Daesoo Lee's Blog

Mathematics

• Wavelet Transform

  Mathematics 2022. 1. 24. 22:56

  Scalogram is the spectral graph by wavelet transform. Good study materials are: https://www.youtube.com/watch?v=ZnmvUCtUAEE https://www.youtube.com/watch?v=MX7ymkYGiZ0 Wavelet transform library: https://github.com/OverLordGoldDragon/ssqueezepy GitHub - OverLordGoldDragon/ssqueezepy: Synchrosqueezing, wavelet transforms, and time-frequency analysis in Python Synchrosqueezing, wavelet transforms, ..

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• Fourier Series

  Mathematics 2021. 4. 28. 11:08

  Fourier Series Explained [1] BRILLIANT, "Fourier Series" (link) [2] White Whale, "푸리에 급수(Fourier Series) 수식 예제" (link) Fourier Series used in N-BEATS (i.e., NN-based Fourier Series) B. Oreshkin et al., 2019, "N-BEATS: Neural basis expansion analysis for interpretable time series forecasting" (link)

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• Odds Ratios and Log(Odds Ratios)

  Mathematics 2021. 3. 23. 09:26

  Odds The chance of my team winning is 0.33, then the chance of my team losing is 0.67. Its odds are computed as $$odds = \frac{0.33}{0.67} = 0.493$$ Odds Ratio "Odds ratio" is different from the odds. It is a ratio of odds. Log(Odds Ratio) If the denominator is larger than the numerator the odds ratio will go from 0 to 1. If the numerator is larger than the denominator, then the odds ratio wil..

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• PCA using Python

  Mathematics 2021. 2. 2. 19:28

  import os import numpy as np from sklearn.decomposition import TruncatedSVD from utils.helper_03 import * # set random-seed np.random.seed(1) # Generate data x = np.arange(-0.5, 0.5, 0.01) y = x + (np.random.rand(len(x))-0.5)*0.2 plt.figure(figsize=(5, 5)) plt.plot(x, y, 'o'); plt.grid(); target_arr = np.stack((x, y)).T # (100x2) u, s, vh = np.linalg.svd(target_arr, full_matrices=True) first_com..

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• Conditional Expectation

  Mathematics 2021. 1. 27. 17:22

  $$E(X | Y)$$ The above equation means the conditional expected value of a random variable $X$ given a condition, $Y$. Example The following equation is from the BYOL paper: $$q^{*} = \mathbb{E}[z_{\xi}^{\prime} | z_{\theta}]$$ It means the conditional expected value of $z_{\xi}^{\prime}$ given $z_{\theta}$.

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• Perpendicular Distance Between a Hyperplane and a Point

  Mathematics 2021. 1. 13. 17:39

  Hyperplane A hyperplane in two dimensions: a line e.g. $\beta_0 + \beta_1 X_1 + \beta_2 X_2 = 0$ A hyperplane in two dimensions: a plane e.g. $\beta_0 + \beta_1 X_1 + \beta_2 X_2 + \beta_3 X_3 = 0$ Perpendicular Distance Between a Hyperplnae and a Point in Two Dimensions Let a point be $(x_1, x_2)$, and the hyperplane be $\beta_0 + \beta_1 X_1 + \beta_2 X_2 = 0$ (line). Then the shortest distanc..

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• Eigendecomposition (spectral decomposition), Singular Value Decomposition (SVD)

  Mathematics 2021. 1. 7. 19:23

  Eigendecomposition (Spectral Decomposition) Fundamental theory of eigenvalues and eigenvectors The definitions of the eigenvalue and eigenvector are expressed in the following equation: $$A \boldsymbol{v} = \lambda \boldsymbol{v}$$ where a non-zero vector $\boldsymbol{v}$ is an eigenvector of a square $N \times N$ matrix $A$ and $\lambda$ is a scalar, termed the eigenvalue corresponding to $\b..

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• Multivariate Gaussian Distribution

  Mathematics 2021. 1. 7. 17:05

  First, a one-dimensional Gaussian distribution is introduced, and then the multivariate Gaussian distribution is introduced. One-dimensional Gaussian Distribution $N(\mu, \sigma^2) = \frac{1}{\sqrt{2 \pi \sigma^2}} \exp{(-\frac{1}{2 \sigma^2} (x - \mu)^2)}$ The corresponding mean and standard deviation are: $\mu = \frac{1}{N} \sum_i{x_i}$ $\sigma^2 = \frac{1}{N} \sum_i{(x_i - \mu)}$ Multivariate..

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