analysis and propagation of errors
DESCRIPTION
Analysis and propagation of errors. Peter Fox GIS for Science ERTH 4750 (98271) Week 8, Tuesday, March 20, 2012. Contents. Error!!! Projects Lab assignment on Friday. Spatial analysis of continuous fields. Possibly more important than our answer is our confidence in the answer. - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Analysis and propagation of errors](https://reader036.vdocument.in/reader036/viewer/2022062301/56813fae550346895daa92cf/html5/thumbnails/1.jpg)
1
Peter Fox
GIS for Science
ERTH 4750 (98271)
Week 8, Tuesday, March 20, 2012
Analysis and propagation of errors
![Page 2: Analysis and propagation of errors](https://reader036.vdocument.in/reader036/viewer/2022062301/56813fae550346895daa92cf/html5/thumbnails/2.jpg)
Contents• Error!!!
• Projects
• Lab assignment on Friday
2
![Page 3: Analysis and propagation of errors](https://reader036.vdocument.in/reader036/viewer/2022062301/56813fae550346895daa92cf/html5/thumbnails/3.jpg)
Spatial analysis of continuous fields
• Possibly more important than our answer is our confidence in the answer.
• Our confidence is quantified by uncertainties as discussed earlier.
• Once we combine numbers, we need to be able to assess how the uncertainties change for the combination.
• This is called propagation of errors or more correctly the propagation of our understanding/ estimate of errors in the result we are looking at…
3
![Page 4: Analysis and propagation of errors](https://reader036.vdocument.in/reader036/viewer/2022062301/56813fae550346895daa92cf/html5/thumbnails/4.jpg)
Types of errors• Mistakes
• Natural variation
• Systematic and random equipment problems
• Data collection methods
• Observer diligence
• Locations errors/accuracy
• Rasterizing and digitizing
• Mismatch of data collected by different methods (e.g., seafloor bathymetry)
4
![Page 5: Analysis and propagation of errors](https://reader036.vdocument.in/reader036/viewer/2022062301/56813fae550346895daa92cf/html5/thumbnails/5.jpg)
Bathymetry
5
![Page 6: Analysis and propagation of errors](https://reader036.vdocument.in/reader036/viewer/2022062301/56813fae550346895daa92cf/html5/thumbnails/6.jpg)
Cause of errors?
6
![Page 7: Analysis and propagation of errors](https://reader036.vdocument.in/reader036/viewer/2022062301/56813fae550346895daa92cf/html5/thumbnails/7.jpg)
Resolution
7
![Page 8: Analysis and propagation of errors](https://reader036.vdocument.in/reader036/viewer/2022062301/56813fae550346895daa92cf/html5/thumbnails/8.jpg)
Reliability• Changes in data over time• Non-uniform coverage• Map scales• Observation density• Sampling theorem (aliasing)• Surrogate data and their relevance• Round-off errors in
computers
8
![Page 9: Analysis and propagation of errors](https://reader036.vdocument.in/reader036/viewer/2022062301/56813fae550346895daa92cf/html5/thumbnails/9.jpg)
Error propagation• Errors arise from data quality, model quality
and data/model interaction.
• We need to know the sources of the errors and how they propagate through our model.
• Simplest representation of errors is to treat observations/attributes as statistical data – use mean and standard deviation.
9
![Page 10: Analysis and propagation of errors](https://reader036.vdocument.in/reader036/viewer/2022062301/56813fae550346895daa92cf/html5/thumbnails/10.jpg)
Analytic approaches
10
Addition and subtraction
![Page 11: Analysis and propagation of errors](https://reader036.vdocument.in/reader036/viewer/2022062301/56813fae550346895daa92cf/html5/thumbnails/11.jpg)
Multiply, divide, exponent, log
11
![Page 12: Analysis and propagation of errors](https://reader036.vdocument.in/reader036/viewer/2022062301/56813fae550346895daa92cf/html5/thumbnails/12.jpg)
Monte Carlo simulation• If a new attribute U is given by U = f (A1, A2, A3, ….
An) where the A’s are attributes and f represents some function combining them, then we want to know what is the standard deviation of the combination U and how does the standard deviation of each A contribute to it?
• By MC simulation we look at the statistical distribution of a lot of realizations (random samples) of U.
12
![Page 13: Analysis and propagation of errors](https://reader036.vdocument.in/reader036/viewer/2022062301/56813fae550346895daa92cf/html5/thumbnails/13.jpg)
MC (ctd)• A single realization of U is Ui = f (R1, R2, R3,
…. Rn) where each Rn is a random sample of its corresponding attribute An based on the statistical properties (mean and standard deviation, for example) of An.
• The probability functions for the attributes themselves need not be Gaussian and could even be taken from histograms of observed values.
13
![Page 14: Analysis and propagation of errors](https://reader036.vdocument.in/reader036/viewer/2022062301/56813fae550346895daa92cf/html5/thumbnails/14.jpg)
Recall…• The mean and standard deviation of U is
estimated by– m = N-1 SUM i=1,N (Ui)
– s2 = (N-1)-1 SUM i=1,N (Ui - m)2
• where N is a very large number of realizations (hundreds or thousands).
14
![Page 15: Analysis and propagation of errors](https://reader036.vdocument.in/reader036/viewer/2022062301/56813fae550346895daa92cf/html5/thumbnails/15.jpg)
When to use?• MC simulation is most useful when the
function relating the attributes is complex or the statistical distribution is known only empirically (from a histogram, for example).
• For simpler combinations of attributes, there are easier, direct (analytical) ways to estimate the new uncertainties from the attribute uncertainties.
15
![Page 16: Analysis and propagation of errors](https://reader036.vdocument.in/reader036/viewer/2022062301/56813fae550346895daa92cf/html5/thumbnails/16.jpg)
Generating pseudo random numbers
• For the Monte Carlo simulation, you will want to generate a series of random numbers with a normal (bell-curve) distribution.
• There are 2 ways to do this in Excel.
• In older versions of Excel, you can use the Tools > Data Analysis > Random number generation > Normal distribution to generate a sequence of random numbers. 16
![Page 17: Analysis and propagation of errors](https://reader036.vdocument.in/reader036/viewer/2022062301/56813fae550346895daa92cf/html5/thumbnails/17.jpg)
Second way• Or, you can take advantage of the central limit
theorem that states that under certain conditions, random samples of any distribution will have a normal distribution.
• The Excel function RAND generates a uniformly distributed random number, that is, the probability is the same for any number between 0 and 1 to be generated.
• To get a normally distributed random sample with mean of 0 and standard deviation of 1 we can simply add 12 uniformly distributed random numbers and subtract 6.
17
![Page 18: Analysis and propagation of errors](https://reader036.vdocument.in/reader036/viewer/2022062301/56813fae550346895daa92cf/html5/thumbnails/18.jpg)
• To get a normally distributed random sample with mean of m and standard deviation of s we use:
• [ SUM i=1,12 RAND() - 6 ] * s + m
• In Matlab – RAND
• In R – randu, seed, sample
18
![Page 19: Analysis and propagation of errors](https://reader036.vdocument.in/reader036/viewer/2022062301/56813fae550346895daa92cf/html5/thumbnails/19.jpg)
Tip• Because this expression is quite long in Excel
you can create a macro to facilitate using it again and again.
• To record a macro, select Tools > Macro > Record new macro > give name to the macro > ok > type in expression > Stop recording.
• You can refer to re-named cells from within a macro, so you might want to use variable names for the mean and standard deviation to keep your macro general.
19
![Page 20: Analysis and propagation of errors](https://reader036.vdocument.in/reader036/viewer/2022062301/56813fae550346895daa92cf/html5/thumbnails/20.jpg)
Shortcuts • You can also specify a Control-key to run the
macro from the worksheet. Otherwise, to run the macro, go to Tools > Macro > Macros > select the macro name and press Run.
• Once the macro is run in a cell, you can drag the expression to other cells using the drag handle in the lower-right corner of the cell.
20
![Page 21: Analysis and propagation of errors](https://reader036.vdocument.in/reader036/viewer/2022062301/56813fae550346895daa92cf/html5/thumbnails/21.jpg)
Statistical ‘tests’• F-test: test if two distributions with the same
mean are the same or different based on their variances and degrees of freedom.
• T-test: test if two distributions with different means are the same or different based on their variances and degrees of freedom
21
![Page 22: Analysis and propagation of errors](https://reader036.vdocument.in/reader036/viewer/2022062301/56813fae550346895daa92cf/html5/thumbnails/22.jpg)
F-test
22
F = S12 / S2
2
where S1 and S2 are the
sample variances.
The more this ratio deviates from 1, the stronger the evidence for unequal population variances.
![Page 23: Analysis and propagation of errors](https://reader036.vdocument.in/reader036/viewer/2022062301/56813fae550346895daa92cf/html5/thumbnails/23.jpg)
T-test
23
![Page 24: Analysis and propagation of errors](https://reader036.vdocument.in/reader036/viewer/2022062301/56813fae550346895daa92cf/html5/thumbnails/24.jpg)
Variability
24
![Page 25: Analysis and propagation of errors](https://reader036.vdocument.in/reader036/viewer/2022062301/56813fae550346895daa92cf/html5/thumbnails/25.jpg)
Dealing with errors• In analyses:
– report on the statistical properties– does it pass tests at some confidence level?
• On maps:– exclude data that are not reliable (map only
subset of data)– show additional map of some measure of
confidence
25
![Page 26: Analysis and propagation of errors](https://reader036.vdocument.in/reader036/viewer/2022062301/56813fae550346895daa92cf/html5/thumbnails/26.jpg)
Elevation map
26
meters
![Page 27: Analysis and propagation of errors](https://reader036.vdocument.in/reader036/viewer/2022062301/56813fae550346895daa92cf/html5/thumbnails/27.jpg)
Larger errors ‘whited out’
27
m
![Page 28: Analysis and propagation of errors](https://reader036.vdocument.in/reader036/viewer/2022062301/56813fae550346895daa92cf/html5/thumbnails/28.jpg)
Elevation errors
28
meters
![Page 29: Analysis and propagation of errors](https://reader036.vdocument.in/reader036/viewer/2022062301/56813fae550346895daa92cf/html5/thumbnails/29.jpg)
Contaminants
29
![Page 30: Analysis and propagation of errors](https://reader036.vdocument.in/reader036/viewer/2022062301/56813fae550346895daa92cf/html5/thumbnails/30.jpg)
Regions with errors ‘whited out’
30
![Page 31: Analysis and propagation of errors](https://reader036.vdocument.in/reader036/viewer/2022062301/56813fae550346895daa92cf/html5/thumbnails/31.jpg)
Map of errors
31
![Page 32: Analysis and propagation of errors](https://reader036.vdocument.in/reader036/viewer/2022062301/56813fae550346895daa92cf/html5/thumbnails/32.jpg)
Summary• Topics for GIS (for Science)
– Estimating, propagating and displaying error considerations
• For learning purposes remember:– Demonstrate proficiency in using geospatial applications and tools
(commercial and open-source).
– Present verbally relational analysis and interpretation of a variety of spatial data on maps.
– Demonstrate skill in applying database concepts to build and manipulate a spatial database, SQL, spatial queries, and integration of graphic and tabular data.
– Demonstrate intermediate knowledge of geospatial analysis methods and their applications.
32
![Page 33: Analysis and propagation of errors](https://reader036.vdocument.in/reader036/viewer/2022062301/56813fae550346895daa92cf/html5/thumbnails/33.jpg)
Friday Mar. 23• Lab assignment session – three problems, up
on ~ Wednesday
• Complete them in class, get signed off before leaving
• 10% of grade
33
![Page 34: Analysis and propagation of errors](https://reader036.vdocument.in/reader036/viewer/2022062301/56813fae550346895daa92cf/html5/thumbnails/34.jpg)
Reading for this week• http://www.chemtopics.com/aplab/errors.pdf
• http://www.nuim.ie/staff/dpringle/gis/gis11.pdf
34
![Page 35: Analysis and propagation of errors](https://reader036.vdocument.in/reader036/viewer/2022062301/56813fae550346895daa92cf/html5/thumbnails/35.jpg)
Next classes
• Friday, March 23 – lab with material from week 7 (lab assignment 10%)
• Tuesday, March 27, Using uncertainties, working with discrete entity types
• Note March 30 – open lab (no assignment, work on your projects, get help from Max), attendance will be taken
35