Vector Functions of Real Variable

Funciones Vectoriales de Variable Real

• A function that maps real numbers to vectors in $\mathbb{R}^n$.
• Denoted as $\overrightarrow{r}: \mathbb{R} \rightarrow \mathbb{R}^n$.

Componentes

• Vector functions can be expressed in terms of component functions: $\overrightarrow{r}(t) = \langle f_1(t), f_2(t), \dots, f_n(t) \rangle$.
• $f_i(t)$ are real functions of the real variable $t$.

Límite

• The limit of $\overrightarrow{r}(t)$ as $t$ approaches $a$ is a vector: $\lim_{t \to a} \overrightarrow{r}(t) = \langle \lim_{t \to a} f_1(t), \lim_{t \to a} f_2(t), \dots, \lim_{t \to a} f_n(t) \rangle$.
• Exists if each component function's limit exists.

Continuidad

A vector function $\overrightarrow{r}(t)$ is continuous at $t=a$ if:

• $\overrightarrow{r}(a)$ is defined.
• $\lim_{t \to a} \overrightarrow{r}(t)$ exists.
• $\lim_{t \to a} \overrightarrow{r}(t) = \overrightarrow{r}(a)$.

Derivada

• Derivative of $\overrightarrow{r}(t)$ is defined as $\overrightarrow{r}'(t) = \lim_{h \to 0} \frac{\overrightarrow{r}(t + h) - \overrightarrow{r}(t)}{h}$.
• Equivalently, $\overrightarrow{r}'(t) = \langle f_1'(t), f_2'(t), \dots, f_n'(t) \rangle$.

Integral

• Integral of $\overrightarrow{r}(t)$ is found component-wise: $\int \overrightarrow{r}(t) dt = \langle \int f_1(t) dt, \int f_2(t) dt, \dots, \int f_n(t) dt \rangle + \overrightarrow{C}$.
• $\overrightarrow{C}$ is a constant vector of integration.

Interpretación Geométrica

• $\overrightarrow{r}(t)$ traces a curve in space as $t$ varies.
• $\overrightarrow{r}'(t)$ is tangent to the curve at $\overrightarrow{r}(t)$, indicating instantaneous direction.

The Scientific Method

• A structured approach to scientific inquiry.

The Six Steps

• Step 1: Ask a Question
• Step 2: Do Background Research
• Step 3: Construct a Hypothesis
• Step 4: Test Your Hypothesis by Doing an Experiment
• Step 5: Analyze Your Data and Draw a Conclusion
• Step 6: Communicate Your Results

Vocabulary

• Hypothesis: An educated guess about how things work.
• Theory: A hypothesis tested with a significant amount of data.
• Law: A proven theory.
• Control Group: Does not receive experimental treatment in an experiment.
• Independent Variable: Deliberately changed variable.
• Dependent Variable: Measured variable.

1. 2 Exponential Functions

Exponential Functions

• For all real numbers $x$, the exponential function with base $a$ is $f(x) = a^x$.
• Applies where $a > 0$ and $a \neq 1$.
• Examples: $f(x) = 3^x$, $g(x) = 10^x$, $h(x) = (\frac{1}{2})^x$.

Graph of Exponential Functions

• Case 1: If $a > 1$, $f(x) = a^x$ is an increasing function with $\lim_{x \to -\infty} a^x = 0$ and $\lim_{x \to \infty} a^x = \infty$.
• Case 2: If $0 < a < 1$, $f(x) = a^x$ is a decreasing function with $\lim_{x \to -\infty} a^x = \infty$ and $\lim_{x \to \infty} a^x = 0$.

Algorithmic Trading

• Definition: Automated execution of orders using pre-programmed instructions.
• Variables: Price, timing and volume.
• Other Names: Automated trading, black-box trading, algo-trading.

How it Works

• Algorithm Development: Convert a trading strategy into computer code.
• Testing: Conduct back-testing using historical data.
• Execution: Algorithm is deployed on trading platforms.
• Monitoring: Continuous monitoring and adjustments.

Advantages

• Speed & Efficiency: Execute trades faster than humans.
• Reduced Emotional Bias: Removes emotional influence.
• Backtesting: Evaluate with historical data.
• Diversification: Ability to trade across markets and instruments.
• Cost Reduction: Lowers transaction costs.
• Improved Order Execution: Execute sizable orders without impacting market prices.

Disadvantages

• Technical Issues: System or connectivity failures.
• Over-Optimization: Poor live trading results from optimization based on historical data.
• Market Volatility: Algorithms may perform poorly.
• Monitoring & Maintenance: Requires ongoing monitoring and adjustments.
• Complexity: Sophisticated algorithms necessitate specific skills.

Types of Algorithmic Trading Strategies

• Trend Following: Capitalize on existing market trends (e.g., moving averages, MACD).
• Mean Reversion: Capitalize on prices reverting to average value (e.g., Bollinger Bands, RSI).
• Arbitrage: Exploit price differences in different markets (e.g., triangular, statistical arbitrage).
• Market Making: Provide liquidity by placing buy/sell orders, capturing bid/ask spread (e.g., HFT firms).
• Statistical Arbitrage: Use statistical models to identify pricing anomalies (e.g., pairs trading, factor models).
• Execution Algorithms: Optimize execution of large orders (e.g., TWAP, VWAP).

Tools and Technologies

• Programming Languages: Python, Java, C++.
• Trading Platforms: MetaTrader, TradingView.
• Data Feeds: Bloomberg, Reuters.
• Libraries: Pandas, NumPy, SciKit-Learn.

Skills Required

• Programming: Expertise in languages like Python, Java, or C++.
• Financial Knowledge: Understand financial market.
• Mathematics & Statistics: Essential for statistical analysis.
• Data Analysis: Analyze and interpret large datasets.
• Risk Management: Mitigate risks associated with trading.

Machine Learning: Unsupervised Learning

• Focuses on Clustering, involving grouping data without prior labels.

Types of Clustering

• Partitioning Clustering: Divides data into non-overlapping subsets.
  • Examples: k-means, k-medoids.
• Hierarchical Clustering: Creates a tree-like structure of clusters.
  • Types: Agglomerative (bottom-up) and Divisive (top-down).
• Density-based Clustering: Identifies clusters as dense regions.
  • Examples: DBSCAN, OPTICS.

K-Means

• Algorithm:
  • Choose the number of clusters $K$.
  • Randomly initialize $K$ cluster centroids.
  • Assign each data point to nearest centroid based on distance.
  • Recalculate the centroids of the clusters.
  • Repeat steps until convergence.
• Objective Function: Minimize Within-Cluster Sum of Squares (WCSS).

Choosing Optimal K

• Elbow Method: Plot WCSS for different $K$ values.
• Silhouette Score: Measures similarity within one's own cluster.

Pros

• Simple to implement
• Efficient for large datasets.

Cons

• Sensitive to initial centroid positions.
• Assumes clusters are spherical and equally sized.
• Requires specifying $K$ in advance.

Hierarchical Clustering

• Types:
  • Agglomerative (Bottom-Up): Start with each data point as a cluster, merge until one remains.
  • Divisive (Top-Down): Start with all data points in one cluster, divide into smaller ones.
• Dendrogram: Tree-like diagram showing the hierarchy.

Choosing Number of Clusters

• Cut the dendrogram at the desired level.

Pros

• Does not require $K$ in advance.
• Provides cluster hierarchy.

Cons

• Computationally expensive.
• Sensitive to noise and outliers.

DBSCAN (Density-Based Spatial Clustering of Applications with Noise)

• Parameters:
  • Eps (ε): Radius to search for neighbors.
  • MinPts: Minimum points within $\epsilon$ radius to consider an area dense.
• Point Types:
  • Core Point: At least MinPts neighbors within Eps radius.
  • Border Point: Not a core point, but has one within Eps radius.
  • Noise Point: Neither core nor border point.
• Algorithm:
  • Start with a data point and retrieve neighbors within Eps radius.
  • If enough neighbors, it's a core point, and a new cluster is formed.

Pros

• Does not require specifying the number of Clusters
• Does arbitrary shapes
• Robust to Outliers

Cons

• Sensitive to Parameter Setting
• May have difficulty with varying densities due to parameters

Evaluation Metrics

• Silhouette Score: ratio between distace between clusters, intra-cluster
• Davies-Bouldin Index: σ_i = average distance
• Calinski-Harabasz Index: ratio of between cluster variance

Economics

• The study of how societies use scarce resources to produce valuable goods and services.
• Studies how these resources are used among individuals.

Tow Key ideas

• Resources are scarce
• The society should use the resources in an efficient manner

Microeconomics

• The study about particular, individual components
• such as industries, companies and housholds.

Macroeconomics

• The study about an economy's general operation,
• GDP, Interest Rates, Jobs, Inflation

Three Fundamental Questions:

• what goods and services are produced
• how are the goods produced
• for whom are the goods made for

Economic Systems:

• Market: Decisions are made in markets between individuals/businesses normally through exchanges of money.
• Planned: Decisions are made by a central authority, employees are directed by the government
• Mixed: Elements of market and planned economies

Technological Possibilities of Society:

• Inputs such as are used to produce outputs
• A PPF shows the maximum amount of goods and service that an ecomony can produce

Opportunity cost:

• the cost of a decision is given by the next best alternative opportunity
• Productive efficiency:
  • when an economy cannot produce more of one, without producing less of the other
  • meaning an economy is on the PPF

Lecture 4. 1: Concept of Limit of a Function

Intuitive Idea

• Consider the function f(x)=(x^2-1)/(x-1)
• The Funtion is not defined at x=1
• What happens to the values as x approaches 1

Lecture 16

Numerical Integration

• Newcon-Cotes Integration Formula
• Uses function values at equaly spaced points
• Constructed to integrate polynomials of degree n exactly, where n+1 number of points

Basic Numerical Formulas

• Midpoint rule
• Trapezoid Rule
• Simpson's Rule

Composite Numerical Integration

• Divide (a,b) into n subintervals of length h