Course Syllabus
OT.1x Observation theory: Estimating the Unknown
Index
1. Introduction
1.1 Learning objectives
1.2 Course activities and resources
1.3 What we expect from you
1.4 What you can expect from us/the course team
2. Course structure & Release dates
3. Assessment
4. Resources & Tools
5. Certificate
1. Introduction
The course introduces a standardized approach for parameter estimation, using a functional model (relating the observations to the unknown parameters) and a stochastic model (describing the quality of the observations). Using the concepts of least squares and best linear unbiased estimation (BLUE), parameters are estimated and analyzed in terms of precision and significance. The course ends with the concept of overall model test, to check the validity of the parameter estimation results using hypothesis testing. Emphasis is given to develop a standardized way to deal with estimation problems. Most of the course effort will be on examples and exercises from different engineering disciplines, especially in the domain of Earth Sciences.
Prerequisites: This course is aimed towards Engineering and Earth Sciences students at Bachelor’s, Master’s and postgraduate level. In this course we are using MATLAB, but don't worry if you don't know how to use it. Throughout the course you will have access to tutorials which will help you to know how to work with the tool. In addition, there is a module on Pre-knowledge Mathematics available from the beginning and which you may consult throughout the course.
1.1 Learning objectives
By the end of this course you will have learnt:
- how to translate real-life estimation problems to easy mathematical models
- practical understanding of least squares estimation and best linear unbiased estimation, and how to apply these methods
- how to assess and describe the quality of your estimators in the form of precision and confidence intervals
- how to check the validity of your estimation results.
1.2 Course activities and resources
This is a 6 module course with an estimated workload of 4 to 6 hours per module, holding several learning activities and educational resources for you to study. The course is organized in a consistent way so you know what to expect in each week.
- Warming up. Each module starts with a general introduction to the problem to be discussed, based on a real-life example.
- Learning modules. Each module consists of 2-3 video lectures and quizzes. The first video will be an introduction, while the others will give you additional examples or work out something in more detail. The quizzes are exercises or MATLAB problems related to the video's. A summary of the learning contents of each module is provide in the form as reading material.
- Assessment. Each module is finished with a graded quizzes and MATLAB exercises.
- Forum. There is a Q&A forum in which you can pose your questions, and answers can be provided by fellow learners or the course team.
Additional sections available at the beginning and throughout the course:
- Pre-knowledge mathematics module: review of most important concepts from linear algebra, statistics and calculus that will be used throughout the course.
- MATLAB learning content: tutorials with an introduction to MATLAB.
1.3 What we expect from you
As an online student we expect you to be an active participant, contributing to a positive atmosphere by questioning, sharing and helping out others using the Discussion Forum. Your active contribution is even more important since this is a self-paced version. Together you create a learning community!
This course is meant to be a save place where you learn with and from others. In this sense, we'd like you to be respectful to other participants. When communicating with your fellow learners and course team using the Discussion Forum make sure you take into account the Communication and Collaboration guidelines.
1.4 What you can expect from us/the course team
This is a self-paced course. That means that all course content is available from the start and it is up to you to decide on your pace. The course team will monitor and answer questions on a regular but not that frequent basis in the Discussion Forum. We rely on you to help each other out! We'll try to have at least a weekly presence in the forum, but as you might understand we cannot promise to quickly attend everyone, due to the thousands of learners participating in this course over many months. But we'll do our best ;-)
2. Course structure
0. Getting started
Familiarise yourself with the system, and take the opportunity to study the materials in the Pre-Knowledge mathematics module and the MATLAB tutorials.
1. Introduction
Introduction on what is “estimation” and when do we need it? What are the generic sources of uncertainty in observations, and what concepts are needed, e.g. deterministic vs. stochastic parameters, random vs. systematic errors, precision vs. accuracy, bias, and the probability distribution function as a metric of randomness. All the concepts are explained by various practical examples.
2. Mathematical models
Learn how to develop a systematic approach to translate real-life problems into mathematical models in the form of observation-equation system including four fundamental blocks: vector of observations, vector of unknown parameters, linear (or linearized) functional relation between observations and unknowns, and stochastic characteristics of observations in the form of dispersion (or covariance matrix) of the observation vector. As well as discussion on different concepts, such as linear vs. nonlinear models, functional vs. stochastic models, consistent vs. inconsistent models, over/under –determined models, redundancy, and solvability of observation-equation systems. All the aforementioned concepts are explained by various practical examples.
3. Least Squares Estimation (LSE)
Given a mathematical model, how to find an estimate that predicts the observations as close as possible? Introduction to (weighted) least squares estimation (WLSE), its mathematical logic and its main properties. Different applications of WLSE are demonstrated via practical examples, as well as discussion on numerical/computational aspects of applying WLSE.
4. Best Linear Unbiased Estimation (BLUE)
How to find the most precise and accurate estimate in linear models? Introduction to the concept of Best Linear Unbiased Estimation (BLUE), its theory and implication, and its relation to other estimators such as WLSE, maximum likelihood, and minimum variance estimators. The concept of BLUE and its application in various real problems are demonstrated by examples and exercises.
5. Non-linear least squares?
Up till this point, the theory and examples have been applied to linear functional models. However, in many practical situations this will not be the case: the relation between the observed and unknown parameters is then non-linear. In this week, students learn how to solve such non-linear problems using the concepts of least squares and BLUE by including a linearisation step. In this week, a more involved assignment will be given, allowing to apply all knowledge gained so far to a real-world problem (volcano modelling using deformation measurements).
6. Quality assessment (February 28, 12 UTC, 2019)
Discussion on how different errors in observations propagate to the estimates, applying the concept of error propagation. Students will learn how to infer confidence intervals or statistical tolerance levels of the results, and describe the expected variability of the estimation results. The interpretation of covariance matrices and confidence intervals is discussed and clarified via different examples and exercises.
Next, an introduction to a probabilistic decision making process (or statistical hypothesis testing) in validating the results of estimation in order to avoid wrong decisions/interpretation of the results. Students learn how to verify the validity of a chosen mathematical model, and how to detect and identify model misspecification/errors. The concepts are explained by various practical examples.
3. Assessment
Each of the six modules is finished with graded assignments, available to verified learners only (this is in order for edX to be able to continue providing freely accessible course contents). Note that:
- you may drop one out of the six assessments;
- each assessment comprises of two parts: one with exercises for which you don't need Matlab, and one with a Matlab exercise.
To complete this course you have to score at least 60 points of the total mark of 100. For each graded exercise the number of points is mentioned. You can see your progress in the Progress tab.
4. Resources & Tools
All educational resources will be available in the course. They consist of short videos and reading materials to support you in the completion of each module's learning activities: quizzes and MATLAB exercises.
MATLAB
We will be making use of the computing environment / programming language MATLAB (freely available for you to use during the course) - you don't need to be familiar with this. Tutorials are available and more examples will be provided in the modules. You may already familiarize yourself by looking at the tutorials and practicing in the online environment of module MATLAB Learning Content.
5. Certificate
If you're interested in a certificate you can upgrade to a Verified Certificate. These certificates will indicate you have successfully completed the course, but will not include a specific grade. Certificates will be issued by edX under the name of DelftX, designating the institutions from which the course originated.
Generating an ID verified certificate
If you have enrolled for a verified Certificate of Achievement you can generate your own certificate at any time after you have qualified to pass the course. To qualify for a certificate, you must achieve a total grade of 60% or higher. You can check your grade at any time under the course’s Progress page. An ID verified Certificate of Achievement is available for $50. You can Upgrade on your edX Dashboard to Verified during the course .
To request your certificate, select "Request Certificate" on the Progress Tab. After that, you can download your certificate directly from the Progress Tab, or from your Student Dashboard (look for the Download button next to the name of our course).
For more information on how to generate and download your certificates, explore edX’s documentation on how to generate your certificate.
Remember, you have until 28 February 2019, 12 UTC to finish all the modules and tests.
Once produced, a certificate cannot be reissued, hence it is very important that you verify the way in which your name appears. Check that, in your edx.org account, your name is correctly spelled, since it will appear on the final certificate. Please note that no Honor Code certificates will be given out by edX for this course.
***
LICENSE
Unless otherwise specified the Course Materials of OT.1x are Copyright Delft University of Technology and are licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
***
Click the button to print the course syllabus.