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Fundamentals of Data Modeling (190.021)

Telefon: +43 3842 402 - 1901 �Email: cps@unileoben.ac.at

Univ.-Prof. Dr. Elmar Rueckert�

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WO AUS FORSCHUNG ZUKUNFT WIRD

Chair of Cyber-Physical-Systems

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Learning Goals

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  • Define in a real-world experiment the core components of a data modeling problem.

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  • Implement and understand the core components of a data modeling problem.

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  • Students can define the core components of data modeling for an independent invented problem.

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Let’s start with an Data Modeling Gummi Bear Experiment!

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Research Question / Goal

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  • Aristotle (384-322 BCE) proposed that objects fall to the ground because that is their natural state. He also asserted that heavier objects of the same material fall faster than lighter ones. �
  • Galileo Galilei (1554-1641) suspected that all objects, regardless of their mass, fall at the same rate in the absence of air resistance.�
  • We (2024-2025) want to proof or disproof that Gummi Bears bound together fall faster compared to a single Gummi Bear.

AI generated sketch of the two scientists Aristotle and Galileo Galilei, both studied objects at free fall but at different times.

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The Experiment

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  • Setup: Gummi Bears jump from a platform and we measure the falling time and the falling height. �
  • Assumptions (prior knowledge)
    • time-speed law of accelerated motion 𝑣 = 𝑎𝑡�
    • distance traveled can be computed by ℎ = ½ 𝑎 𝑡2�
    • we compute the accelerations 𝑎𝑖 = 2ℎ/𝑡2𝑖 for an experiment denoted by the subscript 𝑖.�
  • We (2024-2025) want to proof or disproof that Gummi Bears bound together fall faster compared to a single Gummi Bear.�
    • Our hypothesis: acceleration depend on the object’s mass: 𝑎𝑖(𝑚𝑖) = ℎ/𝑡2

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Small Quiz

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  • Why could Aristotle not measure the falling time? �
  • Assumptions (prior knowledge)
    • The largest building might have been about 100m height, e.g. the ‘Lighthouse of Alexandria’ with 118m (284-246 BCE).�
    • Given the distance traveled by ℎ = ½ 𝑎 𝑡2.�
    • Assume a = 10m/s.�
  • Compute the falling time!
    • Which time measurement systems were used at that time?
    • What was the problem that Aristotle faced?

Lighthouse of Alexandria’ with 118m (284-246 BCE), Wikipedia 2024.

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The Machine Learning Model

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  • Based on a number of 𝑖 = 1 . . . 𝑁 experiments:
    • We measure i weights and falling times and ‘store’ it in a table. �
    • We model the falling times ti via a simple line equation: ti = c1 + c2 mi. �
    • You might remember it as y = kx + d.�
    • Our machine learning goal is to determine the two parameters c1 and c2.�

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The Machine Learning Model

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ti

mi

ti = c1 + c2 mi

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Solution for Two Measurements

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t1 = c1 + c2 m1��t2 = c1 + c2 m2

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2)

3)

4)

using:

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Solution for Two Measurements

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t1 = c1 + c2 m1��t2 = c1 + c2 m2

1)

2)

3)

4)

using:

See page 69 in the DM Book!

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Solution for N Measurements

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1)

3)

2)

See pages 70-71 in the DM Book for a complete derivation!

solved via https://matlab.mathworks.com

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Solution for N Measurements

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1)

3)

2)

See pages 70-71 in the DM Book for a complete derivation!

solved via https://matlab.mathworks.com

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Obtained Solutions for C1 and c2

  • Two Measurement Model:

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  • N Measurements Model:

Blackboard Content

Blackboard Content

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Reproduction of the falling times using the model

Blackboard Content

  • What did we find out?
    • Is the model accurately predicting the measurements?�
    • If not, can you explain why?

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Review: What did we learn so far?

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Recap: Research Question / Goal

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  • Aristotle (384-322 BCE) proposed that objects fall to the ground because that is their natural state. He also asserted that heavier objects of the same material fall faster than lighter ones. �
  • Galileo Galilei (1554-1641) suspected that all objects, regardless of their mass, fall at the same rate in the absence of air resistance.�
  • We (2024-2025) want to proof or disproof that Gummi Bears bound together fall faster compared to a single Gummi Bear.

AI generated sketch of the two scientists Aristotle and Galileo Galilei, both studied objects at free fall but at different times.

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Recap: Reproduction of the falling times using the model

However, this just explains the data that we know already.

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Can we predict unknown future data?

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Let’s measure some more Gummi Bears

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With the goal of testing the model capabilities to:�

  • interpolate
  • extrapolate

Blackboard Content

  • What did we find out?
    • Is the model accurately predicting the measurements?�
    • If not, can you explain why?

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Thank you for your attention!

Univ.-Prof. Dr. Elmar Rückert

Chair of Cyber-Physical-Systems

Montanuniversität Leoben

Franz-Josef-Straße 18,

8700 Leoben, Austria

Phone: +43 3842 402 – 1901 (Sekretariat CPS)

Email: cps@unileoben.ac.at

Web: https://cps.unileoben.ac.at

Youtube: https://youtube.com/@CPSAustria

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Disclaimer: The lecture notes posted on this website are for personal use only. The material is intended for educational purposes only. Reproduction of the material for any purposes other than what is intended is prohibited. The content is to be used for educational and non-commercial purposes only and is not to be changed, altered, or used for any commercial endeavor without the express written permission of Professor Rueckert.

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