This document provides an overview and agenda for a presentation on robust design and variation reduction using the DiscoverSim software tool. The presentation will cover Monte Carlo simulation, stochastic global optimization, and two case studies - one on robust design of a shut-off valve spring force and another on catapult variation reduction. DiscoverSim allows users to perform Monte Carlo simulation and stochastic optimization to quantify risk and minimize variation when there is uncertainty in input variables. It uses various algorithms like DIRECT, genetic algorithm, and sequential quadratic programming for optimization. The tool helps engineers achieve robust parameter design goals in Design for Six Sigma.
Probability and random processes project based learning template.pdfVedant Srivastava
To understand the concept of Monte –Carlo Method and its various applications and it rely on repeated and random sampling to obtain numerical result.
Developing the computational algorithms to solve the problem related to random sampling.
Objective also contains simulation of specific problem in Matlab Software.
The use of Monte Carlo simulation in quantitative risk assessment of IT projectsEswar Publications
Estimation of the likely time and cost to complete the project and in line with it, taking into account the likelihood of occurrence and severity of the risks' effect, is one of the main concerns that have busied the organizational project managers. On the other hand, the diversity and sensitivity of information technology risks have caused to proper risk management, bolder than other issues, influences these projects. Therefore, in order to describe the degree of potential consequences and probability of occurrence of incidents accurately, IT project managers benefit from quantitative assessment. One of the most effective tools for quantitative assessment and likely forecasting of risks is Monte Carlo simulation, which by generating random numbers, calculates the individual components of a project and determine the impact of each of them on project. In this study, we tried to offer the functional model of the impact of risks on performance indicators of information technology project and propose proper time and cost for completing the project under the study by doing a case study and use of software functionality of Primavera Risk Analysis in Monte Carlo simulation.
Computer Vision: Feature matching with RANSAC Algorithmallyn joy calcaben
This document discusses feature matching and RANSAC algorithms. It begins by explaining feature matching, which determines correspondences between descriptors to identify good and bad matches. RANSAC is then introduced as a method to determine the best transformation that includes the most inlier feature matches. The document provides details on how RANSAC works including selecting random samples, computing transformations, and iteratively finding the best model. Applications like image stitching, panoramas, and video stabilization are mentioned.
Probabilistic Matrix Factorization (PMF)
Bayesian Probabilistic Matrix Factorization (BPMF) using
Markov Chain Monte Carlo (MCMC)
BPMF using MCMC – Overall Model
BPMF using MCMC – Gibbs Sampling
The document discusses resampling techniques called the jackknife and bootstrap. The jackknife involves deleting each observation from the dataset and recalculating statistics to estimate bias, standard error, and confidence intervals. The bootstrap resamples the dataset with replacement many times to estimate properties of statistics like the mean. Both techniques are used to assess reliability of estimates and account for uncertainty without assumptions about the population distribution. The document provides examples applying these methods to estimate standard deviation, confidence intervals for the median, and properties of regression.
The document discusses applications of machine learning for robot navigation and control. It describes how surrogate models can be used for predictive modeling in engineering applications like aircraft design. Dimension reduction techniques are used to reduce high-dimensional design parameters to a lower-dimensional space for faster surrogate model evaluation. For robot navigation, regression models on image manifolds are used for visual localization by mapping images to robot positions. Manifold learning is also applied to find low-dimensional representations of valid human hand poses from images to enable easier robot control.
Understanding Random Forests: From Theory to PracticeGilles Louppe
This document provides an overview of random forests and their implementation. It begins with motivating random forests as a way to reduce variance in decision trees. It then discusses growing and interpreting random forests through variable importances. The document presents theoretical results on the decomposition and properties of variable importances. It concludes by describing the efficient implementation of random forests in scikit-learn, including its modular design and optimizations for speed.
This document discusses principal component analysis (PCA) and its applications in image processing and facial recognition. PCA is a technique used to reduce the dimensionality of data while retaining as much information as possible. It works by transforming a set of correlated variables into a set of linearly uncorrelated variables called principal components. The first principal component accounts for as much of the variability in the data as possible, and each succeeding component accounts for as much of the remaining variability as possible. The document provides an example of applying PCA to a set of facial images to reduce them to their principal components for analysis and recognition.
Probability and random processes project based learning template.pdfVedant Srivastava
To understand the concept of Monte –Carlo Method and its various applications and it rely on repeated and random sampling to obtain numerical result.
Developing the computational algorithms to solve the problem related to random sampling.
Objective also contains simulation of specific problem in Matlab Software.
The use of Monte Carlo simulation in quantitative risk assessment of IT projectsEswar Publications
Estimation of the likely time and cost to complete the project and in line with it, taking into account the likelihood of occurrence and severity of the risks' effect, is one of the main concerns that have busied the organizational project managers. On the other hand, the diversity and sensitivity of information technology risks have caused to proper risk management, bolder than other issues, influences these projects. Therefore, in order to describe the degree of potential consequences and probability of occurrence of incidents accurately, IT project managers benefit from quantitative assessment. One of the most effective tools for quantitative assessment and likely forecasting of risks is Monte Carlo simulation, which by generating random numbers, calculates the individual components of a project and determine the impact of each of them on project. In this study, we tried to offer the functional model of the impact of risks on performance indicators of information technology project and propose proper time and cost for completing the project under the study by doing a case study and use of software functionality of Primavera Risk Analysis in Monte Carlo simulation.
Computer Vision: Feature matching with RANSAC Algorithmallyn joy calcaben
This document discusses feature matching and RANSAC algorithms. It begins by explaining feature matching, which determines correspondences between descriptors to identify good and bad matches. RANSAC is then introduced as a method to determine the best transformation that includes the most inlier feature matches. The document provides details on how RANSAC works including selecting random samples, computing transformations, and iteratively finding the best model. Applications like image stitching, panoramas, and video stabilization are mentioned.
Probabilistic Matrix Factorization (PMF)
Bayesian Probabilistic Matrix Factorization (BPMF) using
Markov Chain Monte Carlo (MCMC)
BPMF using MCMC – Overall Model
BPMF using MCMC – Gibbs Sampling
The document discusses resampling techniques called the jackknife and bootstrap. The jackknife involves deleting each observation from the dataset and recalculating statistics to estimate bias, standard error, and confidence intervals. The bootstrap resamples the dataset with replacement many times to estimate properties of statistics like the mean. Both techniques are used to assess reliability of estimates and account for uncertainty without assumptions about the population distribution. The document provides examples applying these methods to estimate standard deviation, confidence intervals for the median, and properties of regression.
The document discusses applications of machine learning for robot navigation and control. It describes how surrogate models can be used for predictive modeling in engineering applications like aircraft design. Dimension reduction techniques are used to reduce high-dimensional design parameters to a lower-dimensional space for faster surrogate model evaluation. For robot navigation, regression models on image manifolds are used for visual localization by mapping images to robot positions. Manifold learning is also applied to find low-dimensional representations of valid human hand poses from images to enable easier robot control.
Understanding Random Forests: From Theory to PracticeGilles Louppe
This document provides an overview of random forests and their implementation. It begins with motivating random forests as a way to reduce variance in decision trees. It then discusses growing and interpreting random forests through variable importances. The document presents theoretical results on the decomposition and properties of variable importances. It concludes by describing the efficient implementation of random forests in scikit-learn, including its modular design and optimizations for speed.
This document discusses principal component analysis (PCA) and its applications in image processing and facial recognition. PCA is a technique used to reduce the dimensionality of data while retaining as much information as possible. It works by transforming a set of correlated variables into a set of linearly uncorrelated variables called principal components. The first principal component accounts for as much of the variability in the data as possible, and each succeeding component accounts for as much of the remaining variability as possible. The document provides an example of applying PCA to a set of facial images to reduce them to their principal components for analysis and recognition.
The document discusses Taguchi techniques for robust design and quality engineering. It describes the parameter design procedure which involves: (1) defining controllable parameters, uncontrollable noise factors, and measurable responses; (2) selecting an objective function to optimize; (3) planning experiments such as full factorial, fractional factorial, or orthogonal arrays; (4) running the experiment; (5) analyzing results to identify optimal parameter settings; and (6) selecting setpoints and conducting additional experiments if needed. The goal is to design products and processes that perform well even in the presence of uncontrollable noise factors.
The document discusses the Taguchi Method, an approach to quality engineering developed by Genichi Taguchi. It was used to improve postwar telephone systems in Japan. The method focuses on designing quality into products and services from the beginning to eliminate non-random errors. It quantifies quality losses using quality loss functions and aims to reduce sources of variation. The document outlines the Taguchi experiment process, which involves identifying problems, designing experiments, running tests, analyzing results, and confirming improvements. It recommends using expert help to fully apply the method to services.
This document provides an overview of a seminar on the basic design of experiments using the Taguchi approach. The seminar aims to teach participants how to apply experimental design principles to solve production problems and optimize product and process designs. The seminar covers topics such as orthogonal arrays, main effects, interactions, mixed level factors, experiment planning, and uses software demonstrations and hands-on exercises. The goal is to prepare attendees for immediate application of experimental design methods in industry.
Taguchi design of experiments nov 24 2013Charlton Inao
This document provides an overview of Taguchi design of experiments. It defines Taguchi methods, which use orthogonal arrays to conduct a minimal number of experiments that can provide full information on factors that affect performance. The assumptions of Taguchi methods include additivity of main effects. The key steps in an experiment are selecting variables and their levels, choosing an orthogonal array, assigning variables to columns, conducting experiments, and analyzing data through sensitivity analysis and ANOVA.
Genichi Taguchi was a Japanese engineer who developed statistical methods for robust design and quality engineering. Some of his key contributions include developing the Taguchi loss function, promoting the philosophy of off-line quality control through designing products and processes that are robust to external factors, and innovations in the statistical design of experiments. He focused on using orthogonal arrays and analyzing signal-to-noise ratios to optimize settings and minimize a product's sensitivity to noise. While influential, his experimental design and analysis methods have some limitations and inefficiencies.
Chapter 1
1.1 Introduction 5
1.2 Definitions of quality 6
1.2.1 Traditional and Taguchi definition of Quality 7
1.3 Taguchi’s quality philosophy 8
1.4 Objective of Taguchi Methods 10
1.5 8-Steps in Taguchi Methodology 10
Chapter 2 (Loss Function)
2.1 Taguchi Loss Function 11
2.2 Variation of Quadratic Loss function 17 Chapter 3 (Analysis of Variation)
3.1 Understanding Variation 19
3.2 What is ANOVA 19
3.2.1 No Way ANOVA 19
3.2. 1.1 Degree of Freedom 20
3.2.2 One Way ANOVA 24
3.2.3 Two Way ANOVA 30
3.3 Example of ANOVA 36
Chapter 4 (Orthogonal Array)
4.1 What is Array 46
4.2 History of Array 46
4.3 Introduction of Orthogonal Array 47
4.3.1 Intersecting many factor- A case study 49
4.3.1.1 Example of Orthogonal Array 50
4.3.2 A Full factorial Experiment 58
4.4 Steps in developing Orthogonal Array 60
4.4.1 Selection of factors and/or interactions to be evaluated 60
4.4.2 Selection of number of levels for the factors 60
4.4.3 Selection of the appropriate OA 62
4.4.4 Assignment of factors and/or interactions to columns 63
4.4.5 Conduct tests 65
4.4.6 Analyze results 66
4.4.7 Confirmation experiment 69
4.5 Example Experimental Procedure
Taguchi Method is a new engineering design optimisation methodology that improves the quality of existing products and processes and simultaneously reduces their costs very rapidly, with minimum engineering resources and development man-hours
Genichi Taguchi was a Japanese engineer and statistician known for his contributions to quality engineering. He developed methods for robust design and applying statistics to manufacturing processes. Some of his key contributions included developing loss functions to quantify the costs of poor quality, advocating for off-line quality control through system, parameter and tolerance design, and using orthogonal arrays in design of experiments to efficiently isolate noise factors affecting production. Taguchi influenced manufacturing quality efforts in Japan and later in Western companies through his consulting work and collaborations.
Here are the steps to complete the homework:
1. Define the inputs:
- Number of sales reps (num_reps)
- Target sales amount per rep (distributed normally)
- Commission rate if meet target
- Commission rate if exceed target
- Standard deviation of sales amounts
2. Use NumPy to generate random sales amounts based on the inputs
- Generate random numbers based on normal distribution
- Multiply random numbers by target amounts
3. Calculate commissions
- Compare sales to targets
- Apply different commission rates below and above target
- Sum commissions
4. Run simulation multiple times (e.g. 10,000) to get distribution
- Record commissions from each run
- Calculate average
In Machine Learning in Credit Risk Modeling, we provide an explanation of the main Machine Learning models used in James so that Efficiency does not come at the expense of Explainability.
(Contact Yvan De Munck for more info or to receive other and future updates on the subject @yvandemunck or yvan@james.finance)
This document discusses techniques for evaluating and improving statistical models, including regularized regression methods. It covers residuals, Q-Q plots, histograms to evaluate model fit. It also discusses comparing models using ANOVA, AIC, BIC, cross-validation, bootstrapping. Regularization methods like lasso, ridge and elastic net are introduced. Parallel computing is used to more efficiently select hyperparameters for elastic net models.
Six Sigma Methods and Formulas for Successful Quality ManagementIJERA Editor
This document discusses Six Sigma methods and formulas for quality management. It begins by introducing Six Sigma and defining key terms like defects per million opportunities and standard deviation. It then presents the Six Sigma implementation process (DMAIC), outlining the Define, Measure, Analyze, Improve, and Control phases. In the Define phase, projects are chartered and teams assembled. The Measure phase involves collecting data and doing capability analyses. The Analyze phase uses statistical tests like t-tests and ANOVA to find sources of variation. Formulas for measures of central tendency, dispersion, hypothesis testing, and process capability are also provided.
This document provides an introduction to computer simulation. It discusses how simulation can be used to model real systems on a computer in order to understand system behavior and evaluate alternatives. It describes different types of models including iconic, symbolic, deterministic, stochastic, static, dynamic, continuous and discrete models. Monte Carlo simulation is introduced as a technique that uses random numbers. The document outlines the steps in a simulation study and provides examples of systems and their components that can be modeled using simulation.
Using Optimization to find Synthetic Equity Universes that minimize Survivors...OpenMetrics Solutions LLC
There were two main factors leading to this presentation:
The desire / customer request to apply our risk management environment on equities. Which was so far exclusively applied on broad market indices, commodities and currencies.
The confrontation with various backtests that claim to generate huge premiums (10% and more than the benchmark index) through equity selection.
This paper develops a novel approach to characterize the uncertainty in the accuracy of surrogate models. This technique segregates the design domain based on the level of cross-validation errors; the overall framework is called Domain Segmentation based on Uncertainty in the Surrogate (DSUS). The estimated errors are classified into physically meaningful classes based on the user’s understanding of the system and/or the accuracy requirements for the concerned system analysis. In each class, the distribution of the cross-validation errors is estimated to represent the uncertainty in the surrogate. Support Vector Machine (SVM) is implemented to determine the boundaries between error classes, and to classify any new design (point) into a meaningful class. The DSUS framework is illustrated using two different surrogate modeling methods: (i) the Kriging method, and (ii) the Adaptive Hybrid Functions (AHF). We apply the DSUS framework to a series of standard problems and engineering problems. The results show that the DSUS framework can successfully classify the design domain and quantify the uncertainty (prediction errors) in surrogates. More than 90% of the test points could be accurately classified into its error class. In real life engineering design, where we use predictive models with different levels of fidelity, the knowledge of the level of error and uncertainty at any location inside the design space is uniquely helpful.
Applications of Machine Learning in High Frequency TradingAyan Sengupta
Machine learning techniques can be applied to high frequency trading by developing predictive models from large datasets capturing market microstructure features at fine granularities. However, this presents challenges due to the lack of understanding how low-level data relates to trading outcomes and lack of intuitions about how order book distributions impact prices. The study compares various machine learning strategies applied to data from Bloomberg Terminal to design an effective high frequency trading strategy.
Two methods are described for optimizing cognitive model parameters: differential evolution (DE) and high-throughput computing with HTCondor. DE is a genetic algorithm that uses a population of models to explore the parameter space in parallel. It is well-suited for models with few parameters or short run times. HTCondor allows running a population of models over a computer network, making it suitable for larger, more complex models or simulating many participants. Examples of using each method with an ACT-R paired associate model are provided.
Finding Robust Solutions to Requirements Modelsgregoryg
The document presents research on finding robust solutions to requirements models. It introduces the KEYS2 algorithm, which incrementally sets mitigation variables to balance solution quality and robustness. KEYS2 is faster than other algorithms like simulated annealing and A* search. It generates well-behaved results that allow exploring the decision neighborhood through decision ordering diagrams to assess robustness. The researcher concludes that KEYS2 is well-suited for optimizing requirements models as it is fast, produces stable solutions, and enables robustness analysis.
M ESH S IMPLIFICATION V IA A V OLUME C OST M EASUREijcga
We develop a polygonal mesh simplification algorithm based on a novel analysis of the mesh
geometry.
Particularly, we propose first a characterization of vertices as hyperbolic or non
-
hyperbolic depend
-
ing
upon their discrete local geometry. Subsequently, the simplification process computes a volume cost for
each non
-
hyperbolic vertex, in anal
-
ogy with spherical volume, to capture the loss of fidelity if that vertex
is decimated. Vertices of least volume cost are then successively deleted and the resulting holes re
-
triangulated using a method based on a novel heuristic. Preliminary experiments i
ndicate a performance
comparable to that of the best known mesh simplification algorithms
Monte Carlo simulation is a computerized mathematical technique to
generate random sample data based on some known distribution for
numerical experiments.
• The Law of large numbers ensures that the relative frequency of
occurrence of a possible result of a random variable converges to the
theoretical or expected outcome as the number of experiments
increases.
• The essence of Monte Carlo simulation is to sample random variables
significant number of times so that the relative frequency converges to
the theoretical probability with greatest reliability
The document discusses Taguchi techniques for robust design and quality engineering. It describes the parameter design procedure which involves: (1) defining controllable parameters, uncontrollable noise factors, and measurable responses; (2) selecting an objective function to optimize; (3) planning experiments such as full factorial, fractional factorial, or orthogonal arrays; (4) running the experiment; (5) analyzing results to identify optimal parameter settings; and (6) selecting setpoints and conducting additional experiments if needed. The goal is to design products and processes that perform well even in the presence of uncontrollable noise factors.
The document discusses the Taguchi Method, an approach to quality engineering developed by Genichi Taguchi. It was used to improve postwar telephone systems in Japan. The method focuses on designing quality into products and services from the beginning to eliminate non-random errors. It quantifies quality losses using quality loss functions and aims to reduce sources of variation. The document outlines the Taguchi experiment process, which involves identifying problems, designing experiments, running tests, analyzing results, and confirming improvements. It recommends using expert help to fully apply the method to services.
This document provides an overview of a seminar on the basic design of experiments using the Taguchi approach. The seminar aims to teach participants how to apply experimental design principles to solve production problems and optimize product and process designs. The seminar covers topics such as orthogonal arrays, main effects, interactions, mixed level factors, experiment planning, and uses software demonstrations and hands-on exercises. The goal is to prepare attendees for immediate application of experimental design methods in industry.
Taguchi design of experiments nov 24 2013Charlton Inao
This document provides an overview of Taguchi design of experiments. It defines Taguchi methods, which use orthogonal arrays to conduct a minimal number of experiments that can provide full information on factors that affect performance. The assumptions of Taguchi methods include additivity of main effects. The key steps in an experiment are selecting variables and their levels, choosing an orthogonal array, assigning variables to columns, conducting experiments, and analyzing data through sensitivity analysis and ANOVA.
Genichi Taguchi was a Japanese engineer who developed statistical methods for robust design and quality engineering. Some of his key contributions include developing the Taguchi loss function, promoting the philosophy of off-line quality control through designing products and processes that are robust to external factors, and innovations in the statistical design of experiments. He focused on using orthogonal arrays and analyzing signal-to-noise ratios to optimize settings and minimize a product's sensitivity to noise. While influential, his experimental design and analysis methods have some limitations and inefficiencies.
Chapter 1
1.1 Introduction 5
1.2 Definitions of quality 6
1.2.1 Traditional and Taguchi definition of Quality 7
1.3 Taguchi’s quality philosophy 8
1.4 Objective of Taguchi Methods 10
1.5 8-Steps in Taguchi Methodology 10
Chapter 2 (Loss Function)
2.1 Taguchi Loss Function 11
2.2 Variation of Quadratic Loss function 17 Chapter 3 (Analysis of Variation)
3.1 Understanding Variation 19
3.2 What is ANOVA 19
3.2.1 No Way ANOVA 19
3.2. 1.1 Degree of Freedom 20
3.2.2 One Way ANOVA 24
3.2.3 Two Way ANOVA 30
3.3 Example of ANOVA 36
Chapter 4 (Orthogonal Array)
4.1 What is Array 46
4.2 History of Array 46
4.3 Introduction of Orthogonal Array 47
4.3.1 Intersecting many factor- A case study 49
4.3.1.1 Example of Orthogonal Array 50
4.3.2 A Full factorial Experiment 58
4.4 Steps in developing Orthogonal Array 60
4.4.1 Selection of factors and/or interactions to be evaluated 60
4.4.2 Selection of number of levels for the factors 60
4.4.3 Selection of the appropriate OA 62
4.4.4 Assignment of factors and/or interactions to columns 63
4.4.5 Conduct tests 65
4.4.6 Analyze results 66
4.4.7 Confirmation experiment 69
4.5 Example Experimental Procedure
Taguchi Method is a new engineering design optimisation methodology that improves the quality of existing products and processes and simultaneously reduces their costs very rapidly, with minimum engineering resources and development man-hours
Genichi Taguchi was a Japanese engineer and statistician known for his contributions to quality engineering. He developed methods for robust design and applying statistics to manufacturing processes. Some of his key contributions included developing loss functions to quantify the costs of poor quality, advocating for off-line quality control through system, parameter and tolerance design, and using orthogonal arrays in design of experiments to efficiently isolate noise factors affecting production. Taguchi influenced manufacturing quality efforts in Japan and later in Western companies through his consulting work and collaborations.
Here are the steps to complete the homework:
1. Define the inputs:
- Number of sales reps (num_reps)
- Target sales amount per rep (distributed normally)
- Commission rate if meet target
- Commission rate if exceed target
- Standard deviation of sales amounts
2. Use NumPy to generate random sales amounts based on the inputs
- Generate random numbers based on normal distribution
- Multiply random numbers by target amounts
3. Calculate commissions
- Compare sales to targets
- Apply different commission rates below and above target
- Sum commissions
4. Run simulation multiple times (e.g. 10,000) to get distribution
- Record commissions from each run
- Calculate average
In Machine Learning in Credit Risk Modeling, we provide an explanation of the main Machine Learning models used in James so that Efficiency does not come at the expense of Explainability.
(Contact Yvan De Munck for more info or to receive other and future updates on the subject @yvandemunck or yvan@james.finance)
This document discusses techniques for evaluating and improving statistical models, including regularized regression methods. It covers residuals, Q-Q plots, histograms to evaluate model fit. It also discusses comparing models using ANOVA, AIC, BIC, cross-validation, bootstrapping. Regularization methods like lasso, ridge and elastic net are introduced. Parallel computing is used to more efficiently select hyperparameters for elastic net models.
Six Sigma Methods and Formulas for Successful Quality ManagementIJERA Editor
This document discusses Six Sigma methods and formulas for quality management. It begins by introducing Six Sigma and defining key terms like defects per million opportunities and standard deviation. It then presents the Six Sigma implementation process (DMAIC), outlining the Define, Measure, Analyze, Improve, and Control phases. In the Define phase, projects are chartered and teams assembled. The Measure phase involves collecting data and doing capability analyses. The Analyze phase uses statistical tests like t-tests and ANOVA to find sources of variation. Formulas for measures of central tendency, dispersion, hypothesis testing, and process capability are also provided.
This document provides an introduction to computer simulation. It discusses how simulation can be used to model real systems on a computer in order to understand system behavior and evaluate alternatives. It describes different types of models including iconic, symbolic, deterministic, stochastic, static, dynamic, continuous and discrete models. Monte Carlo simulation is introduced as a technique that uses random numbers. The document outlines the steps in a simulation study and provides examples of systems and their components that can be modeled using simulation.
Using Optimization to find Synthetic Equity Universes that minimize Survivors...OpenMetrics Solutions LLC
There were two main factors leading to this presentation:
The desire / customer request to apply our risk management environment on equities. Which was so far exclusively applied on broad market indices, commodities and currencies.
The confrontation with various backtests that claim to generate huge premiums (10% and more than the benchmark index) through equity selection.
This paper develops a novel approach to characterize the uncertainty in the accuracy of surrogate models. This technique segregates the design domain based on the level of cross-validation errors; the overall framework is called Domain Segmentation based on Uncertainty in the Surrogate (DSUS). The estimated errors are classified into physically meaningful classes based on the user’s understanding of the system and/or the accuracy requirements for the concerned system analysis. In each class, the distribution of the cross-validation errors is estimated to represent the uncertainty in the surrogate. Support Vector Machine (SVM) is implemented to determine the boundaries between error classes, and to classify any new design (point) into a meaningful class. The DSUS framework is illustrated using two different surrogate modeling methods: (i) the Kriging method, and (ii) the Adaptive Hybrid Functions (AHF). We apply the DSUS framework to a series of standard problems and engineering problems. The results show that the DSUS framework can successfully classify the design domain and quantify the uncertainty (prediction errors) in surrogates. More than 90% of the test points could be accurately classified into its error class. In real life engineering design, where we use predictive models with different levels of fidelity, the knowledge of the level of error and uncertainty at any location inside the design space is uniquely helpful.
Applications of Machine Learning in High Frequency TradingAyan Sengupta
Machine learning techniques can be applied to high frequency trading by developing predictive models from large datasets capturing market microstructure features at fine granularities. However, this presents challenges due to the lack of understanding how low-level data relates to trading outcomes and lack of intuitions about how order book distributions impact prices. The study compares various machine learning strategies applied to data from Bloomberg Terminal to design an effective high frequency trading strategy.
Two methods are described for optimizing cognitive model parameters: differential evolution (DE) and high-throughput computing with HTCondor. DE is a genetic algorithm that uses a population of models to explore the parameter space in parallel. It is well-suited for models with few parameters or short run times. HTCondor allows running a population of models over a computer network, making it suitable for larger, more complex models or simulating many participants. Examples of using each method with an ACT-R paired associate model are provided.
Finding Robust Solutions to Requirements Modelsgregoryg
The document presents research on finding robust solutions to requirements models. It introduces the KEYS2 algorithm, which incrementally sets mitigation variables to balance solution quality and robustness. KEYS2 is faster than other algorithms like simulated annealing and A* search. It generates well-behaved results that allow exploring the decision neighborhood through decision ordering diagrams to assess robustness. The researcher concludes that KEYS2 is well-suited for optimizing requirements models as it is fast, produces stable solutions, and enables robustness analysis.
M ESH S IMPLIFICATION V IA A V OLUME C OST M EASUREijcga
We develop a polygonal mesh simplification algorithm based on a novel analysis of the mesh
geometry.
Particularly, we propose first a characterization of vertices as hyperbolic or non
-
hyperbolic depend
-
ing
upon their discrete local geometry. Subsequently, the simplification process computes a volume cost for
each non
-
hyperbolic vertex, in anal
-
ogy with spherical volume, to capture the loss of fidelity if that vertex
is decimated. Vertices of least volume cost are then successively deleted and the resulting holes re
-
triangulated using a method based on a novel heuristic. Preliminary experiments i
ndicate a performance
comparable to that of the best known mesh simplification algorithms
Monte Carlo simulation is a computerized mathematical technique to
generate random sample data based on some known distribution for
numerical experiments.
• The Law of large numbers ensures that the relative frequency of
occurrence of a possible result of a random variable converges to the
theoretical or expected outcome as the number of experiments
increases.
• The essence of Monte Carlo simulation is to sample random variables
significant number of times so that the relative frequency converges to
the theoretical probability with greatest reliability
S6 l04 analytical and numerical methods of structural analysisShaikh Mohsin
This document provides an overview of analytical and numerical methods for structural analysis. It begins by explaining the process of structural analysis from the real object to the design model. It then discusses analytical methods like mechanics of materials and numerical methods like the finite element method. The document provides examples comparing analytical and numerical solutions. In summary, it outlines the appropriate uses of both methods and emphasizes the importance of understanding the underlying mechanics rather than solely relying on software tools.
A tour of the top 10 algorithms for machine learning newbiesVimal Gupta
The document summarizes the top 10 machine learning algorithms for machine learning newbies. It discusses linear regression, logistic regression, linear discriminant analysis, classification and regression trees, naive bayes, k-nearest neighbors, and learning vector quantization. For each algorithm, it provides a brief overview of the model representation and how predictions are made. The document emphasizes that no single algorithm is best and recommends trying multiple algorithms to find the best one for the given problem and dataset.
This document describes a senior project to improve the pivot point selection algorithm in the simplex method for linear programming. It provides background on the history and development of the simplex method by George Dantzig in the 1940s. It then explains the current standard process for performing the simplex method, including setting up the linear program, identifying the pivot column and element, and performing the pivot steps. The project aims to identify a flaw in the current pivot rule and develop a new algorithm to increase the efficiency and effectiveness of the simplex method.
Innovations in technology has revolutionized financial services to an extent that large financial institutions like Goldman Sachs are claiming to be technology companies! It is no secret that technological innovations like Data science and AI are changing fundamentally how financial products are created, tested and delivered. While it is exciting to learn about technologies themselves, there is very little guidance available to companies and financial professionals should retool and gear themselves towards the upcoming revolution.
In this master class, we will discuss key innovations in Data Science and AI and connect applications of these novel fields in forecasting and optimization. Through case studies and examples, we will demonstrate why now is the time you should invest to learn about the topics that will reshape the financial services industry of the future!
Topic
- Frontier topics in Optimization
The document provides an overview of machine learning, including definitions, types of machine learning (supervised, unsupervised, reinforcement learning), and evaluation metrics for machine learning models. It discusses classification metrics like accuracy, precision, recall, F1 score, and confusion matrices. For regression problems, it covers metrics like mean absolute error, mean squared error, R2 score. It also provides examples of calculating many of these common metrics in Python.
230208 MLOps Getting from Good to Great.pptxArthur240715
1) MLOps is the process of maintaining machine learning models in production environments. It involves monitoring model performance over time and retraining models if needed due to data or concept drift.
2) The MLOps pipeline includes stages for data engineering, modelling, deployment, and monitoring. Key aspects are ensuring reproducibility, managing data processing pipelines, and defining deployment and monitoring strategies.
3) Successful MLOps requires automating model deployment, monitoring model and data metrics over time, and retraining models when performance degrades to keep models performing well as data evolves in production.
Similar to Robust Design And Variation Reduction Using DiscoverSim (20)
2. Robust Design and Variation
Reduction Using DiscoverSim™:
Agenda
Introduction to DiscoverSim
Monte Carlo Simulation
Stochastic Global Optimization
Case Study: Robust Design of Shut-Off
Valve Spring Force
Case Study: Catapult Variation Reduction
2
3. Introduction to DiscoverSim
Variation reduction and robust design are a vital
part of Design for Six Sigma (DFSS).
While Design of Experiments (DOE) play an
important role in DFSS, in order to achieve
optimal results, one must also employ the tools of
Monte Carlo simulation and optimization.
DiscoverSim is a new, low cost, powerful Excel
add-in tool by SigmaXL that will enable these
improvements (even with complex, non-linear
models).
3
5. Introduction to DiscoverSim
Stochastic Global Optimization can be achieved using a
hybrid methodology of Dividing Rectangles (DIRECT),
Genetic Algorithm, and Sequential Quadratic
Programming.
Simulation and optimization speed are realized using
DiscoverSim's Excel Formula Interpreter.
Utilizes Gauss Engine by Aptech.
Common Object Interface (COI) by Econotron Software to
implement an Excel spreadsheet in Gauss Engine.
The Gauss Random Number Generator is the “KISS+Monster”
random number generator developed by George Marsaglia. This
algorithm produces random integers between 0 and 232 – 1 and
has a period of 108859.
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7. Monte Carlo Simulation
We start with the Y = f(X) Model, also
known as the “Transfer Function”:
X Process Y
The Y = f(X) model should be based on theory, process knowledge, or
the prediction formula of a designed experiment or regression
analysis.
This prediction equation should be validated prior to use in
DiscoverSim: “All models are wrong, some are useful” – George Box.
The results of any Monte Carlo Simulation/Optimization should also be
validated with further experimentation or use of prototypes.
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8. Monte Carlo Simulation
After the Y = f(X) relationship has been validated, an important
question that then needs to be answered is: “What does the
distribution of Y look like when I cannot hold X constant, but have
some uncertainty in X?” In other words, “How can I quantify my risk?”.
Monte Carlo simulation comes in to solve the complex problem of
dealing with uncertainty by “brute force” using computational power.
The Monte Carlo method was coined in the
1940s by John von Neumann, Stanislaw Ulam
and Nicholas Metropolis, while they were
working on nuclear weapon projects in the Los
Alamos National Laboratory. It was named in
homage to Monte Carlo casino, a famous
casino, where Ulam's uncle would often
gamble away his money.
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9. Monte Carlo Simulation
The following diagram illustrates a simple Monte Carlo simulation using
DiscoverSim with three different input distributions (X’s also known as
“Assumptions”)
Y = A1 + A2 + A3
10,000 Replications
A random draw is performed from each input distribution, Y is
calculated, and the process is repeated 10,000 times. The
histogram and descriptive statistics show the simulation results.
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10. Selecting a Distribution
Selecting the correct distribution is a critical step towards
building a useful model.
The best choice for a distribution is one based on known
theory, for example the use of a Weibull Distribution for
reliability modeling.
A common distribution choice is the Normal Distribution,
but this assumption should be verified with data that
passes a normality test with a minimum sample size of 30;
preferably 100.
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11. Selecting a Distribution
If data is available and the distribution is not normal, use
DiscoverSim’s Distribution Fitting tool to find a best fit
distribution.
Alternatively, the Pearson Family Distribution allows you to
simply specify Mean, StdDev, Skewness and Kurtosis.
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12. Selecting a Distribution
In the absence of data or theory,
commonly used distributions are:
Uniform, Triangular and Pert.
Uniform requires a Minimum and
Maximum value, and assumes an
equal probability over the range.
This is commonly used in tolerance
design.
Triangular and Pert require a
Minimum, Most Likely (Mode) and
Maximum. Pert is similar to
Triangular, but it adds a “bell shape”
and is popular in project
management. 12
13. Specifying Correlations
DiscoverSim allows you to specify correlations between
any inputs. DiscoverSim utilizes correlation copulas to
achieve the desired Spearman Rank correlation values.
The following surface plot illustrates how a correlation
copula results in a change in the shape of a bivariate (2
input) normal distribution:
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14. Optimization:
Stochastic Versus Deterministic
Monte-Carlo simulation enables you to quantify risk,
whereas stochastic optimization enables you to minimize
risk.
Deterministic optimization is a commonly used tool to find
a minimum or maximum (e.g., Excel Solver) but it does not
take uncertainty into account.
Stochastic optimization will not only find the optimum X
settings that result in the best mean Y value, it will also
look for a solution that will reduce the standard deviation.
Stochastic optimization looks for a minimum or maximum
that is robust to variation in X, thus reducing the
transmitted variation in Y. This is referred to as “Robust
Parameter Design” in DFSS.
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16. Optimization: Local Versus Global
The following surface plot illustrates a function with local
minima and a global minimum:
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17. Optimization: Local Versus Global
Local optimization methods are good at finding
local minima, use derivatives of the objective
function to find the path of greatest improvement,
and have fast convergence. However they require
a smooth response so will not work with
discontinuous functions.
DiscoverSim uses Sequential Quadratic
Programming (SQP) for local optimization.
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18. Optimization: Local Versus Global
Global optimization finds the global minimum, and is
derivative free, so will work with discontinuous functions.
However because of the larger design space,
convergence is much slower than that of local
optimization.
DiscoverSim uses DIRECT (Dividing Rectangles) and
Genetic Algorithm (GA) for global optimization.
A hybrid of the above methodologies is also available
starting with DIRECT to do a thorough initial search,
followed by GA, and then fine tuning with SQP.
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19. Hiwa, S., T. Hiroyasu and M. Miki. “Hybrid Optimization
Using DIRECT, GA, and SQP for Global Exploration”,
Doshisha University, Kyoto, Japan.
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20. DiscoverSim
Components of Optimization
Input Control: The permissible range for the control is
specified, and the control, which unlike an input
distribution, has no statistical variation.
Think of this as a control knob like temperature.
This is also known as a “Decision Variable”.
An input control can be referenced by a constraint and/or an output
function.
It is possible to have a model that consists solely of controls with
no input distributions. (In this case, the optimization is deterministic,
so the number of replications, n, should be set to 1.)
An input control can be continuous or discrete integer.
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21. DiscoverSim
Components of Optimization
Constraint: A constraint can only be applied to an Input
Control or calculation based on Input Control:
A constraint cannot reference an Input Distribution or Output
Response. (Constraints for Outputs, also known as Requirements,
will be added in Version 2.)
A constraint cannot be a part of the model equation (i.e., an output
cannot reference a constraint).
Constraints can be simple linear or complex nonlinear.
Each constraint will contain a function of Input Controls or
Parameter Monitors on the Left Hand Side (LHS), and a constant
on the Right Hand Side (RHS).
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22. DiscoverSim Optimization Metrics
Optimization Goal: Minimize Maximize
Multiple Output Weighted Sum Deviation from Target Weighted Sum Desirability
Metric:
Statistic: Mean Mean Mean Mean
Median Median
1st quartile 1st quartile
3rd quartile 3rd quartile
Minimum Minimum
Maximum Maximum
Standard Deviation Standard Deviation
Mean Squared Error: Skewness
(Taguchi Loss Function) Kurtosis
Skewness Range
Kurtosis IQR (75-25)
Range Span (95-5)
IQR (75-25) Pp
Non-Normal
Span (95-5) Ppu
Capability
Actual DPM: Ppl
Indices and
(defects per million) Ppk
Actual DPM
Calculated DPM : Cpm
(defects per million assuming %Pp (Percentile Pp)
require a
normal distribution) %Ppu (Percentile Ppu)
minimum of
%Ppl (Percentile Ppl)
10,000
%Ppk (Percentile Ppk) replications.
23. Case Study: Robust New Product
Design - Optimizing Shutoff Valve
Spring Force
This is an example of DiscoverSim stochastic optimization for
robust new product design, adapted from:
Sleeper, Andrew (2006), Design for Six Sigma Statistics:
59 Tools for Diagnosing and Solving Problems in DFSS
Initiatives, NY, McGraw-Hill, pp. 782-789.
Sleeper, Andrew, “Accelerating Product Development with
Simulation and Stochastic Optimization”,
http://www.successfulstatistics.com/
This example is used with permission of the author.
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24. Case Study: Robust New Product
Design - Optimizing Shutoff Valve
Spring Force
The figure below is a simplified cross-sectional view of a
solenoid-operated gas shutoff valve:
The arrows indicate the direction of gas flow.
A solenoid holds the plate (shaded) open when energized. When the
solenoid is not energized, the spring pushes the plate down to shut off
gas flow.
If the spring force is too high, the valve will not open or stay open. If the
spring force is too low, the valve can be opened by the inlet gas pressure. 24
25. Optimizing Shutoff Valve Spring
Force
The method of specifying and testing the spring is shown
below:
The spring force requirement is 22 +/- 2 Newtons.
The spring force equation, or Y = f(X) transfer function, is calculated as
follows:
Spring Length, L = -X1 + X2 - X3 + X4
Spring Rate, R = (X8 - X7)/X6
Spring Force, Y = X7 + R * (X5 - L)
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26. Case Study: Catapult Variation
Reduction
This is an example of DiscoverSim stochastic optimization for
catapult distance variation reduction, adapted from:
John O'Neill, Sigma Quality Management,
www.sixsigmanagement.com.
This example is used with permission of the author.
kx2
y sin cos
mg
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27. Recommended Reading
1. Savage, Sam (2009), The Flaw of Averages:
Why We Underestimate Risk in the Face of
Uncertainty, Hoboken, NJ, Wiley.
2. Sleeper, Andrew (2006), Design for Six Sigma
Statistics: 59 Tools for Diagnosing and Solving
Problems in DFSS Initiatives, NY, McGraw-Hill.
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