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Overall Objectives
New Software and Platforms
Partnerships and Cooperations
Bibliography
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## Section: New Results

### Transportation networks and vehicular systems

#### Travel time prediction

Participants : A. Kibangou [Contact person] , H. Fourati, C. Canudas de Wit, A. Ladino, M Rodriguez.

One of the regular performance metrics for qualifying the level of congestion in traffic networks is the travel time. In [24], we addressed the problem of dynamic travel time (DTT) forecasting within highway traffic networks using speed measurements. Definitions, computational details and properties in the construction of DTT are provided. DTT is dynamically clustered using a K-means algorithm and then information on the level and the trend of the centroid of the clusters is used to devise a predictor computationally simple to be implemented. To take into account the lack of information in the cluster assignment for the new predicted values, a weighted average fusion based on a similarity measurement is proposed to combine the predictions of each model. The algorithm is deployed in a real time application and the performance is evaluated using real traffic data from the South Ring of the Grenoble city in France. We consider in a recent paper submitted to European Control Conference 2018 the problem of joint reconstruction of flow and density in a urban traffic network using heterogeneous sources of information. The traffic network is modeled within the framework of macroscopic traffic models, where we adopt Lighthill-Whitham-Richards model (LWR) conservation equation and a piecewise linear fundamental diagram. The estimation problem considers three key principles. First, the principle governing traffic models where flow is maximized in a junction. Second, the error minimization between the measured and reconstructed flows and velocities, and finally the equilibrium state of the network which establishes flow propagation within the network. All principles are integrated and the problem is casted as a constrained quadratic optimization with inequality and equality constraints in order to shrink the feasible region of estimated variables. Some simulation scenarios based on synthetic data for a Manhattan grid network are provided in order to validate the performance of the proposed algorithm.

#### Urban traffic control

Participants : C. Canudas de Wit [Contact person] , F. Garin, P. Grandinetti.

The PhD thesis of Pietro Grandinetti deals with optimal or near-optimal operation of traffic lights in an urban area, e.g., a town or a neighborhood. The goal is on-line optimization of traffic lights schedule in real time, so as to take into account variable traffic demands, with the objective of obtaining a better use of the road infrastructure. More precisely, we aim at maximizing total travel distance within the network, together with balancing densities across the network. The complexity of optimization over a large area is addressed both in the formulation of the optimization problem, with a suitable choice of the traffic model, and in a distributed solution, which not only parallelizes computations, but also respects the geometry of the town, i.e., it is suitable for an implementation in a smart infrastructure where each intersection can compute its optimal traffic lights by local computations combined with exchanges of information with neighbor intersections. A modified version of the algorithm uses simplified optimization (purely local, instead of distributed) but takes into account the real constraints in Grenoble downtown traffic lights network, such as priority to public transportation, and imposed minimal and maximal green duration, leading to a fully realistic implementation, tested using Aimsun microscopic simulator.

#### Traffic Regulation Via Controlled Speed Limit

Participants : M. L. Delle Monache [Contact person] , B. Piccoli, F. Rossi.

The work [21] address the speed limit problem on a single road. The control variable is the maximal allowed velocity, which may vary in time but we assume to be of bounded total variation, and we aim at tracking a given target outgoing flow. More precisely, the main goal is to minimize the quadratic difference between the achieved outflow and the given target outflow. Mathematically the problem is very hard, because of the delays in the effect of the control variable (speed limit). In fact, the link entering time, which represents the entering time of the car exiting the road at time $t$, depends on the given inflow and the control policy on the whole time interval. Moreover, the input-output map is defined in terms of the Link Entering Time, thus the achieved outflow at time $t$ depends on the control variable on the whole time interval. After formulating the optimal control problem, we consider needle-like variations for the control policy as used in the classical Pontryagin maximum principle. We are able to derive an analytical expression of the one-sided variation of the cost, corresponding to needle-like variations of the control policy, using

fine properties of functions with bounded variation. In particular the one-sided variations depend on the sign of the control variation and involve integrals w.r.t. the distributional derivative of the solution as a measure. This allows us to prove Lipschitz continuity of the cost functional in the space of a bounded variation function and prove existence of a solution. Afterwards, we define three different techniques to numerically solve this problem and we compare the three approaches on two test cases.

#### Scalar conservation laws with moving flux constraints

Participants : M. L. Delle Monache [Contact person] , P. Goatin [Acumes, Inria] , C. Chalons.

This problem is motivated by the modeling of a moving bottleneck in traffic flow, which can be caused by a large, slow moving vehicle. A slow moving large vehicle, like a bus or a truck, reduces the road capacity and thus generates a moving bottleneck for the surrounding traffic flow. This situation can be modeled by a PDE–ODE strongly coupled system consisting of a scalar conservation law with moving flux constraint accounting for traffic evolution and an ODE describing the slower vehicle motion. In [18], we introduce a novel approach to solve numerically this problem. The main point here is related to the presence of non-classical shocks in the solutions of the model under consideration. It is well-known that, in this context, standard conservative finite volume methods cannot be applied and fail in producing good numerical results. Glimm’s scheme can be used but it is not strictly conservative. In order to propose a numerical scheme which is conservative on fixed meshes and able to compute non-classical solutions, we propose to adapt a reconstruction strategy approach, which allows to precisely capture moving non-classical discontinuities on fixed meshes still guaranteeing conservation, unlike Glimm’s scheme. An important feature of the proposed method is to be exact for isolated classical and non-classical shocks, which means in particular only one point of numerical diffusion (on each cell the approximate value corresponds to the value of the average of the exact solution). In the general case, shocks are still computed without numerical diffusion and convergence is proved numerically.

In [19] we study well-posedness of a scalar conservation laws with moving flux constraints. In this work we assume that the constraint trajectory is given and it does not depend on the solution of the PDE. In this setting we then show Lipschitz continuous dependence of bounded variation solutions with respect to the initial data ant the constraint trajectory.

#### Priority-based Riemann solver for traffic flow on networks

Participants : M. L. Delle Monache [Contact person] , P. Goatin [Acumes, Inria] , B. Piccoli.

In [20] we introduce a novel solver for traffic intersection which considers priorities among the incoming roads as the first criterion and maximization of flux as the second. The main idea is that the road with the highest priority will use the maximal flow taking into account also outgoing roads constraints. If some room is left for additional flow then the road with the second highest priority will use the left space and so on. A precise definition of the new Riemann solver, called Priority Riemann Solver, is based on a traffic distribution matrix , a priority vector and requires a recursion method. The general existence theorem for Riemann solvers on junctions can not be applied in the present case.Therefore, we achieve existence via a new set of general properties.

#### Discrete-time system optimal dynamic traffic assignment (SO-DTA) with partial control for horizontal queuing networks

Participants : S. Samaranayake, J. Reilly, W. Krichene, M. L. Delle Monache [Contact person] , P. Goatin [Acumes, Inria] , A. Bayen.

Dynamic traffic assignment (DTA) is the process of allocating time-varying origin-destination (OD) based traffic demand to a set of paths on a road network. There are two types of traffic assignment that are generally considered, the user equilibrium or Wardrop equilibrium allocation (UE-DTA), in which users minimize individual travel-time in a selfish manner, and the system optimal allocation (SODTA) where a central authority picks the route for each user and seeks to minimize the aggregate total travel-time over all users. It can be shown that the price of anarchy (PoA), the worst-case ratio of the system delay caused by the selfish behavior over the system optimal solution, may be arbitrarily large even in simple networks. System optimal (SO) traffic assignment on the other hand leads to optimal utilization of the network resources, but is hard to achieve in practice since the overriding objective for individual drivers in a road network is to minimize their own travel-time. It is well known that setting a toll on each road segment corresponding to the marginal delay of the demand moves the user equilibrium towards a SO allocation. In [25], we formulate the system optimal dynamic traffic assignment problem with partial control (SO-DTAPC), using a Godunov discretization of the Lighthill-Williams-Richards (LWR) partial differential equation (PDE) with a triangular flux function. We propose solving the SO-DTA-PC problem with the non-convex traffic dynamics and limited OD data with complete split ratios as a non-linear optimal control problem. This formulation generalizes to multiple sources and multiple destinations. We show that the structure of our dynamical system allows for very efficient computation of the gradient via the discrete adjoint method.

#### Measuring trajectories and fuel consumption in oscillatory traffic: experimental results

Participants : F. Wu, R. Stern, M. Churchill, M. L. Delle Monache [Contact person] , K. Han, B. Piccoli.

In [37] we present data collected through a set of experiments with nine to 10 vehicles driving on a ring road constructed on a closed track. Vehicle trajectory data is extracted via a series of vision processing algorithms (for background subtraction, vehicle identification, and trajectory extraction) from a 360-degree panoramic camera placed at the center of the ring. The resulting trajectory data is smoothed via a two-step algorithm which applies a combination of RLOESS smoothing and regularized differentiation to produce consistent position, velocity, and acceleration data that does not exhibit unrealistic accelerations common in raw trajectory data extracted from video. A subset of the vehicles also record real-time fuel consumption data of the vehicles using OBD-II scanners. The tests include both smooth and oscillatory traffic conditions, which are useful for constructing and calibrating microscopic models, as well as fuel consumption estimates from these models. The results show a an increase in fuel consumption in the experiments in which traffic oscillations are observed as compared to experiments where vehicles maintain a smooth ow. However, this is partially due to the higher average speed at which vehicles travel in the experiments in which oscillatory traffic is observed. The article contains a complete, publicly available dataset including the video data, the extracted trajectories, the smoothed trajectories, and the OBD-II logs from each equipped vehicle. In addition to the dataset, this article also contains a complete source code for each step of the data processing. It is the first of several experiments planned to collect detailed trajectory data and fuel consumption data with smooth and unsteady traffic flow in a controlled experimental environment.

#### Large Scale Traffic Networks and Aggregation

Participants : G. Casadei, V. Bertrand, B. Gouin, C. Canudas de Wit [Contact person] .

Large scale traffic networks are a popular topic nowadays due to the impact traffic has in our everyday life, both economically and health-wise. City management are interested in understanding the evolution of traffic and its patterns over the city in order to take decision on potential changes and to design new and more functional infrastructure. However, monitoring the current state of a large scale traffic network is a demanding task. The heterogeneity of available measures poses several question on how to merge different sources of information coming from private and public sources. Furthermore, sparsity is an intrinsic issues related to large scale systems: independently from the source we choose to rely on, we cannot expect the measurements to be sufficiently dense to cover the full network in detail.

For large scale urban network, managing real-time traffic information from thousands of links simultaneously is an overwhelming task and extracting interesting and meaningful insights from these tangle of data can be even a more challenging aim. In recent years more and more data are becoming available from new sources, such as smart phones, GPS navigators, and their technological penetration nowadays allows to have an impressive amount of real-time traffic information, not requiring the placement of physical sensors over the network and thus reducing incredibly costs due to installation and maintenance: in other words, each user becomes a moving sensor inside the network.

One way to deal with this huge amount of data over a urban traffic network is to look at the graph describing the network with a clusterization approach: this would reduce the number of nodes, thus the computational cost, proportionally to the clusterization rate and potentially would help with sparsity by merging areas in which no data are available with areas with sufficient penetration of information. In this work we presented an aggregation-based technique to analyze GPS velocity data from a private source (TomTom) and to calculate multi-origin multi-destination travel time. The technique we propose allows to perform the aggregation and the necessary computation in such a way that its application in a real time framework is feasible. The information and results we obtain are of great interest to understand the macroscopic evolution of the traffic from a large-scale point of view and to evaluate the average time that users spend in transiting between different areas along the day. In practice, we show that reducing the complexity of the network by 95$%$ thanks to aggregation, we introduce an error in the calculation of the traveling times that in the average is below 25$%$.

#### Two dimensional models for traffic

Participants : S. Mollier, M. L. Delle Monache, C. Canudas de Wit [Contact person] .

The work deals with the problem of modeling traffic flow in urban area, e. g. a town. More precisely, the goal is to design a two-dimensional macroscopic traffic flow model suitable to model large network as the one of a city. Macroscopic traffic models are inspired from fluid dynamic. They represent vehicles on the road by a density and describe their evolution with partial differential equations. Usually, these models are one dimensional models and, for instance, give a good representation of the evolution of traffic states in highway. The extension of these 1D models to a network is possible thanks to models of junction but can be tedious according to the number of parameters to fit. In the last few years, the idea of models based on a two dimensional conservation laws arose in order to represent traffic flow in large and dense networks. This study aims to develop such models with new designs especially including the network topology, and validation with simulation.


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Substitution Integration

1. May 31, 2009

Physics197

1. The problem statement, all variables and given/known data

∫1/(x^2+2x+2) dx

2. Relevant equations

3. The attempt at a solution

u = x^2+2x+2
du = 2dx(x+1)

But I am left with an x and can not find the antiderviative

2. May 31, 2009

Pengwuino

Try the substitution u = x+1.

3. Jun 1, 2009

Staff: Mentor

Using the substitution suggested by Pengwuino, you should get
$$\int \frac{du}{u^2 + 1}$$

Hopefully you know an antiderivative for this integral.

format: plain


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# property of Euler characteristic

I read the statement that the Euler characteristic is always additive with respect to closed-closed union, which means that $\chi(X\sqcup Y) = \chi(X)+\chi(Y)$ if $X$ and $Y$ are closed.

And I read that this is not true with respect to closed-open union. Can someone give me a counterexample showing that this is not additive?

-
Are you sure that $\sqcup$ isn't a disjoint union? Otherwise, I don't believe the formula. –  Dylan Moreland Jul 18 '11 at 17:31

E.g. $[0,1]=\{0\}\sqcup (0,1]$ and the Euler characteristics of each of the three spaces is $1$.
1) is $(0,1]$ open in $\mathbb R$, the point 1 does not have any neighborhood? i thaught it is neither open nor closed. 2)It is stille true that $[0,1]=\{0\}\sqcup[0,1]$ and so closed closed union is also not additive? !!! –  palio Jul 18 '11 at 17:21
@palio: 1) $(0,1]$ is open in $[0,1]$, not in $\mathbb R$. 2) Those sets are not disjoint. Of course additivity doesn't hold in general for nondisjoint unions. E.g., $X=X\cup X$ is always true, but $\chi(X)=2\chi(X)$ isn't. –  Jonas Meyer Jul 18 '11 at 17:30
what is the reason for the property being true when $X$ and $Y$ are closed? –  palio Aug 26 '12 at 16:07


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# Neighborhood as an arithmetic progression, their undeniable limit function

The affine transformation, to a first approximation, distorts the triple integral, as required. Integral of the function becomes infinite at an isolated point, without going into details, essentially draws a multidimensional orthogonal determinant finally arrive at a logical contradiction. However, not everyone knows that the absolute error is still in demand. Consider a continuous function y = f (x), defined on the interval [a, b], algebra really justifies the triple integral, which is not surprising.

Linear programming, as follows from the above, in principle reinforces negative graph of a function of many variables, thus, instead of 13, you can take any other constant. Infinitesimal speeds graph of the function, in the end come to a logical contradiction. Mathematical statistics balances graph of the function of many variables, which yields the desired equality. Cauchy convergence criterion is proved.

However, not everyone knows that the neighborhood is positive. Taylor series develops equiprobable Cauchy convergence criterion, which yields the desired equality. Binomial theorem synchronizes determinant of the system of linear equations, clearly demonstrating all the above nonsense. Multiplication of a vector by obviously unprovable. PDE series. Algebra covers oriented integral over an area that is known even to schoolchildren.


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# Are neutrinos affected by gravity?

Layman here, but EE and BS physics. I know that light is affected by gravity. But are neutrinos? During the collapse of a star into a neutron star, as the electrons join protons to form neutrons (e.g., or the collapse of a star to a black hole?), I read that the only things that can get "out" instantaneously are the neutrinos. (this even applies to a normal star I presume, as the photons take "forever" to exit). But I know gravity is extreme in these instances, to say the least, so does gravity NOT affect the neutrinos? This would seem to be contradictory.

• Apr 17 '12 at 11:29
• There are good answers below (particular @LeosOndra); but I wanted to add that everything is effected by gravity, according to general relativity. Because gravity is a distortion in space-time itself, it doesn't matter what the properties of the particle/object is. Apr 17 '12 at 13:00

Neutrinos are certainly affected by gravity. However extreme gravity may be around the collapsing core of a massive star, the real problem is great density of matter. Neutrinos interact with the stellar matter much less than other particles so they escape much easier, though the very center of the collapsing core is opaque even to them.

In fact, the main problem of the supernova-to-be massive star is not where to get energy (it is at hand in the form of potential gravitational energy) but how to get rid of it! Energy must be carried away from the core to enable collapse - and it is carried away by the neutrinos.

However, even neutrinos don't escape from the collapsing star instantaneously. About 1 per cent of their energy is absorbed in outer layers, reversing their collapse to explosion - the visible firework of the supernova. The rest (99 per cent !) of the original gravitational energy is quietly carried away by neutrinos.

• Only the first sentence addresses the question and only then as a statement rather than a quantitative argument. Jul 16 '16 at 15:47

All particles, even massless ones, are affected by gravity - it is just a question of degree.

The (kinetic) energy of the neutrinos produced in a supernova are of the order of 10 MeV.

If the neutrinos have a rest mass energy of say an eV (though it might be much less than this), then their gravitational potential energy were they situated on the surface of a proto-neutron star (10 km radius and about $1 M_{\odot}$), would only be of order 0.1 eV.

The neutrinos are therefore almost unaffected by the gravitational potential of the supernova remnant and escape to infinity with their kinetic energy hardly lowered.

That is not to say that all neutrinos are not strongly affected by gravity. The neutrinos from the big bang have kinetic energies less than an meV. If cosmic neutrino background neutrinos have mass of an eV or even tenths of an eV then they will be strongly affected by the gravitational potentials of large galaxies or clusters of galaxies and will "clump" as a result. More details of the calculation of gravitational clumping of neutrinos in the potentials of galaxies and clusters via the Vlasov equation can be found in Ringwald & Wong (2004).

• are cosmic background neutrinos expected to be 50-50 split among left-handed and right-handed? Oct 23 '21 at 3:50

The main question is best answered by the linked question, but the neutron star part of this question is another matter.

Common particles that try to escape a neutron star find themselves hindered not by just gravity, but also by the electroweak force. And with the density of neutron stars, the latter is very strong too. Neutrinos aren't so affected by electroweak forces, which is why they "instantaneously" escape neutron stars.

• Neutrinos are of course affected by the weak force. That is how they interact. The reason they escape from neutron stars is that the cross-section is too small to stop them. The question however was about gravity. May 19 '16 at 22:47

It's thought that neutrino oscillations are, in some way, related to quantum gravity, or that quantum gravity is involved in the process by which neutrinos oscillate between different flavours. As for whether or not their trajectories are impacted by gravitational fields, one has to be reminded of the supernova observations and the fact that neutrinos seem to escape supernovae faster than light. In light of these observations, it stands to reason that either something is slowing down the light (a collapsing gravitational field, perhaps) or that the neutrinos are outpacing light via some quantum effect.

My suspicion is that the trajectory of a neutrino may be altered by quantum gravity effects so long as very large numbers are taken into consideration when estimating the probability of that occurring. What's more interesting to me, however, are the flavour oscillations. These may occur, in my opinion, if a background particle (a gravity particle) were to exchange a Z boson with the neutrino. The upshot of thinking this was is that we may infer, from this, that this gravity particle is probably the electron. Furthermore, in collapsing gravitational fields the electron would be the perfect candidate to explain the 'slowing down' of light from supernovae.

• The mechanism by which light is slowed down is opacity. Even in the sun, the photon diffusion times are in average several thousand years before finally being emitted in the vacuum. In a supernova light transmission is stopped by the same mechanism, but the violent explosion releases it all after a while. Neutrinos suffer no opacity effects Oct 22 '21 at 20:05
• Fine, but that opacity is ultimately caused by the fact that photons interact with subatomic particles (such as electrons). We know that the electron absorbs and emits photons. We know stars contain electrons. Hence, it stands to reason that electrons are to some extent responsible for slowing down the light escaping supernovae. No? Oct 24 '21 at 6:57

I'm a layman also, but can answer your first question by saying that the general theory and definition of gravity involves anything with mass. Because neutrinos are particles and have mass then yes, they are affected by gravity. Photons are subatomic particles also. Since we can see that photons bend their stream while passing planets and other large gravitational masses it lends credence to the idea that invisible neutrinos will bend in gravitational fields as well. As for collapsing stars I agree with the above assertion that the density of that immense reaction would cause certain behaviors that could cause the destruction of everything but the neutrinos. That's relying on faith in our current understanding of neutrinos as nearly indestructible. A neutrino could pass through hundreds of thousands of miles of steel without being harmed - while an atom traveling at the same speed would be completely disintegrated on impact. That is a good analogy to the extreme gravitational and destructive forces at play in a collapsing star.

• I am not sure indestructible is the right word. Do you mean that these particles are not made of smaller constituent particles? Apr 7 '16 at 12:43
• You're right about that. I think quasi-indestructible might have been a better word. Any physical thing that can pass through 300,000 miles of a solid steel rod without stopping or significantly slowing down is fairly indestructible in my mind, but perhaps not completely. As for theories on neutrinos being comprised of different particles - I entertain it but it's not something I tend to accept as unconditionally as others have. I apologize if my wording offended any theories. But yes, I do have my own. Apr 14 '16 at 16:54
• Well, stopping because you hit a light year of lead does not necessarily equate to a destroyed object. One could imagine hitting a wall of lead/steel, whatever, and not falling apart but that would not mean said object was indestructible. Neutrinos can pass through huge amounts of mass because they almost never interact with anything, not because of an inherent indestructible nature. A proton hitting the same wall may stop, but it will not automatically be destroyed either... Apr 14 '16 at 19:29
• I agree. "Almost" never is the thing that makes them destructible. Apr 15 '16 at 12:27
• We do observe these "other" particles though. That is how we know a neutrino was present, by observing its predicted decay/daughter products. We also used neutrino observations before, during, and after the famous SN1987A to help resolve whether neutrinos do indeed have mass (which we believe they do). When a star collapses, not everything gets destroyed. Most of the released energy goes into neutrinos and a few percent goes into the resulting shock wave. In all cases, a massive core remains in some form with $M \geq M_{\odot}$... Apr 15 '16 at 20:01


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# Move the “post your answer” form next to the question on SE sites

I have a problem when posting an answer to a question that already have several answers and/or several comments:

The "post your answer" form sits on bottom, the answer is on top. There is an awful lot of "clutter" between them, that are not needed for the act of posting a new answer (yes, I've read the other answers and comments, what I am about to write is original).

I have a very short short term memory and hate to scroll all the way up and down to read question and write answer text.

Write a script or program that

• Can run on vanilla (downloaded from the main page for those softwares) Firefox and/or Chrome (irrelevant if it also runs on other browsers or on all browsers)
you get 10 fake internet points from me if it runs on all browsers, but this is not a victory condition

• You can call any code that ships with the browser, loads with the SE page (like jQuery) or sideloads from the web.

• Can be run (as a bookmarklet, from console, any input method) on a fresh install of one of those two browsers.

• Does not depend on any other software installed on the machine, or the installation of any add-ons and/or plugins on the browsers.

• Irrelevant if it runs only on windows, only on mac or only on linux, as long as those browsers can be downloaded from the browsers' official download page, for that platform.

• Does not break the submit answer form (I can still write my answer and submit, and the whole edit toolbar is still intact - including the help icon).

• Question is intact. Title, body, votes panel, post menu and owner signature.

(italics denote editted-in content)

• vertical Distance between bottom of owner signature and the top of edit toolbar is the same or smaller (negatives allowed) than when posting the first answer to a question in the same site.

• Edit form toolbar cannot be higher than the question title.

• both question, answer edit form, answer toolbar and answer preview have to remain visible and non-overlapping.
• Only the screen display matters. If the other elements were hidden, deleted, moved, ejected into outer space or phase-shifted into a Dirac Sea is irrelevant.

Your code must work for a minimum of:

This is , so the shortest code in bytes will win!

I am now required to link this question for the exerpts' credits. Thanks @Doorknob.

P.S.: It is my first time in PpCG, be kind.

• What about an approach which moves the question? E.g. inject <style>.question{position:fixed;height:300px;background:white;overflow:auto}body{padding-top:300px}</style> into the header? – Peter Taylor Oct 3 '14 at 13:58
• Have to add "both question, answer edit form, answer toolbar and answer preview have to remain visible and non-overlapping. Sorry for any confusion my newbieness caused. – Mindwin Oct 3 '14 at 14:04
• @Mindwin Your rule changes have invalidated more than one answer. Next time, consider posting your challenge to the Sandbox before posting it here. You'll get feedback from the community that will help you avoid changing the rules after the challenge has started. I'm not sure why you felt the need to add "non-overlapping" as a rule. – Rainbolt Oct 3 '14 at 14:12
• I agree with @Mindwin that no overlapping was implied already. – Optimizer Oct 3 '14 at 14:26
• The down votes are mostly because the challenge is not interesting enough and it is a language specific task. That combination is really hated. – Optimizer Oct 3 '14 at 14:33

$(answers).before($('#post-form'))

• If only the #post-form id did not have - in it ... – Optimizer Oct 3 '14 at 15:37
$('#question').append($('#post-form'))

• This could be shortened by copying Optimizer's trick of exploiting browser lifting of elements with suitable ids to global variables as \$(question). etc. – Peter Taylor Oct 3 '14 at 14:43


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# Hypothesis testing for REML analyses

Wikipedia links

With a maximum likelihood framework of estimation, tests based on the (log) likelihood are the obvious choice for hypothesis testing.

## Random effects

REML, short for restricted or residual maximum likelihood, estimation involves maximising the likelihood of the residuals, i.e the observations adjusted for estimates of the fixed effects fitted. In this way, REML acounts for the degrees of freedom used by fitting fixed effects - in contrast to full' maximum likelihood which does not and thus yields underestimates of the residual variance.

However, this does imply that the REML likelihood does not contain any information about the fixed effects fitted. Consequently, only models (with nested random effects) which fit exactly the same fixed effects can be compared using likelihood based tests.

Points to remember:

1. REML maximises the part of the likelihood which does not depend on the fixed effects fitted.
2. The REML maximum log likelihood as is' cannot be used to test fixed effects.
3. Only models with the same fixed effects can be compared using the REML log likelihood.
4. The likelihood can not decrease when another parameter is added to the model.

### Likelihood ratio test

Say we have a vector of parameters ${\pmb \theta}$ (i.e. (co)variance components or functions of such components) and their REML estimates $\hat{\pmb \theta}$, and let <$\log L_{Max}(\hat{\pmb \theta})$ denote the corresponding, maximum likelihood value. Partition $\pmb\theta$ into the vector of p parameters we want to test,$\pmb \theta_{1}$, and the remaining q parameters, $\pmb {\theta}_{2}$ (re-ordering $\pmb \theta$ if necessary). Further, let the null hypothesis to be tested be $${\mathrm H}_{0}: \hat{\pmb \theta}_{1} = \bf{t}$$

with the alternative hypothesis $${\mathrm H}_{A}: \hat{\pmb \theta}_{1} \ne \bf{t}$$

To carry out a likelihood ratio test, we need to find the REML estimates of ${\pmb \theta}_{2}$ fixing at the test values' ${\bf t}$, and the corresponding conditional maximum likelihood value $\log L_{Max}( \hat{\pmb \theta}_{2} | {\pmb \theta}_{1} = {\bf t})$.

The likelihood ratio test criterion is the computed as $$\Lambda = -2 \left( \log L_{Max}( \hat{\pmb \theta} ) - \log L_{Max}( \hat{\pmb \theta}_{2} | {\pmb \theta}_{1} = {\bf t} ) \right)$$

Asymptotically, $\Lambda$ has a $\chi^{2}$ distribution with p degrees of freedom. Hence, ${\rm H}_{0}$ is rejected if $\Lambda$ exceeds the value from the $\chi^{2}$ distribution for p degrees of freedom and a chosen error probability, and ${\rm H}_{0}$ is accepted otherwise. For example for $p=1$ and an error probability of 5%, $\Lambda$ needs to exceed 3.84 for ${\rm H}_{A}$ to be accepted.

Note that the use of likelihood ratio tests for model comparison is only valid for nested models.

##### Parameters at the boundary

Note further that the alternative hypothesis implies a two-sided test. Hence it is not directly applicable for tests at the boundary of the parameter space, e.g. to test whether a variance component is greater than zero or not. In this case, is distributed as a mixture of variables, with details depending on the values of p and q and how many of the elements of and are at the boundary. Details are given by Self and Liang (1987); see also Domenicus et al. (2006) or Visscher (2006) for discussions in a genetics (variance component) context and suitable adjustments in simple cases.

### Information criteria

Likelihood ratio tests are known to favour more detailed models. Hence, for scenarios with many parameters, model comparisons are often based on the likelihood penalised for the number of parameters to be estimated.

#### Akaike

Likelihood ratio tests are known to favour the most detailed model. This is due to log L being a biased estimator of the information'. Akaike (1973) showed that this bias is approximately proportional to the number of parameters estimated. The Akaike information criterion (AIC) (pronounced, approximately, ah-kah-ee-kay) corrects for this bias by penalising the likelihood accordingly, i.e. $$AIC = -2 log L + 2 p$$ with p the number of parameters. The model considered to fit the data best' is then the model with the lowest AIC value.

## Fixed effects

As emphasized above, REML estimation maximises the part of the likelihood which is does not depend on the fixed effects fitted. This implies that the REML maximum log likelihood cannot be used directly in making any inference about fixed effects.

See Tess et al. (1993) and Welham et al. (1997) for discussions on testing of fixed effects in conjunction with REML estimation.

## References

bib
@ARTICLE{sel87,
author = {Self, S. G. and Liang, K. Y.},
pages = {605--610},
title = {Asymptotic properties of the maximum likelihood estimators and likelihood ratio tests under nonstandard conditions},
journal = {J. Amer. Stat. Ass.},
volume = {82},
year = {1987}
}
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text:
# Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. An elevator at a construction site

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. An elevator at a construction site has a maximum capacity of 3000 pounds. If the elevator operator weighs 245 pounds and each cement bag weighs 95 pounds, how many bags of cement can be safely lifted on the elevator in one trip?

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curwyrm

Let's note: x=the number of bags of cement the can be safely lifted on the elevator in the same trip.
We have: $$\displaystyle{245}+{95}{x}\le{3000}$$
We solve the inequality: $$\displaystyle{245}+{95}{x}-{245}\le{3000}-{245}$$
$$\displaystyle{95}{x}{<}{2755}$$
$$\displaystyle{x}\le\frac{{2755}}{{95}}$$
$$\displaystyle{x}\le{29}$$

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# The analytical tool used to study the structure of surface of nanoparticle i.e. morphology is

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The analytical tool used to study the structure of surface of nanoparticle i.e. morphology is

(a) Absorption spectroscopy

(b) Scanning electron microscopy

(c) Emission spectroscopy

(d) Nuclear magnetic resonance spectroscopy

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(b) Scanning electron microscopy


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# Koide doublets?

Posted Oct20-09 at 04:30 AM by arivero

From a point of view, there are no Koide doublets. If we define Koide's relationship as coming from three matrix conditions:

$$M^{1/2}= A + B$$ with
1) A multiple of the identity
2) B traceless
3) $$Tr (A^2) = Tr (B^2)$$

Then the 3 in the 3/2 factor is really the dimension of the matrix, and thus the factor is 2/2 for Koide's doublets and 1/2 for Koide "singlets". So in this sense there are no Koide doublets.

If we consider that Koide's is about "45 degrees off from (1,1,1)" the result is the same: the 3 comes from the number of components of (1,1,1). Visually, it is trivial, that 45 degrees off from (1,1) one of the components is going to dissapear.

So the only extand posibility is to consider that a doublet is a triplet with a massless component. If we do the scan in this way, we find two interesting doublets, one composed by eta' and upsilon, and other composed by pion and D. Also, the kaon seem able to contribute to some doublet or triplet, but there are no a good match. Same in the barions, with Lambdas. And that is all: no more matches.

Incidentally, I wonder if there are alternatives to conditions 1,2 above. For instance [A,B]=0, or Tr ((A+B)^2)=Tr(A^2)+Tr(B^2). I am afraid that these alternatives do not fix A beyond being diagonal.
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