In Open3D library there is a function which calculates the information matrix, it uses 2 clouds, a transformation matrix (output of a registration algorithm) and a distance. I would like to understand the meaning of a information matrix in the context of 3D point cloud registration, for what is it used for?
I know how to calculate it, but just because I read the Wikipedia article. And I read some articles, but there's nothing to guide me.
The derivation of the information matrix comes from equation 8 in the paper:
Robust Reconstruction of Indoor Scenes, CVPR, 2015 by Choi et.
Basically it's used for computing the sum of squared distances between all correspondences, and the paper did an approximation to construct the informaiton matrix.
By the way, the approximation is also used as virtual pair approach in a 1996 paper:
Multiview Registration for Large Data Sets by Kari Pulli. Pay attention to figure 6 if you're interested.
As pointed by #Jing Zhao, the derivation comes from the Robust Reconstruction of Indoor Scenes, but yet a more detailed one can be found on Observability, Covariance and Uncertainty of ICP Scan Matching by Barczyk, Bonnabel and Goulette, especially for the transformation linearization parts.
Elith et al. [1] describe a method of measuring the dissimilarity between the values used in model fitting and the values used in making predictions. In the context of species distribution modelling (ecological niche modelling) that prediction is a 'projection'. The method is called 'multivariate environmental similarity surface (MESS) analysis. There is a function in the dismo package to estimate it (as well as a function built into the MAXENT java program).
q1: Does anyone know what units are reported by the dismo::mess function?
The dismo::mess function reports not only a MESS for each predictor (received and reported as a raster), but also reports a layer named 'rmess'. In the help file it is described as " an additional layer with the MESS values".
q2: How are the MESS values calculated?
q3: What is the rmess layer a measure of?
Thanks for your help!
[1] Elith, J., Kearney, M. & Phillips, S. 2010 The art of modelling range-shifting species. Methods in Ecology and Evolution 1, 330-342. (doi:10.1111/j.2041-210X.2010.00036.x).
You can see what dismo does by typing
dismo::mess
It calls .messi3, which you can see with
dismo:::.messi3
(I found the answer in the appendices...)
I was trying to create some R code from my thesis. Contextualizing my situation I've divided the 505 time series' components of S&P500 with dtw algorithm in 10 clusters. Initially I've created 100 portfolios randomly taking 1 stock for each cluster diversifying the portfolios. Then I've created a code with which I can assign each portfolio' stock a weight with Genetic Algorithm maximizing the Sharpe Ratio.
I was wondering if is there a solution for taking stocks for each cluster maximizing the sharpe ratio to obtain an optimized solution.
You could use a method from the local-search family, such as Stochastic Local Search or Simulated Annealing. For R examples, see Asset selection with Local Search or the more recent Optimization Heuristics: A Tutorial. (Disclosure: I am the maintainer of the NMOF package, which is used in the examples.)
I'm reading a lot about Precision-Recall curves in order to evaluate my image retrieval system. In particular I'm reading this article about feature extractors in VLFeat and the wikipedia page about precision-recall.
I understand that this curve is useful to evaluate our system performance w.r.t. the number of elements retrieved. So we repeatedly compute precision-recall retrieving the top element, then top 2, top 3 and so on...but my question is: when do we stop?
My intuition is: we stop when our list of retrieved elements has recall equal to 1, so we retrieve all the relevant elements (i.e. there are no false negatives, only true positives).
Same question is for average precision: how many elements should be present in the retrieved result for computing it? If my previous intuition is correct, then we just need to find out what is the smallest list s.t. recall is 1 and use it for compute it AP.
I wonder why all the libraries for computing p-r curve don't show how this is implemented?
An information retrieval system with recall 1 means a perfect system which doesn't seem possible in practice! Precision-Recall curves are good when you need to compare two or more information retrieval systems. Its not about stopping when recall or precision reaches some value. Precision-Recall curve shows pairs of recall and precision values at each point (consider top 3 or 5 documents). You can draw the curve upto any reasonable point.
Curves close to the perfect Precision-Recall curve have a better performance level than the ones closes to the baseline. In other words, a curve above the other curve has a better performance level. Two Precision-Recall curves represent the performance levels of two IR systems: A and B. System A clearly outperforms system B according to the following figure.
Remember: Precision-Recall curve not only used for evaluating IR systems. It can be used to show how much good your classifier is! For example, you can compute precision, recall for a binary classification task and plot the Precision-Recall curve that can give you a good estimate of the performance of your classifier.
For example:
I would encourage you to see this tutorial from Coursera. I believe your idea will become more clear about Precision-Recall curve.
I'm not looking for a general discussion on if math is important or not for programming.
Instead I'm looking for real world scenarios where you have actually used some branch of math to solve some particular problem during your career as a software developer.
In particular, I'm looking for concrete examples.
I frequently find myself using De Morgan's theorem when as well as general Boolean algebra when trying to simplify conditionals
I've also occasionally written out truth tables to verify changes, as in the example below (found during a recent code review)
(showAll and s.ShowToUser are both of type bool.)
// Before
(showAll ? (s.ShowToUser || s.ShowToUser == false) : s.ShowToUser)
// After!
showAll || s.ShowToUser
I also used some basic right-angle trigonometry a few years ago when working on some simple graphics - I had to rotate and centre a text string along a line that could be at any angle.
Not revolutionary...but certainly maths.
Linear algebra for 3D rendering and also for financial tools.
Regression analysis for the same financial tools, like correlations between financial instruments and indices, and such.
Statistics, I had to write several methods to get statistical values, like the F Probability Distribution, the Pearson product moment coeficient, and some Linear Algebra correlations, interpolations and extrapolations for implementing the Arbitrage pricing theory for asset pricing and stocks.
Discrete math for everything, linear algebra for 3D, analysis for physics especially for calculating mass properties.
[Linear algebra for everything]
Projective geometry for camera calibration
Identification of time series / statistical filtering for sound & image processing
(I guess) basic mechanics and hence calculus for game programming
Computing sizes of caches to optimize performance. Not as simple as it sounds when this is your critical path, and you have to go back and work out the times saved by using the cache relative to its size.
I'm in medical imaging, and I use mostly linear algebra and basic geometry for anything related to 3D display, anatomical measurements, etc...
I also use numerical analysis for handling real-world noisy data, and a good deal of statistics to prove algorithms, design support tools for clinical trials, etc...
Games with trigonometry and AI with graph theory in my case.
Graph theory to create a weighted graph to represent all possible paths between two points and then find the shortest or most efficient path.
Also statistics for plotting graphs and risk calculations. I used both Normal distribution and cumulative normal distribution calculations. Pretty commonly used functions in Excel I would guess but I actully had to write them myself since there is no built-in support in the .NET libraries. Sadly the built in Math support in .NET seem pretty basic.
I've used trigonometry the most and also a small amount a calculus, working on overlays for GIS (mapping) software, comparing objects in 3D space, and converting between coordinate systems.
A general mathematical understanding is very useful if you're using 3rd party libraries to do calculations for you, as you ofter need to appreciate their limitations.
i often use math and programming together, but the goal of my work IS the math so use software to achive that.
as for the math i use; mostly Calculus (FFT's analysing continuous and discrete signals) with a slash of linar algebra (CORDIC) to do trig on a MCU with no floating point chip.
I used a analytic geometry for simple 3d engine in opengl in hobby project on high school.
Some geometry computation i had used for dynamic printing reports, where was another 90° angle layout than.
A year ago I used some derivatives and integrals for store analysis (product item movement in store).
Bot all the computation can be found on internet or high-school book.
Statistics mean, standard-deviation, for our analysts.
Linear algebra - particularly gauss-jordan elimination and
Calculus - derivatives in the form of difference tables for generating polynomials from a table of (x, f(x))
Linear algebra and complex analysis in electronic engineering.
Statistics in analysing data and translating it into other units (different project).
I used probability and log odds (log of the ratio of two probabilities) to classify incoming emails into multiple categories. Most of the heavy lifting was done by my colleague Fidelis Assis.
Real world scenarios: better rostering of staff, more efficient scheduling of flights, shortest paths in road networks, optimal facility/resource locations.
Branch of maths: Operations Research. Vague definition: construct a mathematical model of a (normally complex) real world business problem, and then use mathematical tools (e.g. optimisation, statistics/probability, queuing theory, graph theory) to interrogate this model to aid in the making of effective decisions (e.g. minimise cost, maximise efficency, predict outcomes etc).
Statistics for scientific data analyses such as:
calculation of distributions, z-standardisation
Fishers Z
Reliability (Alpha, Kappa, Cohen)
Discriminance analyses
scale aggregation, poling, etc.
In actual software development I've only really used quite trivial linear algebra, geometry and trigonometry. Certainly nothing more advanced than the first college course in each subject.
I have however written lots of programs to solve really quite hard math problems, using some very advanced math. But I wouldn't call any of that software development since I wasn't actually developing software. By that I mean that the end result wasn't the program itself, it was an answer. Basically someone would ask me what is essentially a math question and I'd write a program that answered that question. Sure I’d keep the code around for when I get asked the question again, and sometimes I’d send the code to someone so that they could answer the question themselves, but that still doesn’t count as software development in my mind. Occasionally someone would take that code and re-implement it in an application, but then they're the ones doing the software development and I'm the one doing the math.
(Hopefully this new job I’ve started will actually let me to both, so we’ll see how that works out)