The Rigor of Fuzzy Sets

The Rigor of Fuzzy Sets

Perhaps "fuzzy set theory", "fuzzy arithmetic", and "fuzzy rules" could have been named something a bit less mock-worthy. The word "fuzzy" has English synonyms "blurred", "unfocused", and the worst "ill-defined". However, fuzzy set theory is anything but fuzzy. Maybe cuddly and fun, but certainly not ill-defined. We'll take a look in this post at what…

Interventions, not Anomalies

Interventions, not Anomalies

Anomaly Detection is becoming almost universally considered a "hot topic" across industries and business fields. A venture capitalist wants to know if a potential investment is an anomaly that will make billions at the time of IPO. A credit card company wants to know if a particular transaction is anomalous, as it could be potentially fraudulent.…

Defining An Anomaly

Defining An Anomaly

Everyone speaks about anomaly detection and its importance, and for good reason. Excessive credit card charges can signal a stolen card. A large change in errors can signal an impending hard drive failure. Unusually low crop yields can signal a pest infestation. In order to detect these strange occurrences, there are dozens of algorithms with plenty…

Probabilistic Ways to Represent the Lifetime of an Object

Probabilistic Ways to Represent the Lifetime of an Object

Every item, system, person, or animal has a lifetime. For people and animals, we typically just measure the lifetime in years, but we have other options for items and systems. We can measure airplane reliability in flight hours (hours actually flown), or stress test a manufacturing tool in cycles. Regardless of the units we use,…

Vertical Dependency in Sequences of Categorical Random Variables

Vertical Dependency in Sequences of Categorical Random Variables

For the full text, including proofs, download the pdf here. Abstract This paper develops a more general theory of sequences of dependent categorical random variables, extending the works of Korzeniowski (2013) and Traylor (2017) that studied first-kind dependency in sequences of Bernoulli and categorical random variables, respectively. A more natural form of dependency, sequential dependency, is…

Summation Chains of Sequences Part 3: Sequence Chains from Linear Functions

Summation Chains of Sequences Part 3: Sequence Chains from Linear Functions

For the full paper, which includes proofs, click here. Abstract (Editor's note:) This paper represents the third installment of a masters thesis by Jonathan Johnson. The first two can be found here and here. This paper continues the development of the theory of summation chains of sequences. Since summation chains are doubly infinite, it's important to know how little…

Summation Chains of Sequences Part 2: Relationships between Sequences via Summation Chains

Summation Chains of Sequences Part 2: Relationships between Sequences via Summation Chains

For the full paper with proofs,click here. Abstract Editor's note: This paper represents the second installment of a masters thesis by Jonathan Johnson. This paper continues the development of the theory of summation chains of sequences. The concept of sum-related is defined: two sequences are sum-related if one sequence appears in the summation chain of the…

Summation Chains of Sequences Part 1: Introduction, Generation, and Key Definitions

Summation Chains of Sequences Part 1: Introduction, Generation, and Key Definitions

For the full paper, which includes all proofs, download the pdf here Abstract (Editor's note:) This paper represents the first installment of a masters thesis by Jonathan Johnson. This work introduces the notion of summation chains of sequences. It examines the sequence of sequences generated by partial sums and differences of terms in each level of…

A Generalized Multinomial Distribution from Dependent Categorical Random Variables

A Generalized Multinomial Distribution from Dependent Categorical Random Variables

For the full paper, which includes all proofs, download the pdf  here. Abstract Categorical random variables are a common staple in machine learning methods and other applications across disciplines. Many times, correlation within categorical predictors exists, and has been noted to have an effect on various algorithm effectiveness, such as feature ranking and random forests.…