Study Notes for ENGR 1100: Intro to Engineering Problem Solving - Lesson 7

Introduction to Engineering Problem Solving

Lesson 7: Working with Data & Mathematical Modeling

Overview

• Engineers often need to make predictions based on sample data.

A Simple Prediction Problem

• Data Sample: A data sample represented as (x1, y1), (x2, y2), …, (xN, yN).
• Objective: Predict the value of y for a different input value x.

Prediction Process

1. Plot the Data
   • Use graphs to visualize the data sample, aiding in identifying an appropriate mathematical model.
   • Types of Graph Paper:
     • Rectilinear
     • Semi-Log
     • Log-Log
2. Select a Mathematical Model
   • Choose a model that best describes the sample data.
3. Apply a Mathematical Modeling Technique
   • Implement a suitable mathematical modeling approach based on the selected model.

Plotting the Data Sample

Importance of Graphing

• Plotting data is pivotal as it is the first step in prediction.
• Graph Elements:
  • Independent Variable: Plotted on the x-axis.
  • Dependent Variable: Plotted on the y-axis.

Guidelines for Good Graphs

• Ensure the graph is clear and follows standard formatting practices for scientific presentation.

Example 1: Semi-Log Graph

Data Visualization Case

• Context: Experimental data on cortisol concentration in response to a stress event.

Data Presented in Graph:

• X-Axis: Time (min)
• Y-Axis: Cortisol Concentration (nmol/L)
  • Values:
    • 0.1 nmol/L at 2.354 min
    • 0.5 nmol/L at 1.852 min
    • 1.0 nmol/L at 1.372 min
    • 2.0 nmol/L at 0.753 min
    • 5.0 nmol/L at 0.124 min
    • 10.0 nmol/L at 0.006 min

Mathematical Models

Definition

• Mathematical models are expressions that describe various phenomena. They help in making predictions using sample data.

Model Development

• Models can be developed based on:
  • Fundamental Laws: Established scientific principles.
  • Empirical Data: Based on observations and data collected.
  • Combination of Both: Integrating laws and data to formulate a comprehensive model.

Applications of Mathematical Models

• Engineers utilize mathematical models to predict outcomes through:
  • Model Interpolation
  • Model Fitting

Overview of Common Mathematical Models

1. Linear Model

Definition

• A linear model is characterized by being represented as a straight line when plotted on rectilinear graph paper.

Form

• $Y = mX + b$
  • Parameters:
    • m: slope of the line
    • b: y-intercept
    • Independent Variable: X
    • Dependent Variable: Y

2. Power-Law Model

Definition

• A power-law model can also be represented as a straight line but requires log-log transformation.

Form

• Original Untransformed Form:
  • $Y = bX^m$
• Linearized Form:
  • After transformation, this appears in a linear model form.
  • $m imes ext{log}(Y) = ext{log}(b) + m imes ext{log}(X)$

3. Exponential Model

Definition

• The exponential model is represented as a straight line on semi-log graph paper.

Form

• Original Untransformed Form:
  • $Y = b e^{mX}$
• Linearized Form:
  • On a semi-log plot becomes:
  • $mX = ext{log}(Y) - ext{log}(b)$

Model Selection Process

Overview

• The model selection process involves determining the most suitable mathematical model to represent the relationship within a given data sample.

Steps:

1. Plot the Data Sample
   • Utilize rectilinear, log-log, and semi-log graph paper.
2. Select the Best Fit
   • Choose the model that presents as a straight line on one of the graph types:
     • Straight line on rectilinear paper?
     • Straight line on log-log paper?
     • Straight line on semi-log paper?

Example 1 Revisited: Model Selection

Revisiting Graphs

• Assessment of which mathematical model is appropriate given visualizations on Semi-Log and Log-Log graphs.

Conclusion on Selection

• Final model choice must be based on the representation ability seen in visualizations with various graph types, considering clarity in presenting the data relationships.