Inverse Function Theorem¶
Resources: * Wiki * YouTube * Prof Ghist Math - Inverse Function Theorem * The Infinite Looper - Inv Fun Theorem * Professor Leonard - Fundamental Theorem of Calculus | Derivatives of Inverse Functions
Source:
- Mathematics for Machine Learning - Deisenroth (2019)
- Change of Variables: A Precursor to Normalizing Flow - Rui Shu
- Pattern Recognition and Machine Learning - Bishop (2006)
Often we are faced with the situation where we do not know the distribution of our data. But perhaps we know the distribution of a transformation of our data, e.g. if we know that
Let's say we have
Now, we want to know the probability of
So solving for
Now we see that
Probability Integral Transform
Resources * Brilliant
Does a nice example where they talk about the problems with fat-tailed distributions. * Wiki * CrossValidated * How does the inverse transform method work * Help me understand the quantile (inverse CDF) function * Youtube * Ben Hambert - Intro to Inv Transform Sampling * Mathematical Monk * Intro | General Case | Invertible Case * Code Review - Inverse Transform Sampling * R Markdown - Inverse Transform Sampling * Using Chebyshev - Blog | Code * CDFs - Super powerful way to visualize data and also is uniformly distriuted * Histograms and CDFs - blog * Why We Love CDFS so Much and not histograms - Blog * Boundary Issues * Confidence Band from DKW inequality - code * Make Monotonic - code * Matplotlib example of CDF bins versus theoretical (not smooth) - Code * Alternatives * KDE * Statsmodels Implementation - code | Univariate * KDE vs Histograms - blog * Empirical CDF * The Empirical Distribution Function - blog * Plotting an Empirical CDF In python - blog * Scipy histogram - code * Empirical CDF Function - code * ECDFs - notebook
Derivative of an Inverse Function¶
- MathInsight - Link