mathematics Archivi - Paola Elefante

Ch- ch- changes!

Paola Elefante — Wed, 04 May 2016 17:38:03 +0000

I have been neglecting my blog lately as some major changes happened in my life. About three weeks ago I said goodbye to academia and started working at RELEX Solutions, a growing Finnish (& international) software company which offers automatized solutions for managing the supply chain processes.

Fresh from print, my business cards.

Leaving academia - after about 10 years, studying years included - was hard, especially given all the projects I was vetting and those I had myself started. However, I feel it was the right choice for me, given my professional ambitions, my set of skills, and my personal life plans. The recent economic cuts in Finnish academia were an additional push out of there, not for a imminent risk, but because I felt they slimmed my future opportunities in academia even more.

Things at my new job have started great. I have been blessed with the most helpful and friendly colleagues, and a positive and energetic atmosphere at the office. Supply Chain management includes many mathematical challenges, optimization and, of course, handling big data. There are tons of new things to learn and I am putting all my enthusiasm in doing so. Looking forward to present some interesting SC math here soon!

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A small guide to Random Forest - part 2

Paola Elefante — Thu, 17 Mar 2016 13:04:51 +0000

This is the second part of a simple and brief guide to the Random Forest algorithm and its implementation in R. If you missed Part I, you can find it here.

randomForest in R

R has a package called randomForest which contains a randomForest function. If you want to explore in depth this implementation, I suggest to read the support webpage. Here I'd like to show the use of few parameters in the R function. I will here use the Titanic dataset from Kaggle to explore some functions and parameters in randomForest. The problem consists in predicting the survival of passengers, based on some data about them.

library(randomForest)
my_formula <- factor(Survived) ~ Sex + Pclass + Parch + SibSp + Embarked
my_forest <- randomForest(my_formula, data = train, ntree = 400, mtry = 3 )

Here I tuned the number of tree to grow with ntree (standard value is 500). The variable mtry specifies how many random features will be selected to grow a single tree. Here I chose mtry = 3, meaning that three features in the set {Sex, Pclass, Parch, SibSp, Embarked} will be randomly chosen every time a tree is grown. If I type:

my_forest

I get a briefing of the variables and the trained model:

Call:
 randomForest(formula = my_formula, data = train, ntree = 400, mtry = 3) 
 Type of random forest: classification
 Number of trees: 400
No. of variables tried at each split: 3

OOB estimate of error rate: 19.85%
Confusion matrix:
 0 1 class.error
0 442 47 0.09611452
1 110 192 0.36423841

The OOB (out-of-bag) error is complementary to the accuracy, and it's here calculated as the ratio: . Look at the confusion matrix, which summarise how many cases were guessed right from our model. On the principal diagonal we can see the cases which are predicted well from my_forest. Indeed, we get: which corresponds to the OOB error of . This is equivalent to saying that the accuracy of our model is . OOB error is calculated for each tree and you can access to such values by typing:

my_forest$err.rate

Nerdy note: notice that the OOB error of the model is not the mean of my_forest$err.rate. They are calculated differently!
Another nice parameter is sampsize, meaning controlling how many rows of the dataframe will get selected to build a single tree:

my_forest <- randomForest(my_formula, data = train, ntree = 1000, mtry = 2, sampsize = (0.9*nrow(train)), replace=TRUE )

Here I asked that of data is used for each tree. In addition, I set replace = TRUE, meaning that one row may be chosen more than once.

One nice aspect of randomForest is the variable importance, which can turn out very useful in feature engineering. If you type, for instance:

my_forest <- randomForest(my_formula, data = train, ntree = 400, importance = TRUE) 
varImpPlot(my_forest)

you'll get a plot as follows.

The importance of each feature is measured in two ways, as described by documentation:

Here are the definitions of the variable importance measures. The first measure is computed from permuting OOB data: For each tree, the prediction error on the out-of-bag portion of the data is recorded (error rate for classification, MSE for regression). Then the same is done after permuting each predictor variable. The difference between the two are then averaged over all trees, and nor- malized by the standard deviation of the differences. If the standard deviation of the differences is equal to 0 for a variable, the division is not done (but the average is almost always equal to 0 in that case).

The second measure is the total decrease in node impurities from splitting on the variable, averaged over all trees. For classification, the node impurity is measured by the Gini index. For regression, it is measured by residual sum of squares.

Another possible choice is to set localImp = TRUE, and see how much each feature influenced the output of each single row. For example, here is the importance of Sex in our model:

Things to keep in mind

The R package randomForest allows to evaluate variable importance (in randomForest, set the parameter importance = TRUE, save the function output and pass it to varImpPlot()). However, keep in mind the following:

For data including categorical variables with different number of levels, random forests are biased in favor of those attributes with more levels. Methods such as partial permutations and growing unbiased trees can be used to solve the problem. (source: Wikipedia)

How many trees should one grow? In principle, the more the merrier. However, the information gain after a certain number is not worth the additional computational cost. The computational complexity of Random Forest is , where is ntree, is mtry and is sampsize.

I ran randomForest, changing ntree from 100 to 20 000, with a step of 100. After ntree = 5000 the OOB error has a 0.5% oscillation, showing that computational cost is too big compared to a gain in accuracy.

Another question is: how deep should I grow a tree? This is an interesting issue. Growing a superficial tree may lead to underfitting, while a too deep tree may cause overfitting. One idea to test the optimal value is to experiment with some very deep trees and observe how the accuracy behaves on its subsets.

Here I experimented with nodesize = 1, 2, ..., 30 and checked the corresponding OOB error. The minimum is attained at nodesize = 25. Given the oscillations, this is a less trivial choice than the number of trees, but experimenting takes less computational time.

The featured image was found on this webpage.

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A small guide to Random Forest - part 1

Paola Elefante — Fri, 11 Mar 2016 10:23:22 +0000

I've recently started playing with Kaggle and got curious about one of the most famous classification/regression framework, Random Forest. In a problem of classification or regression, several random decision trees (a "forest") are built and at the end the outputs are combined ("bagging"). The intuition is that randomness and a meaningful quantity of trees will avoid over- and underfitting. One possible bagging technique is the majority vote. Take the case of predicting a binary outcome, say a random variable Y which can assume only values in , with respect to some features (occurred events). We assume there exist a correct answer - the "right model" - which we have to predict. The intuition of the majority vote is that if such "divine truth" exists and we build several "quite reasonable" models, most of them will give the right prediction. If the right value is and we make "reasonable" predictions , most of them will be equal to 1 and only a minority will be equal to -1. In mathematical terms, we'll choose the following prediction:

(1)

Bagging is done in other ways, but to me the majority vote example is an easy way to understand the fundamental concept.
The Random Forest framework was introduced by statistician Leo Breiman in 2001 in his seminal paper. Even though implementations have been released in many languages (R, MatLab, Python, Java...), it's important to learn the basics, to be able to tune the parameters well.

Decision trees

The elements of a Random Forest are usually decision trees (there are variants of the framework, though). Assume we have the following database:
training data =
Each column is a sample, each row corresponds to a feature. We consider a binary output: . We now will choose random features (to be able to represent the problem in the plane) and will start building a decision tree. Assume our random sample is:
random sample =
meaning that we randomly selected the features 1 and 3. Let's represent these points on a plane, assigning a different color on the base of the associate output.

Notice the distribution of points in this universal region:

Frequency of output value in the universal region: red corresponds to value 0, blue to 1.

Now an hyperplane is selected (randomly or with some criteria, for instance maximising information gain) and the points are separated into two regions:

Our decision tree starts and we have the following split and new frequency distributions in the two new regions:

START

Now the idea is to iterate this procedure separately on each branch. For instance, we consider only Region 1 () and draw another hyperplane, say :

On the other branch, we draw another hyperplane :

Summing up, we built the following tree.

At this point clearly we can stop. We divided the plane in regions which completely classify our training data.
To summarise, here are the steps of Random Forest:

For k = 1, 2, ..., Ntrees:
--> select a bootstrap sample S from training data
--> grow a decision tree (with a stopping criterion for the depth)
Bagging on

Next, I plan to show the use of some variables and features of the randomForest R package and to make some observations on the algorithm. For instance, how to choose Ntrees? How to determine a reasonable stopping criterion for the tree depth?

The featured image is an excuse to introduce a great visualisation resource for Random Forests: check it out.

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Women in Mathematics in Finland: Amal Attouchi

Paola Elefante — Thu, 10 Mar 2016 11:43:00 +0000

As promised, the series of lectures continues, after the inaugural event hosted by the Department of Mathematics and Statistics of University of Helsinki. This time I'll travel to University of Jyväskylä and the guest speaker will be Amal Attouchi, local postdoctoral researcher. Amal graduated in 2014 at Université Paris XIII, with a thesis on PDEs titled "Local and global behavior for Hamilton- Jacobi equations with degenerate difusion". Her research interests include parabolic and elliptic equations, P-laplacian, studying singularities and problem regularity. She will present a talk on "Optimization problems in Mathematics".
The event is organised on behalf of the association European Women in Mathematics and kindly supported by the Department of Mathematics and Statistics of University of Jyväskylä. The Head of Department Prof. Tero Kilpeläinen has offered to open the event with few words. The appointment is at the Math Department, in room MaA211 at 12 sharp on April, 4th. If you plan to take part, please register at this link, to help me organise the coffee break later. The talk will be addressed to a general audience, to allow undergrad students and people from different fields to actively participate.

The poster of the event: feel free to share!

Why all this? Women are severely underrepresented in mathematics, especially in Scandinavian countries, even though they top charts for gender equality. Recently EWM collected data from the country coordinators to describe the single situations and filed a report. The situation in Nordic countries is too extreme to ignore:

"There are simply too few women in mathematics in Denmark, so keeping up regular meetings seems not to have enough momentum." (Tinne Hoff Kjeldsen, coordinator for Denmark)
"[...] nationally the percentages are not as flattering - less than 5% of all Finnish professors in mathematical sciences are women." (Paola Elefante, coordinator for Finland)
In Sweden and Iceland we do not even have a coordinator. We have one is Norway, but there is no report available.

I am not aware of studies or statistics carried out nationally in Finland, to lean on. There are scattered or old studies to sustain my worried, such as a 2008 report compiled at University of Helsinki (I spoke about it here and at the end of this post). Current numbers prove that women are still excluded from higher positions in mathematics, in academia. I think such low numbers should at least make us wonder why. I hope this series of lectures will help give visibility to women who are contributing to the mathematics panorama in Finland and will help female students relate to competent female professionals, to shape their own future.

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The man who knew infinity: a movie about Ramanujan

Paola Elefante — Thu, 03 Mar 2016 07:31:36 +0000

Thank you Youtube suggestions! Today I came across this trailer of an upcoming movie, "The man who knew infinity".

The movie depicts the story of one of the finest minds of last century and one of the greatest mathematicians of all time, Srinivasa Ramanujan. He was what you'd call a true genius, an independent thinker who could not bend to the typical mathematical formalism nor to social conventions. He was born and lived the first part of his life in (British) India, in the region of Madras. in He showed a lively interest in mathematics already at the age of 11, and by the age of 15 he was able to master advanced notions and carry on his own research, often on known results he was not aware of. Basically he was re-building mathematics from almost scratch! He won a college scholarship, but he was so focused on mathematics that he performed poorly in other subjects and lost it. He tried to enrol in another college, with same results. His thrive to study mathematics and only mathematics led him to extreme poverty and corrupted his health. At age 21, he married and got to search for a real job. He found one in accountancy, a trivial job for his abilities which left him with much spare time to continue his studies.
In 1913, Ramanujan took the initiative and started sending letters to English mathematicians, for support and advice. Among his recipients, was Godfrey Hardy, a brilliant researcher of mathematical analysis and number theory. Ramanujan had no formal education in mathematics. Most of his work relied on epiphanies he had during spiritual meditation. Many mathematicians were skeptic of his true abilities, but came around after making direct contact with him. Hardy was the opposite as the Indian mathematician: he was atheist, while Ramanujan was deeply religious; he championed formalism and pure proofs, while the other basically guessed (right) deep results and did not care about technical details. However, Hardy managed to see past his own style and became Ramanujan's best supporter. He convinced him to travel to England, where he lived the last years of his unfortunately short life. Hardy tried to educate him, without suppressing his unique style, and polished his results so that they would be easy to publish on journals. In 1918 he was elected Fellow of the Royal Society and became one of his youngest members. In the same year he was also appointed Fellow of the Trinity College, in Cambridge, the first Indian to be honoured with the title. One year later, his health forced him to return to India to his wife, and he died in 1920 at the age of 32 from an incurable sickness.

Bust of Ramanujan in the garden of Birla Industrial & Technological Museum (source: Wikipedia).

He left the mathematical community with an incredible treasure. Most of his results needed to be proved (or disproved) and a journal was founded to collect results inspired by his work. In 2011, India declared the day of his birth, December 22nd, as "National Mathematics Day". He also left us the taxicab numbers. His most famous conjecture, named after him, waited for 50 years to be solved by Pierre Deligne.
I am very excited about this upcoming movie, and while waiting for it, I'll make sure to read the book which inspired it.

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Open Data: CT datasets and prototypes

Paola Elefante — Mon, 22 Feb 2016 18:41:22 +0000

In my research work, I often find it difficult to get datasets for X-ray CT for method validation, neither simulated and real data. Of course, there's the classic Shepp-Logan phantom, but in many cases it would save a lot of work to download datasets to test one's methods. As for my knowledge, there is no broad known repository. In the image processing community there are the classical Lena, Barbara pictures, but I am not aware of similar common testing grounds in the X-ray CT one. In many cases the final goal is to apply the developed methods to a particular application case, so datasets are available from a company or a hospital repository. However, I think it would be useful to have a "middle step" and test on simpler phantoms before getting to the real deal. Many applied mathematicians often stop at this middle step, since going further would be out of scope for them. I hope this collection will be useful to others.

Since 3D printing is quite easy and inexpensive nowadays (can be done even in public libraries often), I used a the free software Autodesk 123D to design some 3D printable prototypes for static and dynamic CT. I am here sharing both the ready-made STL file (ready to be printed) and the 123D-project file, in case someone wants to do some personal edits. Anything can be freely use, just please quote the author and the source.

If you are aware of other open data repositories for CT or would like to share suggestions, feel free to comment below. I will update this post and the shared repositories in the future (last update: Feb 22nd, 2016).

Static CT open data

[real data] Tomographic data of a walnut: open dataset from FIPS, authors are indicated at the webpage.

[real data] 3D printable simple phantom prototype: to test contrast agents, geometry preservation of reconstruction method, how different attenuation values are reconstructed. Please quote the author (Paola Elefante) and the link to this post as a source.

[real data] 3D printable blood vessels prototype: to test a realistic static geometry of blood vessels splitting in capillaries. Please quote the author (Paola Elefante) and the link to this post as a source.

The static CT blood vessels prototype.

Dynamic CT open data

[real data] DirLab repository: a open data repository, mostly for image registration researchers.

[real data] 3D printable dynamic blood vessels prototype: to test a realistic dynamic simulation of blood or fluid flowing. In the featured picture you can spot an old version of this prototype. I made some major edits in the design, but I still did not test it. Please quote the author (Paola Elefante) and the link to this post as a source.

Design of the prototype for 2D dynamic CT.

[simulated data] 2D dynamic "Y-phantom": a binary phantom where meaningful topological changes happen, good for interface detection methods (level-set, etc.).

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Science Slam Helsinki 2015: a triumph for Inverse Problems

Paola Elefante — Wed, 13 Jan 2016 08:31:09 +0000

Last November I attended a fun event here in Helsinki: a sort of mix between science and stand-up comedy called Science Slam Helsinki. A Science Slam is a form of science communication to general audience. Each scientist gives a short talk (10-15 mins) to popularise science. Often such events are held in non-academic places, like pubs. Science Slam Helsinki took place on November 6th at Elmun baari.

I am proud to say one of the speakers was my advisor and inverse problems researcher Prof. Samuli Siltanen. His talk was titled "How to defuse a photobomb" and showed some mathematical techniques for image enhancement and manipulation, in particular inpainting. Samuli was claimed winner at the end! You can enjoy his talk right here:

You can browse a photogallery of the event at this link.

Samuli is awarded winner of the 8th edition of Science Slam Helsinki.

It was great to see a mathematics talk winning the hearts of the audience. Congratulations, Samuli!

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Live from Inverse Days 2015: baby on board

Paola Elefante — Wed, 09 Dec 2015 13:39:07 +0000

This week I did something a little crazy, dictated by necessity: I took my 2 year old girl to a conference, namely the Inverse Days in Lappeenranta. We drove from Espoo (bad idea) on Monday evening and will stay until Thursday, cutting at half day to get home not too late.

Baby-wise it went much better than I expected: she played, slept and watched cartoon and I was able to listen to almost all talks. I could even present my own during her afternoon nap (slides to be found here).

Science-wise, I heard about many interesting projects. Inverse Days is the annual meeting of the Inverse Problems research community in Finland. Since it's a great chance to get to know all ongoing projects in Finland, there are often guests from abroad as well. Since 2010, I have been missing only one meeting, because of my parental leave. The dominant theme this year has been EIT, on which many groups in Finland are working at the moment.

For me, the highlight has been a nice presentation by Lauri Harhanen from DTU. He started last May in the huge ongoing project HD-Tomo and he's working on an interesting optimization of gradient descent minimization methods, based on physical modelling of material a priori information in CT. When I will upgrade to real data, this could be a fruitful improvement of my framework.

During the conference, we also had the second meeting of Women in Inverse Problems, a small network started one year ago. We agreed that it would be nice to have some statistics (local or nation-wide) on presence of women in mathematics, and investigate on reasons why the disproportion is created. Let's hope we can follow up on this idea. At the moment our most active action is a small mailing list, where we share interesting events or discussion topics (drop me an email if you want to join, since it's closed).

It's been nice to be at Inverse Days once again and I also realised that this was my first talk at this event. See you soon, Lappeenranta!

Cutest attendee ever.

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Mathematicians Go Hollywood

Paola Elefante — Fri, 27 Nov 2015 07:31:57 +0000

Did you know that in 2008 a mathematician won an Academy Award? Do you know how to model realistic hair for animation movies or a bomb deflagrating for an action movie? Or you simply need some effective active contour segmentation method? All these questions have in common an effective, yet intuitive, mathematical framework: level set methods. I came to know them through my own X-ray tomography research project, which gives you a hint of how wide and inclusive such methods are. Yesterday I gave an introductory and informal talk at the Students' Seminar about them. This post comes as an integration to the slides that you may download from this page.

Let's start from defining what an interface is. I could not find a rigorous definition, but the concept is very intuitive. It is a "boundary" which clearly splits the space in two subsets ("inside" and "outside"). You can imagine a closed (even self-intersecting) curve on the plane, for instance. Or the surface of a ball or a torus in . From now on, let's work with planar interfaces, for better visual intuition. However, everything I will discuss here can be extended to any . Now, imagine we are working with a dynamic interface, meaning that our closed curve, for instance, changes in time.

Representation of our dynamic interface: the red arrows represent the velocity field, which gives information of how the curve will change.

Rigorously speaking, we are given an initial curve, and a velocity field which we assume is normal to the curve at any instant . We would like to determine and parametrise the evolution of the curve, that is . One intuitive idea is the following: let's choose some ordered points on our curve (Fig. A), let's follow their evolution ( will tell us where they are going) and let's complete the curve between any subsequent points and by interpolation. However, it may happen that our curve will split under the action of and Fig. B shows how our method would fail, because we told our algorithm to connect with and with .

How could we explain to our algorithms when the curve splits or merges? It's hard, especially since we are searching for a general method. This is where level set methods come to the rescue.

The idea is very intuitive: what if we would add one extra-dimension (time) and "record" the evolution with a surface? For instance, if our curve is a disc expanding, one candidate surface could be a truncated cone. If our disc would evolve in a "8-shape" and then split, one candidate surface would be some sort of 3-dimensional "Y". In other words, we are looking for a function such that:

Here I denote the "inside" region at the time by . At any time, the zero level set of will detect the interface. In addition, its sign will detect the inside and outside regions. Now, observe that from the previous equation:

By applying the chain rule, we get . We assumed that our velocity field was orthogonal to the interface at any instant. In other words,

Hence, we can write the following evolution equation:

This, in addition to the given initial condition , will define and consequently the interface at any time. Suddenly we are in front of a PDE problem, for which there are many well-developed theoretical and numerical tools. Also, this approach handles perfectly topological changes, such as splitting and merging. Plus, it makes it really easy to compute geometric quantities as the curvature of the interface (simply differentiate ).

This new framework was introduced in 1987 by Stanley Osher and James Sethian. Since then, it has been a thriving topic of research: just know that their seminal paper to date has been cited more than 11 500 times! Level set methods have been applied to an incredible variety of problems and settings: medical imaging, computer vision, image denoising, active contour segmentation, scattering, obstacle detection, and more. It has been widely explored both theoretically and numerically. One its richest areas of application is computer graphics. One of Osher's students, Ron Fedkiw, now full professor at Stanford, won an Academy Scientific and Technical Award in 2008. Fedkiw is a consultant for Industrial Light and Magic, a big name in the special effects industry. He worked on blockbusters as Terminator III, Star Wars Episode III, the Pirates of the Caribbean's saga and some Harry Potter movies. Level set methods are widely used in fluid, fire, hair simulations in animation movies. Think of water, with all his splashes (=topological changes): this framework works very well. One drawback is that this approach does not conserve some physical quantities as the volume. However, there are nowadays many tricks to work around this. For instance, there are hybrid methods that mix level set and volume tracking methods or sometime rendering techniques that fill up for the missing physical properties. You can see many animations at the PhysBAM project page.

If you got curios, I include a selection of references:

Osher – Paragios, “Geometric Level Set Methods in Imaging, Vision and Graphics”, Springer 2003.

Osher – Fedkiw, “Level Set Methods and Dynamic Implicit Surfaces”, Springer 2003.

Links:

http://step.polymtl.ca /~rv101/levelset / explanations

http://www.ams.org/notices/201005/rtx100500614p.pdf

http://physbam.stanford.edu/~fedkiw/papers/stanford2003-04.pdf

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Women in Mathematics in Finland: Kirsi Peltonen

Paola Elefante — Sun, 25 Oct 2015 20:05:30 +0000

I am so excited to announce this upcoming event: the first lecture Women in Mathematics in Finland. I organised the seminar as national coordinator for European Women in Mathematics, since we discussed with the local members that we needed our local network to be more active. Visibility for women in mathematics is much needed in Finland, especially if we consider that only about 4% of math professors (associate, assistant, full, emeritus) are women. I am still in search of why it is so, considering Finland is one of the top countries in Europe regarding gender equality. I owe a thank you to our Department Head Mats Gyllenberg, who did not hesitate to sponsor this first event and will say some opening words.

Kirsi is a force of nature, as the picture she chose for the poster gives away. Her research is mostly in the area of differential geometry, but she goes beyond pure mathematics research. Last year she pioneered an interdisciplinary course at Aalto University about Mathematics, Art and Architecture, that was concluded by a final exhibition of "math-art masterpieces". She is passionate about teaching and never stops searching new ways to improve. You can read her recent interview on Kumpula Women blog, to have a taste of her views before the talk.

The poster of the event.

Only in the few years I spent here, I met amazing women working in science. Maybe I was lucky in my encounters, but I less frequently find the same passion and enthusiasm in men, as I found in many women here. I think of Johanna, who works so hard to improve mathematics teaching at any level; Anne-Maria, who has always been busy with research, training students for Math Olympiads and writing for the magazine Solmu; there's Martina, who's completing her postgrad studies and is a mother of three; Åsa, who does research with one hand and coordinates a doctoral school with the other; Hanne, on whom you can always count to get things done and well. I could go on for hours and these are only few of the amazing women I met at my math department (I apologise to all those who did not appear on the list: it was a brainstorming!).

Maybe I notice their achievements better because I feel I can relate to them, but I really need to feel this connection in my daily life. It gives me great courage to look at these close-by women and see that they are making it. At the same time, I have the privilege of talking plainly with some of them and of learning their difficulties and how they overcome them, another precious contribution.

In the spirit of sharing this opportunities with other girls who are in need of role models to relate better to, I started this series of seminars. Contents will be scientific (no policy making this time!), but accessible to undergrad students. Let's highlight women's contribution to Finnish mathematics!

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