This document discusses using principal component analysis (PCA) to model the signature of manufacturing processes based on machined profile measurements. PCA is presented as a statistical technique that can identify patterns in multivariate data without requiring a parametric model. The document outlines how PCA works and applies it to real roundness measurement data from turned cylindrical parts to investigate using PCA for process signature identification. PCA is shown to effectively describe the variability observed across profiles in a way that summarizes most information with a small number of principal components.
Fault detection based on novel fuzzy modelling csijjournal
The Fault detection which is based on fuzzy modeling is investigated. Takagi-Sugeno (TS) fuzzy model can
be derived by structure and parameter identification, where only the input-output data of the identified system are available. In the structure identification step, Gustafson-Kessel clustering algorithm (GKCA) is used to detect clusters of different geometrical shapes in the data set and to obtain the point-wise membership function of the premise. In the parameter identification step, Unscented Kalman filter (UKF) is
used to estimate the parameters of the premise’s membership function. In the consequence part, Kalman filter (KF) algorithm is applied as a linear regression to estimate parameters of the TS model using the input-output data set. Then, the obtained fuzzy model is used to detect the fault. Simulations are provided to demonstrate the effectiveness of the theoretical results.
2008 "An overview of Methods for analysis of Identifiability and Observabilit...Steinar Elgsæter
The document discusses various methods for analyzing identifiability and observability in nonlinear state and parameter estimation models, which are important concepts in system identification. It describes methods like local sensitivity analysis, empirical observability Gramians, asymptotic analysis, and the Alternating Conditional Expectation algorithm for testing structural and practical identifiability. It also discusses differential algebra approaches for testing observability in nonlinear systems.
ON FEATURE SELECTION ALGORITHMS AND FEATURE SELECTION STABILITY MEASURES: A C...ijcsit
Data mining is indispensable for business organizations for extracting useful information from the huge volume of stored data which can be used in managerial decision making to survive in the competition. Due to the day-to-day advancements in information and communication technology, these data collected from ecommerce and e-governance are mostly high dimensional. Data mining prefers small datasets than high dimensional datasets. Feature selection is an important dimensionality reduction technique. The subsets selected in subsequent iterations by feature selection should be same or similar even in case of small perturbations of the dataset and is called as selection stability. It is recently becomes important topic of research community. The selection stability has been measured by various measures. This paper analyses the selection of the suitable search method and stability measure for the feature selection algorithms and also the influence of the characteristics of the dataset as the choice of the best approach is highly problem dependent.
Statistics is the science of dealing with numbers and data. It involves collecting, summarizing, presenting, and analyzing data. There are four main steps: data collection, summarization by removing unwanted data and classifying/tabulating, presentation with diagrams/graphs/tables, and analysis using measures like average, dispersion, and correlation. Descriptive statistics summarize and describe data, while inferential statistics allow generalizing from samples to populations. Common descriptive statistics include measures of central tendency (mean, median, mode), variability (range, variance, standard deviation), and distribution properties. Inferential statistics techniques like hypothesis testing and ANOVA are used to make inferences about populations based on samples.
applied multivariate statistical techniques in agriculture and plant science 2amir rahmani
This document provides an overview of multivariate statistical techniques that can be used in agriculture and plant science research. It discusses multiple linear regression analysis, which models the relationship between a dependent variable and one or more explanatory variables. The document explains how to determine regression coefficients and test their significance using analysis of variance. It also describes different variable selection techniques for multiple regression like backward elimination, forward selection, and stepwise regression. The goal is to help researchers identify the best predictive model and determine which variables are most important when the number of predictors increases.
This document summarizes the analysis of data from a pharmaceutical company to model and predict the output variable (titer) from input variables in a biochemical drug production process. Several statistical models were evaluated including linear regression, random forest, and MARS. The analysis involved developing blackbox models using only controlled input variables, snapshot models using all input variables at each time point, and history models incorporating changes in input variables over time to predict titer values. Model performance was compared using cross-validation.
Improved Frequent Pattern Mining Algorithm using Divide and Conquer Technique...ijsrd.com
Frequent patterns are patterns such as item sets, subsequences or substructures that appear in a data set frequently. A Divide and Conquer method is used for finding frequent item set mining. Its core advantages are extremely simple data structure and processing scheme. Divide the original dataset in the projected database and find out the frequent pattern from the dataset. Split and Merge uses a purely horizontal transaction representation. It gives very good result for dense dataset. The researchers introduce a split and merge algorithm for frequent item set mining. There are some problems with this algorithm. We have to modify this algorithm for getting better results and then we will compare it with old one. We have suggested different methods to solve problem with current algorithm. We proposed two methods (1) Method I and (2) Method II for getting solution of problem. We have compared our algorithm with the currently worked algorithm SaM. We examine the performance of SaM and Modified SaM using real datasets. We have taken results for both dense and sparse datasets.
This document discusses various methods for selecting optimal input-output pairings for multivariable control systems. It begins with an introduction to the challenges of controlling multivariable systems and the importance of proper input-output pairing. It then reviews several pairing methods including the relative gain array (RGA), relative omega array, dynamic relative gain array, normalized RGA, and relative normalized gain array. It also discusses necessary conditions for decentralized integral controllability and presents rules for eliminating undesirable pairings to achieve this. Overall, the document provides an overview of established and newer techniques for analyzing interactions and selecting input-output pairs for multivariable processes.
Fault detection based on novel fuzzy modelling csijjournal
The Fault detection which is based on fuzzy modeling is investigated. Takagi-Sugeno (TS) fuzzy model can
be derived by structure and parameter identification, where only the input-output data of the identified system are available. In the structure identification step, Gustafson-Kessel clustering algorithm (GKCA) is used to detect clusters of different geometrical shapes in the data set and to obtain the point-wise membership function of the premise. In the parameter identification step, Unscented Kalman filter (UKF) is
used to estimate the parameters of the premise’s membership function. In the consequence part, Kalman filter (KF) algorithm is applied as a linear regression to estimate parameters of the TS model using the input-output data set. Then, the obtained fuzzy model is used to detect the fault. Simulations are provided to demonstrate the effectiveness of the theoretical results.
2008 "An overview of Methods for analysis of Identifiability and Observabilit...Steinar Elgsæter
The document discusses various methods for analyzing identifiability and observability in nonlinear state and parameter estimation models, which are important concepts in system identification. It describes methods like local sensitivity analysis, empirical observability Gramians, asymptotic analysis, and the Alternating Conditional Expectation algorithm for testing structural and practical identifiability. It also discusses differential algebra approaches for testing observability in nonlinear systems.
ON FEATURE SELECTION ALGORITHMS AND FEATURE SELECTION STABILITY MEASURES: A C...ijcsit
Data mining is indispensable for business organizations for extracting useful information from the huge volume of stored data which can be used in managerial decision making to survive in the competition. Due to the day-to-day advancements in information and communication technology, these data collected from ecommerce and e-governance are mostly high dimensional. Data mining prefers small datasets than high dimensional datasets. Feature selection is an important dimensionality reduction technique. The subsets selected in subsequent iterations by feature selection should be same or similar even in case of small perturbations of the dataset and is called as selection stability. It is recently becomes important topic of research community. The selection stability has been measured by various measures. This paper analyses the selection of the suitable search method and stability measure for the feature selection algorithms and also the influence of the characteristics of the dataset as the choice of the best approach is highly problem dependent.
Statistics is the science of dealing with numbers and data. It involves collecting, summarizing, presenting, and analyzing data. There are four main steps: data collection, summarization by removing unwanted data and classifying/tabulating, presentation with diagrams/graphs/tables, and analysis using measures like average, dispersion, and correlation. Descriptive statistics summarize and describe data, while inferential statistics allow generalizing from samples to populations. Common descriptive statistics include measures of central tendency (mean, median, mode), variability (range, variance, standard deviation), and distribution properties. Inferential statistics techniques like hypothesis testing and ANOVA are used to make inferences about populations based on samples.
applied multivariate statistical techniques in agriculture and plant science 2amir rahmani
This document provides an overview of multivariate statistical techniques that can be used in agriculture and plant science research. It discusses multiple linear regression analysis, which models the relationship between a dependent variable and one or more explanatory variables. The document explains how to determine regression coefficients and test their significance using analysis of variance. It also describes different variable selection techniques for multiple regression like backward elimination, forward selection, and stepwise regression. The goal is to help researchers identify the best predictive model and determine which variables are most important when the number of predictors increases.
This document summarizes the analysis of data from a pharmaceutical company to model and predict the output variable (titer) from input variables in a biochemical drug production process. Several statistical models were evaluated including linear regression, random forest, and MARS. The analysis involved developing blackbox models using only controlled input variables, snapshot models using all input variables at each time point, and history models incorporating changes in input variables over time to predict titer values. Model performance was compared using cross-validation.
Improved Frequent Pattern Mining Algorithm using Divide and Conquer Technique...ijsrd.com
Frequent patterns are patterns such as item sets, subsequences or substructures that appear in a data set frequently. A Divide and Conquer method is used for finding frequent item set mining. Its core advantages are extremely simple data structure and processing scheme. Divide the original dataset in the projected database and find out the frequent pattern from the dataset. Split and Merge uses a purely horizontal transaction representation. It gives very good result for dense dataset. The researchers introduce a split and merge algorithm for frequent item set mining. There are some problems with this algorithm. We have to modify this algorithm for getting better results and then we will compare it with old one. We have suggested different methods to solve problem with current algorithm. We proposed two methods (1) Method I and (2) Method II for getting solution of problem. We have compared our algorithm with the currently worked algorithm SaM. We examine the performance of SaM and Modified SaM using real datasets. We have taken results for both dense and sparse datasets.
This document discusses various methods for selecting optimal input-output pairings for multivariable control systems. It begins with an introduction to the challenges of controlling multivariable systems and the importance of proper input-output pairing. It then reviews several pairing methods including the relative gain array (RGA), relative omega array, dynamic relative gain array, normalized RGA, and relative normalized gain array. It also discusses necessary conditions for decentralized integral controllability and presents rules for eliminating undesirable pairings to achieve this. Overall, the document provides an overview of established and newer techniques for analyzing interactions and selecting input-output pairs for multivariable processes.
The Evaluation Model of Garbage Classification System Based on AHPDr. Amarjeet Singh
Based on Shenzhen as an example, the questionnaire was designed in advance to get statistical data. In this paper, the AHP and the linear weighted sum method are used for the weight calculation of each factor, obtaining the long-term cost benefit function of the garbage classification system and the garbage classification pattern grading. Finally, we can choose the better garbage classification mode according to this score.
The document provides an overview of data analysis methods and concepts for graduate fellows. It covers:
1) The objectives of translating research questions into an analysis plan, identifying appropriate data analysis methods and software, and conducting exploratory analysis.
2) Key concepts in data analysis including response and explanatory variables, multi-level data structures, and exploratory versus confirmatory analysis.
3) Guidance on specific exploratory analysis methods and examples of confirmatory analysis options using different statistical models depending on variable types.
Computational Complexity Comparison Of Multi-Sensor Single Target Data Fusion...ijccmsjournal
This document compares the computational complexity of four multi-sensor data fusion methods based on the Kalman filter using MATLAB simulations. The four methods are: group-sensor method, sequential-sensor method, inverse covariance form, and track-to-track fusion. The results show that the inverse covariance method has the best computational performance if the number of sensors is above 20. For fewer sensors, other methods like the group sensors method are more appropriate due to lower computational loads when inverting smaller matrices.
COMPUTATIONAL COMPLEXITY COMPARISON OF MULTI-SENSOR SINGLE TARGET DATA FUSION...ijccmsjournal
Target tracking using observations from multiple sensors can achieve better estimation performance than a single sensor. The most famous estimation tool in target tracking is Kalman filter. There are several mathematical approaches to combine the observations of multiple sensors by use of Kalman filter. An
important issue in applying a proper approach is computational complexity. In this paper, four data fusion algorithms based on Kalman filter are considered including three centralized and one decentralized methods. Using MATLAB, computational loads of these methods are compared while number of sensors
increases. The results show that inverse covariance method has the best computational performance if the number of sensors is above 20. For a smaller number of sensors, other methods, especially group sensors, are more appropriate..
GLMM in interventional study at Require 23, 20151219Shuhei Ichikawa
This document provides an overview of generalized linear mixed models (GLMMs) for medical research:
- GLMMs combine generalized linear models, which handle non-normal data distributions using link functions, and linear mixed models, which incorporate random effects.
- When reporting GLMM analyses, it is important to adequately describe the statistical methods, research design, causal inference approach, model validation, and software/functions used. Previous reviews found much room for improvement in GLMM reporting quality.
- Guidelines for standardized GLMM reporting in medical journals could help ensure the validity of conclusions by documenting the analysis methods generating the results.
Research and Development of Algorithmic Transport Control Systemsijtsrd
The necessity of a systematic approach is shown when considering the problems of transport systems management. The effectiveness of the algorithmic approach is shown for the creation of an automated system for identification and optimization of management processes for complex systems. Experimental statistical methods are proposed for solving the problem in the field of complex systems for the development of modeling control algorithms.The necessity of an algorithmic approach to the development of a methodology for creating an information reference system in the field of transport is shown.The developments are investigated by algorithmic mathematical models of processes in the transport system. Turdibekov Kamolbek Khamidovich | Yakubov Mirjalil Sagatovich | Sulliev Absaid Khurramovich | Soliev Elyor Nigmatovich | Halikov Sarvar Salikhjanovich "Research and Development of Algorithmic Transport Control Systems" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-5 | Issue-6 , October 2021, URL: https://www.ijtsrd.com/papers/ijtsrd46388.pdf Paper URL : https://www.ijtsrd.com/other-scientific-research-area/other/46388/research-and-development-of-algorithmic-transport-control-systems/turdibekov-kamolbek-khamidovich
1) The document describes a method for imputing missing values in large multivariate databases. It proposes modeling variables on bins to address the curse of dimensionality. Variables are selected based on correlation and those with overlapping missingness are excluded. Values are imputed sequentially based on patterns in other variables.
2) Potential improvements discussed include using more sophisticated modeling methods, addressing increased variances from using imputed variables, and handling non-random missingness.
3) An empirical application imputes missing ages in a customer database using various methods. Results show the multivariate method best matches the true distribution of a binary variable compared to mean imputation or deleting missing cases.
The document describes a recursive algorithm for multi-step prediction with mixture models that have dynamic switching between components. It begins by introducing notations and reviewing individual models, including normal regression components and static/dynamic switching models. It then presents the mixture prediction algorithm, first for a static switching model by constructing a predictive distribution from weighted component predictions. For a dynamic switching model, it similarly takes point estimates from the previous time and substitutes them into components to make weighted averaged predictions over multiple steps. The algorithm is summarized as initializing component statistics and parameter estimates, then substituting previous estimates into components to obtain weighted mixture predictions for new data points.
This document provides an introduction and guidelines for linear and multiple regression analyses. It discusses key aspects of each analysis including examining outputs such as model summaries, ANOVA tables, and coefficients. For multiple regression, it recommends a hierarchical approach, entering demographic variables in the first block, extraversion in the second, and narcissism in the third to test if narcissism predicts social media use over and above other factors. The output would show if narcissism explains a significant unique amount of variance in the outcome.
This presentation is based on ``Statistical Modeling: The two cultures'' from Leo Breiman. It compares the data modeling culture (statistics) and the algorithmic modeling culture (machine learning).
An Introduction To Monte Carlo Simulations and Markov Chain Monte CarloMax Yousif
This project serves as an introduction to the field of Monte Carlo Simulations and Markov Chain Monte Carlo.
This project was completed in response to the Monte Carlo Simulations module of the Data Science MSc. The objective was to write a comprehensive dissertation encompassing prominent theoretical and practical themes surrounding Monte Carlo Simulations and MCMC.
This dissertation covers a range of material, some of which is not directly related to Monte Carlo Simulations or MCMC, but was essential to completing the assignment.
The following key-phrases will attempt to summarise the topics covered:
random number generation, rejection-sampling method, inverse-transform method, Monte Carlo simulations, Kolmogorov-Smirnov test, Markov Chains & MCMC, convergence diagnostics, random walks, law of large numbers, central limit theorem, convolution operation, Jarque-Bera test.
Penalized Regressions with Different Tuning Parameter Choosing Criteria and t...CSCJournals
Recently a great deal of attention has been paid to modern regression methods such as penalized regressions which perform variable selection and coefficient estimation simultaneously, thereby providing new approaches to analyze complex data of high dimension. The choice of the tuning parameter is vital in penalized regression. In this paper, we studied the effect of different tuning parameter choosing criteria on the performances of some well-known penalization methods including ridge, lasso, and elastic net regressions. Specifically, we investigated the widely used information criteria in regression models such as Bayesian information criterion (BIC), Akaike’s information criterion (AIC), and AIC correction (AICc) in various simulation scenarios and a real data example in economic modeling. We found that predictive performance of models selected by different information criteria is heavily dependent on the properties of a data set. It is hard to find a universal best tuning parameter choosing criterion and a best penalty function for all cases. The results in this research provide reference for the choices of different criteria for tuning parameter in penalized regressions for practitioners, which also expands the nascent field of applications of penalized regressions.
This document provides an overview of methods for data analysis. It discusses data, descriptive statistics such as measures of central tendency and dispersion, inferential statistics including hypothesis testing and probability, and statistical software packages with a focus on SPSS. SPSS allows users to easily input, manage, and analyze data to obtain summary statistics and perform inferential analyses like t-tests, ANOVA, and regression. Outputs can be copied into reports.
An Influence of Measurement Scale of Predictor Variable on Logistic Regressio...IJECEIAES
Much real world decision making is based on binary categories of information that agree or disagree, accept or reject, succeed or fail and so on. Information of this category is the output of a classification method that is the domain of statistical field studies (eg Logistic Regression method) and machine learning (eg Learning Vector Quantization (LVQ)). The input argument of a classification method has a very crucial role to the resulting output condition. This paper investigated the influence of various types of input data measurement (interval, ratio, and nominal) to the performance of logistic regression method and LVQ in classifying an object. Logistic regression modeling is done in several stages until a model that meets the suitability model test is obtained. Modeling on LVQ was tested on several codebook sizes and selected the most optimal LVQ model. The best model of each method compared to its performance on object classification based on Hit Ratio indicator. In logistic regression model obtained 2 models that meet the model suitability test is a model with predictive variables scaled interval and nominal, while in LVQ modeling obtained 3 pieces of the most optimal model with a different codebook. In the data with interval-scale predictor variable, the performance of both methods is the same. The performance of both models is just as bad when the data have the predictor variables of the nominal scale. In the data with predictor variable has ratio scale, the LVQ method able to produce moderate enough performance, while on logistic regression modeling is not obtained the model that meet model suitability test. Thus if the input dataset has interval or ratio-scale predictor variables than it is preferable to use the LVQ method for modeling the object classification.
A delay decomposition approach to robust stability analysis of uncertain syst...ISA Interchange
This document presents a delay decomposition approach to robust stability analysis of uncertain systems with time-varying delay. It proposes new robust stability criteria for such systems based on Lyapunov stability methods. The criteria are provided in terms of linear matrix inequalities that can be solved efficiently via optimization algorithms. The approach avoids using bounding techniques and model transformations that typically introduce conservatism. Numerical examples demonstrate the proposed method provides less conservative results than existing approaches.
This document presents a study on developing a theoretical heat conduction model for a cold storage using Taguchi methodology. Three control parameters - insulation thickness of side walls, area of side walls, and insulation thickness of roof - were selected as predictor variables, with heat gain in the cold room as the response variable. Using Taguchi's L27 orthogonal array, 27 test runs were conducted and data was analyzed using multiple regression to develop a predictive model. Graphical analysis identified optimal values for the predictor variables that minimized heat transfer to the cold room. The model and optimal parameters can help improve cold storage energy efficiency.
This document discusses data mining techniques for attribute analysis and selection. It describes analyzing attribute relevance by computing a measure to quantify an attribute's relevance to a given class. Attribute selection aims to reduce inputs to a manageable size for processing by choosing the most useful attributes for analysis. Statistical measures of central tendency and dispersion are used to understand data distributions and choose effective implementations. Attribute generalisation and filtering techniques are applied to attributes to reduce complexity and suppress less interesting attributes.
While pumping tail slurry during a 9 5/8" casing cement job, the cementer observed an inability to maintain the required density and noticed the cement silos were empty. The remaining slurry was pumped downhole before dropping the top plug and commencing displacement operations. An investigation found a glove inside the silo had blocked the discharge line. Corrective actions included extra onsite cement, improved silo inspections, and inventory tracking.
Este documento describe cuatro instrumentos de laboratorio: el matraz que se usa para calentar líquidos y preparar soluciones, la pipeta para trasvasar pequeñas cantidades de líquidos, el mortero útil para triturar sólidos, y el vaso de precipitado que se utiliza para recoger y calentar líquidos.
Este documento discute la adicción a las redes sociales. Identifica algunos signos de adicción como revisar las cuentas con frecuencia antes de otras tareas y interrumpir actividades para usar las redes. También sugiere que pasar más de 5 horas al día conectado puede alterar los patrones sociales de una persona y afecta más a los jóvenes con baja autoestima o timidez.
The Evaluation Model of Garbage Classification System Based on AHPDr. Amarjeet Singh
Based on Shenzhen as an example, the questionnaire was designed in advance to get statistical data. In this paper, the AHP and the linear weighted sum method are used for the weight calculation of each factor, obtaining the long-term cost benefit function of the garbage classification system and the garbage classification pattern grading. Finally, we can choose the better garbage classification mode according to this score.
The document provides an overview of data analysis methods and concepts for graduate fellows. It covers:
1) The objectives of translating research questions into an analysis plan, identifying appropriate data analysis methods and software, and conducting exploratory analysis.
2) Key concepts in data analysis including response and explanatory variables, multi-level data structures, and exploratory versus confirmatory analysis.
3) Guidance on specific exploratory analysis methods and examples of confirmatory analysis options using different statistical models depending on variable types.
Computational Complexity Comparison Of Multi-Sensor Single Target Data Fusion...ijccmsjournal
This document compares the computational complexity of four multi-sensor data fusion methods based on the Kalman filter using MATLAB simulations. The four methods are: group-sensor method, sequential-sensor method, inverse covariance form, and track-to-track fusion. The results show that the inverse covariance method has the best computational performance if the number of sensors is above 20. For fewer sensors, other methods like the group sensors method are more appropriate due to lower computational loads when inverting smaller matrices.
COMPUTATIONAL COMPLEXITY COMPARISON OF MULTI-SENSOR SINGLE TARGET DATA FUSION...ijccmsjournal
Target tracking using observations from multiple sensors can achieve better estimation performance than a single sensor. The most famous estimation tool in target tracking is Kalman filter. There are several mathematical approaches to combine the observations of multiple sensors by use of Kalman filter. An
important issue in applying a proper approach is computational complexity. In this paper, four data fusion algorithms based on Kalman filter are considered including three centralized and one decentralized methods. Using MATLAB, computational loads of these methods are compared while number of sensors
increases. The results show that inverse covariance method has the best computational performance if the number of sensors is above 20. For a smaller number of sensors, other methods, especially group sensors, are more appropriate..
GLMM in interventional study at Require 23, 20151219Shuhei Ichikawa
This document provides an overview of generalized linear mixed models (GLMMs) for medical research:
- GLMMs combine generalized linear models, which handle non-normal data distributions using link functions, and linear mixed models, which incorporate random effects.
- When reporting GLMM analyses, it is important to adequately describe the statistical methods, research design, causal inference approach, model validation, and software/functions used. Previous reviews found much room for improvement in GLMM reporting quality.
- Guidelines for standardized GLMM reporting in medical journals could help ensure the validity of conclusions by documenting the analysis methods generating the results.
Research and Development of Algorithmic Transport Control Systemsijtsrd
The necessity of a systematic approach is shown when considering the problems of transport systems management. The effectiveness of the algorithmic approach is shown for the creation of an automated system for identification and optimization of management processes for complex systems. Experimental statistical methods are proposed for solving the problem in the field of complex systems for the development of modeling control algorithms.The necessity of an algorithmic approach to the development of a methodology for creating an information reference system in the field of transport is shown.The developments are investigated by algorithmic mathematical models of processes in the transport system. Turdibekov Kamolbek Khamidovich | Yakubov Mirjalil Sagatovich | Sulliev Absaid Khurramovich | Soliev Elyor Nigmatovich | Halikov Sarvar Salikhjanovich "Research and Development of Algorithmic Transport Control Systems" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-5 | Issue-6 , October 2021, URL: https://www.ijtsrd.com/papers/ijtsrd46388.pdf Paper URL : https://www.ijtsrd.com/other-scientific-research-area/other/46388/research-and-development-of-algorithmic-transport-control-systems/turdibekov-kamolbek-khamidovich
1) The document describes a method for imputing missing values in large multivariate databases. It proposes modeling variables on bins to address the curse of dimensionality. Variables are selected based on correlation and those with overlapping missingness are excluded. Values are imputed sequentially based on patterns in other variables.
2) Potential improvements discussed include using more sophisticated modeling methods, addressing increased variances from using imputed variables, and handling non-random missingness.
3) An empirical application imputes missing ages in a customer database using various methods. Results show the multivariate method best matches the true distribution of a binary variable compared to mean imputation or deleting missing cases.
The document describes a recursive algorithm for multi-step prediction with mixture models that have dynamic switching between components. It begins by introducing notations and reviewing individual models, including normal regression components and static/dynamic switching models. It then presents the mixture prediction algorithm, first for a static switching model by constructing a predictive distribution from weighted component predictions. For a dynamic switching model, it similarly takes point estimates from the previous time and substitutes them into components to make weighted averaged predictions over multiple steps. The algorithm is summarized as initializing component statistics and parameter estimates, then substituting previous estimates into components to obtain weighted mixture predictions for new data points.
This document provides an introduction and guidelines for linear and multiple regression analyses. It discusses key aspects of each analysis including examining outputs such as model summaries, ANOVA tables, and coefficients. For multiple regression, it recommends a hierarchical approach, entering demographic variables in the first block, extraversion in the second, and narcissism in the third to test if narcissism predicts social media use over and above other factors. The output would show if narcissism explains a significant unique amount of variance in the outcome.
This presentation is based on ``Statistical Modeling: The two cultures'' from Leo Breiman. It compares the data modeling culture (statistics) and the algorithmic modeling culture (machine learning).
An Introduction To Monte Carlo Simulations and Markov Chain Monte CarloMax Yousif
This project serves as an introduction to the field of Monte Carlo Simulations and Markov Chain Monte Carlo.
This project was completed in response to the Monte Carlo Simulations module of the Data Science MSc. The objective was to write a comprehensive dissertation encompassing prominent theoretical and practical themes surrounding Monte Carlo Simulations and MCMC.
This dissertation covers a range of material, some of which is not directly related to Monte Carlo Simulations or MCMC, but was essential to completing the assignment.
The following key-phrases will attempt to summarise the topics covered:
random number generation, rejection-sampling method, inverse-transform method, Monte Carlo simulations, Kolmogorov-Smirnov test, Markov Chains & MCMC, convergence diagnostics, random walks, law of large numbers, central limit theorem, convolution operation, Jarque-Bera test.
Penalized Regressions with Different Tuning Parameter Choosing Criteria and t...CSCJournals
Recently a great deal of attention has been paid to modern regression methods such as penalized regressions which perform variable selection and coefficient estimation simultaneously, thereby providing new approaches to analyze complex data of high dimension. The choice of the tuning parameter is vital in penalized regression. In this paper, we studied the effect of different tuning parameter choosing criteria on the performances of some well-known penalization methods including ridge, lasso, and elastic net regressions. Specifically, we investigated the widely used information criteria in regression models such as Bayesian information criterion (BIC), Akaike’s information criterion (AIC), and AIC correction (AICc) in various simulation scenarios and a real data example in economic modeling. We found that predictive performance of models selected by different information criteria is heavily dependent on the properties of a data set. It is hard to find a universal best tuning parameter choosing criterion and a best penalty function for all cases. The results in this research provide reference for the choices of different criteria for tuning parameter in penalized regressions for practitioners, which also expands the nascent field of applications of penalized regressions.
This document provides an overview of methods for data analysis. It discusses data, descriptive statistics such as measures of central tendency and dispersion, inferential statistics including hypothesis testing and probability, and statistical software packages with a focus on SPSS. SPSS allows users to easily input, manage, and analyze data to obtain summary statistics and perform inferential analyses like t-tests, ANOVA, and regression. Outputs can be copied into reports.
An Influence of Measurement Scale of Predictor Variable on Logistic Regressio...IJECEIAES
Much real world decision making is based on binary categories of information that agree or disagree, accept or reject, succeed or fail and so on. Information of this category is the output of a classification method that is the domain of statistical field studies (eg Logistic Regression method) and machine learning (eg Learning Vector Quantization (LVQ)). The input argument of a classification method has a very crucial role to the resulting output condition. This paper investigated the influence of various types of input data measurement (interval, ratio, and nominal) to the performance of logistic regression method and LVQ in classifying an object. Logistic regression modeling is done in several stages until a model that meets the suitability model test is obtained. Modeling on LVQ was tested on several codebook sizes and selected the most optimal LVQ model. The best model of each method compared to its performance on object classification based on Hit Ratio indicator. In logistic regression model obtained 2 models that meet the model suitability test is a model with predictive variables scaled interval and nominal, while in LVQ modeling obtained 3 pieces of the most optimal model with a different codebook. In the data with interval-scale predictor variable, the performance of both methods is the same. The performance of both models is just as bad when the data have the predictor variables of the nominal scale. In the data with predictor variable has ratio scale, the LVQ method able to produce moderate enough performance, while on logistic regression modeling is not obtained the model that meet model suitability test. Thus if the input dataset has interval or ratio-scale predictor variables than it is preferable to use the LVQ method for modeling the object classification.
A delay decomposition approach to robust stability analysis of uncertain syst...ISA Interchange
This document presents a delay decomposition approach to robust stability analysis of uncertain systems with time-varying delay. It proposes new robust stability criteria for such systems based on Lyapunov stability methods. The criteria are provided in terms of linear matrix inequalities that can be solved efficiently via optimization algorithms. The approach avoids using bounding techniques and model transformations that typically introduce conservatism. Numerical examples demonstrate the proposed method provides less conservative results than existing approaches.
This document presents a study on developing a theoretical heat conduction model for a cold storage using Taguchi methodology. Three control parameters - insulation thickness of side walls, area of side walls, and insulation thickness of roof - were selected as predictor variables, with heat gain in the cold room as the response variable. Using Taguchi's L27 orthogonal array, 27 test runs were conducted and data was analyzed using multiple regression to develop a predictive model. Graphical analysis identified optimal values for the predictor variables that minimized heat transfer to the cold room. The model and optimal parameters can help improve cold storage energy efficiency.
This document discusses data mining techniques for attribute analysis and selection. It describes analyzing attribute relevance by computing a measure to quantify an attribute's relevance to a given class. Attribute selection aims to reduce inputs to a manageable size for processing by choosing the most useful attributes for analysis. Statistical measures of central tendency and dispersion are used to understand data distributions and choose effective implementations. Attribute generalisation and filtering techniques are applied to attributes to reduce complexity and suppress less interesting attributes.
While pumping tail slurry during a 9 5/8" casing cement job, the cementer observed an inability to maintain the required density and noticed the cement silos were empty. The remaining slurry was pumped downhole before dropping the top plug and commencing displacement operations. An investigation found a glove inside the silo had blocked the discharge line. Corrective actions included extra onsite cement, improved silo inspections, and inventory tracking.
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approximation of a cone crusher dynamics. Adaptive recursive least squares algorithm is used to
identification of Laguerre model. Various structures of Hammerstein, Wiener, Hammerstein-Wiener models
are tested and the MATLAB simulation results are compared. The mean square error is used for models
validation.It has been found that Hammerstein-Wiener with orthonormal basis functions improves the
quality of approximation plant dynamics. The mean square error for this model is 11% on average
throughout the considered range of the external disturbances amplitude. The analysis also showed that
Wiener model cannot provide sufficient approximation accuracy of the cone crusher dynamics. During the
process it is unstable due to the high sensitivity to disturbances on the output.The Hammerstein-Wiener
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1. Identification of Manufacturing Processes Signature by a Principal Component Based
Approach
Bianca M. Colosimo1
, Andrea N. Intieri
1
, Massimo Pacella
2
1
Dipartimento di Meccanica, Politecnico di Milano, Italy.
2
Dipartimento di Ingegneria dell’Innovazione, Università degli Studi di Lecce, Italy.
Abstract
Machined surfaces and profiles often present a systematic pattern, usually referred to as “the signature” of the
process. Advantages related with identification of process’ signature have been clearly outlined in the
literature. The proposed approaches are mainly based on parametric models, in which the signature is
described as a combination of analytical functions (predictors) that have to be properly chosen by the analyst
depending on the specific case faced. Analytical tools, which do not use parametric model to describe
profiles, were also presented in the so-called “profile monitoring” research field. In particular, the Principal
Component Analysis (PCA), which is a statistical technique utilized to identify patterns in multivariate data,
was successfully applied in chemiometrics. In this paper, the use of PCA is investigated for process’ signature
modelling in the case of machined profiles. The goal is to describe a general-purpose approach, which
alleviates the analyst from the need to identify a suitable kind of analytical functions for the statistical
description of machined profiles. The illustration of the PCA method is based on real measurements data of
circular items obtained by turning.
Keywords:
Profile, Process’ signature, Principal Component Analysis (PCA), Roundness, Turning.
1 INTRODUCTION
Machined surfaces and profiles often present a
systematic pattern and a superimposed random noise:
the first is mainly due to the process used in specific
operation conditions, the second is due to unpredictable
factors, and is usually called “natural variability”. The
systematic pattern constitutes what we will call “the
signature” of the process.
Advantages related to the identification of process’
signature have been clearly outlined in the literature.
When a model of the signature is available, it can be used
to improve quality monitoring (e.g., quickly detecting
whether process is deviating from its natural behaviour)
and quality control (e.g., deciding appropriate corrective
actions that have to be taken). With reference to
monitoring, the machined signature can be considered
similar to a profile. Hence, approaches recently proposed
in the literature on profile monitoring should be in principle
adopted. As example, in [1] the authors discussed the
statistical control of the functional relationship between
the temperature of a mass flow controller in the
microelectronic industry and the flow of gas released. In
[2] the curves considered were obtained from the spectral
analysis of a chemical solution when the concentration of
a mixture is of interest. In [3] the authors introduced an
approach to monitoring the density profile of engineered
wood boards. In [4] the use of profile monitoring for
calibration applications was discussed.
After the seminal work of Weckenmann et al. [5],
advantages related with identification of the signature also
for manufacturing processes have been widely showed in
the literature. These approaches are mainly based on
parametric models, in which the manufacturing signature
characterizing the profile is described as a linear
combination of analytical functions (predictors). These
predictors have to be properly chosen by the analyst
depending on the specific case faced.
For example in turning operations, roundness observed
on machined items is mainly due to systematic radial
spindle error motions which affect that specific machine
tool, as shown in [6]. In these cases, a commonly
appreciated possibility is to model radial deviations with
periodic functions. Several researchers discussed the
modelling of roundness error by fitting a Fourier series,
i.e. by sinusoidal functions at several frequencies used as
predictors. In other applications, wavelets and splines can
be selected as predictors to model complex signatures,
as reported in [7]. For instance, wavelets functions should
be used instead of sinusoidal ones for modelling step
changes in the profile, as for pockets.
As previously mentioned, when the manufacturing
signature is described by means of parametric models, a
cumbersome activity required to the analyst consists in
selecting the proper type of predictor functions that should
be used. Furthermore, measurement data are most of the
times autocorrelated because they are obtained in similar
condition of the machining process and of the
measurement system. In these cases, an appropriate
model should be also defined to describe the
autocorrelated structure which characterizes the
manufacturing signature.
Different analytical methods, which do not require a
specific parametric description of the profile under study,
were also presented in some applications of profile
monitoring. These approaches make use of Principal
Component Analysis (PCA), a statistical technique aimed
at identifying patterns in multivariate data [8] [9]. PCA is
particularly effective because it does not require to the
analyst the identification of suitable kinds of predictors for
the statistical description of the sampled surface faced. In
particular, PCA was successfully applied in
chemiometrics (a research area that combines data
analysis and multivariate statistics in order to improve
chemical industrial plants). In [10] the authors applied
PCA for monitoring a chemical chromatography process.
Similarly, in [2] the authors applied PCA for monitoring a
profile in a chemical process.
In this paper, the use of PCA is investigated for process’
signature modelling in the case of machined profiles. In
particular, the illustration of the PCA method is based on
Intelligent Computation in Manufacturing Engineering - 5
2. real measurements data of circular items obtained by
turning. The objective of this study is to investigate
advantages related with the use of PCA for process
signature identification.
This paper is organized as follows. In section 2,
properties of the canonical PCA are briefly discussed.
Section 3 presents the experimental work faced in this
research. In section 4, the PCA is applied on the
measurement data, while section 5 discusses the
robustness of the proposed approach with respect to
filtering of the original data. Eventually, the last section
reports the conclusion and some final remarks.
2 PRINCIPAL COMPONENT ANALYSIS
Principal Component Analysis (PCA) is a general
statistical approach that allows to explain the variability
observed in a set of multivariate data by means of a small
number of components, namely the principal components,
which can be obtained as linear combinations of the
original variables. These components are able to
summarize most of the information contained in the
original data and allow one to better identify the different
sources of variation which are affecting the process. PCA
is the first step of the data analysis, which is concerned
with data reduction of high-dimensional data frequently
encountered in chemometrics, computer vision and other
domains. An exhaustive description of PCA can be found
in [11]. A rough sketch of how PCA works is reported in
the following.
Assume to collect n profiles, each of them constituted of
p equally-spaced measurement points. Let jky denote
the k -th point observed on the j -th profile, where
1,...,k p= and 1,...,j n= . The measurements can be
hence summarized in the following n p× data matrix Y :
11 1 1 1
1
1
T
k p
T
j jk jp j
Tn nk np
n
y y y
y y y
y y y
⎡ ⎤⎡ ⎤ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥= =⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦
y
Y y
y
(1)
where 1 2
T
j j j jpy y y⎡ ⎤= ⎢ ⎥⎣ ⎦
y is the (column) vector
containing the j -th profile ( 1,...,j n= ) and hence each
row in (1) contains the data points belonging to a specific
profile.
Denote with y the average profile, given by
1
1
n
j
j
n
=
= ∑y y .
Given y , a new matrix T
n= −Y Y i y can be computed,
where ni is an n dimensional column vector of ones, i.e.
[ ]1 1 1
T
n =i . The matrix Y is thus obtained by
subtracting to each profile jy the average profile y and
allows to represent the deviations of each profile from the
average one. Using this new matrix, the covariance of the
original matrix Y , can be rewritten as:
( )( )1
1
1
1 1
n T
T
j j
j
n n
=
= − − =
− −∑
Y Y
S y y y y (2)
The PCA method consists in finding the matrices L and
U which satisfy the following relationship:
1
T
=U S U L (3)
where the matrix L is a diagonal matrix and contains the
eigenvalues of 1S :
1 0 0
0 0
0 0
k
p
l
l
l
⎡ ⎤
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥= ⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥⎣ ⎦
L (4)
Without loss of generality, the eigenvalues are supposed
ranked in decreasing order (i.e., 1 2 ... 0pl l l> > > > ).
The matrix of vectors 1 k p
⎡ ⎤= ⎢ ⎥⎣ ⎦
U u u u is instead
orthonormal and is composed by the eigenvectors of 1S
( ku , 1,...,k p= ), which form a new orthonormal basis for
the space spanned by Y . It is worth noticing that when
the covariance matrix 1S is singular, a subset of the
eigenvalues will be equal to zero. For example, in the
case of high-dimensional data where the number p of
data points collected on each profile is greater than the
number of sampled profiles n , the covariance matrix will
have at most rank equal to 1n− and hence just the first
1n− eigenvalues will be greater than zero (i.e.,
1 2 1... 0nl l l −> > > > ) while the remaining ones will be all
equal to zero (i.e., ... 0n pl l= = = ). In this case, the
number of principal components will be at most 1n− .
Given the eigenvector matrix U , each profile jy can be
projected into the directions identified by the
eigenvectors, i.e.:
( ) 1
TT T
j j j j jk jpz z z⎡ ⎤= − = = ⎢ ⎥⎣ ⎦
z U y y U y (5)
where 1,...,j n= . The new variables are called principal
components, while the values assumed by these
variables whit reference to the j -th observation, i.e.,
1,..., ,...,j jk jpz z z , are called “scores” and represent the
weight that each new principal component has in
explaining this particular observation. In particular, the
first principal component corresponds to the direction in
which the projected observations have the largest
variance. The second component is orthogonal to the first
one and corresponds to the second direction in which the
variance of the data is significant, etc.
Given the matrix U is orthonormal, one can easily show
that j j= +y y Uz , for 1,...,j n= . In other words, each
original observation can be obtained from its scores:
1 11 1 11
1
1 1 2 2
i.e.
...
j p j
jp p p pp jp
j j j jp p
y u u zy
y y u u z
z z z
⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎡ ⎤
⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥
⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥= +⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥
⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥
⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦⎣ ⎦ ⎣ ⎦ ⎣ ⎦
= + + + +y y u u u
(6)
where 1,...,j n= .
3. The principal components (PCs) obtained are statistically
independent and each PC has a variance equal to the
corresponding eigenvalue. Therefore, we can rank the
PCs, i.e. the eigenvectors, according to the associated
eigenvalue and decide to retain just the most important
PCs (the ones which correspond to greater values of the
associated variance), while discarding the less important
ones (i.e., the ones which are associated with smaller
variance).
When the whole set of p PCs is considered, the original
data are obtained starting from the scores by using
equation (6). When instead just the first K ( K p< ) most
important PCs are considered, the original data can be
just estimated as follows:
( ) 1 1 2 2ˆ ...j j j jK KK z z z= + + + +y y u u u (7)
The selection of the proper number of PC is a critical step
because when the number of PCs retained in the model is
too small, the resulting model will not be able to represent
all the significant variability contained in the original data.
When the number of PCs retained is excessive, the
resulting model will include some random variability
inducing a model that try to explain even the random
noise of the process.
3 EXPERIMENTAL ROUNDNESS PROFILES
Quality of mechanical components is more and more
often related to geometric tolerances, e.g., roundness,
flatness, etc. Among different geometric specifications,
roundness plays a relevant role in circular and cylindrical
parts, where functionality is directly related with rotation of
the component. For instance, roundness is critically
related to the proper functioning of rotating shafts, pistons
and cylinders.
Here, the PCA is applied to measurement data of
roundness profiles obtained by turning. In particular, the
experimental data consists in a set of 100n =
components machined by turning C20 carbon steel
cylinders (which were supplied in 30mm∅ = rolled bars).
The final diameter of 26mm∅ = was obtained by
performing two turning steps (cutting speed=163 m/min,
feed rate=0.2 mm/rev), where for each step the depth of
cut was equal to 1 mm.
The machined surfaces were eventually scanned on a
coordinate measuring machine. According to the standard
(ISO/TC 273, 2003 [13]) each roundness profile was
described by 748p = equally distributed measurements
of the radius.
As discussed in [7], data in each sample have to be pre-
treated to focus just on roundness form error. In
particular, data were rescaled by subtracting the least
squares estimation of the radius, and by re-centring the
profile on the least square estimation of the centre. As a
matter of fact, the out-of-roundness does not depend on
the centre’s position and on the average radius.
Secondly, a further step of profiles alignment was
required. This alignment was needed because the tool
starts machining the profile at a point which is random in
each sample. Profile alignment allows identifying a
common reference system and can be performed by
minimizing phase delays.
By a polar representation, the j -th sampled profile can
be described as a sequence of deviations of the radius
measured by the nominal radius, ( )j kr θ , where
( )2 1k k pθ π= − is the angle position and 1,...,k p= . A
polar representation of experimental data is given in
Figure 1. It can be observed that all the roundness
profiles share a common behaviour, i.e. the turning
process leaves a fingerprint or a signature on the
machined components.
Even if there is a systematic behaviour characterizing all
the profiles obtained, variation can be noted from profile
to profile. As an example, Figure 2 depicts the difference
between the average profile (i.e. the one obtained by
averaging all the profile collected, represented by a bold
line) and one specific profile (in this figure, the first profile
is reported and represented by the dashed line).
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Figure 1: Polar diagram of 100 experimental profiles.
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Figure 2: Polar diagram of average profile (bold line) and
profile no 1 (dashed line).
Indeed, even when the process is used with given control
parameters, one can not expect to observe exactly the
same profile since a profile-to-profile variability is often
present.
As stated in [14]: “Common-cause variation is that
variation considered to be characteristic of the process
and that cannot be reduced substantially without
fundamental process changes. It must be determined how
much of the profile-to-profile variation is part of the
common-cause variation and should be incorporated” into
the model of the signature.
PCA helps identifying this profile-to-profile variation
because it basically describes the main ways in which the
generic profile obtained is varying with respect to the
average profile. Hence the average profile can be
considered as a model of the “mean” or the expected
pattern characterizing the generic profile while the
principal components will model the principal ways in
which a generic profile can vary with reference to this
expected behaviour.
4. 4 APPLICATION OF PCA
In this section, the application of PCA is illustrated on the
profile data set previously described. By setting
( )jk j ky r θ= , with 1,...,748k = and 1,...,100j = , the
measurements are collected in a n by p data matrix Y
where 100n = and 748p = .
In this case of high-dimensional data where p n> , the
covariance matrix of matrix Y has at most rank equal to
1n− and hence at most 1n− significant principal
components can be considered. Usually, among such
principal components only the first few eigenvectors are
associated with systematic variation in the data while the
remaining ones are associated with noise. Noise, in this
case, refers to uncontrolled experimental and
instrumental variations arising from random sources. PCA
models are formed by retaining only the PCs which are
representing systematic variation in the data.
In order to select a proper number of PCs, the eigenvalue
corresponding to each PC can be examined. Without loss
of generality, assume to rank the PCs with respect to a
decreasing order of the corresponding eigenvalue, i.e.,
1 2 1... 0nl l l −> > > > .
Hence, the variability explained by the j -th PC can be
expressed as the ratio between the corresponding
eigenvalue jl and the sum
1
1
n
j
j
l
−
=∑ . Therefore, the
cumulative variability explained by the first k dominant
PCs is given by:
1
1 1
k n
j j
j j
l l
−
= =∑ ∑ (8)
Table 1 illustrates the variability explained by each of the
first ten PCs, as well as the cumulative variability
explained by using these PCs.
It can be noted that, by using the first 10 PCs, the 52.88%
of the total variability observed in the original machined
profiles is described.
PC Variability Explained Cumulative Variability Explained
1 13.87% 13.87%
2 10.41% 24.28%
3 7.10% 31.38%
4 4.77% 36.15%
5 3.78% 39.92%
6 3.31% 43.24%
7 2.79% 46.02%
8 2.63% 48.65%
9 2.22% 50.87%
10 2.00% 52.88%
Table 1: Variability explained by the first ten dominant
PCs on the original data.
In order to identify a proper meaning of these set of
significant PCs, Figure 3 depicts the first six dominant
eigenvectors 1 2 3 4 5 6, , , , ,u u u u u u in a polar diagram. As it
can be observed from Figure 3, the first eigenvector,
which describes the most important component of
variability, represents a bi-lobe error form around the
average profile. The second and third eigenvectors, on
the other hand, present two different three-lobe form
errors with a different orientation. Similarly, the fourth and
fifth eigenvectors present four-lobe form errors.
As discussed in Cho and Tu (2001) such lobe-form errors
are often characterizing roundness profiles obtained by
turning as a result of spindle error motion which are very
common in turning.
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PC no1
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PC no2
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PC no3
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PC no4
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PC no5
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PC no6
Figure 3: Polar diagrams of the first six dominant PCs of
original data.
5 FILTERING
Traditional methods to analyze one-dimensional
measurements usually involve applying a certain filtering
process in order to separate the roughness and waviness
components of the measured profile. After filtering, the
roughness and waviness components could be added
again to represent the original profile.
In this section, the effect of longwave-pass filtering on the
PCA of measurement data is investigated. In particular, a
linear Gaussian filter is used, since this is the current
state-of-the-art in ISO standards.
In Gaussian filtering, a series of Gaussian curves is fit to
the data at each data point by averaging over an interval,
which can be specified by the stylus tip radius, the trace
length, the number of data points collected, and the step
size. This filter produces a mean line through the data
set, or waviness component, which is than subtracted
from the original curve to yield the roughness component.
According to the standard ISO/TS 12181-2:2003 (E) [15]
the Gaussian longwave-pass filter is defined in the
frequency domain by the following attenuation function:
1
0
exp
c
a f
a f
α
π
⎡ ⎤⎛ ⎞⎟⎜⎢ ⎥⎟⎜= − ⎟⎢ ⎥⎜ ⎟⎟⎜⎝ ⎠⎢ ⎥⎣ ⎦
(10)
Where ( )ln 2 0.4697α π= = , 0a is the amplitude of the
sine wave undulation before filtering, 1a is the amplitude
of this sine undulation after filtering; cf is the cut-off
5. frequency (in undulation per revolution – UPR) of the
longwave-pass filter, and finally f is the frequency of the
sine wave (in UPR).
The longwave-pass (shortwave-pass) Gaussian filter is a
so-called “phase-correct filter” since it attenuates the
high-frequency (low-frequency) harmonics of the
measurement data without altering their phases. In this
method, the waviness (low-frequencies harmonics) and
roughness (high-frequencies harmonics) components can
be added back together to recreate the original profile.
Figure 4 depicts the data after passing through a
Gaussian filtering process. In particular a longwave-pass
filtering of the original data are considered with cut–off
frequency equal to 50cf = UPR. If compared to the
original data, filtered profiles appear to be smoothed.
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Figure 4: Polar diagram of 100 experimental profiles
(longwave-pass filtered version with cut-off frequency 50
UPR).
Results obtained by applying PCA to these filtered
profiles are reported in Table 2 and in Figure 5. From the
table, it can be noted that, when the first 10 PCs are
exploited to describe the machined profiles, the ratio of
total variability explained by the first ten PCs is 74.77%.
From the figure, it can be noted that the PCs are
smoothed version of the original ones.
Similar results are obtained by applying a longwave-pass
filtering process to the original data with cut–off frequency
equal to 15cf = UPR. Figure 6 shows that the smoother
effect of the longwave-pass filter is evident.
Table 3 shows that the variability explained by the first
few PCs increases as the data are treated by using a
longwave-pass filter. Therefore, filtering can help in
selecting the most important PCs.
PC Variability Explained Cumulative Variability Explained
1 20.93% 20.93%
2 15.41% 36.34%
3 10.30% 46.63%
4 6.62% 53.26%
5 5.24% 58.50%
6 4.27% 62.76%
7 3.36% 66.12%
8 3.29% 69.41%
9 2.64% 72.05%
10 2.42% 74.47%
Table 2: Variability explained by the first ten dominant
PCs on the longwave-pass filtered data (cut-off frequency
50 UPR).
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PC no1
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PC no2
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PC no3
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PC no4
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PC no5
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PC no6
Figure 5: Polar diagrams of the first six dominant PCs of
the longwave-pass filtered data (cut-off frequency 50
UPR).
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Figure 6: Polar diagram of 100 experimental profiles
(longwave-pass filtered version with cut-off frequency 15
UPR).
PC Variability Explained Cumulative Variability Explained
1 27.61% 27.61%
2 19.65% 47.26%
3 12.77% 60.03%
4 7.62% 67.65%
5 6.07% 73.72%
6 4.23% 77.95%
7 3.57% 81.52%
8 2.36% 83.89%
9 2.10% 85.99%
10 1.69% 87.68%
Table 3: Variability explained by the first ten dominant
PCs on the longwave-pass filtered data (cut-off frequency
15 UPR).
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PC no1
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PC no2
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PC no3
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PC no4
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PC no5
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PC no6
Figure 7: Polar diagrams of the first six dominant PCs of
the longwave-pass filtered data (cut-off frequency 15
UPR).
6 SUMMARY
This work relies on the idea of identifying the “fingerprint”
of the manufacturing process, by means of statistical
techniques. In this paper, a PCA-based method was
investigated to this aim.
PCA was used to explain the variance-covariance
structure of profile data through few principal components
(PCs), which are linear combinations of the original data
collected on each profile. PCA was indeed applied to real
profile data, representing roundness profiles obtained by
turning. In this case, it was shown that the first set of most
important PCs have a clear physical meaning, since they
can be associated to the lobe-form errors which are often
associated to spindle-motion error which left their
fingerprint on the roundness profiles machined [6]. The
effect of data filtering was further investigated and the
PCA results resulted to be robust to data filtering.
Further research on the applications of PCA to process’
signature is in order. Firstly, the PCA approach will be
applied within a monitoring strategy aimed at detecting
out-of-control in an SPC framework.
7 ACKNOWLEDGMENTS
This work was carried out with the funding of the Italian
M.I.U.R. (Ministry of Education, University, and
Research).
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