Sets, Subsets, and Set Operations

Preliminaries - Mathematical Background

• A set constitutes a well-defined collection of objects, referred to as elements or members.
• Notation: $x \in A$ indicates $x$ is an element of set $A$, while $x \notin A$ means $x$ is not in $A$.
• Sets can be defined by listing elements (e.g., $A = {1, 2, 3}$) or by specifying a property (e.g., $B = {x \mid x \text{ is an even integer}}$).
• The empty set, denoted by $\emptyset$, contains no elements.

Subsets and Set Equality

• If every element of $A$ is also in $B$, then $A$ is a subset of $B$, written as $A \subseteq B$.
• $A \subset B$ means $A$ is a proper subset of $B$ (i.e., $A \subseteq B$ and $A \neq B$).
• Two sets $A$ and $B$ are equal ($A = B$) if they contain exactly the same elements.
• $A = B$ holds if and only if $A \subseteq B$ and $B \subseteq A$.

Set Operations

• Union: $A \cup B = {x \mid x \in A \text{ or } x \in B}$, includes all elements in $A$ or $B$ (or both).
• Intersection: $A \cap B = {x \mid x \in A \text{ and } x \in B}$, contains elements in both $A$ and $B$.
• Difference: $A \setminus B = {x \mid x \in A \text{ and } x \notin B}$, includes elements in $A$ but not in $B$.
• Complement: $A^c = {x \mid x \notin A}$, includes all elements not in $A$.

Functions Defined

• A function $f: A \rightarrow B$ maps each element $x$ in $A$ (domain) to a unique element $f(x)$ in $B$ (codomain). The range of $f$ is the set of all $f(x)$ values for $x$ in $A$.
• One-to-one (injective) if $f(x_1) = f(x_2)$ implies $x_1 = x_2$ for all $x_1, x_2 \in A$.
• Onto (surjective) if for every $y \in B$, there exists an $x \in A$ such that $f(x) = y$.
• A bijection is both one-to-one and onto.

Inverse Functions and Graphs

• For a bijection $f: A \rightarrow B$, the inverse function $f^{-1}: B \rightarrow A$ maps each $y$ in $B$ to the unique $x$ in $A$ such that $f(x) = y$.
• The graph of $f: A \rightarrow B$ consists of all ordered pairs $(x, f(x))$ with $x \in A$.

Relations

• A relation on $A$ is a subset of $A \times A$ (the Cartesian product of $A$ with itself).
• Reflexive: $(a, a) \in R$ for all $a \in A$.
• Symmetric: $(a, b) \in R$ implies $(b, a) \in R$ for all $a, b \in A$.
• Transitive: $(a, b) \in R$ and $(b, c) \in R$ implies $(a, c) \in R$ for all $a, b, c \in A$.
• An equivalence relation is reflexive, symmetric, and transitive.

Machine Learning Algorithm Steps Summary

• Steps in a standard Machine Learning model
• From Data collection, to data analysis and maintanence

Data Collection

• collect data from a variety of relevant source
• Ensure the quality and data is correctly represented

Data Preperation

• Cleaning
• Transformation
• Feature Engineering
• Data Splitting

Model Selection

• Choose a suitable statistical or ML model for the type of problem you are working on
• Take into account dat size, dimensionality and ease of interpreation.

Model Training

• Apply selected algorithim to the training data
• Optimize model parameters using optimization techniques
• Monitor performance using validation data.

Model Evaluation

• Evaluate trained performance using test data
• Use metrics such as accuracy, preciion, recall etc.
• compare models and select best models

Hyperparameter optimization

• Improve performance by fine tuning hyperparamters
• Use techniques like grid search , random search etc.

Model Deployment

• Deploy trained model to production environement
• Integrate the model into existing systems
• Monitor model performance and retrain when needed

Model Monitoring and maintanence

• Continuously monitor model's performance in production
• Retrain the model with new data
• Regular maintenance to ensure good performance

Machine Learning Models

• Table summarising different models such as Linear Regression, SVM etc.
• Table summarising whether they are for Supervised or Unsupervised Learning
• Table summarising the Algorithm's advantages and disadvantages

The Wave Equation - Definition and Solutions

• Theorem 1: Establishes conditions for the solution of the wave equation on $\mathbb{R}^n$. Given $n \geq 2$, $f \in C^{\lfloor\frac{n}{2}\rfloor+2}(\mathbb{R}^n)$, and $g \in C^{\lfloor\frac{n}{2}\rfloor+1}(\mathbb{R}^n)$, a $C^2$ function $u(t,x)$ is defined by an explicit integral involving $f$ and $g$.
• This function $u(t,x)$ is proven to satisfy the wave equation $u_{tt} - \Delta u = 0$ for $t \in \mathbb{R}$ and $x \in \mathbb{R}^n$, with initial conditions $u(0,x) = f(x)$ and $u_t(0,x) = g(x)$.
• Here, $\alpha(n) = (n-1)(n-2)V_n$, where $V_n$ is the volume of the unit ball in $\mathbb{R}^n$.
• The integral requires careful interpretation to handle singularities.

Resolving Dimensions

• If a solution is known for $n=3$, it can be extended to $n=2$.
• Let $\tilde{u}(t,x_1,x_2,x_3) = u(t,x_1,x_2)$, $f(x_1,x_2,x_3) = f(x_1,x_2)$, $g(x_1,x_2,x_3) = g(x_1,x_2)$. Then $\tilde{u}(t,x_1,x_2,x_3)$ solves the wave equation in $\mathbb{R}^3$.

Kirchhoff's Formula Explained

• For $n=3$, Kirchhoff's formula provides the solution: $u(t,x) = \frac{1}{4\pi t^2 }\int_{\partial B(x,t)} g(y) dS(y) + \partial_t \left( \frac{1}{4\pi t^2 }\int_{\partial B(x,t)} f(y) dS(y) \right)$.
• $G(t,x)$ is used to denote the initial velocity part, while $F(t,x)$ the initial displacement part.

ANOVA Overview

• Analysis of Variance is used to compare multiple means by partitioning total variance.

ANOVA: Key Concepts

• Factor: Categorical variable defining groups.
• Level: Specific value of a factor.
• Treatment: Factor level combinations.
• Response variable: Continuous measurement.

ANOVA: Assumptions

• Independence, Normality, and Homogeneity of Variance
• $H_0$: All group means are equal; $H_a$: At least one group mean is different.

ANOVA Test Statistic

• $F = \frac{MST}{MSE}$, where MST is variance between groups and MSE is variance within groups.
• ANOVA Table: Source, DF, SS, MS, F, p-value.
  • Treatment DF=k-1, Error DF=N-k, Total DF=N-1.

Formula's for sum of squares in the ANOVA table defined

• k is number of groups
• N is total number of observations
• SST is sum of squares
• SSE is sum of squares
• SSTO os total sum of squares

Two-Way ANOVA

• Used for two factors, examining main and interaction effects.

Two Way Hypotheses

• Factor A, Factor B, Interaction (AxB)

Two Way ANOVA table

• Source, DF, SS, MS, F, p-value.
• Factor A DF=a-1, Factor B DF=b-1 and interaction DF=(a-1)(b-1)

Two way ANOVA formula's defined

• a = number of levels of Factor A.
• b = number of levels of Factor B.
• N = total number of observations.
• SSA = sum of squares due to Factor A.
• SSB = sum of squares due to Factor B.
• SSAB = sum of squares due to the interaction effect.
• SSE = sum of squares due to error.
• SSTo = the total sum of squares.

Post-Hoc Tests

• Performed after ANOVA to identify group difference

Types of Post Hoc tesst

• Tukey's, Bonferroni, Scheffe's, Fisher's LSD

Normal Distribution Overview

• A probability distribution with a number of qualities

Normal qualities

• Density curve is always positive. It also has an area of 1 underneath
• Curve is always symmetric
• Curve is always single-peaked

Formula for Normal distribution

• $N(\mu, \sigma)$
• $\mu$ == mean of the distribution
• $\sigma$ is the standard deviation

Standard normal

• normal distrubution with mean of 0 and SD of 1 Denoted N(0,1)

• Z Score

  • Number of SD a value falls from the mean Formula $z = \frac{x - \mu}{\sigma}$

Table gives the area SD to the left of z.

Normality Application

• If normal and points close, approximate to normal probability
• Apply 68-95-99.7 Rule.

Rule breakdown

• Approximately 68% of the observations fall within 1 standard deviation of the mean.
• Approximately 95% of the observations fall within 2 standard deviations of the mean.
• Approximately 99.7% of the observations fall within 3 standard deviations of the mean.

Applying the Rule

• Express the problem in terms of the variable x
• Standardise x to z score then use table A to work out values.

Algorithmic Trading

• Executing orders by automated and pre programmed trading instructions

Pros and Cons

• Advantage of best price and transaction and error reduction
• Disadvantage with algorithm flaws and mechanical failure etc

Popular Algorithmic Trading Strategies

• Trend following strategies, Mean Reversion Strategies, Arbitrage Strategies*.

Trend following Strategies

Moving averages, Breakout , MACD.

Mean Reversion Strategies

Pairs trading, Bollinger Band

Arbitrage Strategies

Index arbitrage and Triangula arbitrage

Algortithim types

• Volume Weighted Average Price and Time Weighted Average Price
• Percentate of Volujme And Implementation Shortfall

Types defined

• Buy as close to VWAP as possible, divde order to evenly spread time
• Perticipate in volume while beign discrete
• Reduce trading cost