The text considers Simple Linear Regression and ask what is it doing? how good is it? how accurate is it? and how can we use it to create estimates? The book considers all this and looks at how you can exploit this transformability and use the capability of Simple Linear Regression.
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.
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
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.
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