Euler's Method: Solving Initial Value Problems

Euler's Method

• Euler's Method is the simplest method for solving Initial Value Problems (IVPs) numerically.
• IVPs follow the format of $y' = f(t, y), \qquad y(t_0) = y_0$

Introduction to Euler's Method

• Euler's Method illustrates the basic ideas behind the numerical approximation of solutions of differential equations.
• The initial value is $y(t_0) = y_0$ and the solution is found at times $t_1 = t_0 + h, t_2 = t_0 + 2h, \dots, t_n = t_0 + nh$, where $h$ is a suitably chosen step size.
• At time $t_i$, the solution at time $t_{i+1} = t_i + h$ is approximated using the tangent line to the solution at $t_i$.
• Tangent Line Approximation: $y(t_{i+1}) \approx y(t_i) + y'(t_i)(t_{i+1} - t_i) = y(t_i) + f(t_i, y(t_i)) \cdot h$
• $y_i$ represents the approximation of $y(t_i)$.

Algorithm for Euler's Method

• Select a step size $h > 0$.
• Define initial conditions: $t_0 = t_0, y_0 = y_0$.
• For $i = 0, 1, 2, \dots$ compute the following:
• $t_{i+1} = t_i + h$
• $y_{i+1} = y_i + f(t_i, y_i) \cdot h$

Example Application of Euler's Method

• Problem: Use Euler's method with step size $h = 0.1$ to approximate the solution of the IVP $y' = y - t^2, \qquad y(0) = 1$ at the points $t = 0.1, 0.2, 0.3, 0.4,$ and $0.5$
• $f(t, y) = y - t^2$, $t_0 = 0$, $y_0 = 1$, and $h = 0.1$

Solution of Example Problem

• $t_1 = t_0 + h = 0 + 0.1 = 0.1$
• $y_1 = y_0 + f(t_0, y_0) \cdot h = 1 + (1 - 0^2) \cdot 0.1 = 1.1$
• $t_2 = t_1 + h = 0.1 + 0.1 = 0.2$
• $y_2 = y_1 + f(t_1, y_1) \cdot h = 1.1 + (1.1 - 0.1^2) \cdot 0.1 = 1.209$
• $t_3 = t_2 + h = 0.2 + 0.1 = 0.3$
• $y_3 = y_2 + f(t_2, y_2) \cdot h = 1.209 + (1.209 - 0.2^2) \cdot 0.1 = 1.3259$
• $t_4 = t_3 + h = 0.3 + 0.1 = 0.4$
• $y_4 = y_3 + f(t_3, y_3) \cdot h = 1.3249 + (1.3249 - 0.3^2) \cdot 0.1 = 1.45249$
• $t_5 = t_4 + h = 0.4 + 0.1 = 0.5$
• $y_5 = y_4 + f(t_4, y_4) \cdot h = 1.45249 + (1.45249 - 0.4^2) \cdot 0.1 = 1.597739$
• Approximate Solution: $y(0.1) \approx 1.1, y(0.2) \approx 1.209, y(0.3) \approx 1.3259, y(0.4) \approx 1.45249, y(0.5) \approx 1.597739$

Varying Step Sizes in Euler's Method

• Problem: Using Euler's method, approximate the solution of the IVP $y' = y - t^2, y(0) = 1$ at $t = 2$ with step sizes $h = 0.1, 0.05, 0.025, 0.0125$.
• Results: Varying approximations with each given h value

Approximate Results from Different Step Sizes

• h = 0.1, y(2) ≈ 6.42714718
• h = 0.05, y(2) ≈ 7.32772349
• h = 0.025, y(2) ≈ 7.87428417
• h = 0.0125, y(2) ≈ 8.17966091

Discussion on Accuracy of Euler's Method

• The exact solution of $y' = y - t^2, y(0) = 1$ is $y = t^2 + 2t + 2 - e^t$
• Thus: $y(2) = 2^2 + 2(2) + 2 - e^2 = 10 - e^2 \approx 2.6109439$
• Approximations are not very good, even with a small step size due to it not being a very accurate method in general.
• Euler's method uses the tangent line at the beginning of the interval to approximate the solution over the entire interval, leading to significant errors, especially if the solution is changing rapidly.

Bayes' Theorem

• The definition of Bayes' Theorem is the probability of an event based on prior knowledge of conditions that might be related to the event.
• It is used in probability theory and statistics.
• $P(A|B) = \frac{P(B|A)P(A)}{P(B)}$
• $P(A|B)$ is the posterior probability of A, given B is true.
• $P(B|A)$ is the probability of B, given that A is true.
• $P(A)$ is the prior probability of A.
• $P(B)$ is the prior probability of B.

Medical Diagnosis Example

• A disease affects 1 in 10,000 people.
• A test detects the disease with 99% accuracy but has a 5% false positive rate.
• If the test is positive, what is the probability of actually having the disease?
• A = patient has the disease
• B = patient tests positive
• $P(A) = 0.0001$ (prior probability of having the disease)
• $P(B|A) = 0.99$ (probability of positive test given the disease)
• $P(B|¬A) = 0.05$ (probability of positive test given no disease)
• To Find: $P(A|B)$ (probability of having the disease given a positive test)

Solving the Medical Diagnosis Example

• $P(B) = P(B|A)P(A) + P(B|¬A)P(¬A)$
• $P(B) = (0.99 \cdot 0.0001) + (0.05 \cdot 0.9999) = 0.000099 + 0.049995 = 0.050094$

Conclusion and Final Calculation of Example

• $P(A|B) = \frac{P(B|A)P(A)}{P(B)} = \frac{0.99 \cdot 0.0001}{0.050094} = \frac{0.000099}{0.050094} \approx 0.001976$
• Even with a positive test, the probability of actually having the disease is only 0.1976% due to disease rarity and the relatively high false positive rate of the test.

Machine Learning

• Definition: Algorithms that learn patterns from data without explicit programming.
• Example: Teaching a dog tricks; dog (algorithm) learns from treats (data).

Types of Machine Learning

• Supervised Learning
• Unsupervised Learning
• Reinforcement Learning

Definition: Supervised Learning

• Using labeled data ("treats" with names).
• Examples: Classification, Regression.

Definition: Unsupervised Learning

• Uses unlabeled data ("treats' without names).
• Examples: Clustering, Dimensionality Reduction.

Definition: Reinforcement Learning

• Learning through trial and error with rewards.
• Examples: Game playing, Robotics.

Supervised Learning

• Goal: Predict a label ((y)) based on input features ((x)).
• Inputs: (x) = Input features (e.g., email text)
• Outputs: (y) = Label (e.g., spam or not spam).

Types of Supervised Learning

• Classification: Predicting discrete categories. Example: Spam detection (yes/no).
• Regression: Predicting continuous values. Example: Stock prices.

Common Algorithms and Their Use

• Linear Regression (Regression): Predicting house prices
• Logistic Regression (Classification): Predicting Customer Churn
• Decision Trees (Both): Credit Risk Assessment
• Support Vector Machines (Both): Image Classification
• Neural Networks (Both): Natural language processing

Unsupervised Learning

• Goal: Discover hidden patterns in unlabeled data.

Types of Unsupervised Learning

• Clustering: Grouping similar data points. Example: Customer segmentation
• Dimensionality Reduction: Reducing the number of variables. Example: Feature extraction.

Common Algorithms and Their Use: Unsupervised Learning

• K-Means Clustering (Clustering): Customer Segmentation
• PCA (Dimensionality Reduction): Reducing noise in data
• Association Mining (Association): Market Basket Analysis

Reinforcement Learning

• Goal: An agent learns to make decisions in an environment to maximize a reward.

Key Components of a Reinforcement Learning System:

• Agent: The learner
• Environment: The world the agent interacts with
• Reward: Feedback for the agent's actions

Common Reinforcement Learning Algorithms and their Applications

• Q-Learning: Learning a Q-Function to make decisions - Game Playing
• SARSA: On-Policy learning algorithm - Robotics
• Deep Q-Networks: Using neural networks for Q-Learning - Complex Game Playing

Considerations for Machine Learning Applications

• Data Quality: Garbage in, garbage out.
• Overfitting: Algorithm learns noise instead of the signal.
• Bias: Systematic errors in the data.

Conclusion for Machine Learning Applications

• Can give many powerful tools for data analysis and prediction.
• Understanding the different types of algorithms and their applications is crucial for success.