# Articles by Aaron Schlegel

### Tukey’s Test for Post-Hoc Analysis

September 7, 2018 |

After a multivariate test, it is often desired to know more about the specific groups to find out if they are significantly different or similar. This step after analysis is referred to as 'post-hoc analysis' and is a major step in hypothesis testing. One common and popular method of post-hoc ... ### Kruskal-Wallis One-Way Analysis of Variance of Ranks

September 3, 2018 |

The Kruskal-Wallis test extends the Mann-Whitney-Wilcoxon Rank Sum test for more than two groups. The test is nonparametric similar to the Mann-Whitney test and as such does not assume the data are normally distributed and can, therefore, be used when the assumption of normality is violated. This example will employ ... [Read more...]

### Combined Linear Congruential Generators with R

September 7, 2017 |

Part of 3 in the series Random Number GenerationCombined linear congruential generators, as the name implies, are a type of PRNG (pseudorandom number generator) that combine two or more LCGs (linear congruential generators). The combination of two or more LCGs into one random number generator can result in a marked... The ... ### Multiplicative Congruential Generators in R

August 31, 2017 |

Part 2 of 2 in the series Random Number GenerationMultiplicative congruential generators, also known as Lehmer random number generators, is a type of linear congruential generator for generating pseudorandom numbers in . The multiplicative congruential generator, often abbreviated as MLCG or MCG, is defined as a recurrence relation similar to... The post Multiplicative ... [Read more...]

### Linear Congruential Generator in R

August 24, 2017 |

Part of 1 in the series Random Number GenerationA Linear congruential generator (LCG) is a class of pseudorandom number generator (PRNG) algorithms used for generating sequences of random-like numbers. The generation of random numbers plays a large role in many applications ranging from cryptography to Monte Carlo methods. Linear congruential... The ... [Read more...]

### Simpson’s Rule for Approximating Definite Integrals in R

August 17, 2017 |

Part 9 of 9 in the series Numerical AnalysisSimpson’s rule is another closed Newton-Cotes formula for approximating integrals over an interval with equally spaced nodes. Unlike the trapezoidal rule, which employs straight lines to approximate a definite integral, Simpson’s rule uses the third Lagrange polynomial, to approximate the definite integral... ... ### The Trapezoidal Rule of Numerical Integration in R

August 10, 2017 |

Part of 8 in the series Numerical AnalysisThe Trapezoidal Rule is another of Closed Newton-Cotes formulas for approximating the definite integral of a function. The trapezoidal rule is so named due to the area approximated under the integral representing a trapezoid. Although there exist much more accurate quadrature methods, the... The ... ### Numerical Differentiation with Finite Differences in R

August 3, 2017 |

Part 1 of 7 in the series Numerical AnalysisNumerical differentiation is a method of approximating the derivative of a function at particular value . Often, particularly in physics and engineering, a function may be too complicated to merit the work necessary to find the exact derivative, or the function itself... The post Numerical ... ### Divided Differences Method of Polynomial Interpolation

July 27, 2017 |

Part of 6 in the series Numerical AnalysisThe divided differences method is a numerical procedure for interpolating a polynomial given a set of points. Unlike Neville’s method, which is used to approximate the value of an interpolating polynomial at a given point, the divided differences method constructs the interpolating polynomial... ... ### Neville’s Method of Polynomial Interpolation

July 19, 2017 |

Part 1 of 5 in the series Numerical AnalysisNeville’s method evaluates a polynomial that passes through a given set of and points for a particular value using the Newton polynomial form. Neville’s method is similar to a now-defunct procedure named Aitken’s algorithm and is based on the divided differences ... [Read more...]

### Lagrangian Polynomial Interpolation with R

July 13, 2017 |

Part 2 of 4 in the series Numerical AnalysisPolynomial interpolation is the method of determining a polynomial that fits a set of given points. There are several approaches to polynomial interpolation, of which one of the most well known is the Lagrangian method. This post will introduce the Lagrangian method... The post ... ### Set Theory Ordered Pairs and Cartesian Product with R

July 6, 2017 |

Part 5 of 5 in the series Set TheoryOrdered and Unordered Pairs A pair set is a set with two members, for example, , which can also be thought of as an unordered pair, in that . However, we seek a more a strict and rich object that tells us... The post Set Theory ... [Read more...]

### Algebra of Sets in R

June 29, 2017 |

Part 4 of 4 in the series Set TheoryThe set operations, union and intersection, the relative complement and the inclusion relation (subsets) are known as the algebra of sets. The algebra of sets can be used to find many identities related to set relations that will be discussed later. We... The post ... ### Set Theory Arbitrary Union and Intersection Operations with R

June 22, 2017 |

Part 3 of 3 in the series Set TheoryThe union and intersection set operations were introduced in a previous post using two sets, and . These set operations can be generalized to accept any number of sets. Arbitrary Set Unions Operation Consider a set of infinitely many sets: It would... The post Set ... [Read more...]

### Set Operations Unions and Intersections in R

June 15, 2017 |

Part 2 of 2 in the series Set TheoryThe set operations of unions and intersections should ring a bell for those who’ve worked with relational databases and Venn Diagrams. The ‘union’ of two of sets and represents a set that comprises all members of and (or both). One of the... The ... ### Introduction to Set Theory and Sets with R

June 8, 2017 |

Sets define a ‘collection’ of objects, or things typically referred to as ‘elements’ or ‘members.’ The concept of sets arises naturally when dealing with any collection of objects, whether it be a group of numbers or anything else. Conceptually, the following examples can be defined as a ‘set’: {1, 2,... The post ... [Read more...]

### QR Decomposition with Householder Reflections

April 13, 2017 |

The more common approach to QR decomposition is employing Householder reflections rather than utilizing Gram-Schmidt. In practice, the Gram-Schmidt procedure is not recommended as it can lead to cancellation that causes inaccuracy of the computation of , which may result in a non-orthogonal matrix. Householder reflections are another method of... The ... [Read more...]

### QR Decomposition with the Gram-Schmidt Algorithm

March 23, 2017 |

QR decomposition is another technique for decomposing a matrix into a form that is easier to work with in further applications. The QR decomposition technique decomposes a square or rectangular matrix, which we will denote as , into two components, , and . Where is an orthogonal matrix, and is... The post QR ... [Read more...]

### Hierarchical Clustering Nearest Neighbors Algorithm in R

March 9, 2017 |

Hierarchical clustering is a widely used and popular tool in statistics and data mining for grouping data into ‘clusters’ that exposes similarities or dissimilarities in the data. There are many approaches to hierarchical clustering as it is not possible to investigate all clustering possibilities. One set of approaches to hierarchical... ... 