Advanced Calculus: Vector Algebra

Advanced Calculus: Vector Algebra

• An n-dimensional vector is an ordered n-tuple of real numbers: $\mathbf{x} = (x_1, x_2,..., x_n)$ where $x_i \in \mathbb{R}$ for $i = 1, 2,..., n$.
• $\mathbb{R}^n$ is the set of all n-dimensional vectors.

Vector Operations

• If $\mathbf{x} = (x_1, x_2,..., x_n)$ and $\mathbf{y} = (y_1, y_2,..., y_n)$, then $\mathbf{x} + \mathbf{y} = (x_1 + y_1, x_2 + y_2,..., x_n + y_n)$.
• If $\mathbf{x} = (x_1, x_2,..., x_n)$ and $c \in \mathbb{R}$, then $c\mathbf{x} = (cx_1, cx_2,..., cx_n)$.

Properties of Vector Operations

• For all $\mathbf{x}, \mathbf{y}, \mathbf{z} \in \mathbb{R}^n$ and $a, b \in \mathbb{R}$:
• Commutativity: $\mathbf{x} + \mathbf{y} = \mathbf{y} + \mathbf{x}$.
• Associativity: $(\mathbf{x} + \mathbf{y}) + \mathbf{z} = \mathbf{x} + (\mathbf{y} + \mathbf{z})$.
• Additive Identity: $\exists \mathbf{0} \in \mathbb{R}^n$ such that $\mathbf{x} + \mathbf{0} = \mathbf{x}$.
• Additive Inverse: $\exists -\mathbf{x} \in \mathbb{R}^n$ such that $\mathbf{x} + (-\mathbf{x}) = \mathbf{0}$.
• Associativity of Scalar Multiplication: $a(b\mathbf{x}) = (ab)\mathbf{x}$.
• Distributivity: $a(\mathbf{x} + \mathbf{y}) = a\mathbf{x} + a\mathbf{y}$.
• Distributivity: $(a + b)\mathbf{x} = a\mathbf{x} + b\mathbf{x}$.
• Multiplicative Identity: $1\mathbf{x} = \mathbf{x}$.

Dot Product

• If $\mathbf{x} = (x_1, x_2,..., x_n)$ and $\mathbf{y} = (y_1, y_2,..., y_n)$, then $\mathbf{x} \cdot \mathbf{y} = \sum_{i=1}^{n} x_i y_i = x_1 y_1 + x_2 y_2 +... + x_n y_n$.

Properties of Dot Product

• For all $\mathbf{x}, \mathbf{y}, \mathbf{z} \in \mathbb{R}^n$ and $c \in \mathbb{R}$:
• Commutativity: $\mathbf{x} \cdot \mathbf{y} = \mathbf{y} \cdot \mathbf{x}$.
• Distributivity: $\mathbf{x} \cdot (\mathbf{y} + \mathbf{z}) = \mathbf{x} \cdot \mathbf{y} + \mathbf{x} \cdot \mathbf{z}$.
• Associativity: $(c\mathbf{x}) \cdot \mathbf{y} = c(\mathbf{x} \cdot \mathbf{y}) = \mathbf{x} \cdot (c\mathbf{y})$.
• Positive Definiteness: $\mathbf{x} \cdot \mathbf{x} \geq 0$, and $\mathbf{x} \cdot \mathbf{x} = 0$ if and only if $\mathbf{x} = \mathbf{0}$.

Euclidean Norm (Magnitude)

• $||\mathbf{x}|| = \sqrt{\mathbf{x} \cdot \mathbf{x}} = \sqrt{\sum_{i=1}^{n} x_i^2}$.

Cauchy-Schwarz Inequality

• $|\mathbf{x} \cdot \mathbf{y}| \leq ||\mathbf{x}|| \cdot ||\mathbf{y}||$.

Angle Between Vectors

• $\cos{\theta} = \frac{\mathbf{x} \cdot \mathbf{y}}{||\mathbf{x}|| \cdot ||\mathbf{y}||}$.

Triangle Inequality

• $||\mathbf{x} + \mathbf{y}|| \leq ||\mathbf{x}|| + ||\mathbf{y}||$.

Lines and Planes

Lines in $\mathbb{R}^n$

• Parametric Form: $\mathbf{x} = \mathbf{p} + t\mathbf{v}$, where:
  • $\mathbf{x}$ is a general point on the line
  • $\mathbf{p}$ is a specific point on the line
  • $\mathbf{v}$ is the direction vector
  • $t \in \mathbb{R}$ is a parameter

Planes in $\mathbb{R}^3$

• Normal Form: $\mathbf{n} \cdot (\mathbf{x} - \mathbf{p}) = 0$, where:
  • $\mathbf{n}$ is the normal vector to the plane
  • $\mathbf{x}$ is a general point on the plane
  • $\mathbf{p}$ is a specific point on the plane
• General Form: $ax + by + cz = d$, where $\mathbf{n} = (a, b, c)$ is the normal vector.

Cross Product in $\mathbb{R}^3$

• If $\mathbf{x} = (x_1, x_2, x_3)$ and $\mathbf{y} = (y_1, y_2, y_3)$, then $\mathbf{x} \times \mathbf{y} = (x_2 y_3 - x_3 y_2, x_3 y_1 - x_1 y_3, x_1 y_2 - x_2 y_1)$.

Properties of Cross Product

• For all $\mathbf{x}, \mathbf{y}, \mathbf{z} \in \mathbb{R}^3$ and $c \in \mathbb{R}$:
  • Anti-commutativity: $\mathbf{x} \times \mathbf{y} = -(\mathbf{y} \times \mathbf{x})$.
  • Distributivity: $\mathbf{x} \times (\mathbf{y} + \mathbf{z}) = (\mathbf{x} \times \mathbf{y}) + (\mathbf{x} \times \mathbf{z})$.
  • Associativity: $(c\mathbf{x}) \times \mathbf{y} = c(\mathbf{x} \times \mathbf{y}) = \mathbf{x} \times (c\mathbf{y})$.
  • $\mathbf{x} \times \mathbf{x} = \mathbf{0}$.
  • $||\mathbf{x} \times \mathbf{y}|| = ||\mathbf{x}|| \cdot ||\mathbf{y}|| \cdot \sin{\theta}$, where $\theta$ is the angle between $\mathbf{x}$ and $\mathbf{y}$.
  • $\mathbf{x} \times \mathbf{y}$ is orthogonal to both $\mathbf{x}$ and $\mathbf{y}$.

Geometric Interpretation

• $||\mathbf{x} \times \mathbf{y}||$ is the area of the parallelogram formed by $\mathbf{x}$ and $\mathbf{y}$.

Scalar Triple Product

• Definition: $\mathbf{x} \cdot (\mathbf{y} \times \mathbf{z})$.
• Geometric Interpretation: $|\mathbf{x} \cdot (\mathbf{y} \times \mathbf{z})|$ is the volume of the parallelepiped formed by $\mathbf{x}, \mathbf{y}, \mathbf{z}$.

Tension

• This is a pulling force carried along the length of a medium.
• The tension force is directed along the length of the wire. It pulls equally on objects from opposite ends.

Tension in a Rope

• If a block of mass m hangs by a rope, the forces on the block are:
  • Gravitational force, $\overrightarrow{w}$ (down).
  • Tension, $\overrightarrow{T}$ (up).
  • $\overrightarrow{T}$ is the force the rope exerts on the block

Forces on the block

• Sum of forces: $\sum{\overrightarrow{F}} = m\overrightarrow{a}$.
• In equilibrium (at rest): $\overrightarrow{a} = 0$.

Tension in an Accelerating System

• Sum of forces: $\sum{\overrightarrow{F}} = m\overrightarrow{a}$.
• Tension is greater than the weight of the block: $T = ma + w = ma + mg = m(a + g)$.

Multiple Ropes

• If several ropes support a load, the tension in each rope depends on the angle and number of ropes.

Executive Summary

Problem

• Manual inventory management leads to errors, inefficiency, unreliable information, and delayed decisions.

Objective

• Implement an automated inventory management system.

Specific Objectives

• Evaluate inventory management needs.
• Select an appropriate inventory management software.
• Implement and configure the software.
• Train personnel in software use.
• Evaluate system effectiveness.

Scope

• Includes evaluation, selection, implementation, and training on inventory management software.
• Manages references for both finished products and raw materials.

Justification

• Automation will reduce operational costs, improve customer satisfaction, optimize planning, and increase profitability.

Methodology

• Analysis of Requirements: Gather info on current inventory processes.
• Software Selection: Evaluate and select software that fits the company needs.
• Implementation and Configuration: Install and configure software, migrate data, and customize settings.
• Training: Train staff on software usage and new inventory management processes.
• Evaluation: Measure results and compare with objectives.

Cronograma

• Analysis of Requisitos: 2 weeks
• Selection of Software: 2 weeks
• Implementation and Configuración: 4 weeks
• Capacitación: 2 weeks
• Evaluación: 2 weeks

Budget

• Software: $5,000
• Hardware: $2,000
• Consultoría: $3,000
• Capacitación: $1,000
• Otros (Imprevistos 10%): $1,100
• Total: $12,100

Expected Results

• Reduce inventory errors by 50%.
• Increase inventory management efficiency by 30%.
• Improve inventory information accuracy by 95%.
• Reduce decision-making wait times by 40%.

Univariate Descriptive Statistics

Objectives

• Calculate and interpret univariate descriptive statistics.
• Build and interpret frequency tables.
• Construct and interpret graphs.

Definitions

Population

• Set of individuals or objects under study.

Sample

• Subset of the population.

Variable

• Characteristic or property measured or observed on each individual or object.

Qualitative (Categorical) Variables

Nominal

• Categories cannot be ordered (e.g., eye color, sex).

Ordinal

• Categories can be ordered (e.g., satisfaction level, appreciation).

Quantitative (Numerical) Variables

Discrete

• Possible values are integers (e.g., number of children, number of rooms).

Continuous

• Possible values can take any value in an interval (e.g., height, weight, temperature).

Frequency Tables

Absolute Frequency

• Number of observations for each value or category.

Relative Frequency

• Proportion of observations for each value or category.

Cumulative Frequency

• Sum of absolute or relative frequencies for values less than or equal to a given value.

Graphs

Qualitative Variables

• Bar chart
• Pie chart

Discrete Quantitative Variables

• Stem-and-leaf diagram

Continuous Quantitative Variables

• Histogram
• Frequency curve

Descriptive Statistics

Central Tendencies

Mean (Average)

• Sum of values divided by the number of observations.
• Formula: $\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}$

Median

• Value that separates data into two equal parts.

Mode

• Most frequent value.

Dispersion

Range

• Difference between max and min values.

Variance

• Average of the squared differences from the mean.
• Formula: $s^2 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n-1}$

Standard Deviation

• Formula: $\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}$
• Square root of the variance.
• Formula: $s = \sqrt{s^2}$

Coefficient of Variation

• Formula: $s^2 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n-1}$
• Standard deviation divided by the mean.
• Formula: $CV = \frac{s}{\bar{x}}$

Quantiles

Values dividing data into equal parts (quartiles, deciles, centiles).

Quartiles

• Divide data into 4 equal parts.

Interquartile Range (IQR)

Formula: $s = \sqrt{s^2}$ Difference between the third and first quartiles.

Outliers

• Values distant from the others
• Typically values $\geq Q1-1.5xIQR$ or $\leq Q3 + 1.5 X IQR$

Shape of Distribution

Symmetric

• Distribution symmetrical about the mean.

Right-skewed

• Distribution has a long tail on the right.

Left-skewed

• Distribution has a long tail on the Left.

Skewness Coefficient

• Measures the asymmetry of a distribution

Kurtosis Coefficient

• Measures the peakedness of a distribution

Algorithmic Game Theory

• Theory: Mathematical framework for analyzing strategic interactions between multiple decision-makers (players).

Basic Elements of a Game

• Players: decision-makers.
• Actions: choices available to each player.
• Payoffs: outcomes or rewards for each player.

Types of Games

Cooperative vs. Non-cooperative

- Cooperative: players can form coalitions and make binding agreements.
- Non-cooperative: players act independently; agreements are not enforceable.

Symmetric vs. Asymmetric

- Symmetric: all players have the same set of actions and payoffs.
- Asymmetric: players have different sets of actions or payoffs.

Zero-Sum vs. Non-Zero Sum

Zero-sum: One player gains is another player loses. Non-zero-sum: total payoff can vary.

• All players can gain or lose.

Perfect vs. Imperfect Information games

- Perfect information games, all players know the actions taken by other players.
- Imperfect information, players have incomplete information about the actions taken by other players.

Static vs. Dynamic Games

- Static: players act simultaneously or without knowledge of each other's actions.
- Dynamic: players act sequentially, with the knowledge of prior actions.

Algorithmic Game Theory

• Involves computational aspects of games.
  • Computing Equilibria: Finding stable states in games
  • Mechanism Design: Incentivizing players to behave a certain way
  • Game Dynamics: Analyze how games evolve as players learn and adapt.

Applications

• Economics
• Political Science
• Computer Science
• Biology

Algorithmic Trading

• Computer programs execute trades following instructions at high speed and frequency.

Advantages to Algorithmic Trading

• Reduced transaction costs
• Improved order execution
• 24/7 trading
• Reduced emotional influence

Disadvantages

• System failure
• Model breakdown
• Requires Technical Skills
• Market microstructure risks

Trend Following Strategies

• Buy assets trending upwards/Sell assets trending downwards
• Moving averages, MACD, and trend lines.

Mean Reversion Strategies

• Buy assets below average/sell assets assets above average price
• Bollinger Bands, RSI, and statistical analysis.

Arbitrage Strategies

• Requires real-time data and fast execution.
• Exploit price differences for the same asset in different markets.

Market Making Strategies

• Requires sophisticated risk management techniques.
• Place both buy and sell orders in the market to capture spread.

Statistical Arbitrage

• Includes:Cross sectional arbitrage, Pairs trading, Index arbitrage, and Triangular arbitrage

Tools

• Languages: Python, R, C++, Java
• Platforms: MetaTrader, TradingView, Bloomberg, and Interactive Brokers
• Libraries: Pandas, NumPy, Scikit-learn, Statsmodels

Backtesting

• Testing a strategy on historical data to evaluate performance.

Risk Management

• Determine the size of each trade based on risk and volatility.
• Fixed fractional position sizing
• Kelly Criterion

Regulations

• Stay up to date and ethical
• Ensure applicable with all laws and rules.

Explore n-dimensional vectors and their properties. Learn vector operations, including addition and scalar multiplication. Understand commutativity, associativity, and additive identities in vector algebra.