Mechanics: Units and Dimensional Analysis

Mecánica: Sistemas de Unidades

• S.I. uses meter (m) for length, kilogram (kg) for mass, and second (s) for time.
• C.G.S. uses centimeter (cm) for length, gram (g) for mass, and second (s) for time.
• Inglés uses foot (ft) for length, slug for mass, and second (s) for time.

Factores de Conversión

• 1 inch (pulg) = 2.54 cm
• 1 foot (ft) = 30.48 cm
• 1 meter (m) = 3.28 ft
• 1 mile (milla) = 1.609 km
• 1 slug = 14.59 kg
• 1 minute (min) = 60 s
• 1 hour (h) = 3600 s

Análisis Dimensional

• Dimension denotes the physical nature of a quantity.
• Dimensions can be treated algebraically.
• Only quantities with the same dimensions can be added or subtracted.
• Dimensions are abbreviated with square brackets:
  • [L] for Length
  • [M] for Mass
  • [T] for Time

Cifras Significativas (Significant Figures)

• Use known digits with certainty + 1 estimated digit.
• Leading zeros are never significant.
• Zeros between digits are always significant.
• Trailing zeros can be significant, use scientific notation when in doubt.

Órdenes de Magnitud (Orders of Magnitude)

• Express quantities by using the nearest power of 10 to approximate them.
• Useful for approximating and revising results.

Tension (T)

• Tension is a pulling force acting along a taut string.
• Tension is uniform throughout the string if the string is light.
• The relationship between tension (T), mass (m), and acceleration (a) is: a = T/m

Angle

• When a string pulls a block at an angle θ, the tension (T) resolves into horizontal (Tcosθ) and vertical (Tsinθ) components.
• Horizontally: Tcosθ = ma
• Vertically: R + Tsinθ = mg (R is the normal reaction force)

Guía rápida de Markdown

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• Énfasis:

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• Negrita* or Negrita

• Negrita Cursiva*

• Listas: - Listas no ordenadas: ```markdown

• Item 1

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-Citas: markdown > Esto es una cita.

• Líneas horizontales:

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Lecture 14: October 26

Dimensionality Reduction

• Dimensionality reduction aims to reduce data dimension while preserving information
• Motivations: easier visualization, increased computation speed, and removal of redundant features for noise reduction.
• Approaches: feature selection and feature extraction.

Feature Selection

• How to measure the relevance of a feature?
  • Variance: low variance = not informative.
  • Correlation: highly correlated = redundant.
  • Mutual Information: highly correlated with target = relevant.

Feature Extraction

• Feature extraction builds new ones from scratch.
• build new features from linear or non-linear from old ones
• Examples:
  • PCA
  • LDA

Principal Component Analysis (PCA)

• PCA finds orthogonal axes that capture maximal variance in the data.
• The first principal component captures the most variance.
• Subsequent components are orthogonal to the previous ones, each capturing the maximum remaining variance.

How to find the principal components?

• Center the data.
• Compute the covariance matrix: Σ = (1/n)X^T X.
• Find the eigenvectors and eigenvalues: Σvi = λivi.
• Sort the eigenvectors by eigenvalue.

How to reduce the dimensionality of the data?

• Project the data onto the first k principal components: Xreduced = XW.
• Reduced data has k dimensions.

How to choose the number of principal components k?

• Find k such as percentage of variance explained is above a set threshold (eg; 90%).

Singular Value Decomposition (SVD)

• SVD decomposes a matrix into three matrices: U, Σ, V^T.
• X = U Σ V^T
• X is an m × n matrix
• U is an m × m orthogonal matrix
• Σ is an m × n diagonal matrix with non-negative real numbers on the diagonal
• V is an n × n orthogonal matrix

How to compute the SVD?

• Covariance matrix: X^T X
• The right singular vectors are the eigenvectors of the covariance matrix: $V = [v_1, v_2,..., v_n]$
• Compute the singular values: $\sigma_i = \sqrt{\lambda_i}$
• The singular values are the diagonal elements of the matrix $\Sigma$: $\Sigma = diag(\sigma_1, \sigma_2,..., \sigma_n)$

How to use SVD for dimensionality reduction?

Project the data by using the first singular values and vectors $X_k = U_k \Sigma_k V_k^T$

• $X_k$ is the best rank-$k$ approximation of $X$.

Relation between PCA and SVD

PCA and SVD are closely related. - The principal components are the right singular vectors of the data matrix. - The eigenvalues of the covariance matrix are the squares of the singular values.

Pros and cons of PCA

-   Pros:
    -Simple and efficient
    -Finds the directions of maximum variance
-   Cons:
    -Assumes that the data is linearly correlated
    -Sensitive to outliers

Algorithmic Game Theory

What is Game Theory?

Game theory is a mathematical framework for strategic interactions between decision-makers.

• Applications
  • Economics
  • Political Science
  • Biology
  • Computer Science

Key concepts

• Players The decision-makers involved in the game.
• Strategies A strategy is a plan of action that a player can take in the game. -Pure Strategy: A deterministic plan to choose an action. -Mixed Strategy: A probability distribution over pure strategies.
• Payoffs The outcome or reward that a player receives after all players have chosen their strategies. $(s_1^, s_2^,..., s_n^*)$

Types of Games

• Cooperative vs. Non-Cooperative Games -Cooperative Games: Players can form coalitions and coordinate their strategies to achieve a common goal. -Non-Cooperative Games: Players act independently, aiming to maximize their individual payoffs
• Static vs. Dynamic Games -Static Games: Players make decisions simultaneously without knowledge of the other players' actions. -Dynamic Games: Players make decisions sequentially, with the knowledge of the previous actions of other players.
• Complete vs. Incomplete Information Games -Complete Information Games: All players have full knowledge of the game structure, including the payoffs, strategies and preferences -Incomplete Information Games: Players have private information that is not known to others.

Solution Concepts

-Dominant Strategy A strategy that yields the payoffs for a player regardless of what other players do. -Nash Equilibirum A set of strategies, one for each player that can unilaterally improve it

• Pareto Optimality An outcome is Pareto optimal if it is impossible to make any player better off without making at least one other player worse off.

Algorithmic Trading

Module 1: Introduction

-What is Algorithmic Trading?- Computer programs execute orders using price, volume, timing, and other parameters. Automatic trading.

Module 2: Getting Started with Python

- Use libraries like NUMPY, PANDAS, yFinance.
- Python is used because of extensive libraries, simplicity, and versatility

Module 3: Data Analysis with Pandas

-   Pandas helps with DataFrames
- **Visualization**
     -   Visualize price change overtime

Module 4: Developing Trading Strategies

-   Rolling window mean allows for trading signals, position signals
- Key tools include: Evaluate performance, returns, drawdowns, risk returns
- Setting a stop loss or position size

Module 5: Executing Trades

-   Brokerage APIs provide execution
- Alpaca credentials allow execution for order parameters, handling, symbol, and order types. Real time tracking of performance

Rules of Inference

-   Help build arguments
- modus ponens and tollens include: P,Q and R for symbols

Overview of unit systems (SI, CGS, English) and conversion factors. Introduces dimensional analysis, treating dimensions algebraically, and significant figures to represent measurements accurately. Learn about performing calculations with proper units.