ENGI 4421: Beam Deflection & Algorithmic Trading

• This document outlines a series of study notes for a student
• The course is ENGI 4421
• The student is A. Student, with ID 1234567, lab section 02
• The notes are dated for July 13, 2023
• The study notes cover Lab 7 Report, Lecture 24, Algorithmic Trading, Statics, and Advanced Data Analysis and Statistical Modelling I

Lab 7 Report: Beam Deflection Experiment

• Conducted on July 13, 2023.
• Experimental deflection compared to theoretical predictions under different loads.
• Objective is to determine beam deflection under different loads and compare with theoretical values.
• Analyze error sources.

Theoretical Background of Beam Deflection

• Deflection ($\delta$) formula for a simply supported beam with point load at the center: $\delta = \frac{PL^3}{48EI}$
• $P$: Applied load
• $L$: Beam length
• $E$: Modulus of elasticity
• $I$: Area moment of inertia
• Area moment of inertia ($I$) for a rectangular beam: $I = \frac{bh^3}{12}$
• $b$: Width
• $h$: Height

Experimental Setup

• Steel beam: Specified dimensions ($L$, $b$, $h$).
• Dial gauge: Measure deflection.
• Setup: Simply supported condition.

Procedure

• Measure: Steel beam dimensions.
• Setup: Beam on supports.
• Position: Dial gauge at the center.
• Apply & Record: Incremental loads and deflection for multiple load values.

Results

• Raw data is shown in a table showing load (N) versus Deflection (mm).
• Beam dimensions and Material Properties
• Length ($L$) = 1 m
• Width ($b$) = 0.025 m
• Height ($h$) = 0.05 m
• Modulus of Elasticity ($E$) = 200 GPa
• Area Moment of Inertia:
• $I = \frac{0.025 \times 0.05^3}{12} = 2.604 \times 10^{-7} m^4$
• Theoretical Deflection Calculation (for P = 100 N):
• $\delta = \frac{100 \times 1^3}{48 \times 200 \times 10^9 \times 2.604 \times 10^{-7}} = 0.001 m = 1 mm$
• The experiment results were very accurate

Discussion

• Measurement Errors: Account for measurement inaccuracies.
• Support Conditions: Account for deviations from simple support condition.
• Dial Gage Accuracy: Account for limited precision of the dial gauge.

Error Mitigation

• Use precise measuring instruments and calibrate the dial gauge.
• Ensure supports closely approximate simply supported conditions.

Conclusion

• Experimental results align with theoretical values.
• Potential sources of error identified with mitigation methods.

Lecture 24: The Lorentz Transformation

• Discusses Geometry, Derivation, and Consequences of Lorentz Transformation
• Reference materials: ISL Chapter 4, HTF Chapter 4.4
• The laws of physics are the same in all inertial reference frames
• The speed of light in a vacuum is the same for all observers
• Coordinate transformation that leaves space and time invariant are Lorentz transformations

Minkowski Space

• Define $w = ct$
• This is invariant under Lorentz transformations
• This is not a conventional distance because it can be negative
• $s^2 > 0$ "space-like" interval
• $s^2 < 0$ "time-like" interval
• $s^2 = 0$ "light-like" interval

Lorentz Transformation

• $x' = \gamma (x - vt)$
• $t' = \gamma (t - \frac{vx}{c^2})$
• where $\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}$
• When $v << c$, $\gamma \approx 1$

Algorithmic Trading

• Employs algorithms to execute trades
• Different strategies include:
• Trend Following
• Arbitrage
• Mean Reversion
• Market Making
• Statistical Arbitrage
• Execution Algorithms

HFT

• High-Frequency Trading (HFT) is a subset of algorithmic trading characterized by extremely high speeds
• Uses specialized infrastructure and co-location to gain a competitive edge
• Controversial due to its potential impact on market stability and fairness

Advantages

• Speed and Efficiency, Reduction in Emotional Bias, Backtesting Capabilities, Scalability, Precision, Reduced Transaction Costs and Diversification

Disadvantages

• Technical and Market Issues, Model Decay, Risk of Automation Bias, Over-Optimization
• Must account for Monitoring and Maintenance, Market Complexity, Data Dependency and Regulatory Scrutiny

Programming Languages

• Python, R, Java, C++, MATLAB

Statics: Vectors

• Scalar: magnitude
• Vector: magnitude and direction

Vector Operations

• Multiplication/Division by a Scalar: scales or reverses direction.
• Parallelogram Law: vector addition.
• Triangle Rule: vector addition.
• Addition is Commutative: $\vec{A}+\vec{B}=\vec{B}+\vec{A}$
• Addition is Associative: $\vec{A}+(\vec{B}+\vec{C})=(\vec{A}+\vec{B})+\vec{C}$
• Subtraction: $\vec{A}-\vec{B}=\vec{A}+(-\vec{B})$

Cartesian Vectors

• Right-hand coordinate system.
• Unit vectors: $\hat{\imath}, \hat{\jmath}, \hat{k}$
• Vector Representation: $\vec{A} = A_x\hat{\imath} + A_y\hat{\jmath} + A_z\hat{k}$
• Magnitude: $A = \sqrt{A_x^2 + A_y^2 + A_z^2}$
• Direction: Defined by angles $\alpha$, $\beta$, $\gamma$.
• Dot Product: $\vec{A} \cdot \vec{B} = AB\cos\theta$

Generalized Linear Models (GLMs)

• The exponential family is a distribution $f(y; \theta, \phi)$

GLM

• Random Component, Systematic Component and Link Function are required for GLM
• Link Functions are used for common choices:
• Identity: $g(\mu) = \mu$
• Log: $g(\mu) = log(\mu)$
• Inverse: $g(\mu) = 1/\mu$
• Logit: $g(\mu) = log(\frac{\mu}{1 - \mu})$
• Probit: $g(\mu) = \Phi^{-1}(\mu)$
• Complementary log-log: $g(\mu) = log(-log(1 - \mu))$

Logistic Regression

• Binary Logistic Regression: $Y_i \overset{ind}{\sim} Bernoulli(\mu_i)$, where $\mu_i = P(Y_i = 1)$.

Poisson Regression

• the Poisson regression is $Y_i \overset{ind}{\sim} Poisson(\mu_i)$, where $\mu_i > 0$ is the mean.
• Link Function: $log(\mu_i) = x_i^T\beta$
• $\mu_i = exp(x_i^T\beta)$