Intro to Machine Learning

Machine Learning

• Machine learning designs algorithms enabling computers to learn from data
• Learning relies primarily on data inputs

The Necessity of Machine Learning

• It facilitates example-based problem-solving for tasks defying definition
• Allows system adaptation to evolving environments
• Enables machines to extract and apply knowledge from raw data
• Offers a path to replace manually-coded expert systems
• Addresses complicated problems where manual programming is not feasible such as speech recognition

Supervised Learning

• Uses a data set {xᵢ, yᵢ}ᵢ=₁ᴺ, where xi represents the data point and yi is the label
• Employs N example input-output pairs (x₁, y₁),..., (x_N, y_N) to discover an approximation function h
• Requires 'y=f(x)' generated by an initially unknown function, where 'x' has d dimensions: x = (x₁, x₂,..., x_d)
• $y$ is continuous in regression
• $y$ is discrete in classification

Unsupervised Learning

• Structure discovery in an input data set x₁, …, x_N
• Involves clustering to discover groups of similar examples
• Includes dimensionality reduction, to discover a low-dimensional data representation

Reinforcement Learning

• Focuses on learning through experience, without direct supervision
• Relies on a clear, but often delayed, reward signal indicating success or failure
• Applications in game playing and control systems

Different Machine Learning Types

• Supervised learning uses training data that includes desired outputs
• Unsupervised learning uses training data without desired outputs
• Recommender systems predict user preferences for items, as used by Netflix and Amazon
• Reinforcement learning utilizes training data that provides occasional rewards or punishments

Steps in Developing a Learning System

• Collect data
• Clean and prepare data
• Model selection
• Model training
• Model evaluation
• Tuning of internal parameters
• Predicting results from the trained model

Choosing a Machine Learning Algorithm

• Key determinates include target function nature (regression/classification), training data volume, accuracy needs
• Requires balancing accuracy with transparency, training time, and prediction time

Curvas Planas

• A vector function maps real numbers to vectors

Representations of Vector Functions

• For a plane curve, 𝑟(𝑡) = <𝑓(𝑡), 𝑔(𝑡)> = 𝑓(𝑡) i + 𝑔(𝑡) j
• For a space curve, 𝑟(𝑡) = <𝑓(𝑡), 𝑔(𝑡), ℎ(𝑡)> = 𝑓(𝑡) i + 𝑔(𝑡) j + ℎ(𝑡) k

Vector Function Example

• The vector 𝑟(𝑡) = <𝑡, 𝑡²> represents a parabola if it is the position of a particle at time $t$

Vector Function Example

• The vector $\mathbf{r}(t) = \langle 2 \cos t, 3 \sin t \rangle \quad 0 \leq t \leq 2\pi$ traces an ellipse

Dervivatives of Vector Functions

• Given a vector function $\mathbf{r}(t)$ defined by $\mathbf{r}(t) = f(t) \mathbf{i} + g(t) \mathbf{j}$, where 𝑓 and 𝑔 are differentiable
• The derivative is calculated as $\mathbf{r}'(t) = f'(t) \mathbf{i} + g'(t) \mathbf{j}$ if 𝑓′(𝑡) and 𝑔′(𝑡) exist

Gas Pressure (5.1)

• Pressure equals force divided by the area to which the force is applied

Pressure Units

• 1 pascal (Pa): 1 N/m²
• 1 atmosphere (atm): 760 mmHg (exact)
• 1 atm: 101,325 Pa
• 1 atm: 14.7 psi (pounds per square inch)
• 1 bar: 10⁵ Pa

Gas Laws (5.2)

Boyle's Law

• Volume of gas varies inversely with its pressure
• $V ∝ 1/P$ at constant $n$ and $T$
• Expressed as $P_1V_1 = P_2V_2$

Charles's Law

• The volume of a gas is directly proportional to its absolute temperature, assuming constant n and P.
• $V ∝ T$ at constant $n$ and $P$
• Expressed as $\frac{V_1}{T_1} = \frac{V_2}{T_2}$

Avogadro's Law

• Volume and Number of moles are directly proportional when Pressure and Temperature are held constant
• $V ∝ n$ at constant $P$ and $T$
• Expressed as $\frac{V_1}{n_1} = \frac{V_2}{n_2}$

Ideal Gas Law (5.3)

Ideal Gas Law formula

• $PV = nRT$
• $R = 0.08206 \frac{L \cdot atm}{mol \cdot K} = 8.314 \frac{J}{mol \cdot K}$

Standard Conditions for Gases

• $T = 0 \degree C = 273.15 K$
• $P = 1 atm$

Molar Volume at STP

• 22.4 L

Calculating Gas Density and Molar Mass

• $d = m/V = \frac{PM}{RT}$
• $M = \frac{dRT}{P}$, where $d$ is density and $M$ is molar mass

Gas Mixtures (5.4)

• Dalton's Law describes Partial Pressures via $P_T = P_1 + P_2 + P_3 +...$

Mole Fraction

• $X_1 = \frac{n_1}{n_T}$
• $P_1 = X_1P_T$

Molecular View (5.5)

Kinetic Molecular Theory: 5 Principles

• Gases are composed of particles in constant, random motion.
• Particle volume is negligible compared to total gas volume.
• Inter-particle forces are negligible.
• Energy is transferred between particles during collisions, but average kinetic energy remains constant.
• Average kinetic energy is proportional to absolute temperature, as $KE = \frac{1}{2}mv^2$

Root Mean Square Speed

• $v_{rms} = \sqrt{\frac{3RT}{M}}$
• Where $R$ is the gas constant (8.314 J/mol⋅K), $T$ is the temperature in Kelvin, $M$ is the molar mass in kg/mol

Effusion and diffusion (5.6)

Graham's Law of Effusion

• Gas effusion rate varies inversely with root of molar mass: $\frac{Rate_A}{Rate_B} = \sqrt{\frac{M_B}{M_A}}$

Proof of Cauchy's Integral Formula

• If 𝑓 is analytic on 𝐷, and γ is a closed rectifiable curve in 𝐷 such that 𝑛(γ, 𝑤) = 0 for all 𝑤 ∉ 𝐷, then for 𝑎 ∈ 𝐷 ∖{γ}, 𝑓(𝑎) ⋅ 𝑛(γ; 𝑎) = (1 / 2𝜋𝑖) ∫γ (𝑓(𝑧) / 𝑧−𝑎) 𝑑𝑧

Cauchy's Integral Formula Proof

• $g(z) = \frac{f(z) - f(a)}{z-a}$ for $z ≠ a$ and $g(z) = f’(a)$ for $z = a$, then g is continuous on $D$ and analytic on $D$
• The Generalized Cauchy Theorem can be applied to find the result

Kirchhoff's Circuit Laws

• Used to describe electrical behavior
• Two laws relate current and voltage in an electric circuit

Kirchhoff’s Current Law

• Sum of currents entering equals sum of currents leaving a node: $\sum I_{entrada} = \sum I_{salida}$
• $I_1 + I_2 = I_3$ is the simplest example

Kirchhoff’s Voltage Law

• Net change in potential around any closed circuit is zero, $\sum V = 0$
• Increase is positive, a decrease is negative

Kirchhoff’s Voltage Law example

• For $V - V_1 - V_2 = 0$, this is the case when one power source and 2 resistors are in a loop

Using Kirchhoff’s Laws

• Analyzing resistive circuits: find voltages/currents with multiple resistors/sources
• Circuit design: calculating component relationships is the goal
• Analyzing circuits with dependent sources

Linear Algebra: Vectors

• A vector (denoted as $\vec{v}$) is defined by a direction (a line), a sense (an orientation), and by it size/norm (denoted $|\vec{v}|$
• Two vectors can be added when starting the first one at the origin
• A vector can be multiplied by a scalar which multiplies the norm of the vector by the scalar
• In an $n$ dimension space, a vector has $n$ coordinates: $\vec{v} = (v_1, v_2,..., v_n)$
• In $ℝ^2$: $\vec{v} = (x, y)$
• In $ℝ^3$: $\vec{v} = (x, y, z)$

Linear Algebra: Vector Operations

• Given $\vec{u} = (u_1, u_2,..., u_n)$ and $\vec{v} = (v_1, v_2,..., v_n)$
• Addition: $\vec{w} = \vec{u} + \vec{v} = (u_1 + v_1, u_2 + v_2,..., u_n + v_n)$
• Scalar Multiplication: $\vec{w} = \lambda \vec{v} = (\lambda v_1, \lambda v_2,..., \lambda v_n)$
• Norm: $|\vec{v}| = \sqrt{v_1^2 + v_2^2 +... + v_n^2}$

Linear Algebra: More Dot Products

• $\vec{u} \cdot \vec{v} = u_1v_1 + u_2v_2 +... + u_nv_n = |\vec{u}| |\vec{v}| \cos(\theta)$
• A measure of the angle, where θ is the angle between vectors
• Vector cross product (in dimensions $ℝ^3$)
• $\vec{w} = \vec{u} \times \vec{v} = (u_2v_3 - u_3v_2, u_3v_1 - u_1v_3, u_1v_2 - u_2v_1)$
• Where $|\vec{w}| = |\vec{u}| |\vec{v}| \sin(\theta)$
• w is orthogonal to both vector u and vector v

Linear Algebra: Equations of lines and planes

• Vector formula is $\vec{v} = \vec{p} + t\vec{d}$
• Point p is in the vector
• d is a directional vector
• $t$ belongs to all Real numbers: $t \in \mathbb{R}$
• Cartesian form (in $ℝ^2$) can be calculated by $ax + by = c$
• No unique solution for equation in $ℝ^3$, but a line can be an intersection of two planes

Linear Algebra: Planes

• With formula $\vec{v} = \vec{p} + s\vec{u} + t\vec{v}$
• p is a point in the plane
• u and v are 2 directional vectors
• parameters $s$ and $t \in \mathbb{R}$
• Can be expressed in Cartesian form as $ax + by + cz = d$ Where $\vec{n} = (a, b, c)$ is the normal vector to the plane

Radiative Processes: Emission

• Consider a cavity at temperature T with energy density $u_v(T)$

Kirchhoff's Law

• An object is placed emitting radiation ($j_v$) and absorption (αv
• At equillibrium $4\pi j_v = c\alpha_v u_v(T)$
• Energy density is equal to $u_v(T) = B_v(T) = \frac{2hv^3}{c^2}\frac{1}{e^{hv/kT}-1}$
• This means $j_v = \frac{c}{4\pi}\alpha_v B_v(T)$ ( a good absorber is an emitter)

Radiative Processes: BlackBody Emission

• This is emmission from a perfect blackbody

BlackBody Equations

• Assuming αv = 1 leads to emission = $j_v = \frac{c}{4\pi} B_v(T)$
• Flux emitted is equal to $F_v = \pi B_v(T)$
• Assuming F= flux and σ = Stefan-Boltzmann we have $F = \sigma T^4$
• With Luminosity equal to $L = 4\pi R^2 \sigma T^4$

Radiative Processes: GreyBody Emission

• This is similar to blackbody, but they emit a fraction of all radiation

GreyBody Equations

• Assuming αv = ϵ which is a constatnt we end up with the equation $j_v = \epsilon \frac{c}{4\pi} B_v(T)$
• The Flux can be expressed as $F_v = \epsilon \pi B_v(T)$
• The total flux can be expressed as $F = \epsilon \sigma T^4$
• Meaning the Luminosity has to be equal to $L = 4\pi R^2 \epsilon \sigma T^4$

Radiative Processes: Opacity

• The absorption coefficient $α_v$ is the opacity, and measure how well radiation is absorbed by something

Radiative Processes: Optical Depth

• Optical Depth: amount radiation is absorbed by a material: $\tau_v = \int \alpha_v ds$
• Material with $\tau_v < 1$ is optically thin, material with $\tau_v > 1$ is optically thick

Radiative Processes: Transfer Equations

• The Radiative Transfer Equation describes how the intensity of radiation ($I_v$) changes as it travels through a material
• It can be desribed with this equation: $\frac{dI_v}{ds} = -\alpha_v I_v + j_v$
• -αvIv represents absorption, and +jv represents emission
• Rewritten as $\frac{dI_v}{d\tau_v} = -I_v + S_v$, with Sv known as the source function: $S_v = \frac{j_v}{\alpha_v}$

Solutions to the Radiative Transfer Equation

• If Sv = 0, then intensity equals $I_v = I_{v,0} e^{-\tau_v}$
• If Sv = constant, then intensity equals $I_v = I_{v,0} e^{-\tau_v} + S_v(1 - e^{-\tau_v})$
• If the material is thick, then $I_v = S_v$
• The source function is equal to blackbody equations when the material is in equilibrium, meaning $I_v = B_v(T)$

HII and Molecular Radiative Regions

• HII Region - regions of ionized H around massive stars
• Molecular Region - the birthplaces of the stars, regions of gas and dust

Chemical Engineering Thermodynamics: Vapor-Liquid Equilibrium (VLE)

• Vapor-Liquid Equilibrium is covered in chapter 10

Thermo Determines

• The first thing we can see is WHAT ARE THE CRITERIA FOR EQUILIBRIUM
• Next we see WHAT IS THE EXTENT OF THE REACTIONS/PHASE TRANSFER
• Lastly, how the conditions affect it

Chapter 10 - Phase Equilibrium

• Multiple phase systems at equilibrium exist, when the rate of a species transfers from phase (α) to phase (β), as its rate from phase (β) to phase (α)

Open Multiphase System

• An example system is at a constant temperature T at pressure P
• nαi indicates the number of moles in phase alpha α
• nβi indicates the number of moles in phase beta β

Equilibrium Requirements

• Tα = Tβ = T
• Pα = Pβ = P
• Master Equation = Fαi = Fβi = Fi with F=Fugacity

Chapter 10 -The Phase Rule

• It is developed by J Willard Gibbs
• Degrees of freedom allows users to specify INTENSIVE state via: F = 2 - π + N F = DOF/Degrees of Freedom π is the number of phases N is the number of chemical species

Examples of System DOF

• Water triple threat: has N = 1 so only water, which also contains π = 3 for its 3 physical state The equation has a result of 0
• Propane VLE: has N = 1 and π = 2 . F = 1, T or P
• Propance-Butane VLE has N = 2 and π = 2. `F = 2, T&X1, or P&X1, T&P

Txy diagrams

• Top Curve = Dew curve, and Bottom curve is the bubble zone
• Above dew = vapor, and Below bubble is physical liquid
• Flash drum is important due to input of zi and we must determine xi, yi, V

Game Theory

• Mathematical analysis of strategic agent interactions with fields in computer science, economics, politics

Prisoner Dilemma

• Confessing will be best option to decrease their jail time, even when its not the ideal situation

Nash Equilibrium in Game Theory

• Players will stick to what they know how to win

Algorithmic Game Theory

• Includes computing NE, designing incentives, complexity of games
• VCG is a well known mechanism, the VickreyClarkeGroves

Algorithmic Game Theory: Mechanism Designs

• Design the game around inducing participants to behave in a desirable manner

Explore machine learning algorithms, their necessity, and learning from data inputs. Understand supervised and unsupervised learning techniques, including data sets and approximation functions. Differentiates between regression and classification.