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pclasso

Refer to pcLasso: a new method for sparse regression

set.seed(1234)
n = 100; p = 10
X = matrix(rnorm(n * p), nrow = n)
y = rnorm(n)
library(pcLasso)
fit <- pcLasso(X, y, theta = 10)

predict(fit, X[1:3, ])[, 5]

groups = list(1:5, 6:10)
fit = pcLasso(X, y, theta = 10, groups = groups)

fit = cv.pcLasso(X, y, theta = 10)
predict(fit, X[1:3,], s = "lambda.min")

Emojis in scatterplot

References:

  1. Using emojis as scatterplot points
  2. dill/emoGG
library(ggplot2)
library(emoGG)
data("ToothGrowth")
p1 <- geom_emoji(data = subset(ToothGrowth, supp == "OJ"),
                aes(dose + runif(sum(ToothGrowth$supp == "OJ"), min = -0.2, max = 0.2),
                   len), emoji = "1f34a")
p2 <- geom_emoji(data = subset(ToothGrowth, supp == "VC"),
                 aes(dose + runif(sum(ToothGrowth$supp == "OJ"), min = -0.2, max = 0.2),
                     len), emoji = "1f48a")

ggplot() +
    p1 + p2 +
    labs(x = "Dose (mg/day)", y = "Tooth length")

Medians in high dimensions

Refer to Medians in high dimensions

  • marginal median
  • geometric median
  • medoid
  • centerpoint
  • Tukey median

Laplace distribution as a mixture of normal distributions

Refer to Laplace distribution as a mixture of normals

\int_0^\infty f_{X\mid W=w}(x)f_W(w)dw=\frac{1}{2b}\exp\Big(-\frac{\vert x\vert}{b}\Big)\,.

Gradient descent as a minimization problem

Refer to Gradient descent as a minimization problem

put gradient decent into the optimization framework, then derive

  • projected gradient descent
  • proximal gradient methods

Coordinate descent doesn’t always work for convex functions

Refer to Coordinate descent doesn’t always work for convex functions

A counterexample:

z=\max(x,y)+\vert x-y\vert

Solution to a sgn equation

Refer to Soft-thresholding and the sgn function

Give a proof of the solution of

ax-b+c\mathrm{sgn}(x)=0

where a>0 and c\ge 0.

Horvitz–Thompson estimator

Refer to Horvitz–Thompson estimator

Perform an inverse probability weighting to (unbiasedly) estimate the total T=\sum X_i.

Illustration of SCAD penalty

Refer to The SCAD penalty

The dotted line is the y=x line. The line in black represents soft-thresholding (LASSO estimates) while the line in red represents the SCAD estimates.

Leverage in Linear regression

Refer to Bounds/constraints on leverage in linear regression

The leverage of data point i is the i-th diagonal entry of the hat matrix.

Modification to fundamental sampling formula

Refer to Inverse transform sampling for truncated distributions

We can draw sample X\sim F conditional on X\ge t.

Borel’s Paradox

Retire Statistical Significance

A petition

EM estimation for Weibull distribution

f_k(x) = k x^{k-1} e^{-x^k} \quad x >0

Refer to EM maximum likelihood estimation for Weibull distribution

A little confused about the answer

Power method for top eigenvector

Power method for obtaining the top eigenvector

Generalized Beta Prime

This distribution, characterized by one scale and three shape parameters, is incredibly flexible in that it can mimic behavior of many other distributions.

GB2 exhibits power-law behavior at both front and tail ends and is a steady-state distribution of a simple stochastic differential equation.

Julia Set

在复杂动力学里,Julia集是个著名的“混沌”行为的集(与之对应的是Fatou集,“非混沌”集)

References: