Matrix norms are an extension of vector norms to matrices and are used to define a measure of distance on the space of a matrix. The most commonly occurring matrix norms in matrix analysis are the Frobenius, \(L_1\), \(L_2\) and \(L_\infty\) norms. The following will investigate these norms, along with some Python implementations of the calculation of the matrix norm.
Similar to the real line concerning two real scalars and the distance between them, vector norms allow us to get a sense of the distance or magnitude of a vector. In fact, a vector of length one is simply a scalar. Norms are often used in regularization methods and other machine learning procedures, as well as many different matrix and vector operations in linear algebra.
The Games-Howell test is a nonparametric post hoc analysis approach for performing multiple comparisons for two or more sample populations. The Games-Howell test is somewhat similar to Tukey's post hoc test. Still, unlike Tukey's test, it does not assume homogeneity of variances or equal sample sizes. Thus, the Games-Howell test can be applied in settings when the assumptions of Tukey's test do not hold. The Games-Howell test and Tukey's test will often report similar results with data that is assumed to have equal variance and equal sample sizes.
Bartlett's test, developed by Maurice Stevenson Bartlett, is a statistical procedure for testing if \(k\) population samples have equal variances. Equality of variances in population samples is assumed in commonly used comparison of means tests, such as Student's t-test and analysis of variance. Therefore, a procedure such as Bartlett's test can be conducted to accept or reject the assumption of equal variances across group samples.
Levene's test is a statistical procedure for testing equality of variances (also sometimes called homoscedasticity or homogeneity of variances) between two or more sample populations. Several commonly used statistical routines such as the t-test and analysis of variance assume the populations have equal variances. Therefore Levene's test is often employed to test this assumption before performing these tests.
The Van der Waerden test is a non-parametric test for testing the hypothesis that \(k\) sample distribution functions are equal. Van der Waerden's test is similar to the Kruskal-Wallis one-way analysis of variance test in that it converts the data to ranks and then to standard normal distribution quantiles. The ranked data is known as the 'normal scores'. Hence, the Van der Waerden test is sometimes referred to as a 'normal scores test'.
Due to the number of different extensions and options on possible underlying assets, a generalized Black-Scholes model was created to simplify computations by significantly reducing the number of equations. In this post, we will explore several of the Black-Scholes option pricing models for different underlying assets and then introduce the generalized Black-Scholes pricing formula.
The Austin Animal Center provides its animal intake and outcome datasets on Socrata. When an animal is taken into the shelter, it is given a unique identifier that is also used in the outcomes dataset. We have already investigated and performed exploratory data analysis on the Austin Animal Center's intakes and animal outcomes individually and found several interesting facets of information. In this analysis, we merge the intakes and outcomes dataset using pandas to enable us to perform exploratory data analysis on the merged data. With the data merged, we will be able to explore in more depth the transition from intake to outcome.
In this example, we will walk through a possible use case of the nasapy library by extracting the next 10 years of close-approaching objects to Earth identified by NASA's Jet Propulsion Laboratory's Small-Body Database. The close_approach method of the nasapy library allows one to access the JPL SBDB to extract data related to known meteoroids and asteroids within proximity to Earth. Setting the parameter return_df=True automatically coerces the returned JSON data into a pandas DataFrame.
In this example, we will go through one possible use of the nasapy library by extracting a decade of fireball data from the NASA API and visualizing it on a map. Using the nasapy library, we can extract the last 10 years of fireball data as a pandas DataFrame by calling the fireballs function. The fireballs method does not require authentication to the NASA API, so we can go straight to getting the data.
Integration by parts is another technique for simplifying integrands. As we saw in previous posts, each differentiation rule has a corresponding integration rule. In the case of integration by parts, the corresponding differentiation rule is the Product Rule. This post will introduce the integration by parts formula as well as several worked-through examples.
L'Hospital's Rule allows us to simplify the evaluation of limits that involve indeterminate forms. An indeterminate form is defined as a limit that does not give enough information to determine the original limit. In this post, we explore several examples of indeterminate forms and how to calculate their limits using L'Hospital's Rule. We also leverage Python and SymPy to verify our answers.
The Fundamental Theorem of Calculus is a theorem that connects the two branches of calculus, differential and integral, into a single framework. We saw the computation of antiderivatives previously is the same process as integration; thus we know that differentiation and integration are inverse processes. The Fundamental Theorem of Calculus formalizes this connection. The theorem is given in two parts, which we will explore in turn along with Python examples to verify our results.
As we noted in the previous sections on the Fundamental Theorem of Calculus and Antiderivatives, indefinite integrals are also called antiderivatives and are the same process. Indefinite integrals are expressed without upper and lower limits on the integrand, the notation \(\int f(x)\) is used to denote the function as an antiderivative of \(F\). Therefore, \(\int f(x) \space dx = F^\prime(x)\).
The Substitution Rule is another technique for integrating complex functions and is the corresponding process of integration as the chain rule is to differentiation. The Substitution Rule is applicable to a wide variety of integrals, but is most performant when the integral in question is similar to forms where the Chain Rule would be applicable. In this post, the Substitution Rule is explored with several examples. Python and SymPy are also used to verify our results.
Antiderivatives, which are also referred to as indefinite integrals or primitive functions, is essentially the opposite of a derivative (hence the name). More formally, an antiderivative \(F\) is a function whose derivative is equivalent to the original function \(f\), or stated more concisely: \(F^\prime(x) = f(x)\). The Fundamental Theorem of Calculus defines the relationship between differential and integral calculus. We will see later that an antiderivative can be thought of as a restatement of an indefinite integral. Therefore, the discussion of antiderivatives provides a nice segue from the differential to integral calculus. The process of finding an antiderivative of a function is known as antidifferentiation and is the reverse of differentiating a function.
Newton's method, also known as Newton-Raphson, is an approach for finding the roots of nonlinear equations and is one of the most common root-finding algorithms due to its relative simplicity and speed. The root of a function is the point at which \(f(x) = 0\). This post explores the how Newton's Method works for finding roots of equations and walks through several examples with SymPy to verify our answers.
An implicit function defines an algebraic relationship between variables. In this post, implicit differentiation is explored with several examples including solutions using Python code.
The chain rule is a powerful and useful derivation technique that allows the derivation of functions that would not be straightforward or possible with the only the previously discussed rules at our disposal. The rule takes advantage of the "compositeness" of a function. In this post, we will explore several examples of the chain rule and will also confirm our results using the SymPy symbolic computation library.
A function limit, roughly speaking, describes the behavior of a function around a specific value. Limits play a role in the definition of the derivative and function continuity and are also used in the convergent sequences. In this post, we will explore the definition of a function limit and some other limit laws using examples with Python.
- Linear Algebra
- Machine Learning