Scalars and Vectors

Scalars

• Scalar quantity entirely specified by magnitude
• Examples of scalars include length, area, volume, time, mass, density, temperature, speed, and energy

Vectors

• Vector quantity defined with both magnitude and direction
• Examples of vectors includes position, displacement, velocity, acceleration, force, and moment
• Vectors are graphically represented by arrows, where the length is proportional to magnitude, the angle defines direction, and the head indicates sense.

Vector Algebra

• Vector addition: $\vec{R} = \vec{A} + \vec{B}$
• Commutative Law: $\vec{A} + \vec{B} = \vec{B} + \vec{A}$
• Associative Law: $\vec{A} + (\vec{B} + \vec{C}) = (\vec{A} + \vec{B}) + \vec{C}$

Vector Subtraction

• Vector subtraction is defined as: $\vec{A} - \vec{B} = \vec{A} + (-\vec{B})$

Scalar Multiplication

• $\vec{B} = a\vec{A}$ implies the magnitude of $\vec{B}$ is scaled by $a$
• $\vec{B}$ has the same direction as $\vec{A}$ if $a$ is positive, and the opposite direction if $a$ is negative

Vector addition for forces

• The resultant force $\vec{F}_R$ can be found by the equation $\vec{F}_R = \vec{F}_1 + \vec{F}_2$

Parallelogram Law

• Used to find the resultant force.

Procedure for Analysis

• To find the resultant force using the parallelogram law: sketch the vectors, find the magnitude, and find the direction.

Magnitude

• The Magnitude of the resultant force $\vec{F_R}$ is determined from the Law of Cosines or Sines

Direction

• The direction of the resultant force $\vec{F_R}$ is defined with respect to a fixed axis or with respect to one of the two component forces by the Law of Sines

The Law of Cosines

• $C = \sqrt{A^2 + B^2 - 2AB \cos{c}}$

The Law of Sines

• $\frac{A}{\sin{a}} = \frac{B}{\sin{b}} = \frac{C}{\sin{c}}$

Algorithmes gloutons

Principe

• An algorithm that makes locally optimal choices at each step to find a global optimum
• It does not guarantee the optimal solution
• It may be efficient and simple to implement

Structure générale

• Définir la fonction objectif à optimiser, identify the choices possible at each step, select the most promising choice (locally optimal), reduce the problem to solve, and repeat until a solution

Exemples classiques

• Problème du sac à dos fractionnaire, rendu de monnaie, ordonnancement de tâches

Avantages

• Simple to understand and implement
• Peu Couteux en termes de ressources (temps et mémoire)

Inconvénients

• Ne garantit pas toujours la solution optimale.
• Nécessite une preuve de l'optimalité dans certains cas.

Preuve d'optimalité

• Il est crucial de prouver que l'algorithme glouton fournit bien la solution optimale.
• Techniques de preuve courantes: argument d'échange, induction

Conclusion

• Les algorithmes gloutons are an interesting approach to solving optimization problems, but it is essential to understand their limits and prove their optimality when necessary

Economía

• Economics is the study of how society manages its scarce resources.

Scarcity

• Society has limited resources and cannot produce all goods and services people wish to have.

Economies is the study of:

• How people make decisions about where they work, what they buy, how much they save, how they invest
• How people interact with each other
• Analysis of trends affecting the economy as a whole

Ten Principles of Economics

• People Face Trade-offs
• The cost of something is what you give up to get it
• Rational people think at the margin
• People respond to incentives.
• Trade can make everyone better off.
• Markets Are Usually a Good Way to Organize Economic Activity.
• Governments can sometimes improve market outcomes
• A country's standard of living depends on its ability to produce goods and services.
• Prices rise when the government prints too much money.
• Society faces a short-run trade-off between inflation and unemployment.

People Face Trade-offs

• You can't eat your cake and continue to have it.
• Decision making requires trading one goal for another.

Examples include:

• Students distributing their time
• Parents spending the family income
• Society's "Guns Vs butter" trade off

Efficiency

• Society getting the benefits from its scarce resources

Equity

• Benefits of those resources are distributed uniformally in society.

The Cost of Something Is What You Give Up to Get It

• Because people face trade-offs, decision-making requires comparing the costs and benefits of alternate courses of action.

Opportunity Cost

• What you must sacrifice to obtain something.

People think at the margins

• Comparing the benefits and costs

Rational Person

• Someone systematically works to achieve goals from a plan of action

Marginal changes

• Small adjustments and incremental adjustments to an existing plan of action.

People Respond to Incentives

• Rational people make decisions by comparing costs and benefits to respond to incentives.

Incentive

• Motivates a person to act, such as a reward or punishment.

Trade can make everyone better off

• Allows each person to specialize in activities they perform best
• Can purchase a variety of goods and services for less costs.

Markets Are Usually a Good Way to Organize Economic Activity

• Markets assign households with the means to goods and services through interactions

Market Economy

• Decided on what to buy and where to work, while firms decide who to hire and what to produce.

Adam Smith observed that homes and business interact as if led by an "invisible hand'

• Leads to markets obtaining desired results

The Government Can Sometimes Improve Market Outcomes

• The invisible hand leads markets to efficiently assign resources

Market Failure

• When the market does not efficiently assign resources

Externality

• Impact of a person's action on the wellfare of a stranger

Market Power

• Ability for an economic actor to influence prices

Productivity

• The number of goods and services produced by each unit of work.

Inflation

• An increase in the level of prices in the economy.
• When the government creates a large amount of money, the value decreases

Phillips Curve

• Illustrates the trade off between Inflation and unemployment

Algorithmes gloutons

Objectifs

• Describe the characteristics of greedy algorithms
• Design greedy algorithms to solve problems
• Analyze the advantages and disadvantages of greedy algorithms
• Identify the problems that can be solved using greedy algorithms
• Apply greedy algorithms to real problems.

Introduction

• A greedy algorithm can be used to solve optimization problems
• the idea is to make the most promising choice at each step, without worrying about future consequences.

Characteristics of Greedy Algorithms

• Optimalité locale, simplicité, efficacité, absence de retour en arrière

Conception d'algorithmes gloutons

• Définir le problème d'optimisation, determining the selection, and verifying if the solution is optimal

Avantages

• Simple to understand and implement
• Time efficient
• Capacity to quickly find a solution

Inconvénients

• Ne garantissent pas toujours une solution optimale globale, Peuvent conduire à des solutions sous-optimales, Nécessitent une preuve d'optimalité dans certains cas pour garantir que la solution obtenue est la meilleure possible.

Exemples d'algorithmes gloutons

• Algorithme de Dijkstra, Algorithme de Kruskal, Algorithme de Huffman, probléme du sac à dos, problème du sac à dos.

Quand utiliser les algorithmes gloutons

• Le problème peut être divisé en sous-problèmes plus petits, Une solution optimale locale à chaque sous-problème conduit à une solution optimale globale, Il n'est pas nécessaire de revenir sur les décisions prises précédemment.

Conclusion

• Les algorithmes gloutons are a tool for solving and finding a global optima.

Algorithmic Trading

Regression

• Linear Regression: $y = X\beta + \epsilon$ and $\hat{\beta} = (X^TX)^{-1}X^Ty$
• Polynomial Regression: $y = X\beta + \epsilon$ with $X = \begin{bmatrix} 1 & x_1 & x_1^2 & \dots & x_1^n \ 1 & x_2 & x_2^2 & \dots & x_2^n \ \vdots & \vdots & \vdots & \ddots & \vdots \ 1 & x_m & x_m^2 & \dots & x_m^n \end{bmatrix}$
• Support Vector Regression: $\min \frac{1}{2} ||w||^2 + C \sum_{i=1}^{n} (\xi_i + \xi_i^)$, $y_i - w^T \phi(x_i) - b \le \epsilon + \xi_i$, $w^T \phi(x_i) + b - y_i \le \epsilon + \xi_i^$, and $\xi_i, \xi_i^* \ge 0$
• Gaussian Process Regression: $y \sim N(0, K)$ and $f_* | X_, X, y \sim N(K(X_, X)K(X, X)^{-1}y, K(X_, X_) - K(X_, X)K(X, X)^{-1}K(X, X_))$

Classification

• Logistic Regression: $P(y=1|x) = \frac{1}{1 + e^{-x^T \beta}}$
• Support Vector Machine: $\min \frac{1}{2}||w||^2 + C \sum_{i=1}^{n} \xi_i$, $y_i(w^T \phi(x_i) + b) \ge 1 - \xi_i$, and $\xi_i \ge 0$
• Gaussian Process Classification: $p(y|f) = \Phi(yf)$ and $p(f|X) = N(0, K(X, X))$

Time Series

• ARIMA: $x_t = c + \phi_1 x_{t-1} + \phi_2 x_{t-2} + \dots + \phi_p x_{t-p} + \epsilon_t + \theta_1 \epsilon_{t-1} + \theta_2 \epsilon_{t-2} + \dots + \theta_q \epsilon_{t-q}$
• Kalman Filter: State Transition $x_t = F_t x_{t-1} + B_t u_t + w_t$ and Observation $z_t = H_t x_t + v_t$

Execution

• Order Book: Limit Order Book (LOB), Market Order, Limit Order
• Market Impact: Temporary Market Impact and Permanent Market Impact

Optimal Execution

• VWAP: Minimize $\sum_{i=1}^{T} |x_i - v_i|$ subject to $\sum_{i=1}^{T} x_i = X$
• TWAP: Minimize $\sum_{i=1}^{T} |x_i - \frac{X}{T}|$ subject to $\sum_{i=1}^{T} x_i = X$
• Almgren-Chriss Model: Minimize $\sum_{i=1}^{N} \left( \gamma \sigma^2 x_i^2 \delta t + \lambda \frac{x_i}{\delta t} \right)$ subject to $\sum_{i=1}^{N} x_i = X$

Risk Management

• Value at Risk (VaR) methods include Historical Simulation, Variance-Covariance Method ($VaR = \mu + \sigma z_{\alpha}$), and Monte Carlo Simulation
• Expected Shortfall (ES): $ES = E[X | X \le VaR]$
• Sharpe Ratio: $Sharpe = \frac{E[R_p - R_f]}{\sigma_p}$

Backtesting Metrics

• Maximum Drawdown is the maximum loss from a peak to a trough
• Information Ratio: $\frac{E[R_p - R_b]}{\sigma_{p-b}}$
• Alpha and Beta: $R_p = \alpha + \beta R_b + \epsilon$

Regime Detection

• Hidden Markov Models (HMM): $p(s_t | s_{t-1}) = A_{s_{t-1}, s_t}$ and $p(o_t | s_t) = B_{s_t, o_t}$
• Switching Regression: $y_t = X_t \beta_{s_t} + \epsilon_t$

Overfitting strategies

• Cross-Validation
• Walk-Forward Analysis
• Regularization (L1 and L2)

Cascade control system

• Determining $K_1$ and $K_2$ allows cascade control to perform better than a single loop control.

Inner Loop Formula

• $G_{CL} = \frac{G_{c2}G_{p2}}{1+G_{c2}G_{p2}} = \frac{3K_2}{5s+1+3K_2} = \frac{\frac{3K_2}{1+3K_2}}{\frac{5}{1+3K_3}s+1}$
• We want the time constant, $𝜏$, to be small.

Outer loop

• Used in finding $ω_𝑛$ and $ξ$ where $ω_n = \sqrt{\frac{1+1.934K_1}{1.61}}$
• $ξ = \frac{10.161}{2\sqrt{1.61(1+1.934K_1)}}$

Without Cascade Control

• $ω_𝑛$ can be found by the equation $ω_n = \sqrt{\frac{1+6K_1}{50}}$

Choose $K_1$ such that

• $ξ = \frac{15}{2\sqrt{50(1+6K_1)}}$

The benefits of choosing the system with the lower $ξ$

• Can be more stable than single loop control

The measurable disturbance with cascade control

• Cascade control can reject it

If there is $|G_p G_c| >> 1$

• The disturbance variable is $\approx 0$

Cascade Control for Disturbance Rejection

• Used to determine the disturbance variable

If $|G_{p1}G_{p2}G_{c2}G_{c1}| >>1$

• Then the distrubance variables is approximately 0.

$G_{c2}$ design

• Such that $|G_{p2}G_{c2}|$ is large enough for disturbance rejection. Increase $G_{c1}$ until the system is unstable, then reduce it a little bit.

Frequency Response

• Steady-state response of a system to a sinusoidal input signal
• It is useful for:
  • Predict output of a system
  • Characterize a system
  • Enables design in the "frequency domain"

Magnitude and Phase

• For a system with transfer function $H(s)$, the frequency response is given by $H(j\omega)$, where $\omega$ is the frequency of the input signal. $H(j\omega)$ is a complex number that can be expressed in polar form as:
  • $H(j\omega) = |H(j\omega)|e^{j\phi(\omega)}$

Bode Plots

• Visualizes frequency response
• Magnitude plot shows $|H(j\omega)|$ in decibels (dB) versus $\omega$ on a logarithmic scale
• Phase plot shows $\phi(\omega)$ in degrees versus $\omega$ on a logarithmic scale

The Magnitude in dB

• Calculates as $|H(j\omega)|{dB} = 20\log{10}|H(j\omega)|$

First and Second Order Systems

• Equations and important points for sketching

Système de santé canadien

• Le Canada se classe régulièrement parmi les 10 meilleurs pays au monde en termes de qualité des soins de santé. Cependant, d'autres pays obtiennent de meilleurs résultats en matière d'indicateurs clés tels que l'espérance de vie et les décès évitables.

Objectif

• Comparer le système de santé du Canada avec ceux d'autres pays développés afin d'identifier les forces et les faiblesses du système canadien et former des recommandations.

Méthodologie

• Examen and interviews with those in healthcare

Key Takeaways

• Le Canada dépense beaucoup en soins de santé
• Les temps d'attente sont longs
• Prévention des maladies chroniques

Recommandations

• Améliorer l'accès aux soins et prévenir les maladies chroniques en investissant davantage dans les soins de santé.

Conclusion

• Canada can learn from other countries in order to improve their system

Measuring wellbeing

Three aspects to consider

• Subjective wellbeing
• Objective wellbeing

Subjective wellbeing

• Asking people how they feel and using questionnaires.

Objective wellbeing

• Measures not based on asking people how they feel examples include levels of stress hormones, heart rate variability, activity levels, and brain activity
• A problem to consider is that they are affected by other factors

Improving wellbeing

• Consider the Hawthorne affect, where people improve due to being measured
• Multi faceted measurements are needed to ensure the system is sound and accurate