There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

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Using Multivariate Statistics

THE DESIGN AND ANALYSIS OF EXPERIMENTS. By Oscar Kempthorne. New York, John Wiley and Sons, Inc., 1952. 631 pp. $8.50. This book by a teacher of statistics (as well as a consultant for \"experimenters\") is a comprehensive study of the philosophical background for the statistical design of experiment. It is necessary to have some facility with algebraic notation and manipulation to be able to use the volume intelligently. The problems are presented from the theoretical point of view, without such practical examples as would be helpful for those not acquainted with mathematics. The mathematical justification for the techniques is given. As a somewhat advanced treatment of the design and analysis of experiments, this volume will be interesting and helpful for many who approach statistics theoretically as well as practically. With emphasis on the \"why,\" and with description given broadly, the author relates the subject matter to the general theory of statistics and to the general problem of experimental inference. MARGARET J. ROBERTSON

The Design and Analysis of Experiments

Machine Learning is the study of methods for programming computers to learn. Computers are applied to a wide range of tasks, and for most of these it is relatively easy for programmers to design and implement the necessary software. However, there are many tasks for which this is difficult or impossible. These can be divided into four general categories. First, there are problems for which there exist no human experts. For example, in modern automated manufacturing facilities, there is a need to predict machine failures before they occur by analyzing sensor readings. Because the machines are new, there are no human experts who can be interviewed by a programmer to provide the knowledge necessary to build a computer system. A machine learning system can study recorded data and subsequent machine failures and learn prediction rules. Second, there are problems where human experts exist, but where they are unable to explain their expertise. This is the case in many perceptual tasks, such as speech recognition, hand-writing recognition, and natural language understanding. Virtually all humans exhibit expert-level abilities on these tasks, but none of them can describe the detailed steps that they follow as they perform them. Fortunately, humans can provide machines with examples of the inputs and correct outputs for these tasks, so machine learning algorithms can learn to map the inputs to the outputs. Third, there are problems where phenomena are changing rapidly. In finance, for example, people would like to predict the future behavior of the stock market, of consumer purchases, or of exchange rates. These behaviors change frequently, so that even if a programmer could construct a good predictive computer program, it would need to be rewritten frequently. A learning program can relieve the programmer of this burden by constantly modifying and tuning a set of learned prediction rules. Fourth, there are applications that need to be customized for each computer user separately. Consider, for example, a program to filter unwanted electronic mail messages. Different users will need different filters. It is unreasonable to expect each user to program his or her own rules, and it is infeasible to provide every user with a software engineer to keep the rules up-to-date. A machine learning system can learn which mail messages the user rejects and maintain the filtering rules automatically. Machine learning addresses many of the same research questions as the fields of statistics, data mining, and psychology, but with differences of emphasis. Statistics focuses on understanding the phenomena that have generated the data, often with the goal of testing different hypotheses about those phenomena. Data mining seeks to find patterns in the data that are understandable by people. Psychological studies of human learning aspire to understand the mechanisms underlying the various learning behaviors exhibited by people (concept learning, skill acquisition, strategy change, etc.).

Machine learning

Preface 1. Introduction Overview A Brief History of LMIs in Control Theory Notes on the Style of the Book Origin of the Book 2. Some Standard Problems Involving LMIs. Linear Matrix Inequalities Some Standard Problems Ellipsoid Algorithm Interior-Point Methods Strict and Nonstrict LMIs Miscellaneous Results on Matrix Inequalities Some LMI Problems with Analytic Solutions 3. Some Matrix Problems. Minimizing Condition Number by Scaling Minimizing Condition Number of a Positive-Definite Matrix Minimizing Norm by Scaling Rescaling a Matrix Positive-Definite Matrix Completion Problems Quadratic Approximation of a Polytopic Norm Ellipsoidal Approximation 4. Linear Differential Inclusions. Differential Inclusions Some Specific LDIs Nonlinear System Analysis via LDIs 5. Analysis of LDIs: State Properties. Quadratic Stability Invariant Ellipsoids 6. Analysis of LDIs: Input/Output Properties. Input-to-State Properties State-to-Output Properties Input-to-Output Properties 7. State-Feedback Synthesis for LDIs. Static State-Feedback Controllers State Properties Input-to-State Properties State-to-Output Properties Input-to-Output Properties Observer-Based Controllers for Nonlinear Systems 8. Lure and Multiplier Methods. Analysis of Lure Systems Integral Quadratic Constraints Multipliers for Systems with Unknown Parameters 9. Systems with Multiplicative Noise. Analysis of Systems with Multiplicative Noise State-Feedback Synthesis 10. Miscellaneous Problems. Optimization over an Affine Family of Linear Systems Analysis of Systems with LTI Perturbations Positive Orthant Stabilizability Linear Systems with Delays Interpolation Problems The Inverse Problem of Optimal Control System Realization Problems Multi-Criterion LQG Nonconvex Multi-Criterion Quadratic Problems Notation List of Acronyms Bibliography Index.

/pdf/linear-matrix-inequalities-in-system-and-control-theory-2q5cf0pqm2.pdf

Linear Matrix Inequalities in System and Control Theory

The objective of this paper is to propose an approach to harmonise noisy spatial data acquired by different operators using (low-cost) hand-held sensors over the same spatial domain. In such cases, datasets need to be harmonised before to be comparable before decision making. This work proposes a methodology to address this issue in the case of nested and noisy spatial data. First, it proposes the implementation of a non-parametric test of Kolmogorov-Smirnov to determine if harmonisation is needed. Then, it proposes an aspatial harmonization method based on a standardization. The method was applied on grape sugar content datasets collected by 2 hand-held spectrometers. Results showed that harmonizing a less confident dataset with respect to a more trustworthy one is interesting solely if the size of the trusted dataset is too small.

/pdf/investigating-the-harmonization-of-highly-noisy-4wxlif1gl3.pdf

Investigating the harmonization of highly noisy heterogeneous datasets hand-collected over the same study domain

Rainer Gob, Kristina Lurz and Antonio Pievatolo (hereinafter GLP) address a very important issue in power systems management—load forecasting. Generally, load forecasting is concerned with the accurate prediction of the electric load (or demand) for specific geographical locations and over the different periods of the planning horizon. However, as in the discussed paper, the quantity of interest is usually the hourly total load, and the planning horizon is short-term—it ranges from a few minutes to a few weeks. Short-term load forecasting has become increasingly important since the rise of competitive energy markets. Electric utilities (called ‘energy vendors’ by GLP) are the most vulnerable as they typically cannot pass costs directly to the retail consumers. When electricity sectors were regulated, utility monopolies used short-term load forecasting to ensure the reliability of supply (prevent overloading, reduce occurrences of equipment failures, etc.). Nowadays, the costs of over-contracting/under-contracting and then selling/buying power in the balancing (or real-time) market are typically so high that they can lead to huge financial losses of the utility and bankruptcy in the extreme case. Load forecasting has become the central and integral process in the planning and operation not only of electric utilities (as GLP clearly point out) but also of energy suppliers, system operators and other market participants.

A variety of methods and ideas have been tried for load forecasting, with varying degrees of success. Following Weron [1], GLP classify them into two broad categories: (i) statistical approaches, including similar-day (or naive), exponential smoothing, regression and time series methods; and (ii) artificial intelligence-based techniques. Among them, exponential smoothing stands out as a simple yet powerful approach, whereby the prediction is constructed from a weighted average of past observations with exponentially smaller weights being assigned to older observations. More complex variants—such as the Holt-Winters method—have been developed to model time series with seasonal and trend components [1, 2]. Application of exponential smoothing to hourly electricity load data requires further generalization to accommodate the prevailing seasonalities (daily, weekly and possibly annual) and weather-related exogenous variables (called ‘covariates’ by GLP).

This discussion paper offers comments in three sections. The first section discusses the model and its position within the exponential smoothing literature on load forecasting. The second section comments on the empirical part of the paper. The final section offers recommendations for further research.

Discussion on ‘Electrical load forecasting by exponential smoothing with covariates’

Are ruptured and unruptured aneurysms different? [2] (multiple letters)

We present a new expert-system "front end" or design advisor for implementing systems (DAIS) for use in conjunction with a commercial digital control system environment, e.g., the Elsag Bailey INFI 90 System. The objective of DAIS is to make it substantially easier for applications engineers to make effective use of the broad spectrum of capabilities of this and similar hardware and software systems for industrial controls implementation. This concept is of quite general applicability for industrial controls implementation environments.

An intelligent front end for control system implementation

Background A combined forecast from multiple models is typically more accurate than an individual forecast, but there are few examples of studies of combining in infectious disease forecasting. We investigated the accuracy of different ways of combining interval forecasts of weekly incident and cumulative coronavirus disease-2019 (COVID-19) mortality. Methods We considered weekly interval forecasts, for 1- to 4-week prediction horizons, with out-of-sample periods of approximately 18 months ending on 8 January 2022, for multiple locations in the United States, using data from the COVID-19 Forecast Hub. Our comparison involved simple and more complex combining methods, including methods that involve trimming outliers or performance-based weights. Prediction accuracy was evaluated using interval scores, weighted interval scores, skill scores, ranks, and reliability diagrams. Results The weighted inverse score and median combining methods performed best for forecasts of incident deaths. Overall, the leading inverse score method was 12% better than the mean benchmark method in forecasting the 95% interval and, considering all interval forecasts, the median was 7% better than the mean. Overall, the median was the most accurate method for forecasts of cumulative deaths. Compared to the mean, the median’s accuracy was 65% better in forecasting the 95% interval, and 43% better considering all interval forecasts. For all combining methods except the median, combining forecasts from only compartmental models produced better forecasts than combining forecasts from all models. Conclusions Combining forecasts can improve the contribution of probabilistic forecasting to health policy decision making during epidemics. The relative performance of combining methods depends on the extent of outliers and the type of models in the combination. The median combination has the advantage of being robust to outlying forecasts. Our results support the Hub’s use of the median and we recommend further investigation into the use of weighted methods.

James Taylor

Papers

Investigating the harmonization of highly noisy heterogeneous datasets hand-collected over the same study domain

Discussion on ‘Electrical load forecasting by exponential smoothing with covariates’

Are ruptured and unruptured aneurysms different? [2] (multiple letters)

An intelligent front end for control system implementation

Interval forecasts of weekly incident and cumulative COVID-19 mortality in the United States: A comparison of combining methods