Why do we Need a Mathematical Model? SAMSI/CRSC Undergraduate Workshop 2006 Moustapha Pemy.

Slides:



Advertisements
Similar presentations
What Could We Do better? Alternative Statistical Methods Jim Crooks and Xingye Qiao.
Advertisements

Modeling of Data. Basic Bayes theorem Bayes theorem relates the conditional probabilities of two events A, and B: A might be a hypothesis and B might.
Experiments and Variables
Some terminology When the relation between variables are expressed in this manner, we call the relevant equation(s) mathematical models The intercept and.
Hazırlayan NEURAL NETWORKS Least Squares Estimation PROF. DR. YUSUF OYSAL.
A Physics Toolkit Chapter Physics Energy, matter and their relationship Understanding the physical world Careers –Scientists, astronomers, engineers,
Read Chapter 17 of the textbook
MAE 552 Heuristic Optimization
WFM 5201: Data Management and Statistical Analysis
The Basics of Regression continued
Linear and generalised linear models
Linear and generalised linear models
Basics of regression analysis
Linear and generalised linear models Purpose of linear models Least-squares solution for linear models Analysis of diagnostics Exponential family and generalised.
Using Origami to Find Rational Approximations of Irrational Roots Jeremy Lee Amherst Regional High School Hudson River Undergraduate Mathematics Conference.
A prediction or a forecast is a statement about the way things will happen in the future, often but not always based on experience or knowledge. While.
name___________________________ World of Physical Science
1 Dr. Jerrell T. Stracener EMIS 7370 STAT 5340 Probability and Statistics for Scientists and Engineers Department of Engineering Management, Information.
Ch 8.1 Numerical Methods: The Euler or Tangent Line Method
Prediction concerning Y variable. Three different research questions What is the mean response, E(Y h ), for a given level, X h, of the predictor variable?
SCI Scientific Inquiry The Big Picture: Science, Technology, Engineering, etc.
Mechatronic group at UiA 15 full time employed in teaching and labs. Mechatronic profile at UiA characterized by: High power / Power Mechatronics Dynamic.
Stats for Engineers Lecture 9. Summary From Last Time Confidence Intervals for the mean t-tables Q Student t-distribution.
MATH 224 – Discrete Mathematics
CISE301_Topic41 CISE301: Numerical Methods Topic 4: Least Squares Curve Fitting Lectures 18-19: KFUPM Read Chapter 17 of the textbook.
Discrete Mathematics Tutorial 11 Chin
Chapter 7 Models for Wave and Oscillations Variable Gravitational acceleration –Lunar Lander –Escape velocity Wave motion and interaction –Pendulum without.
Astrophysics ASTR3415: Homework 4, Q.2. Suppose there existed Velman cosmologists who were observing the CMBR when the light we now see from the supernova.
Regression Regression relationship = trend + scatter
Scientific Processes Mrs. Parnell. What is Science? The goal of science is to investigate and understand the natural world, to explain events in the natural.
< BackNext >PreviewMain Chapter 2 Data in Science Preview Section 1 Tools and Models in ScienceTools and Models in Science Section 2 Organizing Your DataOrganizing.
Measurement Metric units and what they mean to you…
FORECASTING. Minimum Mean Square Error Forecasting.
Scientific Method. My 4 out of 5 Rule If you make an observation.
Scale Factor and the relationship to area and volume GLE SPI:
9.3 and 9.4 The Spatial Model And Spatial Prediction and the Kriging Paradigm.
Chapter 1: The Science of Physics Section 1: An Intro to Physics.
Tuesday, August 28, 2012PHYS 1444, Dr. Andrew Brandt 1 PHYS 1444 – Section 003 Lecture #2 Tuesday August 28, 2012 Dr. Andrew Brandt 1.Introduction (longish)
Principles of Extrapolation
What is a Hypothesis? A hypothesis is a claim (assumption) about the population parameter Examples of parameters are population mean or proportion The.
Copyright © 2014, 2010, and 2006 Pearson Education, Inc. 5-1 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Chapter 5 Polynomials and Factoring.
The Nature of Science The Methods of Science Scientific Measurements Graphing.
Chapter 2 Measurements and Calculations. 2-1 Scientific Method The scientific method is a logical approach to solving problems by observing and collecting.
Bounded Error Parameter Estimation for Delay Differential Equations Adam Childers and John Burns Interdisciplinary Center for Applied Mathematics Virginia.
The inference and accuracy We learned how to estimate the probability that the percentage of some subjects in the sample would be in a given interval by.
Descriptive Statistics Used in Biology. It is rarely practical for scientists to measure every event or individual in a population. Instead, they typically.
Copyright © Cengage Learning. All rights reserved. 8 9 Correlation and Regression.
Copyright © by Holt, Rinehart and Winston. All rights reserved. Section 1 The Nature of Science Objectives  Describe the main branches of natural science.
Colorado Center for Astrodynamics Research The University of Colorado 1 STATISTICAL ORBIT DETERMINATION Statistical Interpretation of Least Squares ASEN.
Chapter 4 Some basic Probability Concepts 1-1. Learning Objectives  To learn the concept of the sample space associated with a random experiment.  To.
Calibration of Photomultiplier Arrays for Medical Imaging Applications Eric Kvam Engineering Physics Undergraduate.
Inference: Conclusion with Confidence
Robert Anderson SAS JMP
Nature of Science.
What makes a question testable?
Chapter 5 STATISTICS (PART 4).
Chapter 4: The Nature of Regression Analysis
Introduction to science
Statistical Methods For Engineers
Regression Models - Introduction
Inference for Proportions
Nanotechnology/Nanoscience Education Connections
Product moment correlation
Measurements in Science & Scientific Notation
Chapter 6 Section 4.
Financial Econometrics Fin. 505
Gold: Important concept. Write this down.
Chapter 4: The Nature of Regression Analysis
Chapter 3 Techniques of Differentiation
CORRELATION & REGRESSION compiled by Dr Kunal Pathak
Presentation transcript:

Why do we Need a Mathematical Model? SAMSI/CRSC Undergraduate Workshop 2006 Moustapha Pemy

1. The Reality, the experiment and the model. Let us denote by y true the true displacement of the beam, y data the data collected in the lab, and y the solution of the beam model.

1. The Reality, the experiment and the model. Let us denote by y true the true displacement of the beam, y data the data collected in the lab, and y the solution of the beam model. Question: Is it always possible to find a mathematical formula to express y true as function of time?

1.The Reality, the experiment and the model. 1.Let us denote by y true the true displacement of the beam, y data the data collected in the lab, and y the solution of the beam model. Question: Is it always possible to find a mathematical formula to express y true as function of time? How can we compare y true and y data ? How can we compare y true and y ? How can we compare y data and y ?

The Reality, the experiment and the model. 1. Answer: It is not always possible to find a mathematical expression of the reality. Due to measurement errors the data collected always differ from the reality.

The Reality, the experiment and the model. 1. Answer: It is not always possible to find a mathematical expression of the reality. Due to measurement errors the data collected always differ from the reality. In a mathematical model it is impossible to take into account all parameters of the experiment, for example in the beam model we do not take into account temperature of lab and any other gravitational forces that exist in the lab. This is why y true differs from y. As we have seen throughout this week, y data differs from y. This leads to the errors analysis in the model.

2. The data and the black box Assume we do not have a model. Can we just with data collected in the CRSC lab derive a good understanding of the system?

2.The data and the black box Assume we do not have a model. Can we just with data collected in the CRSC lab derive a good understanding of the system? Yes, just with data collected in CRSC lab we can interpolate the data using the least square approach or any other interpolation technique to derive functional relationship between the displacements of the beam and the times.

2.The data and the black box Assume we do not have a model, and that we have interpolated the data from the Beam in the CRSC lab and obtained a functional relationship between the displacements and the times.

2.The data and the black box Assume we do not have a model, and that we have interpolated the data from the Beam in the CRSC lab and obtained a functional relationship between the displacements and the times. Can we use this functional relationship to study a larger beam?

2.The data and the black box Assume we do not have a model, and that we have interpolate the data from the Beam in the CRSC lab and obtain a functional relationship between the displacements and the times. Can we use this functional relationship to study a larger beam? No, without a model we cannot use the functional relationship of a smaller beam to study a larger beam.

3. Interdependence of the model and the data Is it possible, just with experiments to obtain all necessary or desirable data of a physical system?

3. Interdependence of the model and the data Is it possible, just with experiments to obtain all necessary or desirable data of a physical system? No, there are certain data that are not observable, we cannot obtain them just with experiments. In fact, we need experiments and models in order to filter out unobservable data.

3. Interdependence of the model and the data Can we fit the model without experiments?

3. Interdependence of the model and the data Can we fit the model without experiments? No, in order to calibrate and validate the model we need experiments

3. Interdependence of the model and the data Why do scientists use both models and observed data?

3. Interdependence of the model and the data Why do scientists use both models and observed data? Control and design Navigation (Space shuttles, Satellites, Rockets) Predictions and Forecasting etc…

4. Models without data Give examples from science and industry where it is extremely difficult to collect data, and scientists mainly rely on mathematical models?

4. Models without data Give examples from science and industry where it is extremely difficult to collect data, and scientists mainly rely on mathematical models. Astrophysics: due to the large scale.

4. Models without data Give examples from science and industry where it is extremely difficult to collect data, and scientists mainly rely on mathematical models. Astrophysics: due to the large scale. Nanotechnology, quantum physics (very small scale).

4. Models without data Give examples from science and industry where it is extremely difficult to collect data, and scientists mainly rely on mathematical models. Astrophysics: due to the large scale. Nanotechnology, quantum physics (very small scale). The design and the testing of nuclear weapons.

4. Models without data Give examples from science and industry where it is extremely difficult to collect data, and scientists mainly rely on mathematical models. Astrophysics: due to the large scale. Nanotechnology, quantum physics (very small scale). The design and the testing of nuclear weapons. In Biomedical sciences, certain measurements in vivo can be destructive

4. Models without data Give examples from science and industry where it is extremely difficult to collect data, and scientists mainly rely on mathematical models. Astrophysics: due to the large scale. Nanotechnology, quantum physics (very small scale). The design and the testing of nuclear weapons. In Biomedical sciences, certain measurements in vivo can be destructive In the design and the development of jet engines and big airplanes like the Airbus A380 etc…