The PaleoMath Archive
These essays on various data-analysis and statistical procedures were published originally in the Palaeontological Association (PalAss) Newsletter from 2004 to 2013. They were also available on the PalAss website as individual PDF files until 2025 when the PalAss Council approved a plan to redesign their website and move its management to a commercial provider. The newsletters containing these essays are still available on the PalAss website, but the association now charges £1.00 to non-members for downloading each issue. In order to save the larger paleo. community members that expense, and because the PaleoMath essays were written originally on the understanding they would be provided to that community free-of-charge, I have ensured free-of-charge access to the original PDF versions by moving them to PaleoNet. Movement of the PaleoMath essays to PaleoNet also provides me with the opportunity to add new essays to this collection covering additional procedures useful in paleontological/paleobiological data analysis.
The original PaleoMath essay series explored aspects of quantitative analysis in paleontological contexts with an emphasis on practicality, understanding of the issues involved in planning a data analysis strategy, and the implications of making choices between available data-analysis techniques. Each essay was written for the novice data analyst, especially those who want to gain knowledge of this subject, but never had the opportunity to take formal courses in statistics, linear algebra and multivariate analysis and haven't managed to make much progress through self-education. The series was divided into two parts. Part 1 (essays 04-01 - 08-01) focused on standard bivariate (linear) regression and multivariate data-analysis methods, including statistical tests that can be implemented in evaluate their results. Part 2 (essays 08-02 - 13-03) focused on the methods and procedures used to analyze morphometric data. This series takes the reader through techniques used in multivariate morphometrics, geometric morphometrics, outline morphometrics, the morphometric characterization/comparison of surfaces and the direct comparison of specimen images.
In addition to basic descriptions of the mathematics, along with presentation of example-application results, all calculations involved in the example analyses were illustrated in a series of MS-Excel spreadsheets containing the various formulas, procedures and intermediate results in the form of worked examples. These detailed, worked examples were created as an encouragement to the reader to explore use of these methods on small sets of their own data. In my own training I found considerable value in working through small example analyses myself in terms of understanding exactly what particular procedures were doing to my data and how the various options available for each technique could be used, to improve the match between the data, the data-analysis method and the question(s) being asked.
The original PaleoMath essay series proved quite popular, not only with the paleontological community, but with students of other natural history disciplines. Throughout their creation my primary goal was to give interested readers access to the sort of instructional resource I wish I had when I was first learning about, and trying to understand, the concepts, the mathematics and the application(s) of these procedures. Morphometricians like to refer to the set of data-analysis techniques they use as their “toolkit”. The metaphor is apt. In any data-analysis situation, some of these tools will be ideal for the job at hand, others will be useful, and others serviceable, yet prone to various difficulties. Some will lead you to the answer(s) you seek quickly while others will do so in a more roundabout manner. Some will find an answer while others will highlight a range of possible answers for consideration. Moreover, most of these tools are flexible; they can be modified to alter their assumptions and expand their ranges of utility.
Mathematics has a reputation for being a complex, abstract, but somewhat dry topic. In my experience it is nothing of the sort. Indeed, mathematics has a deep connection to what we do as paleontologists and natural historians. Most textbooks will tell you paleontology is the study of fossils. How absurd! I can sit and look at fossils for hours, marveling their intrinsic beauty and wondering about the origins, uses and meaning. But appreciating fossils from an aesthetic point-of-view is not paleontology. Paleontology is the study of patterns in fossils, in their morphologies, in their occurrences (in time and space), in the environments in which they lived as organisms, in the behaviors of those organisms, in their ecologies, etc. To be done effectively paleontology requires a comparative, pattern analysis-based approach.
By the same token, mathematics is not really the study of numbers. Ask most people what mathematics is and they'll tell you its arithmetic combined, perhaps, with a bit of geometry - the aspects of mathematics studied by secondary school students. But these are just component topics within a far larger plexus. What is mathematics? It’s not a list of topics or procedures, It’s the study of patterns in numbers. This deep similarity between the goals of mathematics and paleontology – with the goals of all natural history – explains why mathematical treatments of natural history topics are so attractive. The pattern-recognition tools of mathematics are not just useful in the study of nature, they are essential.
The original PaleoMath essay series explored aspects of quantitative analysis in paleontological contexts with an emphasis on practicality, understanding of the issues involved in planning a data analysis strategy, and the implications of making choices between available data-analysis techniques. Each essay was written for the novice data analyst, especially those who want to gain knowledge of this subject, but never had the opportunity to take formal courses in statistics, linear algebra and multivariate analysis and haven't managed to make much progress through self-education. The series was divided into two parts. Part 1 (essays 04-01 - 08-01) focused on standard bivariate (linear) regression and multivariate data-analysis methods, including statistical tests that can be implemented in evaluate their results. Part 2 (essays 08-02 - 13-03) focused on the methods and procedures used to analyze morphometric data. This series takes the reader through techniques used in multivariate morphometrics, geometric morphometrics, outline morphometrics, the morphometric characterization/comparison of surfaces and the direct comparison of specimen images.
In addition to basic descriptions of the mathematics, along with presentation of example-application results, all calculations involved in the example analyses were illustrated in a series of MS-Excel spreadsheets containing the various formulas, procedures and intermediate results in the form of worked examples. These detailed, worked examples were created as an encouragement to the reader to explore use of these methods on small sets of their own data. In my own training I found considerable value in working through small example analyses myself in terms of understanding exactly what particular procedures were doing to my data and how the various options available for each technique could be used, to improve the match between the data, the data-analysis method and the question(s) being asked.
The original PaleoMath essay series proved quite popular, not only with the paleontological community, but with students of other natural history disciplines. Throughout their creation my primary goal was to give interested readers access to the sort of instructional resource I wish I had when I was first learning about, and trying to understand, the concepts, the mathematics and the application(s) of these procedures. Morphometricians like to refer to the set of data-analysis techniques they use as their “toolkit”. The metaphor is apt. In any data-analysis situation, some of these tools will be ideal for the job at hand, others will be useful, and others serviceable, yet prone to various difficulties. Some will lead you to the answer(s) you seek quickly while others will do so in a more roundabout manner. Some will find an answer while others will highlight a range of possible answers for consideration. Moreover, most of these tools are flexible; they can be modified to alter their assumptions and expand their ranges of utility.
Mathematics has a reputation for being a complex, abstract, but somewhat dry topic. In my experience it is nothing of the sort. Indeed, mathematics has a deep connection to what we do as paleontologists and natural historians. Most textbooks will tell you paleontology is the study of fossils. How absurd! I can sit and look at fossils for hours, marveling their intrinsic beauty and wondering about the origins, uses and meaning. But appreciating fossils from an aesthetic point-of-view is not paleontology. Paleontology is the study of patterns in fossils, in their morphologies, in their occurrences (in time and space), in the environments in which they lived as organisms, in the behaviors of those organisms, in their ecologies, etc. To be done effectively paleontology requires a comparative, pattern analysis-based approach.
By the same token, mathematics is not really the study of numbers. Ask most people what mathematics is and they'll tell you its arithmetic combined, perhaps, with a bit of geometry - the aspects of mathematics studied by secondary school students. But these are just component topics within a far larger plexus. What is mathematics? It’s not a list of topics or procedures, It’s the study of patterns in numbers. This deep similarity between the goals of mathematics and paleontology – with the goals of all natural history – explains why mathematical treatments of natural history topics are so attractive. The pattern-recognition tools of mathematics are not just useful in the study of nature, they are essential.
PaleoMath-1 (Data Analysis)
PaleoMath-2 (Morphometrics)
- 04-01 Regression 1 - Ordinary Least Squares (OLS) Regression
- 04-02 Regression 2 - Reduced Major Axis (RMA) Regression
- 04-03 Regression 3 - Major Axis (MA) Regression
- 05-01 Going Multivariate - Multivariate Regression
- 05-02 Principal Components Analysis (PCA)
- 05-03 Factor Analysis (FA)
- 06-01 Rs & Qs - Principal Coordinates Analysis (PCoord)
- 06-02 Rs & Qs II - Correspondence Analysis (CA)
- 06-03 Data Blocks - Partial Least Squares Analysis (PLSA)
- 07-01 Groups I - Linear Discriminant Analysis (LDA)
- 07-02 Groups II - Canonical Variates Analysis (CVA)
- 07-03 Groups III - Cluster Analysis
- 08-01 Multidimensional Scaling (MDS)
PaleoMath-2 (Morphometrics)
- 08-02 - Distances, Landmarks & Allometry
- 08-03 - Size & Shape Coordinates
- 09-01 - Who is Procrustes?
- 09-02 - Shape Theory
- 09-03 - Form & Shape Models
- 10-01 - Shape Models II (The Thin Plate Spline)
- 10-02 - Principal & Partial Warps
- 10-03 - Relative Warps
- 11-01 - Semilandmarks & Radial Fourier Analysis
- 11-02 - Centroids, Complex Outlines & Shape Functions
- 11-03 - Zahn & Roskies (Z-R) Fourier Analysis
- 12-01 - Elliptic Fourier Analysis
- 12-02 - Eigenshape Analysis I (Standard)
- 12-03 - Eigenshape Analysis II (Extended)
- 13-01 - Sliding Semilandmarks
- 13-02 - Eigensurface Analysis
- 13-03 - Eigenimage Analysis
If anyone has any problems downloading any of these essays please feel free to contact me directly: N.MacLeod9@gmail.com.
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