## A Seriously Slow Fibonacci Function

I recently wrote an article which was ostensibly about the Fibonacci series but was really about optimization techniques. I wanted to follow up on its (extremely moderate) success by going in the exact opposite direction: by writing a Fibonacci function which is as slow as possible.
This is not as easy as it sounds: any program can trivially be made slower, but this is boring. How can we make it slow in a fair and interesting way?

## ML From Scratch, Part 5: Gaussian Mixture Models

Consider the following motivating dataset:
Unlabled Data
It is apparent that these data have some kind of structure; which is to say, they certainly are not drawn from a uniform or other simple distribution. In particular, there is at least one cluster of data in the lower right which is clearly separate from the rest. The question is: is it possible for a machine learning algorithm to automatically discover and model these kinds of structures without human assistance?

## Adaptive Basis Functions

Today, let me be vague. No statistics, no algorithms, no proofs. Instead, we’re going to go through a series of examples and eyeball a suggestive series of charts, which will imply a certain conclusion, without actually proving anything; but which will, I hope, provide useful intuition.
The premise is this:
For any given problem, there exists learned featured representations which are better than any fixed/human-engineered set of features, even once the cost of the added parameters necessary to also learn the new features into account.

## ML From Scratch, Part 4: Decision Trees

So far in this series we’ve followed one particular thread: linear regression -> logistic regression -> neural network. This is a very natural progression of ideas, but it really represents only one possible approach. Today we’ll switch gears and look at a model with completely different pedigree: the decision tree, sometimes also referred to as Classification and Regression Trees, or simply CART models. In contrast to the earlier progression, decision trees are designed from the start to represent non-linear features and interactions.

## A Fairly Fast Fibonacci Function

A common example of recursion is the function to calculate the \(n\)-th Fibonacci number:
def naive_fib(n):
if n < 2:
return n
else:
return naive_fib(n-1) + naive_fib(n-2)
This follows the mathematical definition very closely but it’s performance is terrible: roughly \(\mathcal{O}(2^n)\). This is commonly patched up with dynamic programming. Specifically, either the memoization:
from functools import lru_cache
@lru_cache(100)
def memoized_fib(n):
if n < 2:
return n
else:
return memoized_fib(n-1) + memoized_fib(n-2)
or tabulation:

## ML From Scratch, Part 3: Backpropagation

In today’s installment of Machine Learning From Scratch we’ll build on the logistic regression from last time to create a classifier which is able to automatically represent non-linear relationships and interactions between features: the neural network. In particular I want to focus on one central algorithm which allows us to apply gradient descent to deep neural networks: the backpropagation algorithm. The history of this algorithm appears to be somewhat complex (as you can hear from Yann LeCun himself in this 2018 interview) but luckily for us the algorithm in its modern form is not difficult - although it does require a solid handle on linear algebra and calculus.

## ML From Scratch, Part 2: Logistic Regression

In this second installment of the machine learning from scratch we switch the point of view from regression to classification: instead of estimating a number, we will be trying to guess which of 2 possible classes a given input belongs to. A modern example is looking at a photo and deciding if its a cat or a dog.
In practice, its extremely common to need to decide between \(k\) classes where \(k > 2\) but in this article we’ll limit ourselves to just two classes - the so-called binary classification problem - because generalizations to many classes are usually both tedious and straight-forward.

## ML From Scratch, Part 1: Linear Regression

To kick off this series, will start with something simple yet foundational: linear regression via ordinary least squares.
While not exciting, linear regression finds widespread use both as a standalone learning algorithm and as a building block in more advanced learning algorithms. The output layer of a deep neural network trained for regression with MSE loss, simple AR time series models, and the “local regression” part of LOWESS smoothing are all examples of linear regression being used as an ingredient in a more sophisticated model.

## ML From Scratch, Part 0: Introduction

Motivation
“As an apprentice, every new magician must prove to his own satisfaction, at least once, that there is truly great power in magic.” - The Flying Sorcerers, by David Gerrold and Larry Niven
How do you know if you really understand something? You could just rely on the subjective experience of feeling like you understand. This sounds plausible - surely you of all people should know, right? But this runs head-first into in the Dunning-Kruger effect.

## Visualizing Multiclass Classification Results

Introduction Visualizing the results of a binary classifier is already a challenge, but having more than two classes aggravates the matter considerably.
Let’s say we have $k$ classes. Then for each observation, there is one correct prediction and $k-1$ possible incorrect prediction. Instead of a $2 \times 2$ confusion matrix, we have a $k^2$ possibilities. Instead of having two kinds of error, false positives and false negatives, we have $k(k-1)$ kinds of errors.

## Craps Variants

Craps is a suprisingly fair game. I remember calculating the probability of winning craps for the first time in an undergraduate discrete math class: I went back through my calculations several times, certain there was a mistake somewhere. How could it be closer than $\frac{1}{36}$?
(Spoiler Warning If you haven’t calculated these odds for yourself then you may want to do so before reading further. I’m about to spoil it for you rather thoroughly in the name of exploring a more general case.

## Complex Numbers in R, Part II

This post is part of a series on complex number functionality in the R programming language. You may want to read Part I before continuing if you are not already comfortable with the basics.
In Part I of this series, we dipped our toes in the water by explicitly creating some complex numbers and showing how they worked with the most basic mathematical operators, functions, and plots.
In this second part, we’ll take a more in-depth look at some scenarios where complex numbers arise naturally – where they are less of a choice an more of a necessity.

## Complex Numbers in R, Part I

R, like many scientific programming languages, has first-class support for complex numbers. And, just as in most other programming languages, this functionality is ignored by the vast majority of users.
Yet complex numbers can often offer surprisingly elegant formulations and solutions to problems. I want to convince you that familiarizing yourself with R’s excellent complex number functionality is well worth the effort and will pay off in two different ways: first by showing you how they are so amazingly useful you’ll want to go out of your way to use them, and then by showing you how they are so common and fundamental to modern analysis that you couldn’t avoid them if you wanted to.

## So, Apparently I'm an iPad Developer Now

Last week my boss stopped by and dropped a brand spanking new iPad on my desk. "Make our application work on this," he commanded. "You have two days before we demo it at the trade show." Madness? No, these are web apps! You see, for the last couple years we've been working exclusively on AJAX applications: web pages stuffed with so much JavaScript they look and feel like desktop apps. It's harder than writing desktop software, but if you pull it off you get an application that can be run anywhere, instantly.

## Deep Copy in JavaScript

Update 2017-10-23: This article and code library have not kept up with the rapidly changing JavaScript landscape and are now hopelessly out of date. First came non-enumerable properties, and with ES2015 came the introduction of classes, proxies, symbols, and anonymous functions, all of which break the below logic. I'm afraid I no longer know how to fully copy the full menagerie of JavaScript objects while preserving relative references, and it's quite possible that no one else knows either.

## Semantic Code

se-man-tic (si-man’tik) adj. 1. Of or relating to meaning, especially meaning in language.
Programming destroys meaning. When we program, we first replace concepts with symbols and then replace those symbols with arbitrary codes — that’s why it’s called coding.
At its worst programming is write-only: the program accomplishes a task, but is incomprehensible to humans. See, for example, the story of Mel. Such a program is correct, yet at the same time meaningless.