Meaningful measurements for the Millennium
Once upon a time, measurements were taken with simple scales or meters and resulted in explicit quantities such as weight, distance, or voltage. With the ubiquity of low-cost computing power, we now have the ability to measure more complex and dynamic systems, using implicit parameters that are not directly accessible to physical measuring devices, and which are often highly interdependent.
The Art of Measurement: Theory and Practice is the first book to bring together the practical tools of standard measurement with the philosophy and theory behind probability and statistical estimation. The result is a book that empowers users to find the best approach for each measurement they make, and to learn how to evaluate and balance the impact of the inevitable inaccuracies that occur.
In keeping with its practical approach, The Art of Measurement: Theory and Practice uses numerous case studies to give concrete illustrations of theoretical concepts. Drawn from the fields of electrical and mechanical engineering, control systems, and satellite navigation technologies, these examples offer a broad variety of applications for all users.
The book's appendices offer a review of basic mathematical terminology and tools, but to get the most out of this book some higher-level math is required, in particular a familiarity with Fourier and Laplace transforms, matrix algebra, multi-dimensional vector spaces, probability theory, statistics, and estimation theory. Scientists and engineers in a variety of fields will find The Art of Measurement: Theory and Practice an invaluable addition to their reference libraries.
Click here for a sample chapter for this book: 0130261742.pdf
1. A Philosophical Viewpoint.
The “Ultimate” Measurement. The Mindset of Measuring. Measurement Quality. Attributes of the Measurement Model. Bibliography on Our World Geometry.
The Optimum Number of Parameters. Various Parameter Attributes. Example: DC Battery Resistance. Example: Control System Loop Gain. Example: Modes of Vibration. Example: GPS Navigation System. Summary. Bibliography on Modeling Examples.
Prologue on Uncertainty. Introduction to Probability Theory. Probability Theory—Extended. Example: A Noisy Measurement. Example: A Noisy Measurement with Bias. Discrete or Quantized Random Variables. PDFs for Mixed Random Variables. Some Matrices in Probability Theory. Summary. Bibliography on Probability Theory.
To Measure Is to Estimate. The Concept of Abstract Metric Spaces. Example: Single-Pole Transfer Function. The Least-Squares Estimate. Maximum Likelihood Estimation. Estimation Using Bayes Theorem. Sequential Estimation. The Kalman Filter. General Comments on Estimation Methods. Summary of Estimation Techniques. Bibliography on Estimation Theory.
Preview of Four Examples. Measuring the DC Resistance of a Battery. Closed-Loop Control System Measurements. Vibration Modes of Mechanical Structures. GPS Navigation System. Summary of Four Measurement Examples.
Review of Basic Mathematical Tools. Preview. The Fourier Transform. The Laplace Transform. Matrix Algebra. Abstract Vector Spaces. Bibliography on Mathematical Tools.
Derivation of GPS Navigation Equations. Glossary of Technical Words and Concepts.
This is a book about measuring and is consequently aimed at an audience interested in the quantitative aspects of the world around them. It is dedicated especially to those who feel they do not understand a physical system until they can fully measure its parameters, including the statistics on those parameters. The book is general in scope and does not attempt to cover particular fields of measurement in any detail. It is largely up to you, the reader, to adapt these ideas and techniques to your own particular area of interest.
Before the advent of microprocessors, a measurement generally involved explicit quantities such as distance or voltage, determined by reading a scale or a meter. Most early books about measuring techniques tended to emphasize the "best" methods for getting good results from this approach. There has always been a need to measure the characteristics of more complicated physical systems, but it was very difficult to do. Now that microprocessors are so ubiquitous, we have the ability to measure much more complicated systems, involving various implicit parameters, which are not directly accessible to a scale or meter. The new approach is to collect quantities of data from whatever points are physically accessible, and then to estimate the parameters of interest within the system that would cause this particular set of data. This estimation procedure is centered around a mathematical model of the physical system, in which there are "free" parameters. The measurement task is to estimate these parameter values such that the observed data "matches" the corresponding data from the model.
In addition, a measured parameter value expressed as a single number has little meaning and conveys only a minuscule amount of information when standing alone, unless many similar measurements have already been made, or unless a thorough error analysis of the measurement model has been completed. Is the number within 50% of the correct value, or is it within a part in a million? Is it perhaps a "wild" value that might occur only once out of many tries? Is it biased in one direction by some (unknown) amount? The probable errors in each measured parameter must be included as an integral part of the measurement, and all conditions and assumptions associated with the results must be clearly stated.
There are numerous, predominately theoretical, books on probability and statistics, Laplace and Fourier transforms, matrix algebra, and estimation procedures, which have little discussion of practical applications. Conversely, there are books that describe specific measurement tasks, but with minimal acknowledgment of the rationale or logical framework behind each one. What we really need is an approach that places one foot in the theory and the other foot in the practice. There are abundant practical measurement tasks that confront us, but the tools we need for these tasks reside in the theory of measuring. No one can be expected to do a professional quality job in any field without using the best available tools, applied in the proper manner.
In some respects this is a "how to . . ." book for people who make measurements, analogous to books that are available to the home handyman in endeavors like carpentry, plumbing, and wiring. There are well defined tasks to be performed, and there is a set of available tools to be used. The difference is that the tools here are mathematical procedures or algorithms, and the end products are numbers or parameter estimates in a mathematical model of the physical system of interest.
Although this is not meant to be an academic textbook on the subject, the treatment herein is more than superficial. This topic cannot be adequately discussed and explored without the help of mathematical concepts and relationships, since measuring is a quantitative activity. The required level of mathematical ability is somewhat difficult to define, but you will get the most out of the book if you have a background in such areas as Fourier and Laplace transforms, matrix algebra, multidimensional vector spaces, probability theory, statistics, and estimation theory. Even though some degree of technical expertise is a definite plus, nontechnical readers may still find the ideas interesting and understandable in a general way. You may simply skip the equations and read the text.
Many of the technical and mathematical "buzzwords" that appear herein are explained, although it is not possible to include a complete exposition on these subjects in a book of this size. We review some basic mathematical tools in Appendix A, and we include a Glossary in the back, where some of the words and concepts are briefly defined. If you run across a word or concept that is unfamiliar, then you might consider that as an invitation to investigate the subject more fully in other books.
The goals of this book are: First, to lay the groundwork for making meaningful measurements, by discussing the philosophy and abstract thinking that is required. The mindset or viewpoint of the measurer is so important that we emphasize the philosophy of measuring as a necessary starting point. Second, to address the many potential errors and error mechanisms that always seem to exist. It is sometimes very tedious to get decent results from a measurement because there are so many sources of error. The task of making a set of reliable measurements is primarily a task of system error analysis. Third, to review various theoretical procedures that are needed to do the "best" possible job in estimating the parameter values. At the end of the measuring process, we must decide where the measured parameter values actually fall within the range of all possible values, as well as on the degree of confidence we can place in these final values.
The first four chapters address the many broad theoretical and philosophical aspects of making good quantitative measurements, but it is equally important to illustrate the practical side of this activity. To this end, a few representative measurement examples are discussed throughout the first chapters, and are more fully explored in chapter 5. Four such examples have been selected, covering a broad spectrum of measurement applications. These are: (1) measuring the DC resistance of a battery or power supply, (2) estimating the loop gain of a closed-loop control system, (3) measuring the modal parameters of a vibrating mechanical structure, and (4) determining position coordinates on the Earth, using the Global Positioning System (a satellite navigation technique, abbreviated GPS). With these examples as guides, it should be possible for the reader to adapt these ideas and techniques to his or her own particular measurement project.
We should set the highest standards for each measurement task. The quality of measured parameters is just as important as the estimated values of the parameters themselves, and both of these depend on a quality measurement model. The consequences of incomplete, incorrect, or misunderstood measurements can be quite serious, and can also be very costly. Hopefully, after reading this book, you will be especially critical of measurements in general, whether made by yourself or by other people.
One of the graphical tools that we use rather freely in this book is the stereoscopic image pair, in which a separate image is drawn for each eye. By suitably combining these image pairs, you can see a three-dimensional view of an object or mathematical function. These images are more than just entertaining diversions, because they help convey ideas and concepts to our brain that might otherwise be hard to understand from flat drawings on paper. They enable us to actually see into a three-dimensional observation space, and to see one and two dimensional parameters spaces embedded in this three-space. For example, we can see two-dimensional joint probability density functions, and other two-dimensional surfaces as subspaces of a three-dimensional parent space. We can also see clouds of observation points around sets of parameter coordinates.
If you have not already trained your eyes to view these stereo images, you will be greatly rewarded when you master this technique. The object is to get each eye to look at (and focus on) only the image drawn for that eye. The left image is for the left eye and the right image is for the right eye. There are several ways to accomplish this task. One technique is to hold the paper relatively close to your face, while staring through the paper to infinity (by relaxing your eyes). Ignore poor focus in the beginning. You should see four images, comprising the two original images along with two more in between the originals. Concentrate on bringing the two interior images together by relaxing your eyes and imagine looking at the far wall of the room, through the paper. Once the images coalesce into one, slowly move the paper away from your face until it is easy to focus on the stereo figure. Some people like to hold a postcard or some larger opaque surface between their eyes, perpendicular to the paper. This seems to help decouple the eyes from one another. Don't worry about any sort of damage to your eye-tracking ability! The more practice, the easier this procedure becomes. If you get desperate, buy yourself a pair of stereo viewing glasses used for viewing pairs of aerial photographs. Good luck.