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Course Catalog 2010-2011
MAT-45807 Mathematics for Positioning, 4 cr |
Person responsible
Henri Pesonen, Robert Piche, Simo Ali-Löytty
Lessons
| Study type | P1 | P2 | P3 | P4 | Summer | Implementations | Lecture times and places |
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Requirements
Exam, approved week exercises (at least 34%) and at least half of returnable exercises need to be passed before the exam.
Completion parts must belong to the same implementation
Principles and baselines related to teaching and learning
The course consists of lectures, returnable exercises and week exercises. Solving week exercises at home gives bonus points (0-6 points) for exam.
Learning outcomes
Upon completing the required coursework, the student is able to analyze mathematically and apply different position methods. He/she can also apply linear algebra and the probability theory to real world problem. After the course the student has the necessary skills to use basic closed form static positioning methods, nonlinear Kalman filters, Gaussian mixture filters and particle filters. Moreover, the student is able to apply these methods to new position application.
Content
| Content | Core content | Complementary knowledge | Specialist knowledge |
| 1. | Review of linear algebra and probability theory: Schur decomposition, square root of matrix, overdetermined system of equations, non-singular and singular multivariable Gaussian (normal) distribution and expectation of random variable. | Independence of random variables; conditional probability density function; conditional expectation; expectation of affine transformation; Generation of the sample from the Gaussian distribution; Uniform distribution | rank theorem; null space; orthogonal matrix; properties of idempotent matrix; definiteness of a matrix; Gaussian mixture (sum) distribution; visualization of probability density function of the Gaussian distribution; chi-square distribution |
| 2. | Static positioning: measurement equations, iterative least squares method (Gauss-Newton method), maximum likelihood method and Bayesian method. | Weighted Gauss-Newton method; sensitivity analysis; closed form positioning formulas; Bayesian error analysis; Confidence intervals; Chebyshev inequality. | Bancroft method; Dilution of Precision (DOP)-numbers: GDOP, PDOP, HDOP, VDOP and TDOP; n-dimensionla Chebyshev inequality; coordinate systems. |
| 3. | Filtering: different variations of Kalman filter, general Bayesian filter. Optimal estimators: posterior mean, maximum a posteriori (MAP) and Best Linear Unbiased Estimator (BLUE); Nonlinear Kalman filters: Extended Kalman Filter (EKF) and Unscented Kalman Filter (UKF); particle filters: Sequential Importance Sampling (SIS), Sampling Importance Resampling (SIR) | Derivation of Kalman filter; constant velocity model; Monte Carlo integration; number of effective samples; systematic resampling; filters MatLab implementations | Stochastic process; white noise; Gaussian mixture filter; law of large number; different proposal distributions |
Evaluation criteria for the course
The final grade is based on the combined points from exercises and final exam. The exam will be "open book" style, meaning you can bring your pocket calculator and any written material you wish. Student have to get at least two bonus points and half of returnable exercises need to be passed before the final exam.
Assessment scale:
Numerical evaluation scale (1-5) will be used on the course
Partial passing:
Study material
| Type | Name | Author | ISBN | URL | Edition, availability, ... | Examination material | Language |
| Summary of lectures | Mathematics for Positioning | Simo Ali-Löytty et al. | download from course home page | English |
Prerequisites
| Course | Mandatory/Advisable | Description |
| MAT-20501 Todennäköisyyslaskenta | Mandatory | |
| MAT-33311 Tilastomatematiikka 1 | Mandatory | |
| MAT-34000 Tilastomatematiikka 2 | Mandatory |
Prerequisite relations (Requires logging in to POP)
Correspondence of content
| Course | Corresponds course | Description |
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Additional information
This course MAT-45807 is applying to the students whose main subject is mathematics. Course MAT-45806 is applying to the students whose main subject is not mathematics.
Courses MAT-45806 and MAT-45807 have common lectures.
Suitable for postgraduate studies
More precise information per implementation
| Implementation | Description | Methods of instruction | Implementation |
| Lectures Excercises Practical works |
Contact teaching: 0 % Distance learning: 0 % Self-directed learning: 0 % |