Data Structures and Algorithm Analysis

Analyse des algorithmes

• Complexité temporelle measures execution time as input size increases.
• Asymptotic behavior is evaluated as n approaches infinity.
• Notations O, Ω, and Θ describe upper, lower, and exact bounds, respectively.
• Complexité spatiale measures memory usage relative to input size.
• It also analyzes the algorithm's use of extra memory.

Structures de données

• Abstact Data Types (ADT) define a data structure's behavior through operations like lists, stacks, and trees.
• Tableaux offer direct access, but have a fixed size.
• Listes chaînées are good for insertions/deletions, but require sequential access.
• Piles use LIFO (last in, first out) with push, pop, and peek operations.
• Files use FIFO (first in, first out) with enqueue and dequeue operations.

Algorithmes de tri

• Tri par insertion is simple and efficient for small inputs.
• Tri par fusion uses divide and conquer with O(n log n) complexity.
• Tri rapide is divide and conquer, O(n log n) on average, O(n^2) worst case.
• Tri par tas uses heap data structure, O(n log n) complexity.
• Algorithms are compared using time and space complexity and stability.

Algorithmes de recherche

• Recherche linéaire scans data sequentially with O(n) worst-case complexity.
• Recherche binaire requires sorted data, halves the search space each step, O(log n).

Arbres

• Arbres binaires have each node with at most two children.
• Common traversals include prefix, infix, suffix, and breadth-first.
• Arbres binaires de recherche (BST) have a left subtree with node values less than the current node's and a right subtree with values greater.
• Operations include search, insertion, and deletion.
• Arbres équilibrés like AVL and Rouge-Noir remain O(log n) in height.
• These balanced operations are more complex but improve performance.

Graphes

• Graphes has a set of nodes and edges and can be oriented or non-oriented.
• They can be represented as an adjacency matrix or adjacency list.
• Parcours en largeur (BFS) uses a FIFO file.
• Parcours en profondeur (DFS) uses a LIFO pile (recursion).
• Shortest path algorithms for graphs include Dijkstra and Bellman-Ford.
• Minimum spanning tree algorithms include Prim and Kruskal.

Techniques de conception d'algorithmes

• Diviser pour régner tecnique breaks the problem into smaller sub-problems, solves them recursively, then combines results.
• Examples include Tri par fusion and Tri rapide.
• Programmation dynamique solves sub-problems once, stores the results for larger problems, optimizing overlapping sub-problems.
• An example is calculating Fibonacci numbers.
• Algorithmes gloutons make the optimal choice at each step.
• These do not guarantee the global optimum.

Table de hachage

• Fonctions de hachage converts a key to array index.
• Good key distribution to avoid collisions.
• Gestion des collisions with separate chaining, each array cell points to list.
• Adressage ouvert seeks empty array cell which can be linear, quadratic, or double hashing.

Gestion de la mémoire

• Allocation et libération includes static versus dynamic allocation.
• Gestion manuelle is for memory management (C, C++).
• Ramasse-miettes is garbage collection (Java, Python).
• Fuites de mémoire is forgetting to free allocated memory, requires leak detection and prevention.

Algorithmic Trading

• The execution of buy/sell orders is done using automated instructions.
• Variables such as price, timing, and volume are taken into account.

Benefits

• Algorithmic trading trades at the best possible price.
• It also reduces the impact of human emotion and is able to conduct simultaneous automated checks.
• Reduced possibility of manual errors.

Trend Following Strategies

• Taking positions is done in the same direction as a current trend.
• Moving average crossover is a simple trend-following strategy.
• It uses moving averages to identify trend direction.
• Breakout strategy says price breaks key support or resistance level.
• Price is likely to continue moving in that direction.

Mean Reversion Strategies

• Assumes asset prices tend to revert to their average over time.
• Pairs Trading involves 2 stocks with high correlation and taking a long over short position, waiting for prices to coverage.
• Bollinger Band uses upper/lower bands where prices tend to stay.
• A high or low price is likely to revert to the mean.

Arbitrage Strategies

• Simultaneously buying and selling an asset across markets profits from price differences.
• Triangular Arbitrage exploits currency differences between three different currencies in the foreign exchange market.
• Statistical Arbitrage uses statistical models to spot pricing discrepancies.
• Positions profit from expected price convergence.

Tools

• Programming Languages such as Python and C++ may be used.
• Data Analysis Packages such as Pandas and NumPy may be used.
• A wide variety of brokerage APIs or cloud computing services may be used.
• Backtesting Platforms and Data Visualization Tools like matplotlib or seaborn may be used.

Example code for Algorithmic trading

import alpaca_trade_api as tradeapi
api = tradeapi.REST('YOUR_API_KEY', 'YOUR_SECRET_KEY', api_version='v2')

symbol = 'AAPL'
qty = 1

api.submit_order(symbol, qty, 'buy', 'market', 'day')

Espacios vectoriales y subespacios

• A vector space is a non-empty set $V$ of objects, called vectors.
• Two operations, addition and scalar multiplication, are defined in these non-empty sets of objects.

Definition of vector space

• These operations are subject to the given 10 axioms.
• If $\mathbf{u}$ and $\mathbf{v}$ are in $V$, then $\mathbf{u} + \mathbf{v}$ is in $V$.
• $\mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u}$
• $(\mathbf{u} + \mathbf{v}) + \mathbf{w} = \mathbf{u} + (\mathbf{v} + \mathbf{w})$
• There exists a zero vector $\mathbf{0}$ in $V$ such that $\mathbf{u} + \mathbf{0} = \mathbf{u}$
• For each $\mathbf{u}$ in $V$, there exists a vector $-\mathbf{u}$ in $V$ such that $\mathbf{u} + (-\mathbf{u}) = \mathbf{0}$
• If $k$ is a scalar and $\mathbf{u}$ is in $V$, then $k\mathbf{u}$ is in $V$.
• $k(\mathbf{u} + \mathbf{v}) = k\mathbf{u} + k\mathbf{v}$
• $(k + l)\mathbf{u} = k\mathbf{u} + l\mathbf{u}$
• $k(l\mathbf{u}) = (kl)\mathbf{u}$
• $1\mathbf{u} = \mathbf{u}$

Subspace Definition

• A subspace of a vector space $V$ is a subset $H$ of $V$ that satisfies these three conditions:
• The zero vector $\mathbf{0}$ is in $H$.
• If $\mathbf{u}$ and $\mathbf{v}$ are in $H$, then $\mathbf{u} + \mathbf{v}$ is in $H$.
• If $k$ is a scalar and $\mathbf{u}$ is in $H$, then $k\mathbf{u}$ is in $H$.

Theorem 1

• If $\mathbf{v}_1, \mathbf{v}_2,..., \mathbf{v}_r$ are vectors in a vector space $V$, then:
• $H = gen{\mathbf{v}_1, \mathbf{v}_2,..., \mathbf{v}_r}$ is a subspace of $V$.

Algorithmic Game Theory

• Studies the way rational agents interact.

Selfish Agents

• Game theory studies the interactions of rational but selfish agents.
• Agents Act in order to maximize its own and only cares about its own utility.

Strategic Decisions

• Decision making can be very useful to others.

Examples:

• Auctions and network routing
• Election and social networks

What is Algorithmic Game Theory?

• An area with the intersection of game theory and algorithm design.

Equilibrium Computation

• How hard is it to compute an equilibrium?
• What are the properties of equilibria?

Mechanism Design

• How can games achieve a desired outcome?

Price of Anarchy

Ratio between a Wardrop equilibrium over a social optimum.

Example

Selfish Routing

• The model
• A network of $n$ nodes and $m$ edges.
• Each edge $e$ has a cost function $c_e(x)$ which depends on the amount of flow $x$ on the edge.
• There is a set of $k$ players, each of whom wants to route traffic from a source node $s_i$ to a target node $t_i$. Each player can choose any path from $s_i$ to $t_i$.
• The cost to a player is the sum of the costs of the edges on its path.

Wardrop Equilibrium

• A Wardrop equlibrium is a routing of traffic of minimum cost path.
• I.E no player has incentive to unilaterally deviate to another path.

Social Optimum

• A social optimum is a routing of traffic that minimizes the total cost to all players.
• I.E minimizes $\sum_{e} x_e c_e(x_e)$, where $x_e$ is the amount of flow on edge $e$.

Price of Anarchy

ratio between cost of wardrop equilibrium over the cost of social optimum. $$PoA = \frac{\text{Cost of Wardrop Equilibrium}}{\text{Cost of Social Optimum}}$$

Example

Prisoner's Dilemma

Two suspects are arrested for a crime. They are held in separate cells and cannot communicate. The police offer this deal:

• If you confess and your accomplice does not, you go free and your accomplice gets 10 years.
• If you both confess, you each get 5 years.
• If you do not confess and your accomplice does, you get 10 years and your accomplice goes free.
• If neither of you confess, you each get 1 year.

The payoff Matrix

The Nash Equilibrium

• A set of strategies with one for each player such that no player has incentive to deviate to another strategy.

Matrix Algebra

• Matrix algebra is the manipulation of matrices where rules can be defined for arithemitic operations.

Definitions

• A set of numbers arranged in rows and columns.
• Numbers are called elements of the matrix.
• i.e Matrix A below has three rows and four columns.

$A = \begin{bmatrix} a_{11} & a_{12} & a_{13} & a_{14} \ a_{21} & a_{22} & a_{23} & a_{24} \ a_{31} & a_{32} & a_{33} & a_{34} \end{bmatrix}$

Properties

• Size of the matrix is based on rows and columns.
• Matrix A is a 3x4 Matrix
• Rows is denoted first and then columns. Elements can be identified by two sub-indices, row and then column.

Square Matrix

• The same number of rows and columns.
• Ensamble: mxn where m=n For example:

$A = \begin{bmatrix} 1 & 2 & 3 \ 4 & 5 & 6 \ 7 & 8 & 9 \end{bmatrix}$

• A 3x3 matrix

Row Vector

• Is a matrix that has a single row.
• A 1xn matrix.
• For example:

$A = \begin{bmatrix} 1 & 2 & 3 & 4 \end{bmatrix}$

a 1x4 row vector

Column Vector

• The transpose of a row vector.
• A matrix that has one column in it.
• A mx1 vector
• For example:

$A = \begin{bmatrix} 1 \ 2 \ 3 \end{bmatrix}$

a 3x1 column vector

Diagonal Matrix

• A square matrix where all elements outside of the diagonal is 0.
• Diagonal goes from the upper-left to the bottom-right.
• For example:

$A = \begin{bmatrix} 1 & 0 & 0 \ 0 & 5 & 0 \ 0 & 0 & 9 \end{bmatrix}$

3x3 diagonal matrix

Identity matrix

• A diagonal matrix where all elements are 1.
• For example:

$I = \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix}$

• A 3x3 identity matrix, denoted with I.
• AXI = A IxA = A

Zero Matrix

• A matrix in which all the elements are zero.
• For example:

$A = \begin{bmatrix} 0 & 0 & 0 \ 0 & 0 & 0 \ 0 & 0 & 0 \end{bmatrix}$

• a 3x3 zero matrix, denoted by 0.
• A matrix where elements above diagonal are zero.
• For example:

$A = \begin{bmatrix} 1 & 2 & 3 \ 0 & 5 & 6 \ 0 & 0 & 9 \end{bmatrix}$

a 3x3 upper triangular matrix.

Lower Triangular Matrix

• A matrix square in size that elements below are zero:
• For example:

$A = \begin{bmatrix} 1 & 0 & 0 \ 4 & 5 & 0 \ 7 & 8 & 9 \end{bmatrix}$

Matrix Symmetric

• A square matrix equivalent to the transpose, ATM = A
• Transpose is is the exchange of rows and columns.
• For example, the transpose of a matrix

$A = \begin{bmatrix} 1 & 2 & 3 \ 4 & 5 & 6 \ 7 & 8 & 9 \end{bmatrix}$

$A^T = \begin{bmatrix} 1 & 4 & 7 \ 2 & 5 & 8 \ 3 & 6 & 9 \end{bmatrix}$

• The matrix A IS NOT symmetric because A=/ $A^T$
• However, the matrix

$A = \begin{bmatrix} 1 & 2 & 3 \ 2 & 5 & 6 \ 3 & 6 & 9 \end{bmatrix}$

a symmetric matrix because A= $A^T$

Anti-symmetric matrix

• The transpose of the matrix has it's negative.
• For example:

$A = \begin{bmatrix} 0 & 2 & -3 \

• 2 & 0 & -6 \ 3 & 6 & 0 \end{bmatrix}$

• an example of a antisymettric matric that follows A=-$A^T$

1. Background

• AI Timeline: 1950: Alan Turing asks "Can Machines Think?" and 1956: Dartmouth Workshop - Birth of AI.
• Further timeline: 1966: ELIZA - First chatbot and 1972: MYCIN - First expert system.
• 1980s: Expert systems boom and 1997: Deep Blue beats Kasparov.
• 2011: Watson wins Jeopardy! and 2012: AlexNet - Deep learning revolution.
• 2014: AlphaGo beats Lee Sedol and 2018: GPT-2 - Large language models
• More timeline: 2020: GPT-3 - Even larger language models and 2022: ChatGPT - AI for everyone.

1. Objectives

• This course will provide students with a solid understanding and real-world application ability of data science and AI techniques.
• Course will also have hand-on experience with state-of-the0art technologies and tools and understand the ethical ramifications.

1. Intended Learning Outcomes

• Completion of the course nets AI understanding, the ability to solve real world problems, apply state-of-the-art AI tech.
• The course will provide a view into ethical implications, a means to critically evaluate results, and communicate effectively.

1. Course Information

• Course Instruction information is listed
• Teaching Assistant Names are provided
• A course website exists
• Course Schedule for a 14 week program is listed
• A grading system is listed including Assignments(30%), Midterm Exam(30%), Final Project(40%)
• A textbook is indicated
• Course prequisites listed: [Prerequisite 1], [Prerequisite 2], [Prerequisite 3]

1. Course Policies

• Attendance is expected for lectures and resitations.
• A late policy of 10% per day is incurred.
• Assignments are not accepted after 1 week.
• Full academic integrity abidance is expected and violation results in failure.
• Support for students with disabilities available.
• Students who need absence due to religious observance should notify the instructor.

1. Course Content

• The curse covers an Introduction to Data science and AI, what is it, and the history.
• This provides a foundation to Data Collection and Processing which entails integration and cleaning.

Exploratory Data Analysis

• Descriptive statistics
• Data visualization
• Hypothesis testing

Supervised learning which uses Regression which includes many techniques

• Linear regression
• Polynomial regression
• Logistic regression
• Support vector machines
• Decision trees

Supervised Learning is a classification which includes:

• Naive Bayes, k-nearest neighbors, SVMS, Decision trees, Ensemble methods

Unsupervised Learning including Clustering.

• k-means clustering
• Hierarchical clustering
• DBSCAN

Unsupervised learning including dimentionality reduction

• Principal component analysis and Linear discriminant analysis
• t-distributed stochastic neighbor embedding

Deep learning introduction

• Includes neural networks
• Includes Activation functions and Backpropagation algos.
• Also includes Optimization Algorithms

Deep learning Convolutional Neural Nets

• Includes Convolutional Layers and Pooling Layers
• Also includes Recurrent Neural Networks

More Deep learning

• Includes Recurrent Neural Networks and Long short-term memory and Gated recurrent unit

Many items under natural Language Processing including:

• Text processing, Sentiment analysis, Machine translation

Reinforcement Learning

• There is a Markov decision process
• Q-learning
• Deep reinforcement learning

The final section is dedicated to the Ethical and Societal implications of AI:

Bias, fairness, transparency and accountability in AI are examined.

Final Project

• Course concludes with a final project.

Project Description

• Students solve real-world problems by working in AI groups.

Project proposal

• Must submit by set deadline.
• Must include title, description, goals, data sources, methods and timelines.

Project Presentation

• A presentation must be given during allocated time.

Project Report

• All must submit a final report including similar information to a proposal as well as results and discussion sections.

Disclaimer

• A disclaimer that this can be changed by the institution.

Heat Capacity

• Heat capacity is amount of heat required to change a substance by one degree, usually measured in joules per Kelvin.

There are two distinct types of heat capacity:

1. "specific heat capacity"
2. "molar heat capacity"

Specific Heat Capacity

• To raise one degree Celsius on one gram of substance.
• Dependent on Mol structure and phase.
• Specific heat is $q = mc\Delta T$ Where:
• q is he heat transfered (J)
• M is mass
• C is specific heat
• Change in temperature is T

Molar Heat Capacity

• Raise one mole of a substance by single degree Celsius
• $q == nC\Delta T$ Where:
• q is the amount of heat energy transferred
• n is the amount of substance (moles)
• C is the molar heat capacity of the substance (J/(mol·°C))
• $\Delta T$ is the change in temperature (°C or K)

Relation between

Molar heat capacity at consistent pressure is bigger than at constant volume: $C_p = C_v + nR$ where:

• Cp is molar specificity.
• N is Moles
• R is gas constant

Degrees of Freedom

Molecule can move. Has translational, rotational, and vibrational

Monotonic Gas

3 DOF translation: $Cv =3/2 R$

Polytonic gas

5 DOF: Translation rotation is normal temp. $Cv= 5/2 R$ VIBRational at high temp.

Polyatomic

six degrees of freedom

Quick Start Guide

Software for Consultation Management

Welcome to Medilink

• Medilink is a cloud-based consultation managements with key features for payments, invoicing, patient records

How to Start

1. Register on the website
2. Register consultation
3. Add the patients
4. Schedule appts and send reminders 5.Manage invoicing and payments.

Key Features include:

• Appointment managment, appt reminders, patient storage
• Invoices payments
• Reports generations and mob app access via mobile.

Function Logratithm

The logarthm is noted in LN It is defined between .0 and infinity which has function

Logarthm Consequences

• Log is a drivable function
• (LN/x) = 1x ln the derivate.

Function

For all postive and integers:

• ln(ab) = lna + lnb ln=I/b = .Ibn ln(a2) 112ln a

Derivates

Foruma

ln is derivable over one interal l

First Chapter of Algebra and Analytic Geometry: Systems of Linear Equations

Introduction:

• a system can be defined in terms on N variables.
• Solutions must satify all n variables.

Elements of Systems

• operations define a system of numbers based on multplication, interchanges, adding constant.
• Theorem: Systems change due to systems theory.

Systems defined.

the coefficient augmented matrices have zero columns in a reduced Echelon system.

• Pivot, principale, lefre variable

Solutions Types

• A system can be compatible and inclompatible.
• Theorem: linear system.
• Theorem: Infinite solution exists.
• System: all conside rates are zero

Solution

• Has all solutions or 0.
• theorem Homogenous linear exists if infitine solutions exist.

System Requirements

If homogeneous requirements is less it will h ave infinite solutions.