Best Fit Lines & Curves: And Some Mathe-Magical Transformations, 1st Edition (Hardback) book cover

Best Fit Lines & Curves

And Some Mathe-Magical Transformations, 1st Edition

By Alan R. Jones


498 pages

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Hardback: 9781138065000
pub: 2018-09-10
eBook (VitalSource) : 9781315160085
pub: 2018-10-09
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Best Fit Lines and Curves, and Some Mathe-Magical Transformations (Volume III of the Working Guides to Estimating & Forecasting series) concentrates on techniques for finding the Best Fit Line or Curve to some historical data allowing us to interpolate or extrapolate the implied relationship that will underpin our prediction. A range of simple ‘Moving Measures’ are suggested to smooth the underlying trend and quantify the degree of noise or scatter around that trend. The advantages and disadvantages are discussed and a simple way to offset the latent disadvantage of most Moving Measure Techniques is provided.

Simple Linear Regression Analysis, a more formal numerical technique that calculates the line of best fit subject to defined ‘goodness of fit’ criteria. Microsoft Excel is used to demonstrate how to decide whether the line of best fit is a good fit, or just a solution in search of some data. These principles are then extended to cover multiple cost drivers, and how we can use them to quantify 3-Point Estimates.

With a deft sleight of hand, certain commonly occurring families of non-linear relationships can be transformed mathe-magically into linear formats, allowing us to exploit the powers of Regression Analysis to find the Best Fit Curves. The concludes with an exploration of the ups and downs of seasonal data (Time Series Analysis). Supported by a wealth of figures and tables, this is a valuable resource for estimators, engineers, accountants, project risk specialists as well as students of cost engineering.


"In the Working Guides to Estimating and Forecasting Alan has managed to capture the full spectrum of relevant topics with simple explanations, practical examples and academic rigor, while injecting humour into the narrative." Dale Shermon, Chairman, Society of Cost Analysis and Forecasting (SCAF).

"If estimating has always baffled you, this innovative well illustrated and user friendly book will prove a revelation to its mysteries. To confidently forecast, minimise risk and reduce uncertainty we need full disclosure into the science and art of estimating. Thankfully, and at long last the "Working Guides to Estimating & Forecasting" are exactly that, full of practical examples giving clarity, understanding and validity to the techniques. These are comprehensive step by step guides in understanding the principles of estimating using experientially based models to analyse the most appropriate, repeatable, transparent and credible outcomes. Each of the five volumes affords a valuable tool for both corporate reference and an outstanding practical resource for the teaching and training of this elusive and complex subject. I wish I had access to such a thorough reference when I started in this discipline over 15 years ago, I am looking forward to adding this to my library and using it with my team." - Tracey L Clavell, Head of Estimating & Pricing, BAE Systems Australia

"At last, a comprehensive compendium on these engineering math subjects, essential to both the new and established "cost engineer"! As expected the subjects are presented with the author’s usual wit and humour on complex and daunting "mathematically challenging" subjects. As a professional trainer within the MOD Cost Engineering community trying to embed this into my students, I will be recommending this series of books as essential bedtime reading." - Steve Baker, Senior Cost Engineer, DE&S MOD

"Alan has been a highly regarded member of the Cost Estimating and forecasting profession for several years. He is well known for an ability to reduce difficult topics and cost estimating methods down to something that is easily digested. As a master of this communication he would most often be found providing training across the cost estimating and forecasting tools and at all levels of expertise. With this 5-volume set, Working Guides to Estimating and Forecasting, Alan has brought his normal verbal training method into a written form. Within their covers Alan steers away from the usual dry academic script into establishing an almost 1:1 relationship with the reader. For my money a recommendable read for all levels of the Cost Estimating and forecasting profession and those who simply want to understand what is in the ‘blackbox’ just a bit more." - Prof Robert Mills, Margin Engineering, Birmingham City University. MACOSTE, SCAF, ICEAA.

"Finally, a book to fill the gap in cost estimating and forecasting! Although other publications exist in this field, they tend to be light on detail whilst also failing to cover many of the essential aspects of estimating and forecasting. Jones covers all this and more from both a theoretical and practical point of view, regularly drawing on his considerable experience in the defence industry to provide many practical examples to support his comments. Heavily illustrated throughout, and often presented in a humorous fashion, this is a must read for those who want to understand the importance of cost estimating within the broader field of project management." - Dr Paul Blackwell, Lecturer in Management of Projects, The University of Manchester, UK.

"Alan Jones provides a useful guidebook and navigation aid for those entering the field of estimating as well as an overview for more experienced practitioners. His humorous asides supplement a thorough explanation of techniques to liven up and illuminate an area which has little attention in the literature, yet is the basis of robust project planning and successful delivery. Alan’s talent for explaining the complicated science and art of estimating in practical terms is testament to his knowledge of the subject and to his experience in teaching and training." - Therese Lawlor-Wright, Principal Lecturer in Project Management at the University of Cumbria

"Alan Jones has created an in depth guide to estimating and forecasting that I have not seen historically. Anyone wishing to improve their awareness in this field should read this and learn from the best." Richard Robinson, Technical Principal for Estimating, Mott MacDonald

"The book series of ‘Working Guides to Estimating and Forecasting’ is an essential read for students, academics and practitioners who interested in developing a good understanding of cost estimating and forecasting from real-life perspectives". Professor Essam Shehab, Professor of Digital Manufacturing and Head of Cost Engineering, Cranfield University, UK.

"In creating the Working Guides to Estimating and Forecasting, Alan has captured the core approaches and techniques required to deliver robust and reliable estimates in a single series. Some of the concepts can be challenging, however, Alan has delivered them to the reader in a very accessible way that supports lifelong learning. Whether you are an apprentice, academic or a seasoned professional, these working guides will enhance your ability to understand the alternative approaches to generating a well-executed, defensible estimate, increasing your ability to support competitive advantage in your organisation." - Professor Andrew Langridge, Royal Academy of Engineering Visiting Professor in Whole Life Cost Engineering and Cost Data Management, University of Bath, UK.

"Alan Jones’s "Working Guides to Estimating and Forecasting" provides an excellent guide for all levels of cost estimators from the new to the highly experienced. Not only does he cover the underpinning good practice for the field, his books will take you on a journey from cost estimating basics through to how estimating should be used in manufacturing the future – reflecting on a whole life cycle approach. He has written a must-read book for anyone starting cost estimating as well as for those who have been doing estimates for years. Read this book and learn from one of the best." - Linda Newnes, Professor of Cost Engineering, University of Bath, UK.

Table of Contents

Foreword, 1 Introduction and Objectives, 1.1 Why write this book? Who might find it useful? Why Five Volumes? 1.1.1 Why write this series? Who might find it useful? 1.1.2 Why Five Volumes? 1.2 Features you'll find in this book and others in this series, 1.2.1 Chapter Context, 1.2.2 The Lighter Side (humour), 1.2.3 Quotations, 1.2.4 Definitions, 1.2.5 Discussions and Explanations with a Mathematical Slant for Formula-philes, 1.2.6 Discussions and Explanations without a Mathematical Slant for Formula-phobes, 1.2.7 Caveat Augur, 1.2.8 Worked Examples, 1.2.9 Useful Microsoft Excel Functions and Facilities, 1.2.10 References to Authoritative Sources, 1.2.11 Chapter Reviews, 1.3 Overview of Chapters in this Volume, 1.4 Elsewhere in the 'Working Guide to Estimating & Forecasting' Series, 1.4.1 Volume I: Principles, Process and Practice of Professional Number Juggling, 1.4.2 Volume II: Probability, Statistics and other Frightening Stuff, 1.4.3 Volume III: Best Fit Lines & Curves, and some Mathe-Magical Transformations, 1.4.4 Volume IV: Learning, Unlearning and Re-Learning Curves, 1.4.5 Volume V: Risk, Opportunity, Uncertainty and Other Random Models, 1.5 Final Thoughts and Musings on this Volume and Series, References, , 2 Linear and Nonlinear Properties (!) of Straight Lines, 2.1 Basic Linear Properties, 2.1.1 Inter-relation between Slope and Intercept, 2.1.2 The Difference between Two Straight Lines is a Straight Line, 2.2 The Cumulative Value (Nonlinear) Property of a Linear Sequence, 2.2.1 The Cumulative Value of a Discrete Linear Function, 2.2.2 The Cumulative Value of a Continuous Linear Function, 2.2.3 Exploiting the Quadratic Cumulative Value of a Straight Line, 2.3 Chapter Review, References, , 3 Trendsetting with Some Simple Moving Measures, 3.1 Going All Trendy: The Could and The Should, 3.1.1 When should we consider trend smoothing?, 3.1.2 When is trend smoothing not appropriate?, 3.2 Moving Averages, 3.2.1 Use of Moving Averages, 3.2.2 When not to use Moving Averages, 3.2.3 Simple Moving Average, 3.2.4 Weighted Moving Average, 3.2.5 Choice of Moving Average Interval: Is there a better way than guessing? 3.2.6 Can we take the Moving Average of a Moving Average?, 3.2.7 A Creative Use for Moving Averages - A Case of Forward Thinking, 3.2.8 Dealing with Missing Data, 3.2.9 Uncertainty Range around the Moving Average, 3.3 Moving Medians, 3.3.1 Choosing the Moving Median Interval, 3.3.2 Dealing with Missing Data, 3.3.3 Uncertainty Range around the Moving Median, 3.4 Other Moving Measures of Central Tendency, 3.4.1 Moving Geometric Mean, 3.4.2 Moving Harmonic Mean, 3.4.3 Moving Mode, 3.5 Exponential Smoothing, 3.5.1 An Unfortunate Dichotomy, 3.5.2 Choice of Smoothing Constant, or Choice of Damping Factor, 3.5.3 Uses for Exponential Smoothing, 3.5.4 Double and Triple Exponential Smoothing, 3.6 Cumulative Average and Cumulative Smoothing, 3.6.1 Use of Cumulative Averages, 3.6.2 Dealing with Missing Data, 3.6.3 Cumulative Averages with Batch Data, 3.6.4 Being slightly more Creative - Cumulative Average on a Sliding Scale, 3.6.5 Cumulative Smoothing, 3.7 Chapter Review, References, , 4 Simple and Multiple Linear Regression, 4.1 What is Regression Analysis?, 4.1.1 Least Squares Best Fit, 4.1.2 Two Key Sum-to-Zero Properties of Least Squares, 4.2 Simple Linear Regression, 4.2.1 Simple Linear Regression using Basic Excel Functions, 4.2.2 Simple Linear Regression using the Data Analysis Add-in Tool Kit in Excel, 4.2.3 Simple Linear Regression using Advanced Excel Functions, 4.3 Multiple Linear Regression, 4.3.1 Using Categorical Data in Multiple Linear Regression, 4.3.2 Multiple Linear Regression using the Data Analysis Add-in Tool Kit in Excel, 4.3.3 Multiple Linear Regression using Advanced Excel Function, 4.4 Dealing with Outliers in Regression Analysis?, 4.5 How Good is our Regression? Six Key Measures, 4.5.1 Coefficient of Determination (R-Square): A Measure of Linearity?!, 4.5.2 F-Statistic: A Measure of Chance Occurrence, 4.5.3 t-Statistics: Measures of Relevance or Significant Contribution, 4.5.4 Regression through the Origin, 4.5.5 Role of Common Sense as a Measure of Goodness of Fit, 4.5.6 Coefficient of Variation as a Measure of Tightness of Fit, 4.5.7 White's Test for Heteroscedasticity … and By Default Homoscedasticity, 4.6 Prediction and Confidence Intervals - Measures of Uncertainty, 4.6.1 Prediction Intervals and Confidence Intervals: What's the Difference?, 4.6.2 Calculating Prediction Limits and Confidence Limits for Simple Linear Regression, 4.6.3 Calculating Prediction Limits and Confidence Limits for Multi-Linear Regression, 4.7 Stepwise Regression, 4.7.1 Backward Elimination, 4.7.2 Forward Selection, 4.7.3 Backward or Forward Selection - Which should we use?, 4.7.4 Choosing the Best Model when we are Spoilt for Choice, 4.8 Chapter Review, References, , 5 Linear Transformation: Making Bent Lines Straight, 5.1 Logarithms, 5.1.1 Basic Properties of Powers, 5.1.2 Basic Properties of Logarithms, 5.2 Basic Linear Transformation: Four Standard Function Types, 5.2.1 Linear Functions, 5.2.2 Logarithmic Functions, 5.2.3 Exponential Functions, 5.2.4 Power Functions, 5.2.5 Transforming with Excel, 5.2.6 Is the Transformation Really Better, or Just a Mathematical Sleight of Hand?, 5.3 Advanced Linear Transformation: Generalised Function Types, 5.3.1 Transforming Generalised Logarithmic Functions, 5.3.2 Transforming Generalised Exponential Functions, 5.3.3 Transforming Generalised Power Functions, 5.3.4 Reciprocal Functions - Special Cases of a Generalised Power Functions, 5.3.5 Transformation Options, 5.4 Finding the Best Fit Offset Constant, 5.4.1 Transforming Generalised Function Types into Standard Functions, 5.4.2 Using the Random-Start Bisection Method (Technique), 5.4.3 Using Microsoft Excel's Goal Seek or Solver, 5.5 Straightening Out Earned Value Analysis … or EVM Disintegration, 5.5.1 EVM Terminology, 5.5.2 Taking a Simpler Perspective, 5.6 Linear Transformation Based on Cumulative Value Disaggregation, 5.7 Chapter Review, References, , 6 Transforming Nonlinear Regression, 6.1 Simple Linear Regression of a Linear Transformation, 6.1.1 Simple Linear Regression with a Logarithmic Function, 6.1.2 Simple Linear Regression with an Exponential Function, 6.1.3 Simple Linear Regression with a Power Function, 6.1.4 Reversing the Transformation of Logarithmic, Exponential and Power Functions, 6.2 Multiple Linear Regression of a Multi-linear Transformation, 6.2.1 Multi-linear Regression using Linear and Linearised Logarithmic Functions, 6.2.2 Multi-linear Regression using Linearised Exponential and Power Functions, 6.3 Stepwise Regression and Multi-Linear Transformations, 6.3.1 Stepwise Regression by Backward Elimination with Linear Transformations, 6.3.2 Stepwise Regression by Forward Selection with Linear Transformations, 6.4 Is the Best Fit Really the Better Fit?, 6.5 Regression of Transformed Generalised Nonlinear Functions, 6.5.1 Linear Regression of a Transformed Generalised Logarithmic Function, 6.5.2 Linear Regression of a Transformed Generalised Exponential Function, 6.5.3 Linear Regression of a Transformed Generalised Power Function, 6.5.4 Generalised Function Transformations: Avoiding the Pitfalls and Tripwires, 6.6 Pseudo Multi-linear Regression of Polynomial Functions, 6.6.1 Offset Quadratic Regression of the Cumulative of a Straight Line, 6.6.2 Example of a Questionable Cubic Regression of Three Linear Variables, 6.7 Chapter Review, References, 7 Least Squares Nonlinear Curve Fitting without the Logs, 7.1 Curve Fitting by Least Squares … without the Logarithms, 7.1.1 Fitting Data to Discrete Probability Distributions, 7.1.2 Fitting data to Continuous Probability Distributions, 7.1.3 Revisiting the Gamma Distribution Regression, 7.2 Chapter Review, References, , 8 The Ups and Downs of Time Series Analysis, 8.1 The Bits and Bats … and Buts of a Time Series, 8.1.1 Conducting a Time Series Analysis, 8.2 Alternative Time Series Models, 8.2.1 Additive/Subtractive Time Series Model, 8.2.2 Multiplicative Time Series Model, 8.3 Classical Decomposition: Determining the Underlying Trend, 8.3.1 See-Saw … Regression Flaw?, 8.3.2 Moving Average Seasonal Smoothing, 8.3.3 Cumulative Average Seasonal Smoothing, 8.3.4 What happens when our world is not perfect? Do any of these trends work?, 8.3.5 Exponential Trends and Seasonal Funnels, 8.3.6 Meandering Trends, 8.4 Determining the Seasonal Variations by Classical Decomposition, 8.4.1 The Additive/Subtractive Model, 8.4.2 The Multiplicative Model, 8.5 Multi-Linear Regression: A Holistic Approach to Time Series?, 8.5.1 The Additive/Subtractive Linear Model, 8.5.2 The Additive/Subtractive Exponential Model, 8.5.3 The Multiplicative Linear Model, 8.5.4 The Multiplicative Exponential Model, 8.5.5 Multi-linear Regression: Reviewing the Options to Make an Informed Decision, 8.6 Excel Solver Technique for Time Series Analysis, 8.6.1 The Perfect World Scenario, 8.6.2 The Real World Scenario, 8.6.3 Wider examples of the Solver Technique, 8.7 Chapter Review, References, Glossary of Estimating Terms, Index

About the Author

Alan R. Jones is Principal Consultant at Estimata Limited, aconsultancy service specialising in Estimating Skills Training. He is a Certified Cost Estimator/Analyst (US) and Certified Cost Engineer (CCE) (UK). Prior to setting up his own business, he enjoyed a 40-year career in the UK aerospace and defence industry as an estimatorAlan is a Fellow of the Association of Cost Engineers and a member of the International Cost Estimating and Analysis Association. Historically (some four decades ago), Alan was a graduate in Mathematics from Imperial College of Science and Technology in London, and was an MBA Prize-winner at the Henley Management College.

About the Series

Working Guides to Estimating & Forecasting

As engineering and construction projects get bigger, more ambitious and increasingly complex, the ability of organizations to work with realistic estimates of cost, risk or schedule has become fundamental.

And working with estimates requires technical and mathematical skills from the estimator but it also requires an understanding of the processes, the constraints, and the context by those making investment and planning decisions. You can only forecast the future with confidence if you understand the limitations of your forecast.

The Working Guides to Estimating introduce, explain and illustrate the variety and breadth of numerical techniques and models that are commonly used to build estimates. Alan Jones defines the formulae that underpin many of the techniques; offers justification and explanations, for those whose job it is to interpret the estimates; advice on pitfalls and shortcomings and worked examples. These are often tabular in form to allow you to reproduce the examples in Microsoft Excel. Graphical or pictorial figures are also frequently used to draw attention to particular points as the author advocates that you should always draw a picture before and after analysing data.

The five volumes in the Series provide expert applied advice for estimators, engineers, accountants, project risk specialists as well as students of cost engineering, based on the author’s thirty-something years’ experience as an estimator, project planner and controller.

Learn more…

Subject Categories

BISAC Subject Codes/Headings:
BUSINESS & ECONOMICS / Project Management
TECHNOLOGY & ENGINEERING / Construction / Estimating


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This complimentary eResource is provided to assist users of the Working Guides to Estimating and Forecasting series of books in recreating the tables and figures from the published books (printed or electronic). It is supplementary to, and is not intended to replace, the material in the published books. The files are available to download in their original colour format.

This material is copyrighted by the author and is provided solely as an aid to learning. Neither the author nor the publisher accepts any liability for its subsequent use or adaptation for any other purpose other than for which it was originally intended.

The figures and tables are provided in either interactive Microsoft Excel spreadsheet files, complete in general with the original calculation formulae, or as static images, where the reader may prefer to view those examples in full colour that were created in Microsoft PowerPoint. Any tables originally created in Microsoft Word have been converted and included in the Microsoft Excel files to reduce the number of download options for the users.

CAVEAT AUGUR: Whilst the author considers it to be good practice to lock calculation cells etc and to protect worksheets and workbooks, this has not been done here; the spreadsheets have been left unlocked and unprotected. Neither have the worksheet examples been independently Validated and Verified, as is also good practice.