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Simple logistic regression is a statistical method that can be used for binary classification problems. In the context of image processing, this could mean identifying whether a given image belongs to a particular class ($y=1$) or not ($y=0$), e.g. "cat" or "not cat". A logistic regression algorithm takes as its input a feature vector $\boldsymbol{x}$ and outputs a probability, $\hat{y} = P(y=1|\boldsymbol{x})$, that the feature vector represents an object belonging to the class. For images, the feature vector might be just the values of the red, green and blue (RGB) channels for each pixel in the image: a one-dimensional array of $n_x = n_\mathrm{height} \times n_\mathrm{width} \times 3$ real numbers formed by flattening the three-dimensional array of pixel RGB values. A logistic regression model is so named because it calculates $\hat{y} = \sigma(z)$ where $$ \sigma(z) = \frac{1}{1+\mathrm{e}^{-z}} $$ is the logistic function and $$ z = \boldsymbol{w}^T\boldsymbol{x} + b, $$ for a set of parameters, $\boldsymbol{w}$ and $b$. $\boldsymbol{w}$ is a $n_x$-dimensional vector (one component for each component of the feature vector) and b is a constant "bias".

For a basic logistic regression image classification exercise, it is usually desirable for the training and test images to have the same dimensions and images are often chosen to be square. However, many datasets provide images with different sizes and aspect ratios for more advanced classification algorithms.

This script demonstrates the relaxation of an ensemble of colliding particles towards the equilibrium, Maxwell-Boltzmann distribution of their speeds. Unlike this previous post, the collision detection and dynamics are handled using NumPy arrays without explicit python loops (except over collision pairs), which improves the performance greatly.

The spherical harmonics are a set of special functions defined on the surface of a sphere that originate in the solution to Laplace's equation, $\nabla^2f=0$. Because they are basis functions for irreducible representations of SO(3), the group of rotations in three dimensions, they appear in many scientific domains, in particular as the angular part of the wavefunctions of atoms (atomic orbitals). They are described in terms of an integer *degree* $l=0,1,2,\ldots$ and *order* $m=-l,-l+1,\ldots,l$.

The last episode of 3Blue1Brown's "lockdown math" series posed the problem: given two numbers, $x$ and $y$, chosen randomly from the uniform distribution between 0 and 1, what is the proportion that their ratio, when rounded down to the nearest integer, is even? That is, what is $P(\lfloor \frac{x}{y} \rfloor \mod 2 = 0)$?