Vector Functions: Definition and Domains

• Functions that map real numbers to vectors are vector functions.
• A vector function is defined as $\vec{r}(t) = \langle f(t), g(t), h(t) \rangle = f(t)\hat{i} + g(t)\hat{j} + h(t)\hat{k}$, with $f(t)$, $g(t)$, and $h(t)$ being real-valued functions.
• The domain of $\vec{r}(t)$ consists of all $t$ values for which the function is defined.
• The domain is the intersection of the domains of its component functions: $\text{Dom}(\vec{r}(t)) = \text{Dom}(f(t)) \cap \text{Dom}(g(t)) \cap \text{Dom}(h(t))$.
• Example: The function $\vec{r}(t) = \left\langle \sqrt{4-t^2}, e^{-3t}, \ln(t+1) \right\rangle$ has a domain of $(-1, 2]$.
• The graph of $\vec{r}(t)$ is traced by the vector $\vec{r}(t)$ as $t$ varies through its domain.
• The graph is all points $(x, y, z)$ such that $x = f(t), y = g(t), z = h(t)$ for some $t$.
• Example: The graph of $\vec{r}(t) = \langle t, t^2 \rangle$ is the parabola $y = x^2$.
• The limit of a vector function is defined as $\lim_{t \to a} \vec{r}(t) = \left\langle \lim_{t \to a} f(t), \lim_{t \to a} g(t), \lim_{t \to a} h(t) \right\rangle$, provided the limits exist.
• Example: $\lim_{t \to 0} \left\langle t^2, \frac{\sin(t)}{t}, \cos(t) \right\rangle = \langle 0, 1, 1 \rangle$.
• A vector function $\vec{r}(t)$ is continuous at $t = a$ if $\vec{r}(a)$ is defined, $\lim_{t \to a} \vec{r}(t)$ exists, and $\lim_{t \to a} \vec{r}(t) = \vec{r}(a)$.
• A vector function is continuous at a point if its components are continuous.
• Example: $\vec{r}(t) = \left\langle t^2, \frac{\sin(t)}{t}, \cos(t) \right\rangle$ is discontinuous at $t = 0$.

Heat Waves

• Heat waves feature abnormally hot weather for extended periods.

Causes of Heat Waves

• High pressure systems trap warm air.
• Climate change increases average temperatures.
• The "urban heat island effect" describes the absorption of heat by buildings and pavement.

Impacts of Heat Waves

• Heat waves can lead to dehydration and heat stroke.
• Existing health conditions are worsened.
• Wildfires and droughts may occur.
• Economic activities can be disrupted.

Preparation for Heat Waves

• Stay informed about weather forecasts.
• Stay hydrated and cool, avoid strenuous activities.
• Check on vulnerable individuals.
• Conserve water.

Heat Wave Safety Tips

• Stay indoors.
• Drink fluids.
• Avoid strenuous activity.
• Wear light clothing.
• Take cool baths.
• Never leave people or pets in cars.
• Check on vulnerable people.
• Seek medical help for heat-related symptoms.

Heat Index Chart

• The heat index combines air temperature and humidity.
• Higher temperatures and humidity result in a higher heat index.
• Higher heat index values correspond to higher risk levels like moderate, high, and extreme.

Example Table

Algorithmic Trading

Introduction

• Algorithmic trading uses computer programs to execute orders based on pre-defined instructions.
• It aims to generate profits at high speeds and frequencies.

Advantages of Algorithm Trading

• Lower transaction costs by optimizing order placement.
• Efficient and accurate order execution through automation.
• Access to diverse markets simultaneously.
• Reduced errors through automation.
• Performance evaluation via backtesting on historical data.
• Elimination of emotional biases for rational decisions.

Disadvantages of Algorithmic Trading

• Model decay, where strategies lose effectiveness as market conditions shift.
• Data overfitting, leading to poor performance on new data.
• Inability to handle unexpected events.
• Potential for system errors causing losses.
• Need for constant monitoring.

Types of Algorithms

• Trend Following
• Mean Reversion
• Arbitrage
• Market Making
• Statistical Arbitrage
• Execution Algorithms including VWAP, TWAP, Implementation Shortfall

Definitions

• Backtesting: Evaluating a strategy's performance on historical data.
• Overfitting: Optimizing the strategy for historical data, resulting in poor performance on new data.
• Walkforward Analysis: Analyzing the algorithm on out of sample data.
• Risk Management: Identifying and managing risks like model decay and system errors.
• Latency: The time take to achieve a certain result.

Resources

• Some books including "Algorithmic Trading: Winning Strategies and Their Rationale" by Dr. Ernest Chan.
• Use Online Courses on Coursera or Udemy.
• Consider software like Python, R, or MetaTrader.
• Check helpful websites like Quantopian or Stack Overflow (Quantitative Finance).

Hypothesis Testing

• A measurement of whether results of a test are caused by chance or represent the population.
• Statistical significance is measured with the p-value.

P-value

• The probability of observing a test statistic as extreme or more extreme, assuming the null hypothesis is true.
• A small p-value means results are unlikely under the null hypothesis.

Drug Testing Example

• Null Hypothesis ($H_0$): The drug has no effect on blood pressure; μ=0.
• Alternative Hypothesis ($H_A$): The drug lowers BP; μ<0.
• Significance Level: Probability of rejecting null when true; α=0.05 typically.
• Test Statistic: T-statistic: $t = \frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}}}$
• Where: $\bar{x}$ is sample mean, μ is population mean under null, s is sample standard deviation, n is sample size
• P-value is the probability of observing a test statistic as extreme or more extreme than the calculated one, assuming the null hypothesis is true.
• Find this using a t-table or calculator
• Decision:
• If p-value < Significance Level (α): Reject Null Hypothesis
• Conclude the drug is effective at lowering blood pressure

Type I Error: Reject Null when True

• false positive
• Probability equals significance level alpha (α)

Type II Error: Fail to Reject Null when False

analogies

• Judge:
• $H_0$ : The defendant is innocent
• $H_A$ : The defendant is guilty
• Type I Error: Convicting an innocent person
• Type II Error: Letting a guilty person go free

Error Table

One-Tailed vs Two-Tailed Tests

• One-Tailed: Testing if parameter is > or < value.
• Two-Tailed: Testing if parameter is different from value.Two options
• If the Alternative is
  • Greater than: right tail
  • Less Than: Left Tail
    • Not Equal To: 2 Tail
      • Example functions
      • Z-test: Use population standard deviation, or when sample size is large ($n \geq 30$).
      • T-test: when population standard deviation is not known and sample size is small ($n \lt 30$).
        • Multiple Hypothesis Testing with example of Bonferroni correction.
        • Divide the Significance Level by the number of tests

Integration Exercices

• Let $f \in \mathcal{C}^{0}([0,1])$, compute $\lim_{n \rightarrow+\infty} \int_{0}^{1} x^{n} f(x) d x$.
• Compute $\lim {n \rightarrow+\infty} \int{0}^{+\infty} \frac{n \sin (x / n)}{x\left(1+x^{2}\right)} d x$.
• Compute $\lim {n \rightarrow+\infty} \int{0}^{+\infty} \frac{d x}{\left(1+x^{2} / n\right)^{n}}$.
• Compute $\lim {n \rightarrow+\infty} \int{0}^{n}\left(1-\frac{x}{n}\right)^{n} d x$.
• Let $f \in \mathcal{C}^{0}([0,1])$, determine $\lim {n \rightarrow+\infty} \int{0}^{1} \frac{n f(x)}{1+n^{2} x^{2}} d x$.
• Compute $\lim {n \rightarrow+\infty} \int{0}^{+\infty} \frac{\cos (x)}{1+x^{n}} d x$.
• Compute $\lim {n \rightarrow+\infty} \int{0}^{1} \frac{n x^{n}}{1+x} d x$.
• Let $f \in \mathcal{C}^{0}([0,1])$, determine $\lim {n \rightarrow+\infty} \int{0}^{1} f\left(x^{n}\right) d x$.
• Compute $\lim {n \rightarrow+\infty} \int{0}^{\pi / 2} \sin ^{n}(x) d x$.
• Compute $\lim {n \rightarrow+\infty} \int{0}^{+\infty} e^{-n x^{2}} d x$.
• Compute $\lim {n \rightarrow+\infty} \int{0}^{1} \frac{e^{-n x}}{\sqrt{x}} d x$.
• Compute $\lim {n \rightarrow+\infty} \int{0}^{+\infty} \frac{n}{1+n^{2} x^{2}} d x$.

Mathematica

• A premier technical computing environment is Mathematica.

Getting Started

• Key Actions includes typing input, then evaluating it through SHIFT+ENTER.
• Palettes provide interface for entering commands and symbols, accessible via the menu.
• Press F1 or ?FunctionName for documentation.

Core Basics

• Numerical Calculations:
• Example: 12345 * 67890 = 838102050.
• Functional Applications:
• Example: Plot Sin[x].

Actions

• Request a free Trial
• Watch the tutorials
• Access Docs
• Join the community

Bioinspired Algorithms for Optimisation

• Bioinspired algorithms are a method of optimization based on nature.

What

• What are bioinspired algorithms
• Examples include biology, such as animals, plants, and cells.

Why

• An alternative to optimisation algorithms.

Features

• are Robust
• Adapted to changes.
• Easy to Implement.

Types

• Algorithmic includes:

  • Genetic - evolving through populations. - Optimistic - swarm patterns.
  • Simulated-temperature levels.

  Collective Learning

  • Occurs amongst indidivuals - through cooperation and communication for difficult tasks.

    • Function thorugh experience - from a single individuals POV versus the larger group. - Bee and ant collonies

    Key Algorithms

    • Ant collonies
    • Particle-based designs
    • Artificial bee-based

Ant Logic

• Start a test and initiate patterns
  • test the graph by choosing new patterns based on factors and probabiliies.

Adv

• Robust & Adaptable

  • Complex
  • East to Implent

  PSO design

  • Start and evaluate each design one at a time. The best gets used. Keep comparing over time.

  ABC algorithm

  • Divide into 3 groups, search, assess and pick best food sourse/pattern. This results in a search for new food.

Chemical Kinetics

Reaction Rate

$aA + bB \rightarrow cC + dD$

• Rate $= -\frac{1}{a} \frac{d[A]}{dt} = -\frac{1}{b} \frac{d[B]}{dt} = \frac{1}{c} \frac{d[C]}{dt} = \frac{1}{d} \frac{d[D]}{dt}$

Rate Law

• Rate $= k[A]^x [B]^y$

• $k : constant

• $x : order with respect to A

• $y : order with respect to B

• $x + y$: overall order

     ####### Different Rate Orders
       Zeroth Order:
  

• $Rate = k$

• $ [A] = [A]_0 - kt$

• $t_{1/2} = \frac{[A]_0}{2k}$

       First Order:
  

• $Rate $= k[A]$

• $[A] = [A]_0 e^{-kt}$

• $ln[A] = ln[A]_0 - kt$

• $t_{1/2} = \frac{0.693}{k}$

        Second Order:
  

• Rate $= k[A]^2$

• $\frac{1}{[A]} = \frac{1}{[A]_0} + kt$

• $t_{1/2} = \frac{1}{k[A]_0}$

Arrhenius Equation

$k = Ae^{-\frac{E_a}{RT}}$

 -*k* : rate constant

• $A$ : frequency factor
• $E_a$: activation energy
• $R$ : gas constant $(8.314 J/mol \cdot K)$
• $T$ : temperature (K)

#$ln(k) = ln(A) - \frac{E_a}{RT}$*

#$\ln(\frac{k_2}{k_1}) = \frac{E_a}{R} (\frac{1}{T_1} - \frac{1}{T_2})$*