1. Derivatives of Logarithmic Functions

    Implicit differentiation can also be employed to find the derivatives of logarithmic functions, which are of the form \(y = \log_a{x}\). In this post, we explore several derivatives of logarithmic functions and also prove some commonly used derivatives. The symbolic computation library SymPy is also employed to verify our answers.

  2. Product, Quotient and Power Rules of Differentiation

    Several rules exist for finding the derivatives of functions with several components such as \(x \space sin \space x\). With these rules and the chain rule, which will be explored later, any derivative of a function can be found (assuming they exist). There are five rules that help simplify the computation of derivatives, of which each will be explored in turn. We will also take advantage of SymPy to perform symbolic computations to confirm our results.

  3. Tukey's Test for Post-Hoc Analysis

    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 analysis is Tukey's Test. The test is known by several different names. Tukey's test compares the means of all treatments to the mean of every other treatment and is considered the best available method in cases when confidence intervals are desired or if sample sizes are unequal.

    Tagged as : R statistics
  4. Kruskal-Wallis One-Way Analysis of Variance of Ranks

    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 the Kruskal-Wallis test on the PlantGrowth dataset as used in previous examples. Although the data appear to be approximately normally distributed as seen before, the Kruskal-Wallis test performs just as well as a parametric test.

    Tagged as : R statistics
  5. Quadratic Discriminant Analysis of Several Groups

    Quadratic discriminant analysis for classification is a modification of linear discriminant analysis that does not assume equal covariance matrices amongst the groups (\(\Sigma_1, \Sigma_2, \cdots, \Sigma_k\)). Similar to LDA for several groups, quadratic discriminant analysis for several groups classification seeks to find the group that maximizes the quadratic classification function and assign the observation vector \(y\) to that group.

  6. Quadratic Discriminant Analysis of Two Groups

    LDA assumes the groups in question have equal covariance matrices (\(\Sigma_1 = \Sigma_2 = \cdots = \Sigma_k\)). Therefore, when the groups do not have equal covariance matrices, observations are frequently assigned to groups with large variances on the diagonal of its corresponding covariance matrix (Rencher, n.d., pp. 321). Quadratic discriminant analysis is a modification of LDA that does not assume equal covariance matrices amongst the groups. In quadratic discriminant analysis, the respective covariance matrix \(S_i\) of the \(i^{th}\) group is employed in predicting the group membership of an observation, rather than the pooled covariance matrix \(S_{p1}\) in linear discriminant analysis.

  7. Linear Discriminant Analysis for the Classification of Several Groups

    Similar to the two-group linear discriminant analysis for classification case, LDA for classification into several groups seeks to find the mean vector that the new observation \(y\) is closest to and assign \(y\) accordingly using a distance function. The several group case also assumes equal covariance matrices amongst the groups (\(\Sigma_1 = \Sigma_2 = \cdots = \Sigma_k\)).

  8. Linear Discriminant Analysis for the Classification of Two Groups

    In this post, we will use the discriminant functions found in the first post to classify the observations. We will also employ cross-validation on the predicted groups to get a realistic sense of how the model would perform in practice on new observations. Linear classification analysis assumes the populations have equal covariance matrices (\(\Sigma_1 = \Sigma_2\)) but does not assume the data are normally distributed.

  9. Discriminant Analysis for Group Separation

    Discriminant analysis assumes the two samples or populations being compared have the same covariance matrix \(\Sigma\) but distinct mean vectors \(\mu_1\) and \(\mu_2\) with \(p\) variables. The discriminant function that maximizes the separation of the groups is the linear combination of the \(p\) variables. The linear combination denoted \(z = a′y\) transforms the observation vectors to a scalar. The discriminant functions thus take the form:

  10. Discriminant Analysis of Several Groups

    Discriminant analysis is also applicable in the case of more than two groups. In the first post on discriminant analysis, there was only one linear discriminant function as the number of linear discriminant functions is \(s = min(p, k − 1)\), where \(p\) is the number of dependent variables and \(k\) is the number of groups. In the case of more than two groups, there will be more than one linear discriminant function, which allows us to examine the groups' separation in more than one dimension.

  11. Building a Poetry Database in PostgreSQL with Python, poetpy, pandas and Sqlalchemy

    In this example, we walk through a sample use case of extracting data from a database using an API and then structuring that data in a cohesive manner that allows us to create a relational database that we can then query with SQL statements. The database we will create with the extracted data will use Postgresql. The Python libraries that will be used in this example are poetpy, a Python wrapper for the PoetryDB API written by yours truly, pandas for transforming and cleansing the data as needed, and sqlalchemy for handling the SQL side of things.

  12. Introduction to Rpoet

    The Rpoet package is a wrapper of the PoetryDB API, which enables developers and other users to extract a vast amount of English-language poetry from nearly 130 authors. The package provides a simple R interface for interacting and accessing the PoetryDB database. This vignette will introduce the basic functionality of Rpoet and some example usages of the package.

    Tagged as : R APIs poetry
  13. Calculating and Performing One-way Analysis of Variance (ANOVA)

    ANOVA, or Analysis of Variance, is a commonly used approach to testing a hypothesis when dealing with two or more groups. One-way ANOVA, which is what will be explored in this post, can be considered an extension of the t-test when more than two groups are being tested. The factor, or categorical variable, is often referred to as the 'treatment' in the ANOVA setting. ANOVA involves partitioning the data's total variation into variation between and within groups. This procedure is thus known as Analysis of Variance as sources of variation are examined separately.

    Tagged as : R statistics
  14. Introduction to poetpy

    The poetpy library is a Python wrapper for the PoetryDB API. The library provides a Pythonic interface for interacting with and extracting information from the PoetryDB database. In this introductory example, we will explore some of the basic functionality of the poetpy library for interacting with the PoetryDB database.

    Tagged as : Python APIs poetpy
  15. Computing Working-Hotelling and Bonferroni Simultaneous Confidence Intervals

    There are two procedures for forming simultaneous confidence intervals, the Working-Hotelling and Bonferroni procedures. Each estimates intervals of the mean response using a family confidence coefficient. The Working-Hotelling coefficient is defined by \(W\) and Bonferroni \(B\). In practice, it is recommended to perform both procedures to determine which results in a tighter interval. The Bonferroni method will be explored first.

    Tagged as : R statistics
  16. PetfindeR, R Wrapper for the Petfinder API, Introduction Part One

    The goal of the PetfindeR package is to provide a simple and straightforward interface for interacting with the Petfinder API through R. The Petfinder database contains approximately 300,000 adoptable pet records and 11,000 animal welfare organization records, which makes it a handy and valuable source of data for those in the animal welfare community. However, the outputs from the Petfinder API are in messy JSON format and thus it makes it more time-consuming and often frustrating to coerce the output data into a form that is workable with R.

    Tagged as : R PetfindeR APIs

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