can be described using the following generalized algorithm. << /Type /XRef /Length 67 /Filter /FlateDecode /DecodeParms << /Columns 4 /Predictor 12 >> /W [ 1 2 1 ] /Index [ 16 48 ] /Info 14 0 R /Root 18 0 R /Size 64 /Prev 96781 /ID [<8a7c60dad2128f758c0ffd96cb0473f8>] >> This paper deals with the parameter estimation problem for multivariable nonlinear systems described by MIMO state-space Wiener models. ��-�� To follow the tread of the book, we start outlining the nature of subspace identification algorithms first for the special case of using step response measurements neglecting errors on the data. You can estimate parameters of AR, ARMA, ARX, ARMAX, OE, or BJ model coefficients using real-time data and recursive algorithms. Estimation of distribution algorithms (EDAs), sometimes called probabilistic model-building genetic algorithms (PMBGAs), are stochastic optimization methods that guide the search for the optimum by building and sampling explicit probabilistic models of promising candidate solutions. The step input response is treated in Section 8.4. Copyright © 2020 Elsevier B.V. or its licensors or contributors. Parameter estimation during hydrologic modelling is usually constrained by limited data and lack of ability to perfectly represent insutu conditions. The characteristics of SAF-SFT algorithm include: (1) After the generalized keystone transform, the first SAF and SFT operations are applied to achieve the range and velocity estimations. Scaled axis labels for confidentiality reasons. The global amount of information that can be obtained from IVGTT for diabetic subjects (Figure 3b) is significantly lower than the one obtained for healthy subjects (Figure 3a), due to the small contributions given to the sensitivities by some parameters. For example, the point estimate of population mean (the parameter) is the sample mean (the parameter estimate). The reproducibility of the model prediction across the different batches which exhibit very different oxygen transfer conditions is very encouraging, and the state estimation has future application as a process monitoring tool. The proposed parameter estimation algorithm can be regarded as the Monte Carlo batch techniques , and it is perfect for estimating parameters of stochastic dynamic systems. 3��p�@�a���L/�#��0 QL�)��J��0,i�,��C�yG�]5�C��.�/�Zl�vP���!���5�9JA��p�^? Apart from the fact that the user has to make a selection on a particular model parametrization, the iterative nature of many of these optimization schemes requires accurate initial estimates. Tailored approaches exist nowadays to strike against certain problems encountered in classical (LSE) parameter estimation. First of all, a PEDR Client can choose to perform either a DR or a PE task. The proposed algorithm provides comparable estimation accuracy compared to the EM-based algorithms x�cbd�g`b`8 $��A,c �x ��\�@��HH/����z ��H��001��30 �v� This paper addresses the problem of parameter estimation for the multi-variate t-distribution. There is very good agreement between the model prediction and the measured data for all variables. The problem of GPE consists of finding the set of all possible parameter values such that the predicted values of model outputs match—do not falsify—the corresponding measurements within prescribed error bounds. 17 0 obj Parameters related to the M3 and M4 submodels are more critical to be estimated. Optimal experiment design has been extensively studied in literature (Franceschini and Macchietto, 2008) as an approach that identifies the best available conditions for the collection of information-rich data from a dynamic system. The arising bilevel program is regularized such that the resulting nonlinear optimization problem with complementarity constraints is well-conditioned. Hence, for this subset of model parameters the information generated by a single IVGTT is not sufficient to achieve a statistically sound estimation. Among these the most prominent place is taken by least-squares estimation (LSE). The 3 scaling parameters, 1 for each Gaussian, are only used for density estimation. We start the chapter by formulating the identification problem considered for general input and perturbation conditions. Batch data obtained from Novozymes A/S with different conditions for headspace pressure, aeration rate and stirrer speed. The efficiency of a GA is greatly dependent on its tuning parameters. Anwesh Reddy Gottu Mukkula, Radoslav Paulen, in Computer Aided Chemical Engineering, 2016. The param_info argument has the same content as in the specific and varietal parameters estimation … endobj 20 0 obj Then, it selects the measured data to be reconciled or used for parameter estimation, the required mathematical model to be used and the appropriate solver for solving the resulting optimization problem. Figure 2. N��"C-B&Wp����s�;��&WF$ Hf�$�ķ�����$� 18 0 obj Photovoltaic Solar Cell Models & Parameters Estimation Methods: One Diode Model, Two Diode Model, Temperature Sensitivity of IV Model Parameters, Other Circuit Models for Photovoltaic Cells, Artificial Bee Colony &Genetic Algorithm for Determining PV Cell Parameters �"ۺ:bRQx7�[uipRI������>t��IG�+?�8�N��h� ��wVD;{heջoj㳶��\�:�%~�%��~y�6�mI� ����-Èo�4�ε[���j�9�~H���v.��j[�� ���+�߅�����1`&X���,q ��+� 3) designed according to the methods that the Manager exposes. A special section, Section 8.6, is devoted to the analysis of perturbations considered in Section 8.2 in a subspace identification context. We propose a new approximate algorithm which is both computationally e cient and incrementally updateable. In the real system, DO was the controlled variable, and feed rate the manipulated variable, however in the model the control action is not simulated since the feed rate is an input to the model. Parameters of BM are normalised with respect to the values reported in [4] to improve numerical robustness. Genetic Algorithm (GA) Parameter Settings. On the one hand, both selections can have a critical influence on the results of the optimization run and hence on the quality of the identified model. Model prediction (grey), offline measured data (black). In the process, GMM uses Bayes Theorem to calculate the probability of a given observation xᵢ to belong to each clusters k, for k = 1,2,…, K. As the expectations of the realization of the measurement noise in LSE are GPE differ, the results are not the same for these two approaches. If the algorithm converged on the parameter values correctly, the set of parameter estimates minimize the sum of squared errors (SSE). The parameter update occurs every hour. The parameters describe an underlying physical setting in such a way that their value affects the distribution of the measured data. Optimal experiment design (OED) for the LSE is, however, not consistent with the OED for the GPE. By continuing you agree to the use of cookies. The objective of parameter estimation is to obtain the parameter estimates of system models or signal models. Guaranteed parameter estimation (GPE) is an approach formulated in the context of parameter estimation that accounts for bounded measurement error (Kieffer and Walter, 2011), contrary to the LSE that assumes normal distribution of error. The Graphical User Interface for the PEDR Manager. Figure 3. Glucose and insulin profiles as predicted by BM model after parameter identification are shown in Figure 2. In this chapter, we highlight the fundamental nature of subspace identification algorithms. A mixture modelis a model comprised of an unspecified combination of multiple probability distribution functions. Glucose and insuline profiles after parameter identification from IVGTT data: (a) healthy subject; (b) subject affected by T2DM. The Gaussian Mixture Model, or GMM for short, is a mixture model that uses a combination of Gaussian (Normal) probability distributions and requires the estimation of the mean and standard deviation parameters for each. Let this parameter set be w∗, hence the estimate for the output density is: P\(y | D) = P(y | w∗,D) i.e. The algorithm starts with a small number (5 by default) of burn-in iterations for initialization which are displayed in the following way: (note that this step can be so fast that it is not visible by the user) Afterwards, the evoluti… [Research Report] RR-2676, INRIA. Run the parameter estimation. Parameter Estimation Techniques: A Tutorial with Application to Conic Fitting Zhengyou Zhang To cite this version: Zhengyou Zhang. This is done in Section 8.3. Model prediction (grey), offline measured data (black). Analytical groundwater flow models were employed to analyze different pumping test records (constant discharge, step-tests and recovery test) and semivariograms and Krigging tools applied to the averaged results to interpolate between the sparsely sampled boreholes, in order to estimate hydraulic parameters in Wakiso and Mpigi districts, Uganda.
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