Statistics on Solarized Sublime Sekai

Finite Population Sampling without Replacement

Fri, 26 Apr 2019 09:46:29 +0200

First moment

Population: $ \Omega = \{ x_1, \dots, x_N \} $
Collection of $n$-samples: $\mathcal{S} = \{ s \in \Omega^n \mid \forall i,j \in s, i \ne j \} $
Collection of $n$-samples containing $x$: $ \mathcal{S}_x = \{ s \in \mathcal{S} \mid x \in s \} $
Observe that $ |\mathcal{S}_x| = \binom{N-1}{n-1} $.
Let population mean be zero. $\mu = 0$, i.e. $ \sum_{i = 1}^N x_i = 0 $
Fix an order for $\mathcal{S}$: $ \mathcal{S} = \{ s_1, s_2, \dots, s_{|\mathcal{S}|} \} $.
$j$-th $n$-sample mean $ m_j = \frac1n \sum_{x \in \mathcal{S}_j} x $
Remark: I don’t use $ \sum s_j $ as in $ \cup \mathcal{T} $ in topology to avoid misreading the $n$-sample $ s_j $ as an element.
mean of $n$-sample mean $ m = \frac{1}{|\mathcal{S}|} \sum_{s_j \in \mathcal{S}} m_j $

What are Dataframes?

Sun, 07 Oct 2018 17:30:49 +0200

Motivation

The books that I read in the past didn’t explain what a dataframe meant.

Definition

Dataframe: A table of data in which the values of each observed variable is contained in the same column.

Counterexample

I’ve difficulty in reading long lines of text like the above definition, so let’s illustrate this definition with a counterexample.

We have carried out repeated experiments with four types of things and obtaine some data. (Say, poured some liquid into an empty cup and take the temperature.)

2018-10-04 Seminar Notes

Thu, 04 Oct 2018 02:16:44 +0200

I jotted down only a few keywords that might be reusable. I didn’t understand any of the talks.

Functional Data Analysis

Goal: predict equipment temperature
Tools: Fourier coefficients (trigo ones), followed by discretisation, min-error estimation, cross-validation 10-folds, $R^2$ adjusted ?, MAE, MSPE
Comparison with non-functional data

Tolérancement

Thème : Traiter les incertitudes sur les dimensions des pièces de l’avion
Objectif :
- établir une modélisation mathématiques
- construire un virtual twin de l’avion
Outils :
- Modèle de variabilité
- Modèle d’assemblage $\text{airbus}: Y = \sum_{i = 1}^n a_iX_i?$
- Notion de risque … calculs des coefficients de convolution

SVM

Multiclass vs structual, hidden Markov model
Plan for this year:
- apply structual SVM for real SVM
- apply structual SVM for deep neural network

Auxiliary information

auxiliary function given in one partition
auxiliary function given in mutiple partitions
bootstrap
law of iterated logarithms
Kullback–Leibler distance
convergence: Donsker class, var, covar
ranking ration method: convergence to Gaussian process, entropy conditions, Telegrandś inequality
- weak convergence: KMT, Berthet-Maison
- strong convergence: ?
  - consequences: Berry-Essen bound, bias & variance estimation of ranking ration method

Euler scheme SDE

I could only write “Toeplitz tape operator”.