Bernoulli Naive Bayes Classifier

Bernoulli Naive Bayes

• The Bernoulli Naive Bayes classifier is suitable for discrete data.
• This classifier is designed for binary or boolean features, unlike multinomial Naive Bayes, which uses occurrence counts.

Example Document Classification

• Documents are classified into "China" or "Not China" based on the occurrence of specific words.
• Used words like "Chinese," "Beijing," "Japan," "Shanghai," and "Tokyo" as features.
• Document #1 (Chinese, Beijing, Chinese) is represented as [1, 1, 0, 0, 0] for [Beijing, Chinese, Japan, Shanghai, Tokyo].

Contingency Table

• The contingency table summarizes word counts for each class, showing how many times a word appears in "China" versus "Not China" documents.
• For example, "Beijing" appears 2 times in "China" documents and 0 times in "Not China" documents.

Bernoulli Naive Bayes Formula

• $P(x_i \mid y) = P(i \mid y)x_i + (1 - P(i \mid y))(1 - x_i)$ is the formula for prediction in Bernoulli Naive Bayes.
• $P(i \mid y)$ is the probability of feature $i$ appearing in class $y$.
• $x_i$ is a binary variable (0 or 1) that indicates the absence or presence of the $i$th feature.

Advantages of Bernoulli Naive Bayes

• Simple and easy to implement.
• Computationally efficient.
• Appropriate for binary or boolean feature datasets.

Disadvantages of Bernoulli Naive Bayes

• Assumes features are independent, which is often untrue.
• Features that are highly correlated may result in poor performance.

Lecture 11: The Poisson Process

• Poisson processes can possess infinitely many states and offer properties for ease of use.
• Poisson processes are utilized for modeling various systems.
• Examples include customer arrivals at a store, phone calls at a call center, car traffic, earthquakes, and DNA mutations.

Definition of Poisson Process

• A Poisson process counts events over time satisfying these qualifications.
• Events occur one at a time.
• The time between events has an exponential distribution with rateλ.
• The number of events in disjoint intervals are independent.
• $N(t)$ is the number of events up to time $t$. ${N(t), t \geq 0}$ has independent increments.
• The number of events in any interval of length $t$ is Poisson distributed with mean $\lambda t$: $P(N(t+s) - N(s) = n) = e^{-\lambda t} \frac{(\lambda t)^n}{n!}, \quad n = 0, 1, 2, \dots$

Inter-arrival Times

• $T_i$ is the time between the $(i-1)$-th and $i$-th event. Then $T_i$ has an exponential distribution with rate λ.
• $T_1, T_2, \dots$ are independent.
• $S_n$ is the time of the $n$-th event where $S_n = \sum_{i=1}^n T_i$, said to have a Gamma distribution.

Memoryless Property

• The Poisson process has the memoryless property.
• Formally for any $s, t \geq 0$, $P(N(t+s) - N(s) = n \mid N(s) = k) = P(N(t+s) - N(s) = n) = e^{-\lambda t} \frac{(\lambda t)^n}{n!}, \quad n = 0, 1, 2, \dots$

Superposition

• The superposition of two independent Poisson processes with rates $\lambda_1$ and $\lambda_2$ is a Poisson process with rate $\lambda_1 + \lambda_2$.
• Provided $N_1(t)$ and $N_2(t)$ are two independent Poisson processes with rates $\lambda_1$ and $\lambda_2$, respectively, $N(t) = N_1(t) + N_2(t)$ is a Poisson process with rate $\lambda = \lambda_1 + \lambda_2$.

Decomposition

• A Poisson process with rate $\lambda$ exists where each event is classified as type 1 with probability $p$ and type 2 with probability $1-p$, independently of everything else.
• If $N_1(t)$ and $N_2(t)$ are the number of type 1 and type 2 events, respectively, then $N_1(t)$ and $N_2(t)$ are independent Poisson processes with rates $\lambda p$ and $\lambda (1-p)$, respectively.

Example 1: Customer Arrivals

• Customers arrive at a store according to a Poisson process with rate $\lambda = 10$ customers per hour.
• The probability that no customers arrive between 9:00 AM and 9:15 AM is approximately 0.082
• The probability that at least two customers arrive between 9:00 AM and 9:15 AM is approximately 0.713
• Given two customers arrive between 9:00 AM and 9:15 AM, the probability that both arrived between 9:00 AM and 9:10 AM is approximately 0.444

Example 2: Bus Arrivals

• Buses arrive with rate $\lambda = 2$ buses per hour according to a Poisson process.
• The probability that it takes more than 30 minutes for the first bus to arrive is approximately 0.368.
• The probability that exactly 5 buses arrive in a 2-hour period is approximately 0.156.
• The probability that exactly 3 full buses arrive in a 2-hour period, given each bus has a $\frac{1}{3}$ chance of being full, is approximately 0.082.

1 Sistemas de ecuaciones lineales

• Systems of linear equations.

1.1 Método de eliminación de Gauss

• Gauss elimination method.

Definición 1.1

• A system of $m$ linear equations with $n$ unknowns $x_1, x_2,..., x_n$ has the form:
  • $\begin{cases}
  • a_{11}x_1 + a_{12}x_2 +... + a_{1n}x_n = b_1 \
  • a_{21}x_1 + a_{22}x_2 +... + a_{2n}x_n = b_2 \ -... \
  • a_{m1}x_1 + a_{m2}x_2 +... + a_{mn}x_n = b_m
  • \end{cases}$
• $a_{ij}$ and $b_i$ are constants, with $i = 1, 2,..., m$ and $j = 1, 2,..., n$.

Ejemplo 1

• An example includes a system of 3 linear equations with 3 unknowns, like:
  • $\begin{cases}
  • 2x_1 + x_2 - x_3 = 1 \
  • x_1 - x_2 + x_3 = 2 \
  • x_1 + 2x_2 + x_3 = 3
  • \end{cases}$

Definición 1.2

• A solution is values for unknowns $x_1, x_2,..., x_n$ satisfying all equations simultaneously.

Ejemplo 2

• For Ejemplo 1, $x_1 = 1$, $x_2 = 1$ and $x_3 = 2$ is a solution.

Definición 1.3

• A consistent system has at least one solution, and an inconsistent system has no solutions.

Definición 1.4

• Equivalent systems have the same solutions.

Ejemplo 3

• The following systems of equations are equivalent:
  • $\begin{cases}
  • x_1 + x_2 = 2 \
  • x_1 - x_2 = 0
  • \end{cases}$
• and
  • $\begin{cases}
  • x_1 = 1 \
  • x_2 = 1
  • \end{cases}$
• Both have the same solution: $x_1 = 1$ and $x_2 = 1$.

1.1.1 Operaciones elementales

• The following operations can be performed to solve a system of linear equations:
  • Interchanging two equations.
  • Multiplying an equation by a nonzero constant.
  • Adding a multiple of one equation to another.
• Systems remain equivalent during these operations.

Biología Celular y Molecular

• Central Dogmas

Replicación

• ADN $\rightarrow$ ADN

Transcripción

• ADN $\rightarrow$ ARN

Traducción

• ARN $\rightarrow$ Proteína

Ingeniería Genética

• ADN recombinante $\rightarrow$ Clonación $\rightarrow$ Secuenciación $\rightarrow$ Organismos modificados

Niveles de Organización

Átomos

• $C, H, O, N, P, S$

Moléculas

• $H_2O$, Proteínas, Lípidos, ADN, ARN, Azúcares

Célula

• Unidad anatómica y funcional de todo ser vivo

Tipos de Células:

• Procariotas
• Eucariotas

Orgánulos

• Núcleo
• Mitocondria
• Ribosomas
• Retículo endoplasmático
• Aparato de Golgi
• Lisosomas
• Peroxisomas
• Plastos (solo en células vegetales)

Tejidos

• Asociación de células similares que realizan una función específica.

Tipos de Tejidos:

• Epitelial
• Conectivo
• Muscular
• Nervioso

Órganos

Estructura compuesta por diferentes tejidos que trabajan juntos para realizar una función común.

• Corazón
• Pulmones
• Cerebro
• Hígado
• Riñones

Sistemas

Grupo de órganos que trabajan juntos para realizar una función compleja y coordinada.

• Sistema circulatorio
• Sistema respiratorio
• Sistema nervioso
• Sistema digestivo
• Sistema excretor

Organismo

• Ser vivo individual, compuesto por todos los niveles de organización trabajando en conjunto.
• Bacteria
• Hongo
• Planta
• Animal

Población

• Grupo de organismos de la misma especie que viven en la misma área geográfica al mismo tiempo.

Comunidad

• Conjunto de poblaciones de diferentes especies que interactúan entre sí en una determinada área geográfica.

Ecosistema

• Comunidad de organismos que interactúan entre sí y con su entorno físico.

Biósfera

• Conjunto de todos los ecosistemas de la Tierra.

Chapitre 1 : Introduction

• Chapter 1: Introduction

1.1 Qu'est-ce que la science des données ?

• 1.1 What is Data Science?
• Data science is an interdisciplinary field extracting knowledge and insights from data.
• Involves scientific methods, processes, algorithms, and systems.
• Data can be structured or unstructured.
• Related to data mining, machine learning, and big data.

1.2 Composants de la science des données

• 1.2 Components of Data Science
• Consists of several main components:
  • Data Collection: Gathering data from databases, files, APIs, sensors, etc.
  • Data Preparation: Cleaning, transforming, and preparing data for analysis.
  • Data Analysis: Applying statistical and machine learning techniques to analyze data.
  • Data Visualization: Creating visualizations to communicate the results.
  • Deployment: Deploying data science models and applications in production.

1.3 Processus de science des données

• 1.3 Data Science Process
• The data science process is iterative. Consists of:
  • Problem Definition: Defining the business problem.
  • Data Collection: Collecting relevant data.
  • Data Preparation: Cleaning and preparing the data.
  • Data Analysis: Analyzing data to extract insights.
  • Data Visualization: Creating visualizations.
  • Deployment: Deploying models and applications.
  • Evaluation: Assessing the impact on the business.

1.4 Applications de la science des données

• 1.4 Applications of Data Science
• Wide range of applications:
  • Marketing: Personalized ads, optimized campaigns, predicting customer behavior.
  • Finance: Fraud detection, risk assessment, portfolio management.
  • Healthcare: Disease diagnosis, new treatments, patient care improvement.
  • Manufacturing: Optimized production, equipment failure prediction, quality improvement.
  • Retail: Price optimization, inventory management, customer experience improvement.

1.5 Outils de science des données

• 1.5 Tools for Data Science
• Many tools are available:
  • Programming Languages: Python, R, SQL
  • Libraries: NumPy, pandas, scikit-learn, TensorFlow, PyTorch
  • Visualization Tools: Matplotlib, Seaborn, Tableau, Power BI
  • Big Data Platforms: Hadoop, Spark

1.6 Conclusion

• 1.6 Conclusion
• Data science is a powerful field that can solve a range of problems.