The Well-Built City-优快云博客

原创 How Does Adjusted R Squared Avoid Model Overfitting (in Linear Regression)

Radj2R^2_{adj}Radj2 FormulaRadj2R^2_{adj}Radj2 is a statistical measure that represents the proportion of the variance for a dependent variable (YYY) that’s explained by an independent variable (XXX) in a regression model given by E(Y∣X=x)=β0+β1x+β2x2+

2021-09-29 07:38:50 230

原创 Chapter 6-7

2k2^k2k Factorial DesignSuch design provides the smallest number of runs in which kkk factors can be studied in a complete factorial design.AssumptionsIn each replicate of the design, the experiment runs are completely randomizedThe factors are fixed

2021-06-07 13:05:31 598

原创 Absolute Value Inequality

Absolute Value InequalityBase FormVariant 1ProofVariant 2ProofVariant 3x,y∈Rx,y\in\Rx,y∈RBase Form∣x+y∣≤∣x∣+∣y∣|x+y|≤|x|+|y|∣x+y∣≤∣x∣+∣y∣Variant 1∣∣x∣−∣y∣∣≤∣x−y∣||x|-|y|| ≤ |x- y|∣∣x∣−∣y∣∣≤∣x−y∣ProofWe use the technique of “adding 000 wisely”x

2021-05-28 15:15:40 314

原创 (Probability) Understanding Marginal Probability Mass Function

For a bivariate discrete random variable, (X,Y)(X,Y)(X,Y), where X,YX,YX,Y are two discrete random variables, the probability mass function, f(x,y)f(x,y)f(x,y), is defined as the probability P(X=x∩Y=y)P(X=x\cap Y=y)P(X=x∩Y=y).The Marginal Probability Mas.

2021-05-23 18:44:37 440

原创 Central Limit Theorem Overview

We go over the contents and the application of the CLT along with some preliminary introduction to Random Variables and Unbiased Estimators of population parameters. We offer a computational example of CLT in estimating the Sample Mean's distribution.

2021-05-17 16:22:43 525 1

原创 (Reading) Sources of Power

There are two roots to building up your sources of power: dominance and prestige.

2021-05-07 21:27:35 402 1

原创 (Counting & Probability) Fisher-Yates Shuffling & Implementation in Java

This note goes over the Fisher-Yates Shuffling, its proof, and implementation in Java.

2021-04-25 04:45:23 256

原创 Why Can We Compare Alpha and P-Value in Hypothesis Tests?

Hopefully, this note offers a straightforward yet holistic understanding of the significance level, p-value, and their relationship in Hypothesis Tests.

2021-04-13 18:28:06 305

原创 (JAVA) Miscellaneous Topics

Installing JDKGo to this website to download the Java Development Kit (JDK), and we then install it.To check if the JDK is installed correctly, open Terminal on MacOS and runjavac --version, which should show javac <some version number>javac stan

2021-04-04 16:28:12 218

原创 (Counting & Probability) The Counting Principles, Permutation, and Combination

Counting TechniquesMultiplication PrincipleCoin TossPermutation nPnnPnnPnPermutation nPrnPrnPrCombination nCrnCrnCrnCr=nCn−rnCr=nCn-rnCr=nCn−rMore Coin Toss ProblemPermutation With RepetationMultiplication PrincipleIf procedure AAA has aaa (distinct) ou

2021-04-04 04:01:13 517

原创 (Graph Theory) Is Minimum Spanning Tree Unique?

Is the Minimum Spanning Tree Unique?The following problem with two parts explores a condition that always implies a unique minimum spanning tree.

2021-03-29 00:57:31 392

原创 (Graph Theory) Proving Prim‘s Algorithm

This note proves Prim's Algorithm for finding MST, and it shows why an edge with the minimum weight is always in every MST.

2021-03-28 14:33:14 145

原创 (Graph Theory) A Connected Graph has a Spanning Tree

This note goes over a theorem about spanning trees: A Connected Graph has a Spanning Tree

2021-03-27 03:23:14 294

原创 (Graph Theory) Important Exercise

In a connected graph, removing any single edge from a cycle retains connectivity.

2021-03-25 17:06:06 214

原创 (Graph Theory) Some Properties of Trees (with Proof)

This note briefly goes over a theorem that introduces some key properties of trees. This applies to any kind of trees.

2021-03-25 16:33:28 253

原创 (Graph Theory) The Hand Shaking Lemma & Application

This is a short note about the Hand Shaking Lemma and a formula for the sum of degrees of all vertices in a graph. It also provides an example of its application.

2021-03-25 01:46:00 263

原创 (Graph Theory) Graphs and Trees

This note briefly goes over some (i.e. not all) topics about graphs and trees. It also contains links to additional posts about relevant topics written by myself.

2021-03-23 19:13:54 893

原创 (Graph Theory) Adjacency Matrix

This note briefly goes over the Adjacency Matrix and relevant theorems about how the Adjacency Matrix is related to Graph Isomorphism.

2021-03-23 19:07:51 817

原创 Go Over a Complete Example of Parametric Bootstrap Test with R

文章目录QuestionSolution1. State the hypotheses2. Importing and examining the data3. Exploratory analysis of the data4. Paramatrizing5. Executing the bootstrap test6. Draw conclusion (statistical inference)Full CodeBelow I did a complete example of a parametr

2021-03-14 00:34:29 236

原创 Types of Research & Two Sample t, Randomization, and Bootstrapping Tests

文章目录IntroTypes of ResearchRandom Sampling (RS)Random Assignment (RA)Types of researchFour TestsTwo Independent Sample t-test for Sample MeanGoal (When to use)Key value: p-value.ConclusionConditionsImplementation in RRandomization TestGoal (When to use)Key

2021-03-10 18:27:24 992 2

原创 What Will Happen When Adding a New Variable to a Multiple Linear Regression Model

What Will Happen When Adding a New Variable to a Multiple Linear Regression ModelConclusions:This new (different from existing ones) variable will ALWAYS reduce Residual Sum of Squares (RSSRSSRSS)However, a large ppp for the partial t-test for this ne

2021-03-04 05:10:26 216 1

原创 Hypothesis Test Overview

Hypothesis Test1. ComponentsH0H_0H0, Null Hypothesis. Our default assumption of the value of the population parameter of our interest. For example, for simple linear regression, the null hypothesis is the true model’s slope, β1\beta_1β1, equals 000 (i

2021-02-15 03:03:29 508

原创 Vectorization of the Gradient Descent Algorithm

Vectorization of the Gradient Descent AlgorithmProblem SetupGiven a dataset with mmm samples (x(1);x(2);...;x(m))(x^{(1)};x^{(2)};...;x^{(m)})(x(1);x(2);...;x(m)) with each sample having nnn features (so x1(1)x^{(1)}_1x1(1) denotes the value of the

2020-08-15 16:55:21 143

原创 Normal Equation

Deriving the Normal EquationAn alternate of Gradient Descent in Multiple Linear RegressionProblem Setup:Given a dataset with mmm samples (x(1);x(2);...;x(m))(x^{(1)};x^{(2)};...;x^{(m)})(x(1);x(2);...;x(m)) with each sample having nnn features (so x1

2020-08-14 18:22:32 139

Bill_Wang_01的博客