Ryan L Buchanan
Ogden, Utah, USA


Machine Learning Mathematics
Virtual Reality Aesthetics

"Mathematics requires a small dose, not of genius, but of an imaginative freedom which, in a larger dose, would be insanity."
Angus K. Rodgers

Noteworthy Machine Learning Algorithms

  Machine Learning   ⇒   software able to detect patterns, make decisions, predict outcomes, learn from mistakes & optimize own performance without being explicitly programmed to do so

Supervised Learning

Learning a function that maps to an output based on the example of input-output pairs. In other words, training a model on data where the outcome is known, for subsequent application to data where the outcome is not known."
"Present labeled examples to learn from. For instance, when we want to be able to predict the selling price of a house in advance in a real estate market, we can get the historical prices of houses and have a supervised learning algorithm successfully figure out how to associate the prices to the house characteristics.
Using the uppercase letter X we intend to use matrix notation, since we can also treat the y as a response vector (technically a column vector) and the X as a matrix containing all values of the feature vectors, each arranged into a separate column of the matrix. . . . building a function that can answer the question about how X can imply y . . . [with] a functional mapping that can translate X values into y without error or with an acceptable margin of error. . . . to determinate a function of the following kind:" (Massaron, pg 24)

Unsupervised Learning

"Looks for previously undetected patterns in a data set with no pre-existing labels and with a minimum of human supervision"
"[P]resent examples without any hint, leaving it to the algorithm to create a label. For instance, when we need to figure out how the groups inside a customer database can be partitioned into similar segments based on their characteristics and behaviors." WGU MSDA

Reinforcement Learning

"how software agents ought to take actions in an environment in order to maximize the notion of cumulative reward"
"[P]resent examples without labels, as in unsupervised learning, but get feedback from the environment as to whether label guessing is correct or not. For instance, when we need software to act successfully in a competitive setting, such as a videogame or the stock market, we can use reinforcement learning. In this case, the software will then start acting in the setting and it will learn directly from its errors until it finds a set of rules that ensure its success." WGU MSDA

Lovely Deep Learning

"[M]achine learning uses multiple layers of simple, adjustable computing elements." (Russell, p. 26)
"Deep learning solves [the] central problem in representation learning by introducing . . . simpler representations . . . [and] enables the computer to build complex concepts out of simpler concepts . . . breaking the desired complicated mapping into a series of nested simple mappings . . . called "hidden [layers]" because their values are not given in the data." (Bengio, p. 5-6)

Artificial Neural Networks

  ↳ A computing system that consist of a number of simple but highly interconnected elements or nodes, called ‘neurons’, which are organized in layers which process information using dynamic state responses to external inputs, an extremely useful algorithm for finding patterns too complex to be manually extracted

Convolutional Neural Networks

  ↳ A class of deep neural networks, most commonly applied to analyzing visual imagery. CNNs are regularized versions of multilayer perceptrons. Multilayer perceptrons usually mean fully connected networks, that is, each neuron in one layer is connected to all neurons in the next layer.

Natural Language Processing

  ↳ Starts with raw text in whatever format available, processes it, extracts relevant features and builds models to accomplish various NLP tasks


Etymology: The word "mathematics" comes from Ancient Greek máthēma (μάθημα), meaning "that which is learnt," "what one gets to know," hence also "study" and "science". Wikipedia

Intimate Linear Algebra

The study of linear equations & geometric transformations using matrices, vectors spaces & determinants.
"Solving for unknowns within a system of linear equations." Mathematical Foundations of Machine Learning

Fundamental Mathematical Objects

Linear Regression


  ↳ The volume scaling factor of the linear transformation described by the matrix

Geometrical Aspects of Linear Algebra

  ↳ Mathematics to used see through to the governing dynamics of the physical universe

Salient Statistics & Probabilities

Statistics is the art of making numerical conjectures about puzzling questions.
↳ Is statistics a field of mathematics? Some say it is not mathematics but the science of data. Whatever you decide, you must embrace it, my Dear Friends.



Generalizing logic to situations with uncertain outcomes & measurements, & incomplete theories; the possible outcomes of events.
"The formalization of probability, combined with the availability of data, led to the emergence of statistics as a field." Artificial Intelligence: A Modern Approach, pg 8

Descriptive Statistics

"Descriptive statistics are brief descriptive coefficients that summarize a given data set, which can be either a representation of the entire or a sample of a population. Descriptive statistics are broken down into measures of central tendency and measures of variability (spread)." Investopedia
Used to describe data; univariate analysis on a single variable or multivariate analysis when looking at two or more variables in the dataset

Inferential Statistics

Putting foundational statistics to use with samplig to find meaningful statistics that will inform us about a population.
"Scholars interested in human society . . . grasped these ideas and found to their surprise that the variation in human characteristics and behavior often displays the same pattern as the error in measurement . . ." (regarding the application of the standard normal distribution to social science in the early 19th century)
- Leonardo Mlodinow, The Drunkard's Walk: How Randomness Rules Our Lives

Statistical Significance Testing


  ↳ So . . . why is it called "Regression", anyway?

Dynamic Calculus

The mathematics of curves, motion and change, calculus is basically very advanced algebra (finding rates & slopes) & geometry (addition to infinity & finding area).

Three Central Problems of Calculus

Fundamental Theorem of Calculus (FTC)

Shows the relationship between differentiation & integration. If a function is integrated & then differentiated, it is back to the original function. Integration & differentiation are inverse to each other.


The derivative function tells how fast & where the function is increasing or decreasing.


The integral of a function models the area under the graph of a function.

Computer Science Mathematics

Absolute Value Inequalities

Scientific Notation

Any number can be written in scientific notation. It involves shifting the decimal place to the left (positive) or right (negative) until the result is a number with only one place before the decimal point & then multiplying by 10 raised to the number of places shifted.

NP Complete Problem

"We call these problems "nondeterministic polynomial" or NP, because you can't give someone a pre-determined set of steps to solve it (unless that someone is a perfect guesser!), but if someone does happen to solve it, they would only need a polynomial number of steps. . . .
  [W]e can find the answer to any NP problem by solving a related problem in this group. The problems in this group are called "NP-complete" (because solving one of them can solve the complete group of NP problems). If we ever found a fast (i.e. polynomial) way to solve any NP-complete problem, we could find fast ways to solve every NP problem. Then we wouldn't have to talk about NP any more, because they would all just be P (polynomial) problems. That's why we call the problem "P=NP". . . .
  [M]athematicians think that NP-complete problems are not P, because so many people have spent so much time thinking about it that if they were, somebody would have found out how by now (because it's usually easier to find a way to do something than to prove that there is no way)." Reddit ELI5

Marvelous Logarithms

Logarithms were the supercomputers their era.   See the Description of the Marvelous Canon of Logarithms by John Napier

Cordially Discrete Mathematics

Branch of mathematics dealing with discrete (distinct & disconnected) or finite sets of elements rather than continuous or infinite sets of elements. The terms discrete & continuous are analogous to the computer science terms digital & analog.
* Brilliant course from Shawn Grooms at freeCodeCamp: Math for Programmers

Nimble Number Theory

The study of the numbers & their properties.

The Integers

Constructs of the Universe

Mind-expanding & ancient wisdom from A Beginner's Guide to Constructing the Universe - Michael S. Schneider

The Monad | One

"The ancient philosophers conceived that the Monad breathes in the void and creates all subsequent numbers"

The Dyad | Two

The Triad | Three

The unity of the circle manifests as a trinity: center or point, radius or line & circumference.

The Tetrad | Four

Three points define a flat surface, but it takes a fourth to define depth, progress to three dimensions and express geometry as volume.

The Pentad | Five

Pentagonal symmetry is the supreme symbol of life.

The Hexad | Six

The Hexad is sometimes symbolized by the "Pythagorean triangle" or "3-4-5 right triangle" made by the ancient method using a twelve-knotted rope. It displays the sequence from one to six (1 - 6): one right angle (1), two unequal angles (2), sides of three (3), four (4), and five (5), and closing an area of six square units (6).

The Heptad | Seven

Seven is perhaps the most venerated number of the Dekad, the number par excellence in the ancient world

The Octad | Eight

Periodic Renewal & the doubling number

The Ennead | Nine

Composed of a trinity of trinities, the number nine represents the principles of the sacred Triad taken to their utmost expression.
The ancient Greeks called nine "the horizon," as it lies at the edge of the numerical shore before the boundless ocean of numbers that repeat in endless cycles the principles of the first nine digits.

The Decad | Ten

The Decad represents the power to generate numbers beyond itself, toward the infinite. Multiplying any number by ten does not change its essential nature but only acts to expand its power.

Positively Brilliant References

About Ryan L Buchanan

I am training as a Software Developer, Data Analyst & Machine Learning Engineer.  I am currently enrolled in the Software Technology program at Ogden-Weber Technical College.  I am also acquiring certifications as an ML Engineer & Algorithmic Trader from Udacity.   I have a Masters in Data Analytics, an MBA & an MS in Instructional Design.  I have working knowledge of C#, R, SQL, HTML, CSS, Javascript, Java and Python programming languages.

I have a multi-displinary background including military intelligence, psychology, linguistics, economics, virtual reality & educational technology.  I have worked abroad for ten years with military, universities & vocational schools.   I have working knowledge of Arabic, Chinese & French.  I am very mobile, able to relocate quickly, adapt easily to diverse working conditions & have a current passport.

I have a passion for mathematics, statistics & artificial intelligence.  I am enthusiastic, highly self-motivated & enjoy presenting informative data to decision makers.  I am eager to work with dynamic teams to create high quality products & services.