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.

## Analyzing the Next Decade of Earth Close-Approaching Objects with nasapy

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.

## Plot Earth Fireball Impacts with nasapy, pandas and folium

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

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 for Calculating Limits and Indeterminate Forms

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

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.

## Indefinite Integrals

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, \(\int f(x) \space dx = F^\prime(x)\).

## Substitution Rule

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

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 for Finding Equation Roots

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.

## Implicit Differentiation

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 of Differentiation

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.

## Limit of a Function

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.

## 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.

## 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.

## Continuous Functions

In this post, we explore the definition of a continuous function and introduce several examples with Python code.

## 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.

## 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.## 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.

## 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.

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### Recent Posts

- Get All NASA Astronomy Pictures of the Day from 2019
- Analyzing the Next Decade of Earth Close-Approaching Objects with nasapy
- Plot Earth Fireball Impacts with nasapy, pandas and folium
- Integration by Parts
- L'Hospital's Rule for Calculating Limits and Indeterminate Forms
- The Fundamental Theorem of Calculus