text:
## Stream: general

### Topic: detecting terminal goals

#### Scott Morrison (Apr 26 2018 at 10:52):

I have a little tactic that is meant to determine with the current goal is "terminal", that is, no other goals depend on it.

#### Scott Morrison (Apr 26 2018 at 10:52):

Unfortunately it is not working at the moment.

#### Scott Morrison (Apr 26 2018 at 10:53):

(The idea is just that if you know your current goal is terminal, you can be much more aggressive in discharging it, because nothing can go wrong later.)

#### Scott Morrison (Apr 26 2018 at 10:54):

I currently have

meta def metavariables : tactic (list expr) :=
do r ← result,
pure (r.fold [] $λ e _ l, match e with | expr.mvar _ _ _ := insert e l | _ := l end) meta def terminal_goal : tactic unit := do goals ← get_goals, let current_goal := goals.head, other_goals ← metavariables, let other_goals := other_goals.erase current_goal, other_goals.mmap'$ λ g, (do t ← infer_type g, d ← kdepends_on t current_goal,
monad.whenb d $pp t >>= λ s, fail ("This is not a terminal goal: " ++ s.to_string ++ " depends on it."))  #### Scott Morrison (Apr 26 2018 at 10:54): And this works great detecting when the current goal appears in the form ?m_1 in a later goal. #### Scott Morrison (Apr 26 2018 at 10:54): But fails to detect that the current goal appears as something like ?m_1[_] in a later goal. #### Scott Morrison (Apr 26 2018 at 10:55): Q1. Is this already implemented somewhere, better? #### Scott Morrison (Apr 26 2018 at 10:55): Q2. Any suggestions how I fix it? #### Scott Morrison (Apr 26 2018 at 11:00): Here's a MWE: private structure D := ( w : ℕ → Type ) ( x : list (w 0) ) private def test_terminal_goal : D := begin split, swap, success_if_fail { terminal_goal }, -- succeeds, because terminal_goal correctly fails intros, success_if_fail { terminal_goal }, -- fails, because terminal_goal incorrectly succeeds exact ℕ, exact [] end  #### Simon Hudon (Apr 26 2018 at 13:49): I think I would do it as: meta def terminal_goal' : tactic unit := do g :: gs ← get_goals, gs.for_each$ λ g', guard (g'.occurs g)

It works with your example but does it work with your use cases?

#### Scott Morrison (Apr 27 2018 at 04:43):

@Simon Hudon, this doesn't seem to work in other cases. My tests are here: <https://gist.github.com/semorrison/5188f3c3e508148657be4d66f4875d8d> if you want to have a look. I have to run now, but will try to decipher why your version isn't working on the other tests later.

#### Simon Hudon (Apr 27 2018 at 04:50):

Ok, I'll have a look when I wake up :)

#### Scott Morrison (Apr 27 2018 at 04:52):

I don't really understand the logic of your suggestion: surely you meant guard (¬ g.occurs g') not guard (g'.occurs g)? In any case neither of those work. Sleep well. :-)

Thanks :)

#### Scott Morrison (Apr 28 2018 at 02:48):

When I see a goal with a "parametrised metavariable" like ?m_1[0], what does the underlying expr look like? I can't decipher what it should be.

#### Scott Morrison (Apr 28 2018 at 02:50):

Is it just an app of an mvar?

#### Scott Morrison (Apr 28 2018 at 03:29):

Okay --- I've worked this all out. For anyone keeping score:

meta def metavariables : tactic (list expr) :=
do r ← result,
pure (r.fold [] $λ e _ l, match e with | expr.mvar _ _ _ := insert e l | _ := l end) meta def propositional_goal : tactic unit := do goals ← get_goals, let current_goal := goals.head, current_goal_type ← infer_type current_goal, p ← is_prop current_goal_type, guard p meta def subsingleton_goal : tactic unit := do goals ← get_goals, let current_goal := goals.head, current_goal_type ← infer_type current_goal >>= instantiate_mvars, to_expr (subsingleton %%current_goal_type) >>= mk_instance >> skip meta def terminal_goal : tactic unit := propositional_goal <|> subsingleton_goal <|> do goals ← get_goals, let current_goal := goals.head, other_goals ← metavariables, let other_goals := other_goals.erase current_goal, other_goals.mmap'$ λ g, (do t ← infer_type g, t ← instantiate_mvars t, trace t, d ← kdepends_on t current_goal,
monad.whenb d \$ pp t >>= λ s, fail ("This is not a terminal goal: " ++ s.to_string ++ " depends on it."))

#### Simon Hudon (Apr 28 2018 at 04:22):

I don't know if you still need the answer but ?m_1[0] is a regular meta variable. I think [0]signals the context that the variable can see. I don't know how to decode it though.

#### Scott Morrison (Apr 28 2018 at 04:30):

Thanks @Simon Hudon, all sorted out. ?m_1[0] is encoded simply as an application of an mvar on the expressions appearing inside the [ ... ]. It turned out that I was missing an instantiate_mvars, which was preventing successfully detection of these sort of "dependent mvars". I've made a PR for my terminal_goal tactic and some relatives. https://github.com/leanprover/mathlib/pull/125

#### Simon Hudon (Apr 28 2018 at 04:31):

You have to be careful with those variables. They sneak up on you

#### Scott Morrison (Apr 28 2018 at 04:34):

I can't believe that I've written code that successfully uses instantiate_mvars. What it actually means is still voodoo to me.

#### Simon Hudon (Apr 28 2018 at 04:40):

I can try to demystify if you like

#### Simon Hudon (Apr 28 2018 at 04:41):

Have you used get_goals and set_goals?

Yes, many times!

#### Simon Hudon (Apr 28 2018 at 04:42):

Good! And do you know that get_goals returns a list of expressions that are in fact unassigned meta variables?

Yes!

#### Scott Morrison (Apr 28 2018 at 04:43):

(If I were Kevin, I'd be promising to write documentation in exchange for your explanation. :-)

#### Simon Hudon (Apr 28 2018 at 04:45):

(Haha! Where's Kevin when we need him!)

Good. So think of the proof state as being made of two parts (for the sake of this explanation): list of goals which is an arbitrary list of meta variables and the set of all allowed meta variables, some of which are assigned a value.

Great!

#### Simon Hudon (Apr 28 2018 at 04:48):

The relationship between the proof state and whatever you're trying to prove is that, when you enter tactic mode, you create an unassigned variable, put it in the list of goals and that variable is in fact the proof that you're supposed to return. From that point on, the assertion you're trying to prove does not matter. Only the variables and the goals do.

#### Simon Hudon (Apr 28 2018 at 04:49):

I don't know if you've ever had that problem but sometimes, you succeed in leaving a tactic block and when you finish the proof, you're told that your proof contains unassigned variables. Does this ring a bell?

#### Simon Hudon (Apr 28 2018 at 05:08):

Long story short, instantiate_mvar finds all the mvar nodes in your expression and, if the meta variable has a value (or type expr) in the proof state it replaces mvar with it. That means that, if you're pattern matching (using match) on an expression and you're looking for expr.app (expr.app (const eq _) e0) e1 and that you mean mvar instead, even if that variable is assigned to a value that matches exactly what you're looking for, the pattern matching will fail. That's why it's prudent to use instantiate_mvar before matching on a expr of which you don't know for sure that it doesn't contain meta variables.

#### Simon Hudon (Apr 28 2018 at 05:09):

I realize my explanation was a bit round about. I just needed to introduce the idea that the life span of meta variables is a whole proof and that it doesn't disappear once it's assigned.

I see.

#### Scott Morrison (Apr 28 2018 at 05:20):

I've encountered needing it in another strange place, as well, which I think matches with your explanation.

#### Scott Morrison (Apr 28 2018 at 05:20):

I was writing a tool for finding _all_ the possible rewrites of a given expression by a given rule.

#### Scott Morrison (Apr 28 2018 at 05:21):

Unfortunately the built-in rewrite has some problems --- once it has matched some of the parameters of your rewrite rule a particular way, it will subsequently only match the parameters the same way, when looking for further places the rule matches.

#### Scott Morrison (Apr 28 2018 at 05:22):

This even persists between different invocations of rewrite_core, because this information is actually stored in the tactic state.

#### Scott Morrison (Apr 28 2018 at 05:22):

To solve this, I had to write a tactic lock_tactic_state (t : tactic A) : tactic A which just discards any changes to the tactic state after invocation.

#### Scott Morrison (Apr 28 2018 at 05:22):

This throws away any information the tactic state was storing about how rewrite parameters had to match.

#### Scott Morrison (Apr 28 2018 at 05:23):

Unfortunately, the proofs that rewrite_core produced where suddenly full of metavariables!

#### Scott Morrison (Apr 28 2018 at 05:23):

I understand now that this is because these metavariables were _assigned_, but not _instantiated_, so when I discarded the tactic_state those assignments were being lost.

#### Scott Morrison (Apr 28 2018 at 05:24):

Calling instantiate_mvars on the proof term, before discarding the state, saved those assignments.

phew! :-)

#### Scott Morrison (Apr 28 2018 at 05:25):

(Sorry I missed the end of your explanation; family things happening at this end. I'm going to ping you @Simon Hudon as I expect you'll find this other example interesting.)

#### Simon Hudon (Apr 28 2018 at 05:31):

I'd be curious to see how you implemented lock_tactic_state`. Did you deconstruct the tactic value and extract the expected proof?

#### Simon Hudon (Apr 28 2018 at 05:33):

I'm looking forward to your next example. I'll probably see it when I wake up. Incidentally, Australia is way too far! Someone should move it closer to Europe and America, that way our days would overlap!

Last updated: May 16 2021 at 21:11 UTC

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# Pseudoisotopy in low dimensions

Recall that the space $P(M)$ of (smooth) pseudoisotopies of the compact manifold $M$ is defined as the space of all diffeomorphisms $M\times I\to M\times I$ that fix every point in $(M\times 0)\cup (\partial M \times I)$. I have worked on these things in high-dimensional cases, but I realize that there are big gaps in my knowledge in low-dimensions. I am wondering how much of what I don't know is known.

If the dimension of $M$ is less than $2$ then $P(M)$ is contractible. According to the Smale Conjecture $P(D^2)$ (or equivalently $P(S^2)$) is contractible. I have the impression that Hatcher also proved that $P(M)$ is contractible when $M$ is a two-dimensional annulus. Is it perhaps known, or believed, that this holds for all compact $2$-manifolds? Or maybe it's not true. (I could have started by just asking Allen Hatcher or Ryan Budney, but I thought others might be interested in any answers.)

And what is known in the next dimension?

-

When $M$ is a compact $2$-manifold, with or without boundary, $P(M)$ is known. When $M$ is a 3-manifold there's bits and pieces known, especially once you get to more fine detail like pseudo-isotopy embedding spaces. But at the level of $P(M)$ I don't think there's a complete description for a single $3$-manifold. For $4$-manifolds the situation is worse, but again there are some things known for pseudo-isotopy embedding spaces.

As you mention, $P(S^2)$ and $P(D^2)$ are contractible by the Smale Conjecture. I think $P(M)$ is contractible for an arbitrary $2$-manifold. The argument goes like this: You look at locally trivial fiber bundle $P(M) \to Diff(M)$. The fiber over the identity map is $Diff(M \times I)$, the group of diffeomorphisms fixing the entire boundary $\partial M \times I \cup M \times \partial I$. The other fibers are empty because diffeomorphisms are prescribed up to isotopy by their action on $\pi_1 M$. Except in a few cases, $Diff(M)$ has contractible components, so showing $P(M)$ is contractible is equivalent to showing $Diff(M \times I)$ is contractible. But $M \times I$ has incompressible annuli in it, corresponding to $C \times I$ where $C \subset M$ is a closed curve in $M$ that does not bound a disc in $M$. Hatcher's work on spaces of incompressible surfaces kicks in and tells you the space of these incompressible annuli is contractible. So you can reduce studying $Diff(M \times I)$ to things like $Diff(A \times I)$ where $A$ is an annulus or a pair of pants, but then you have vertical incompressible discs you can use and reduce to the point that $A$ is itself a disc.

The above argument works for any surface other than $\mathbb RP^2$, a torus or a klein bottle. But similar arguments cover these cases.

Here's the reference for the results of Hatcher's I'm using, it also defines the terminology like "incompressible annulus, disc" and so on. http://www.math.cornell.edu/~hatcher/Papers/emb.pdf

Allen had a student (Kiralis) who wrote some papers on pseudo-isotopy diffeomorphisms of 3-manifolds. That might be a place to look.

But the kind of things that I remember are mostly along the lines of pseudo-isotopy embeddings of knots and links in $D^3$ and $S^3$. I'm about to hop on an airplane. I'll edit this response sometime in the next few days and add some of these observations if they haven't already been made by someone else.

edit 1: Here are two unrelated observations.

(a) Let $N$ be a co-dimension zero solid torus in $M=\mathbb R^3$ or $M=S^3$. There's the pseudo-isotopy embedding fibration $P(N,M) \to Emb(N,M)$. If $N$ is an unknotted solid torus, the question of what the map the image of the map $\pi_0 P(N,M) \to \pi_0 Emb(N,M)$ is, this is a long-standing hard problem in knot theory. Another way to say it is `which knots in $S^3$ bound a disc in $D^4$?'. These knots are called slice knots. Ralph Fox has the Slice Ribbon Conjecture, which might be described as a hopeful combinatorial answer to the question. There are many useful tools for determining whether or not a given knot is slice,starting with the Alexander module and more recently tools from Heegaard Floer theory.

Two examples:

• Look at the class of knots that are a connect-sum of torus knots. Using the Alexander module Litherland proved such a knot bounds a disc in $D^4$ if and only if in the prime decomposition, the number of times a prime summand appears (like say a right-handed trefoil) is equal to the number of times its mirror-image appears (the left handed trefoil, in my example). Related previous MO thread

• Paolo Lisca has used Heegaard Floer theory to determine when a connect sum of 2-bridge knots in $S^3$ bounds a disc in $D^4$ REF, but in this case the answer is more elaborate.

(b) Just like how diffeomorphism groups $Diff(D^n)$ and spaces of long knots have actions of cubes operads, pseudo-isotopy embedding spaces and diffeomorphism groups also have such actions. I'm pretty sure there's a pseudo-isotopy splicing operad as well, but I haven't written out the details.

edit 2: Rick Litherland generalized a result of Zeeman (Deforming twist-spun knots TAMS 250 (1979) 311--331) showing that "Deform-spun knots" have complements that frequently fibre over $S^1$. This process called "Deform spinning" is just the boundary map in the pseudo-isotopy fibre sequence for spaces of knots. One of the nice things about this is Litherland gives a prescription for what the fibre is. In the case where you're looking at the pseudo-isotopy sequence for long knots in $\mathbb R^3$, it generates long embeddings of $\mathbb R^2$ in $\mathbb R^4$. So the fibre is a 3-manifold with boundary a sphere. This process produced some embeddings of 3-manifolds in the 4-sphere that nobody had known about at the time, like the once-punctured Poincare Dodecahedral Space (which without a puncture does not embed in $\mathbb R^4$, at least, not smoothly, it does admit a tame topological embedding). I got interested in this case largely because it represents sort of an extreme end of the terrain of your dissertation.

edit 3: I forgot to mention, I kept on pushing trying to understand why your dissertation broke down in co-dimension two. In some sense my paper "An obstruction to a knot being deform-spun via Alexander polynomials" is an answer. In a way it's not, since co-dimension two deform-spinning is more of a free loop space construction than a based loop space construction. But I found the exercise informative.

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# Find the first odd multiplicity root of a function

I'm trying to find the "first" (greater than some initial $t_0$) odd root (that is, a root after which the sign of the function changes) of a function $f(t)$, if there is one, that is also less than some max bound $t_{\max}$. I can evaluate derivatives of the function, too, but they get more much more expensive the further I go, so I don't want to derive past the first or second derivatives. (The function is a combination of polynomials and trig functions). The idea is that this would run as part of another algorithm, so I can't visually inspect the function to try and bracket the roots, and the method I use has to be robust.

So far I figure I can walk "downhill" (or "uphill", depending on where we start) from $t_0$ to $t_{\max}$ by using a Taylor expansion's bounds on the max error and iterating by taking the largest "safe" step size that won't cross 0. Eventually, this should converge to the "first" zero of the function. It's not a great algorithm numerically, but let's say I use either this or some other method to find the first root.

If the zero has multiplicity of 1, the first derivative will not be 0 and I can test for that, and then I'm done. However, anything greater than that multiplicity-wise will generate a first derivative that is 0, so I won't be able to determine from that if the multiplicity is even or odd.

So I had the thought that if I could take some step size $h$ and evaluate the function at $f(t+h)$ I could check its sign and see if the function crossed the x axis or just bounced off of it (if the root was odd or even). If it's an even root, I can start hillclimbing away from 0 and continue on with the algorithm. If it's an odd root I'm done.

However for this scheme to work I need to be sure that there aren't any roots in the interval $(t, t+h]$. So I need some way to find a safe step size $h$. I could just use some small epsilon but I was hoping that there was a better symbolic way to approach the problem.

Or even better if there's a way to determine whether a given root is even or odd multiplicity when you have it in hand, that would be even better.

Specifically, annoying test cases I have in mind:

$f(t) = \cos(t)+1$ <-- Infinite number of even roots

$f(t) = (t-3)^3$ <-- Odd root, but first derivative is 0

$f(t) = t^2-1^{-15}$ <-- Odd root, but not necessarily detectable with machine precision. I wouldn't mind counting these cases as even, but I want to be careful throwing epsilons all over the place.

...

Specifically, the function of interest has this form:

$f(t) = \mu^T \Pi^T \upsilon$

where:

$\mu = [\theta_1(t)](r_2-r_1)$

$\upsilon = b(t) - [\theta_0(t)]r_0 + [\theta_1(t)]r_1$

$\Pi$ is the 2x2 rotation matrix that rotates a vector by 90 degrees.

$b(t)$ is a 2 element column vector polynomial (that is, a polynomial with coefficients that are 2D vectors)

$\theta_0(t), \theta_1(t)$ are scalar polynomials

$r_0, r_1, r_2$ are constant vectors.

$[\theta(t)]$ is shorthand for the 2x2 rotation matrix by angle $\theta$.

...

So basically, is there a way to declare some neighborhood around a given root as containing no other roots?

Or alternatively, if anyone has other ideas about how to approach this problem.

-
For finding roots of transcendental equations, there really is no good "general" method; one adapts Newton-Raphson and ilk to the nature of the function in question. I have to say that we might be more helpful if you post your actual function. –  Ｊ. Ｍ. Apr 18 '11 at 23:10
The actual function is... messy. It's basically the time varying dot product of two vectors rotated by a time varying rotation matrix. It's about as well behaved as a polynomial summed with the cosine of another polynomial. –  Jay Lemmon Apr 18 '11 at 23:43
I'd say that in that case, we need all the more to exploit whatever structure there is in your function. If you're going to be solving lots of members of a family of equations, a good analysis is very much worth it. At the very least, take comfort in the fact that you're only dealing with one variable... –  Ｊ. Ｍ. Apr 18 '11 at 23:59
I added the function in the original post... don't say I didn't warn you! –  Jay Lemmon Apr 19 '11 at 0:56
Rather confusing... you use $\theta_0(t)$ but you do not define it anywhere, and you define $[\theta(t)]$ but use it nowhere. Are you sure that's what you have? –  Ｊ. Ｍ. Apr 19 '11 at 2:13


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## Sunday, 13 December 2009

### Duals Duals Duals

Is there a connection between all these "duals" in math?
There seem to be two main classes of duals.
One from linear algebra / functional analysis, and one from optimization theory. (Lagrangian/convex duals)

Well actually there is.

Recall that dual space of a vector space V is simply the set of all linear functionals of V. To put it another way, dual space is simply all the row vectors if the original space is finite dimensional.

Dual norm comes up in convex optimization. Dual norm of z is defined as sup_x { | ||x||<=1}. This is actually the same as the operator norm of the vector z when you think of z as an element of the dual space (row vector). In great generality, we can think of dual norm as INDUCED NORMS ON THE DUAL SPACE.

This really plays a role in Lp duality.  Lp dualtity says that the dual of Lp space is isometric to Lq. Here we are talking about infinite-dimensional spaces. Also the dual norm of an element of the dual space of Lp will be equal to the Lq norm of that linear functional when mapped into Lq. (See any real analysis text)

You can think about the dual cone as one way of generating a INDUCED CONE IN THE DUAL SPACE. (Again, we think of row vectors as elements in the dual space)

I don't see the relation between the Fenchel dual (Legendre transform), Lagrangian dual, and the dual space though.

## Wednesday, 9 December 2009

### Syndicated from another blog.

“男人有 老婆又有情人是男人的本事。请放心，情人不会成为老婆，两个概念不一样。男人的大部分心思还是在你身上，因为你是他老婆，是孩子的妈妈，名正言顺，任何时 候都改变不了。而情人只不是婚姻的一种补充，是成功男人一种精神上的追求和情感上的需要，其实也是挺正常的事情，也不必太认真。”——这是昨天中午我在一 个案宗中看到的，记录着一个成功男人的一段非常有代表性的“经典话语”，这就是他们奉行“家中红旗不倒，外面彩旗飘飘”，今朝有酒今朝醉，自嘲“自己喜新 不厌旧，风流不下流”。

人生 短暂，即使你通过努力拼搏成功了也不要为自己找许多漂亮的借口，而去三番五次的寻找情人当作自己的“标志”。虽然在你的脸上贴上美丽好看的“情人标志”， 但那都啻是你一厢情愿的虚荣心在作怪罢了，人们一定会对你深恶痛绝而讥笑藐视你的，同样你也将会是臭名远洋、遗臭万年的。之所以，你还是做一个洁身自好的 成功男人为好，这样人们就会对你另眼相看而铭记你的忠贞气节，却给以你流芳百世。

现在 和谐的社会需要更多的是老实敦厚、温文尔雅的成功男人，同样也需要更多的是安分守己、贤惠善良的温柔女人。只有男人永远不再做纸醉金迷、拈花惹草的男人， 女人不再做水性扬花、放任自流的女人，那么整个世界将会是更加充满人情味的，而幸福的婚姻家庭就会是永远的牢固不破了。


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(Generated on Fri Feb 19 2016 13:25:10 by 1.8.9.1)
EW

A module for electroweak precision observables. More...

## Files

file  EWObservables.h
A header file for the observables defined in EW module.

## Classes

class  Abottom
An observable class for the left-right asymmetry in $$e^+e^-\to Z\to b\bar{b}$$ at the $$Z$$ pole. More...

class  Acharm
An observable class for the left-right asymmetry in $$e^+e^-\to Z\to c\bar{c}$$ at the $$Z$$ pole. More...

class  AFBbottom
An observable class for the forward-backward asymmetry in $$e^+ e^-\to Z\to b\bar{b}$$ at the $$Z$$ pole. More...

class  AFBcharm
An observable class for the forward-backward asymmetry in $$e^+ e^-\to Z\to c\bar{c}$$ at the $$Z$$ pole. More...

class  AFBlepton
An observable class for the forward-backward asymmetry in $$e^+ e^-\to Z\to \ell^+ \ell^-$$ at the $$Z$$ pole. More...

class  Alepton
An observable class for the left-right asymmetry in $$e^+e^-\to Z\to \ell^+ \ell^-$$ at the $$Z$$ pole. More...

class  Epsilon1
An observable class for the $$\epsilon_1$$ parameter. More...

class  Epsilon2
An observable class for the $$\epsilon_2$$ parameter. More...

class  Epsilon3
An observable class for the $$\epsilon_3$$ parameter. More...

class  Epsilonb
An observable class for the $$\epsilon_b$$ parameter. More...

class  GammaW
An observable class for the total decay width of the $$W$$ boson. More...

class  GammaZ
An observable class for the total decay width of the $$Z$$ boson. More...

class  Mw
An observable class for the $$W$$-boson mass. More...

class  PtauPol
An observable class for the $$\tau$$ polarization in $$e^+e^-\to Z\to \tau^+\tau^-$$ at the $$Z$$ pole. More...

class  Rbottom
An observable class for $$R_b^0=\Gamma(Z\to b\bar{b})/\Gamma(Z\to {\rm hadrons})$$. More...

class  Rcharm
An observable class for $$R_c^0=\Gamma(Z\to c\bar{c})/\Gamma(Z\to {\rm hadrons})$$. More...

class  Rlepton
An observable class for $$R_\ell^0=\Gamma(Z\to {\rm hadrons})/\Gamma(Z\to \ell^+ \ell^-)$$. More...

An observable class for the cross section of $$e^+ e^-\to Z\to \mathrm{hadrons}$$ at the $$Z$$ pole. More...
An observable class for the leptonic effective weak mixing angle $$\sin^2\theta_{\rm eff}^{\rm lept}$$ at the $$Z$$ pole. More...


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# Using {rms} package for multinomial logit

Is it possible to use the rms package to model multinomials logits, or elsewise to model several binary logits to achieve the same effect? I am aware that there are many other packages specifically designed to fit multinomial logits, but I am quite attached to the suite in rms.

• Not the answer you're looking for, but the multinom function in the nnet is superb. Uses the exact same functionality as any glm object (which technically speaking, it is). Aside from preference, what do you need form rms? – AdamO Jul 28 '15 at 19:13
• Bootstrapped covariance, prediction plots (with prediction intervals), influence statistics, and splines are some of the advantages I'm after. I'm actually using nnet and I don't have any issues with it—I think it covers my basic needs very well. – Johan Larsson Jul 28 '15 at 19:18


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Since a number of people weren’t able to make today’s lunch presentation, I thought I’d post the slides so those who missed out could catch up. They are pretty bare-bones, so if you have any questions, please don’t hesitate to ask!

You can navigate through the slides by using the arrow keys on your computer.


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## finite fields

Find two distinct monic irreducible polynomials f(x) and g(x) of degree 3 over the finite field F_3.(for instance, let f(x)=x^3 + 2x + 1 and g(x)= x^3 + 2x + 2.) Let a be a root of f(x) and be be a root of g(x).

i)identify a root of f(x) in F_3(b) (this means we adjoin b to F_3) and similarly identify a root of g(x) in F_3(a). (I didn't understand what this part of the question means.) Then write an explicit isomorphism from F_3(a) to F_3(b).

ii) identify all the three roots of f(x) in F_3(a) and that of g(x) in F_3(b). (Okay, i can do this by trail and error, but is there a better procedure for these type of questions?)

Thanks!


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# Autoincrement field when New Feature added

I know that this is probably a very basic programming but I am brand new to Python.

I have a layer in a geodatabase that I need a unique ID field (for joining). I can't use the ObjectID because that can change. I can't use a GlobalID because the data will be used by databases that can't handle the "{" brackets or dashes.

I have already populated the field. When I add a new feature I want to have the value of the max existing value + 1 for the new feature.

Thanks to Jason here is what I have so far-

import arcpy
gdb_full_path = r'D:\GIS\ZA.gdb\Current'
feat = 'ZA_Merge'
arcpy.env.workspace = gdb_full_path
with arcpy.da.SearchCursor(feat,["ANID"]) as cursor:
for row in cursor:
unique_id_counter = row.count
with arcpy.da.InsertCursor(feat,["ANID"]) as cursor:
cursor.insertRow((unique_id_counter + 1))

Here is the error that I get -

Runtime error Traceback (most recent call last): File "", line 9, in TypeError: unsupported operand type(s) for +: 'builtin_function_or_method' and 'int'

-
Are you working with ArcGIS? If so, which version? –  Jason Apr 18 '13 at 20:48
Yes...sorry. ArcGIS 10.1 –  Ken B Apr 18 '13 at 20:51

Here is a basic example of how you can add a new feature to a feature class with an incremental unique ID based on the last feature in said feature class:

import arcpy

gdb_full_path = r'c:\full\path\to\your\geodatabase.gdb'
feat = 'feature_class'

arcpy.env.workspace = gdb_full_path

# create a search cursor so we can read the rows of the feature class
with arcpy.da.SearchCursor(feat,['name_of_unique_id_field']) as cursor:
for row in cursor:
# for each row, we read the unique ID...once we get to the end our
# new variable, "unique_id_counter", will be equal to the value of the
# last feature
unique_id_counter = row.name_of_unique_id_field

# create an insert cursor to add new features to the feature class
with arcpy.da.InsertCursor(feat,['name_of_unique_id_field']) as cursor:
# insert a new row and populate the unique ID field with a new value
cursor.insertRow( (unique_id_counter+1) )

-
nice and smart! –  MC5 Apr 18 '13 at 21:22
Error that I get- Runtime error Traceback (most recent call last): File "<string>", line 7, in <module> AttributeError: 'tuple' object has no attribute 'ANID' –  Ken B Apr 18 '13 at 22:08
That field exists and I had to tweek a little because I was getting the same problem with this line - with arcpy.da.SearchCursor(feat,['name_of_unique_id_field']) as cursor: –  Ken B Apr 18 '13 at 22:12
@KenB, update your question and include the code that you are using. It looks like you are trying to access data from a field that wasn't read by the cursor object we created. When we create the cursor objects in the example above, we are specifying that the cursor only returns the fields in the list ['name_of_unique_id_field']. You can drop this list entirely and the cursor will return all fields - this way you can access data from other fields. The main advantage of returning specific fields is seen when you have very large tables with many fields that you might not need for a given case. –  Jason Apr 18 '13 at 22:32
@Jason, Sorry, just getting used to the formaatting. –  Ken B Apr 18 '13 at 23:06


text:
# THE QUADRATIC FORMULA REVISITED; THE DISCRIMINANT

• PRACTICE (online exercises and printable worksheets)

The quadratic formula and discriminant were introduced in a prior section in the Algebra II curriculum.
Quickly read through this earlier section and do plenty of the online exercises to make sure that you understand the concepts there.
For your convenience, this current section gives an in-a-nutshell review of the quadratic formula, discriminant, and related concepts.

 REVIEW the quadratic formula, the discriminant, related concepts QUADRATIC EQUATIONS: A quadratic equation is an equation that can be written in the form $\,ax^2 + bx + c = 0\,$, where $\,a\,$, $\,b\,$ and $\,c\,$ are real numbers, with $\,a\ne 0\,$. Thus, a quadratic equation must have an $\,x^2\,$ term; it is allowed to have an $\,x\,$ term and a constant term. STANDARD FORM OF A QUADRATIC EQUATION: The form ‘$\,ax^2 + bx + c = 0\,$’ is called the standard form of the quadratic equation. Thus, in standard form, a zero appears on the right-hand side; on the left, the $\,x^2\,$ term comes first, followed by the (optional) $\,x\,$ term and the (optional) constant term. STANDARD FORM IS NOT UNIQUE: Standard form is not unique. For example, ‘$\,3x^2 - 2x + 1 = 0\,$’ and ‘$\,-3x^2 + 2x - 1 = 0\,$’ are equivalent equations; both are in standard form. If uniqueness is desired, the following additional requirements can be imposed: the greatest common factor of all nonzero coefficients is $\,1\,$ $a > 0$ THE QUADRATIC FORMULA: Let $\,a\,$, $\,b\,$ and $\,c\,$ be real numbers, with $\,a\ne 0\,$. Then, the following two equations are equivalent—they are true at the same time, and false at the same time: $$\begin{gather} ax^2 + bx + c = 0\cr\cr \color{red}{x = \frac{-b\pm\sqrt{b^2 - 4ac}}{2a}}\cr \color{red}{\text{THE QUADRATIC FORMULA}} \end{gather}$$ The quadratic formula is derived here. The plus-or-minus sign ($\,\pm\,$) that appears in the quadratic formula provides a convenient shorthand. Otherwise, the quadratic formula would look like this: $$\color{red}{x = \frac{-b +\sqrt{b^2 - 4ac}}{2a} \qquad \text{ or } \qquad x = \frac{-b -\sqrt{b^2 - 4ac}}{2a}}$$ The quadratic formula gives all complex solutions to the quadratic equation $\,ax^2 + bx + c = 0\,$. THE DISCRIMINANT OF A QUADRATIC EQUATION: See that expression ‘$\,b^2 - 4ac\,$’ under the square root in the quadratic formula? It is called the discriminant. As shown below, it helps us to discriminate between the various types of solutions to a quadratic equation. The graph of every quadratic function $\,y = ax^2 + bx + c\,$,   $\,a\ne 0\,$, is a parabola. The parabola has different $x$-intercepts based on different values of the discriminant: $b^2 - 4ac = 0$ exactly one $x$-intercept $b^2 - 4ac > 0$ two different $x$-intercepts $b^2 - 4ac < 0$ no $x$-intercepts

## NATURE OF SOLUTIONS TO A QUADRATIC EQUATION WITH REAL COEFFICIENTS

There are three types of solution sets possible for a quadratic equation $\,ax^2 + bx + c = 0\,$, $\,a\ne 0\,$.
These solution sets are determined by the discriminant, $\,b^2 - 4ac\,$: whether it is zero, positive, or negative.
The three situations are summarized in the table below.

 SOLUTIONS OF $ax^2 + bx + c = 0\,$,   coefficients are real numbers, $a\ne 0$ value of the discriminant $b^2 - 4ac = 0\$ $b^2 - 4ac > 0\$ $b^2 - 4ac < 0\$ nature of solution set exactly one real number solution two different real number solutions no real number solutions; a complex conjugate pair the solution(s) are: $$\displaystyle\,x = -\frac{b}{2a}$$ $\displaystyle\,x = \frac{-b + \sqrt{b^2 - 4ac}}{2a}$ or $\displaystyle\,x = \frac{-b - \sqrt{b^2 - 4ac}}{2a}$ \begin{align} x &= \frac{-b + i\sqrt{-(b^2 - 4ac)}}{2a}\cr &= -\frac{b}{2a} + i\frac{\sqrt{-(b^2-4ac)}}{2a} \end{align} or \begin{align} x &= \frac{-b - i\sqrt{-(b^2 - 4ac)}}{2a}\cr &= -\frac{b}{2a} - i\frac{\sqrt{-(b^2-4ac)}}{2a} \end{align} Note: Since $\,b^2 - 4ac < 0\,$, we have $\,-(b^2 - 4ac) > 0\,$. Therefore, $\,\frac{\sqrt{-(b^2-4ac)}}{2a} \ne 0\,$, so these two solutions have the same real part and opposite nonzero imaginary parts, hence form a complex conjugate pair.

The quadratic expression $\ ax^2 + bx + c\$ ($\,a \ne 0\,$) is called irreducible if $\ b^2 - 4ac < 0\$.

Equivalently, the quadratic expression $\ ax^2 + bx + c\$ ($\,a \ne 0\,$) is called irreducible if the equation $\,ax^2 + bx + c = 0\,$ has no real number solutions.

In this case, the trinomial $\,ax^2 + bx + c\,$ cannot be factored using real numbers.
It can, however, be factored using the complex conjugate pair that is given by the quadratic formula.
This idea is explored in a future section.

## EXAMPLES: Connection Between Zeroes and Factors

Recall the beautiful relationship between the zeroes of a polynomial and its factors:

$k\,$ is a zero             if and only if             $x-k\,$ is a factor

Indeed, the only factor that the zeroes don't give is a constant factor, which we often need to provide ourselves.
Some examples follow.

EXAMPLE:
Find a quadratic function with zeroes $\,2\,$ and $\,-3\,$;
the coefficient of the $\,x^2\,$ term should equal $\,5\,$.

SOLUTION:
Since $\,2\,$ is a zero, $\,x - 2\,$ is a factor.
Since $-3\,$ is a zero, $\,x - (-3) = x + 3\,$ is a factor.
Since the coefficient of the $\,x^2\,$ term must be $\,5\,$, we need a constant factor of $\,5\,$.
(Note that this constant factor does not change the zeroes.)

The function $\,f\,$ defined below is the desired quadratic function: $$\,f(x) = 5(x-2)(x+3) = \ldots = 5x^2 + 5x - 30\,$$

EXAMPLE:
Find a quadratic function with real coefficients that has exactly one real zero, $x = 4\,$.
The coefficient of the $\,x\,$ term should equal $\,7\,$.

SOLUTION:
Since $\,4\,$ is a zero, $\,x - 4\,$ is a factor.
To get a quadratic function with no different zero, we need two factors of $\,x - 4\,$.
Thus, every function of the form $\,k(x-4)(x-4) = \ldots = kx^2 - 8kx + 16k\,$ satisfies the zero requirement.
For the coefficient of the $\,x\,$ term to equal $\,7\,$, we need $\,-8k = 7\,$, so $k = -\frac{7}{8}\,$.

The function $\,f\,$ defined below is the desired quadratic function: $$\,f(x) = -\frac{7}{8}(x-4)(x-4) = \ldots = -\frac{7}{8}x^2 + 7x - 14\,$$

Master the ideas from this section


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## What to do with excess clothes (updated)

An interesting story from Reuters. What should retailers do with all of the inventory they had for the spring season?

Read Full Post »

## What’s next for just-in-time?

The Financial Times has an interesting set of articles on how the ongoing pandemic impacts supply chains (Trade Secrets: Supply Chain Disruption). These hit on things like toilet paper, firms pivoting to new markets or switching from a business-to-business focus to serving retail customers. The one I want to highlight deals with how any fragility exposed by the pandemic will impact supply chain strategy going forward (Be wary of scapegoating ‘just-in-time’ supply chains, May 27) that links to a post that Gady wrote a few weeks ago.

Here is the gist of the article:

A lot of intellectual momentum is building behind the idea that the Covid-19 pandemic has revealed the foolishness of corporate executives in extending their supply chains without properly assessing the risks. Companies have been thinking of “just-in-time” when they need to be thinking about “just-in-case”. …

The reality is complex, and — a crucial point — differs with each industry. Some, like the car industry, have such sophisticated supply chains involving thousands of different components, some manufactured to extremely low tolerance, that diversifying into different suppliers is totally impractical through effort and cost. Sure, you will have a more resilient supply chain, but you’ll also go bust before the next pandemic arrives.

Read Full Post »

## Production planning for the holidays at Lego

This should not surprise you at all: Christmas is a big deal for Lego. According to the Financial Times, half of the company’s sales come in the month or so before the holiday (Lego makes push to avoid disappointments of Christmas past, Dec 22). But how do they gear up for that big peak in sales? Check out the video below:

Read Full Post »

## What happens when the World Series results in inventory no one needs?

The World Series starts tonight. While everyone in Chicago is focused on the prospect of the Cubs winning the Series, that is not a certainty. The one thing that is certain is that someone is going to lose – and that raises the prospect of a Cub or Chief Wahoo on t-shirt proclaiming that a team won something that they didn’t.

So what happens to t-shirts and other tchotchkes celebrating events that never happened? That was the topic of a recent Chicago Tribune story (Where do losing baseball teams’ postseason T-shirts end up?, October 18). The article itself is a little confused (it very much seems that a paragraph was dropped) but it does layout some options:

Last year, VF Licensed Sports Group required customers who wanted early access to merchandise celebrating a baseball team’s postseason run agree to ship any merchandise with a losing team’s 2015 MLB postseason clinch logos, images or graphics to international nonprofit World Vision. Customers had 24 hours following a loss to get in touch with World Vision to start the donation process, according to a 2015 agreement provided by a retailer. …

Another retailer was sent a revised agreement that replaced the donation requirement with a mandate to ship any items for losing teams back for destruction. …

Retailers who violate an agreement not to sell, advertise or promote the losing team’s merchandise agree to pay $100,000 per breach, according to the 2016 World Series preprinted merchandise agreement. Read Full Post » ## The Cubs might win: A supply chain story for championship gear For those who are not baseball fans, let me give you a quick update: The Chicago Cubs are really good this year. They won over 100 games in the regular season and have now jumped out to a 2 – 0 lead in their best-of-five series with the San Francisco Giants. FiveThirtyEight has them as the favorite to win the World Series. If all of that is news to you, you should also be told that the Cubs have, frankly, sucked for a long, long time. They haven’t won a pennant since 1945 and a World Series since 1908. There is even a short story (The Last Pennant Before Armageddon) tying the Cubs winning a pennant to the end of the world. (To answer the obvious question, the World Series is scheduled to start on October 25th. Barring rain delays, the last possible game would be on November 2nd. The US presidential election is on November 8th.) So if the Cubs make it to the World Series, there will be a lot of excitement around here. If they actually win the Series, Cook County will likely shut down for a month. And that all raises a question: How many Cubs t-shirts can be sold? Read Full Post » ## Zulily, delivery times, and operations strategy About a year ago, we had a post on Zulily and how they managed their order fulfillment. It featured a nifty graphic from the Wall Street Journal showing just how much longer their delivery times were relative to other interet retailers. Now, the Journal has another story — with a spiffy updated graphic — discussing how their delivery times have gotten even worse (Zulily Nips Business Model in the Bud, Mar 23). Read Full Post » ## Does selling over the web offer better margins than selling in stores? So what should be more profitable for a retailer, selling from physical stores or selling over the web? That’s the question that a recent Wall Street Journal article considers (How the Web Drags on Some Retailers, Dec 1). At first glance, the answer seems straightforward. Web sellers don’t need to rent stores or have staff cooling their heels waiting for customers. However, the reality isn’t necessarily so clear, While conventional wisdom holds that online sales should be more profitable, because websites don’t need the pricey real estate and labor necessary to maintain a store network, many retailers actually earn less or even lose money online after factoring in the cost of shipping, handling and higher rates of returns. For retailers that outsource their Web and fulfillment operations, costs can run as high as 25% of sales, industry analysts said. Kohl’s Corp. says its profitability online is less than half what it reaps in its store. Wal-Mart Stores Inc. says it expects to lose money online at least through early 2016 as it invests to build its technology, infrastructure and fulfillment networks. Target Corp. says its margins will shrink as its online sales grow. Best Buy Co. said faster growth on its website will weigh on its profitability at the end of the year. Click here for a video of the reporter discussing her findings. Read Full Post » ## The impact of pooled inventory on branding risk Shop on Amazon.com and you will find a lot of items sold by lots of different sellers. For many of those sellers, Amazon isn’t just handling acting as a store front; it is also handling the logistics of order fulfillment. Now suppose that Amazon has a particular product which both it and several third parties are selling out its warehouses. How should Amazon physically manage the inventory? Should it keep the inventory it is selling physically separate from that offered by third-party sellers? In many instances, Amazon chooses to do just the opposite, allowing for “stickerless, commingled inventory.” Here is an Amazon video explaining just what that means. And here is how the Wall Street Journal explains the benefits of the program (Do You Know What’s Going in Your Amazon Shopping Cart?, May 11). The system has enabled Amazon to make better use of its warehouse space and keep a wide variety of items in stock around the country. The idea is to give Amazon flexibility to ship certain products based on their proximity to customers, speeding delivery times. For third-party sellers, it saves them the trouble of having to label individual items sent to the Amazon warehouse. Read Full Post » ## A road salt supply chain The weather in Chicagoland over the last week has been miserable. I have shoveled the walks way too many times and it now feels like we’ve been transported to Hoth. That has gotten me thinking about road salt. That and a New Yorker article on the Atlantic Salt’s operations on Staten Island (The Mountain, Dec 23). The mountain referenced in the article’s title is a giant pile of salt — a third of a mile long and four stories high. It’s big enough to see on Google Earth. Check out the multicolored tarps. In any event, managing the inventory of road salt is an interesting challenge. (more…) Read Full Post » ## How do you determine how much money to put into a flexible spending account? It is late October and on my to-do list is wading through Open Enrollment options. That inevitably brings up the question of how much money to put into our Flexible Spending Account (FSA). Given that a lot of people face that question at this time of year, I thought I would recycle a post (from Dec 2011) on how to think of funding an FSA as an inventory problem: FSAs allow US tax payers to set aside pre-tax dollars to pay for authorized expenses. One can have separate accounts for healthcare related expenses (think office-visit co-pays or dental work beyond what your insurance covers) and dependent care. Let’s focus on the medical one. Here is how a Forbes blog explained the pros and cons of the program (A Tax Break For Driving To Wal-Mart!, Dec 2). The way you save with an FSA is this: If you divert$5,000 from taxable salary to pay for braces and your combined federal/state income tax rate is 40%, you save $2,000. You can use the money you stash in the account for medical, dental and vision expenses for yourself, your spouse or your kids. … If you don’t spend your FSA money within the plan year (or a 2.5 month grace period in some cases), you lose it. The fear of forfeiture leads folks to underfund these accounts. Not paying attention to how expansive the list of eligible expenses is leads folks into forfeiting money. The average amount employees set aside into a healthcare flexible spending account is$1,500, and about half of participants lose an average of $75 at year-end, according to WageWorks. But even these employees who forfeit$75 are still better off with the FSA, says Dietel. Since the average election is just under $1,500, the employee has saved 25% to 40% of that, or$375 to $600, so they are still$300 to \$525 ahead of where they would have been without a healthcare FSA.

So here’s the question: Do people, in fact, underfund these accounts?

Let me suggest that a simple inventory problem provides some insights into how much to set aside for an FSA. The newsvendor model assumes that one has a single opportunity to purchase inventory before uncertain demand is realized. The analysis is pretty simple. Suppose that F(x) is the known demand distribution. That is, F(x) gives the probability that demand is less than or equal to x. If we are selling newspapers (which is the motivating example that gives the model its name), F(10) = 60% means that 60% of the time 10 or fewer customers show up to buy papers. Note that F(x) will be increasing (i.e.,the chance that demand is less than or equal 11 has to be greater than the chance that it is less than or equal to 10).

Let Q be the quantity ordered. Let MB be the marginal benefit of stocking one more unit. The relevant thought experiment is to suppose that one stocks Q units but realized demand is Q+1 and ask how much one would have earned if one had had that extra unit to satisfy that last unit of demand. In our newspaper example, suppose they retail for 50¢ but we have to pay the publisher 20¢. We then get MB = 50¢ – 20¢ = 30¢. Similarly let MC be the marginal cost of stocking one more unit. The thought experiment here is to suppose that realized is Q-1. If we can recycle papers at 5¢ a piece, we get MC = 20¢ – 5¢ = 15¢.

The model then says we should balance the expected marginal benefit with equal the expected marginal cost. If we stock Q newspapers, the probability we incur the marginal cost MC is F(Q) and the probability that we receive the marginal benefit MB is (1 – F(Q)). That means we want Q to solve:

MC × F(Q) = MB × (1 – F(Q))

The wonders of junior high algebra then give:

$F(Q) = \frac{MB}{MB+MC}$

In words, we can think of the model as saying that the optimal inventory level is driven by picking a probability of covering all demand. In our newspaper example, we get $\frac{MB}{MB+MC} = \frac{30}{30+15} = 0.66$ so our newsvendor should not stockout (i.e., have newspapers left) on two out of three days.

OK, back to FSAs. Note that we’ve got a one-period inventory model. You decide in October how much to set aside for the coming year and dollars set aside for 2011 cannot be used in 2012. We just need to determine MB and MC. The tricky part here is that we need to be clear of when we are talking about pre-tax dollars and when we are talking about post-tax dollars. Let’s work with pre-tax dollars because that is the amount you specify when you enroll.

MC is then easy. If you put one too many pre-tax dollars into your FSA, you lose that dollar. Thus, MC = 1. For MB, if you are short a dollar in your FSA, you need to spend one post-tax dollar. That costs you $\frac{1}{1 - \tau}$ pre-tax dollars, where τ is your marginal tax rate. The only way to avoid this would have been putting another pre-tax dollar in your FSA (note that this is the analog of paying the publisher for the newspaper). We then get $MB = \frac{1}{1 - \tau} - 1 = \frac{\tau}{1 - \tau}.$

Now we just need to calculate that fraction. The optimal FSA amount should satisfy

$F(Q) = \frac{MB}{MB+MC} = \frac{\frac{\tau}{1 - \tau}}{\frac{\tau}{1 - \tau}+1}= \tau.$

So what do we see? First, not too surprisingly, the marginal tax rate matters a lot. The recommendation is that the tax payer should cover all of her qualifying expenses with probability τ. If her marginal tax rate is very low, say, 10%, she should use all of her FSA contributions nine time out of ten. If τ is very high, say, 90%, she will very often have excess cash in her FSA.

That clearly sounds right. The higher the marginal tax rate, the greater value in paying with pre-tax dollars. Said another way, FSAs have a regressive effect since they provide greater benefit to those in higher tax brackets.

This also lets us answer my original question: Do people, in fact, underfund these accounts? I would argue probably not. Marginal tax rates top out around 40% so even the highest paid Americans should max out their FSAs in most year. Indeed, if half of users end up with excess cash in their FSA (as the quote above suggests), they are putting TOO much money in their FSAs.

There is a caveat to all this analysis. I assuming here that customers are just endowed with some distribution of their spending. To some extent this is true. If you have a chronic condition that requires regular medication, you can forecast your co-pays and then add in kids getting ear infections or needing fillings. That is like forecasting demand for newspapers. However, FSAs also covers spending that is essentially discretionary. If you have a few hundred dollars left in your account in early December, you can always go buy new glasses even if your current ones are fine. Thus, as is pointed out in a recent LA Times article (FSAs encourage rather than reduce unnecessary healthcare spending, Nov 20), FSAs aren’t just regressive, they are inefficient. If one considers people buying unnecessary glasses to clear out their FSA and we still have half of users with leftover cash, then clearly average users are not underfunding their FSAs.

A final point, if you don’t like risk — that is, you would prefer greater certainty in your post-tax income — you may want to put some extra pre-tax dollars in your FSA. See here.

Read Full Post »

Older Posts »


text:
# Calculating the mortality rate of a pandemic, e.g. coronavirus 2019-nCov?

When calculating how lethal a virus outbreak is, I've noticed that most sources use

CONFIRMED_DEATHS/CONFIRMED_INFECTIONS

However, given that there are only 2 possible outcomes; death or recovery, wouldn't it be more logical to use:

CONFIRMED_DEATHS/(CONFIRMED_DEATHS + CONFIRMED_RECOVERIES)

Which formula is applicable to which situations?

• do you know that CONFIRMED_DEATHS CONFIRMED_INFECTIONS CONFIRMED_RECOVERIES are time function, and they will change for the time t Nov 23 '20 at 21:21

This may not strike most as a bioinformatics, but getting the key clinical outcome is essential in understanding the molecular basis of pathogenicity.

I think the mortality rate is over-reported. This is not to say the situation of 2019-nCov is not serious - it is very serious.

The two essential factors missing in your equation for 2019-nCov are:

• Age. Mortality is likely skewed towards the old and very young
• Seroprevalence. To identify the number of asymptomatic patients.

Old people have a weakened immune system (e.g. reduced T-cell response). The very young have an underdeveloped immune system. The problem is hospital cases are skewed towards those with severe infection, which is likely the old and very young. In 2019-nCov the first infecteds were middle aged men, but likely to be restricted to those exposed to the "Wuhan meat market".

Problem Asymptomatic "spreaders" are known in influenza and are suspected in 2019-nCov. Asymptomatics will inflate the mortality rate (because we're not detecting them).

Solution(s)

1. One solution is to conduct seroprevalence surveys. Serological surveys screening for antibodies (IgG) against 2019-nCov will identify asymptomatic patients, using the first formula you stated. However, would you seriously go around the general population in a 2019-nCov epidemic taking blood samples? Errhhh no.
2. The bio-statistics approach is to use a catalytic model and critical to this is the age of the patient. Catalytic models are a priori models that incorporates the age-prevelance profile of infection. By introducing the incidence and age of hospitalisation of e.g. the old and the young, the rest of the distribution, i.e. asymptomatics can be calculated by regression. This estimates the total population of infecteds and again you use equation 1 to calculate the mortality rate recorded by the hospital.
3. Mathematical models based on transmission dynamics. The problem here is the rate of transmission is unknown.

I would assume the hospital is the place virus-induced mortality is reasonably accurate.

The second formula you described works very well for Ebolavirus, no-one has ever heard of asymptomatic Ebolavirus. It would also work well for another betacoronavirus Middle Eastern respiratory syndrome. The key problem with 2019-nCov is we don't know the number of asymptomatic patients or patients with such mild infections they never report illness.

Summary In summary, I think the mortality is being over-reported if it is based on your equation 2. Equation 1, however, will also over-report the mortality rate.

• i think that data is changing and no one can report the right mortality Dec 5 '20 at 21:27

To add to Michael G.'s answer, there is a further wrinkle in the case of 2019-nCov that is due to the exponential growth in confirmed infections daily.

That growth affects your second formula CONFIRMED_DEATHS/(CONFIRMED_DEATHS + CONFIRMED_RECOVERIES since deaths probably occur earlier on in the disease trajectory than recoveries do. Since the number of confirmed cases is growing exponentially on a daily basis, the number of deaths will presumably be inflated relative to the number of recoveries at this point.

This feature may have the opposite effect on your first formula CONFIRMED DEATHS/CONFIRMED INFECTIONS, depending on when the infections are confirmed. If the infections are generally confirmed several days before death occurs, then a large portion of confirmed infections are still too early to have ended in death and CONFIRMED DEATHS/CONFIRMED INFECTIONS actually underestimates the actual number of fatalities.

I was facing this problem in January when coming up with a methodology to forecast on the UrbanSurvival.com site.

My solution - using low data collection was to use a daily rolling rate of infections and divide that into the 5-day previous reported death counts.

This was based on the assumption that, for a given cohort, the people who get sick today will "get dead" in about 10-days time.
So half of this (5 days) ought to get me into the ballpark.

In Feb, when CDC was reporting 3.4% mort. rate, we'd been reporting that for a couple of weeks.

As a formula it's not unlike deriving an exponent.

That is: Cases Today (T1) / Deaths 5-Days Ago (T2)

Unfortunately, this is presently running a bit higher mortality rate now (4.5%) but I would expect on the backside of the expansion of cases it will decline.

Here's hopin'...


text:
me: i love type theory
me: *uses a proof assistant*
me, through pained breath: man

oh my god

can you seriously not eliminate prop into set

fuck you *one-point compactifies your space*

hazel boosted

i could solve the halting problem easily, but certain people have been a bitch to me, so i wont

"Secret sigma they keep locked away" - @ionchy

hazel boosted

@hazel "oh look at me I'm half the base of base 10 I'm easy to do arithmetic with and I'm a prime number" ass little bitch. Gonna shove 5 into a locker

i just. i just want the fucking squimsher. give me the fucking Set -> Prop squimsher.

"how do I do X?"
"solve the halting problem"
"god damnit"

a conversation I have had on 5 separate occasions this month

hazel boosted

sorry but if you make it to year 3 of CS and your take is that anything is computable you're a complete fucking moron. it's like being a year 3 geologist and thinking the earth is flat

I think that less numbers should be prime

hazel boosted

Q: what is the loneliest number

A: 5 because it's a dipshit idiot loser. lol

#conservativecomedy

-- Quotient types:
-- I don't entirely understand what these do.

If I leave then Jonathan gets full control of types.pl, or I find a replacement for myself.

This isn't a commitment to leaving, but it's possible.

I really appreciate the community I've built here, thank you all for sticking with a random undergrad's rented Ubuntu server.

If you want to talk, my contact info's public. If I want to talk to you, though, you should already have that information somewhere.

I might leave Mastodon entirely. If I do, I'm not going to shut down types.pl, and I'll continue paying for its hosting.

quick note to all: I haven't posted here in a while because I've pretty much felt exclusively like human garbage for about a week.

It's only been amplified lately because uh. circumstances. whatever.

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User nicholas proudfoot - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T04:27:30Z http://mathoverflow.net/feeds/user/10273 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/122683/fundamental-groups-of-symplectic-leaves Fundamental groups of symplectic leaves Nicholas Proudfoot 2013-02-23T01:26:27Z 2013-03-17T09:55:55Z
Let $X = \operatorname{Spec}(R)$ be a conical Poisson variety over $\mathbb{C}$. This means that $R$ is a non-negatively graded Poisson algebra over $\mathbb{C}$ with $R_0 = \mathbb{C}$ and that the Poisson bracket is homogeneous of negative degree. Assume that $X$ admits a conical symplectic resolution. Examples to keep in mind are:

the nilpotent cone in a simple Lie algebra

a Slodowy slice to a nilpotent orbit inside the nilpotent cone

the symmetric scheme of $n$ points on a Kleinian singularity

a hypertoric variety

a quiver variety.

Let $S$ be a symplectic leaf of $X$.

Q: Is it possible for the fundamental group of $S$ to be infinite?

In the first four examples listed above, the fundamental group of every symplectic leaf is finite. I don't know whether or not this property holds for quiver varieties, or for other examples.
 http://mathoverflow.net/questions/94569/stable-isomorphisms-of-groups Stable isomorphisms of groups Nicholas Proudfoot 2012-04-19T20:43:43Z 2012-04-19T21:26:13Z

Possible Duplicate:
 Stably isomorphic groups

If $G$ and $H$ are two groups (finitely presented, if you wish) with the property that $G\times\mathbb{Z}$ is isomorphic to $H\times\mathbb{Z}$, does that imply that $G$ is isomorphic to $H$?
 http://mathoverflow.net/questions/62544/homotopy-type-of-hilbert-schemes-of-points-of-mathbb-c2/62634#62634 Answer by Nicholas Proudfoot for Homotopy type of Hilbert schemes of points of $\mathbb C^2$ Nicholas Proudfoot 2011-04-22T16:24:52Z 2011-04-22T16:24:52Z
Here's how I understand the argument in Slodowy's book. Sticking with Tamas' notation, let $X$ denote the Hilbert scheme and $D$ the punctual Hilbert scheme. You'd like to define a deformation retraction from $X$ to $D$ by sending every point to its limit under the dilation action. Unfortunately, as Ben points out, $D$ is not fixed by the dilation action, and this function isn't even continuous! So that doesn't work.

Instead, you choose a closed tubular neighborhood $A$ of $D$ such that $D$ is a deformation retract of $A$; this is possible by a basic result in algebraic topology (Slodowy cites Spanier's book). You then use the dilation action to define a deformation retraction from $X$ to $A$. More precisely, send every element $x\in X$ to the first element of $A$ that it hits when you dilate it inward. This is a perfectly well-behaved map, and it does the trick.
 http://mathoverflow.net/questions/55569/the-scheme-theoretic-flow-in-locus The scheme-theoretic flow-in locus Nicholas Proudfoot 2011-02-16T00:45:11Z 2011-02-17T00:16:09Z
Let $R$ be a ring with an $\mathbb{N}\times\mathbb{Z}$-grading. The $\mathbb{N}$-grading allows you to construct the scheme $X = \operatorname{Proj} R$, and the $\mathbb{Z}$-grading defines an action of $T = \mathbb{G}_m$ on $X$.

Let $S$ be an $\mathbb{N}\times\mathbb{Z}$-graded quotient of $R$ such that every element of $R$ whose $\mathbb{Z}$-degree is nonzero vanishes in $S$, and let $Y = \operatorname{Proj} S$. Then we have an inclusion from $Y$ into the fixed locus $X^T$. $T$ acts trivially on the tautological bundle of $Y$, which is equal to the restriction of the tautological bundle of $X$. Assume that $X$ and $Y$ are smooth and that $Y$ is a connected component of $X^T$.

Let $R^+ \subset R$ be the subring generated by elements of non-negative $\mathbb{Z}$-degree, and let $V = \operatorname{Proj} \left(R\;\otimes_{R^+}S\right)\subset X$.

Morally, $V$ should be thought of as the locus of points $x$ with the property that $\lim_{t\to 0}t\cdot x \in Y$, though in fact the support of $V$ may be strictly bigger than this set. For example, let $X= T^*\mathbb{P^1}$, where the action of $T$ on $X$ is induced from the standard action on $\mathbb{P^1}$. Then $X$ has two fixed points which I'll call $N$ and $S$ (for the North and South poles). If we take $Y$ to be $N$, then $V$ will be the cotangent fiber over $N$. If we take $Y$ to be $S$, then $V$ will be the union of the cotangent fiber over $N$ with the zero section (even though the set-theoretic flow-in locus consists only of the zero-section minus $N$). In particular, $V$ need not be irreducible.

More generally, if $X = T^{*}(G/B)$ and $T$ acts on $X$ via a generic cocharacter of $G$, $V$ will be a union of conormal varieties to Schubert strata that appear with various multiplicities. In particular, $V$ need not be reduced.

The example that most interests me is when $X$ is the family over $\mathbb{A}^1$ with special fiber $T^{*}(G/B)$ and general fiber $G/H$, where $H$ is a maximal torus of $G$. This family is $U(1)$-equivariantly diffeomorphic (but not isomorphic) to $T^{*}(G/B)\times \mathbb{A}^1$, where $U(1)\subset T$ acts trivially on $\mathbb{A}^1$. Thus the components of $X^T$ are images of sections of the map from $X$ to $\mathbb{A}^1$. In this example $V$ will always be irreducible and reduced, even though its intersection with $T^{*}(G/B)$ is neither irreducible nor reduced (see the previous paragraph).

What I would like to know is that if I take $X$ to be a variety like this (a twistor families for a "nice" symplectic variety on which $T$ acts hamiltonianly with isolated fixed points), then $V$ will be reduced. In an attempt to get this, I'll try posing the following general question:

Suppose that $V$ is irreducible and generically reduced. Does it follow that $V$ is reduced?

 http://mathoverflow.net/questions/46741/cohomology-of-a-configuration-space/46792#46792 Answer by Nicholas Proudfoot for Cohomology of a configuration space Nicholas Proudfoot 2010-11-21T02:37:13Z 2010-11-21T02:37:13Z
This isn't exactly an answer to your question, but here's how I like to think about the fact that you quoted.

Let's assume for a minute that $n=2$, so that we can think of $X$ as the complement of the braid arrangement in $\mathbb{C}^k$. Let $G = \mathbb{Z}/2\mathbb{Z}$, which acts on $X$ by complex conjugation.

Replace $\mathbb{Q}$ with a field $F$ of characteristic $2$. The $G$-equivariant cohomology ring $H^*_G(X; F)$ is a free module over $H^*_G(pt; F) \cong F[x]$ with the property that specializing at $x=0$ gives $H^* (X; F)$ and specializing at $x=1$ gives $H^*(X^G;F)$.

Thus we have a family of $\Sigma_k$ representations over the $F$-affine line interpolating between $H^* (X; F)$ and $H^*(X^G; F)$. Since the category of $\Sigma_k$ representations is semisimple, these two representations have to be isomorphic. The fact that $H^*(X^G; F)$ is the regular representation is obvious.

This is a good way to see that $H^*(X; F)$ is isomorphic to the regular representation of $\Sigma_k$. I'm not sure how to modify this argument to get $H^* (X; \mathbb{Q})$. I'm also not sure if this will help you find a cyclic vector, since it does not give you an explicit isomorphism between $H^* (X; F)$ and $H^* (X^G; F)$.

By the way, for $n>2$ you can do something similar, where $G$ acts on $\mathbb{R}^n$ by negating the last $n-1$ coordinates.
 http://mathoverflow.net/questions/6186/counting-solutions-to-xp1y4-in-a-finite-field Counting solutions to x^{p+1}=y^4 in a finite field Nicholas Proudfoot 2009-11-19T22:25:16Z 2010-10-25T00:41:31Z
I need to compute the number of solutions to the equation $x^{p+1} = y^4$ in the field with $p^2$ elements (for p sufficiently large). The form of the equation suggests to me that the solution would depend on the congruence class of p mod 4, but I have reason to believe that the answer is a single polynomial in p.

I feel as if this should be easy, and I'm missing an obvious approach. Can anyone help me out?
 http://mathoverflow.net/questions/127495/cohomology-of-configuration-spaces/127506#127506 Comment by Nicholas Proudfoot Nicholas Proudfoot 2013-04-14T18:18:41Z 2013-04-14T18:18:41Z In this paper, Bezrukavnikov cites a paper of Kohno and Oda (1987) in which they prove (among other things) an LCS formula for the Poincare polynomial in question. Assuming that one can compute the ranks of the subquotients in the lower central series of the fundamental group, this completely answers Christin's question. However, Roman points out on page 133 of his paper that there are some incorrect results in the Kohno-Oda paper. Do you know if the Kohno-Oda LCS formula is correct as stated? http://mathoverflow.net/questions/127495/cohomology-of-configuration-spaces Comment by Nicholas Proudfoot Nicholas Proudfoot 2013-04-14T15:59:20Z 2013-04-14T15:59:20Z @Dan Peterson: Totaro's paper does not give a formula for the Poincare polynomial. There's a big gap between being able to write down a DGA in terms of generators and relations and actually having a formula for its Poincare polynomial. http://mathoverflow.net/questions/122683/fundamental-groups-of-symplectic-leaves/124766#124766 Comment by Nicholas Proudfoot Nicholas Proudfoot 2013-03-18T05:31:36Z 2013-03-18T05:31:36Z Thanks, your reference to Namikawa's paper is very helpful! In fact, finiteness of the algebraic fundamental group is sufficient for the application that I have in mind, so this is perfect. http://mathoverflow.net/questions/14841/are-the-strata-of-nakajima-quiver-varieties-simply-connected-do-they-have-odd-c/14896#14896 Comment by Nicholas Proudfoot Nicholas Proudfoot 2013-02-26T04:58:34Z 2013-02-26T04:58:34Z Do there exist any examples where the fundamental groups of these strata are infinite? (See also mathoverflow.net/questions/122683/… for the same question.) http://mathoverflow.net/questions/40488/secondary-fans-and-stanley-reisner-ideals/40755#40755 Comment by Nicholas Proudfoot Nicholas Proudfoot 2012-11-20T20:18:51Z 2012-11-20T20:18:51Z David, do you know of any way to understand the non-coherent refinements of cone(S)? Put differently (if I understand correctly), let X be the affine toric variety associated to cone(S). The cones of the secondary fan classify projective toric maps Y->X that are bijective on orbits of codimension 0 or 1. The non-coherent refinements correspond to such maps that are proper but not projective. Can these be nicely classified? http://mathoverflow.net/questions/94566/equivariant-cohomology-of-the-complement-to-the-arrangment-cup-i-neq-joverrig Comment by Nicholas Proudfoot Nicholas Proudfoot 2012-04-19T20:40:16Z 2012-04-19T20:40:16Z If $V = \mathbb{R}^3$ and you consider not the full group $SO(3)$ but rather the circle subgroup consisting of rotations about some axis in $V$, a presentation is given by Daniel Moseley in Theorem 3.4 of arxiv.org/abs/1110.5369. Whatever the answer to your question is, it should admit lots of natural maps to Moseley's ring (one for every choice of oriented line in $V$). http://mathoverflow.net/questions/55569/the-scheme-theoretic-flow-in-locus Comment by Nicholas Proudfoot Nicholas Proudfoot 2011-02-16T17:23:19Z 2011-02-16T17:23:19Z @Dave continued again: If you choose the $\mathbb{Z}$-degrees of $x$ and $y$ to be 1 and $-1$ as you say, then you get the square of the standard action. Furthermore, you get it with a $T$-structure on the tautological line bundle such that $T$ acts nontrivially on the fibers at both fixed points. Instead, you should assign degrees $(2,0)$ or $(0,-2)$, depending on which of the two fixed points you want to look at. Finally, you're right, it's no coincidence that Ben and I are asking similar questions! http://mathoverflow.net/questions/55569/the-scheme-theoretic-flow-in-locus Comment by Nicholas Proudfoot Nicholas Proudfoot 2011-02-16T17:20:09Z 2011-02-16T17:20:09Z @Dave continued: If you want to get the inclusion of the other fixed point, you should let $x$ have degree 0 and $y$ have degree $-1$, and then define your map from $R$ to $S$ by sending $x$ to $t$ and $y$ to $0$. These two choices of $\mathbb{Z}$-grading on $R$ induce the same $T$-action, but with a different equivariant structure on the tautological line bundle. With my conventions, the restriction of the tautological line bundle from $X$ to $Y$ will have the trivial $T$-action on every fiber. http://mathoverflow.net/questions/55569/the-scheme-theoretic-flow-in-locus Comment by Nicholas Proudfoot Nicholas Proudfoot 2011-02-16T17:16:29Z 2011-02-16T17:16:29Z @Dave: As you say, let $R = k[x,y]$, where $x$ and $y$ both have $\mathbb{N}$-degree 1, so that $X = \operatorname{Proj} R = \mathbb{P}^1$. To define a $T$-action on $X$, we need to choose the $\mathbb{Z}$-degrees of $x$ and $y$. If you want to get the standard action, you should let $x$ have degree 1 and $y$ have degree $0$. Then you can let $S = k[t]$, where $t$ has $\mathbb{N}$-degree 1 and $\mathbb{Z}$-degree 0, and the map from $R$ to $S$ sending $x$ to $0$ and $y$ to $t$ induces the inclusion of one of the two fixed points. http://mathoverflow.net/questions/46741/cohomology-of-a-configuration-space/46792#46792 Comment by Nicholas Proudfoot Nicholas Proudfoot 2010-11-27T14:48:56Z 2010-11-27T14:48:56Z To Semen: When $k=2$, both the regular representation and $H^*(X; F)$ are trivial. (Recall that in my argument $F$ is supposed to have characteristic 2.) http://mathoverflow.net/questions/6186/counting-solutions-to-xp1y4-in-a-finite-field Comment by Nicholas Proudfoot Nicholas Proudfoot 2009-11-20T00:23:35Z 2009-11-20T00:23:35Z This is a small piece of a larger calculation in which I'm trying to understand the action of a particular finite group on the intersection cohomology of a particular complex variety. The intersection cohomology is pure and concentrated in even degrees, so this computation is amenable to finite field techniques. The fact that the answer to my question (as Sivek points out) DOES depend on the residue of p mod 4 means either that this dependence will be magically cancelled out by another part of the calculation, or that I've made a mistake earlier on.


text:
# How can I prevent a new paragraph between two \input commands?

I want to use the command `\input{}` to insert two textfiles, `text1.tex` and `text2.tex`, into my latex working file. I want these two textfiles to produce one single paragraph when I am compiling my working file. However, when I place

``````\input{text1.tex}\input{text2.tex}
``````

in a single line in the working file, the content of the two text files still ends up in two different paragraphs in the compiled text. I have tried to put the command `\nolinebreak` between the two `\input{}` commands without success.

So, how can I prevent a new paragraph to be produced between the two textfiles?

-
it produces one paragraph for me. Helps to add a trailing `%` as in `\input{text1.tex}%`. –  Peter Grill Apr 16 '12 at 15:56
Remove the empty line in `text1.tex` or `text2.tex` which produces the new paragraph. –


text:
# American Institute of Mathematical Sciences

ISSN:
1930-8337

eISSN:
1930-8345

All Issues

## Inverse Problems & Imaging

February 2013 , Volume 7 , Issue 1

Select all articles

Export/Reference:

2013, 7(1): 1-25 doi: 10.3934/ipi.2013.7.1 +[Abstract](2974) +[PDF](4059.5KB)
Abstract:
In this paper we construct a shape space of medial ball representations from given shape training data using methods of Computational Geometry and Statistics. The ultimate goal is to employ the shape space as prior information in supervised segmentation algorithms for complex geometries in 3D voxel data. For this purpose, a novel representation of the shape space (i.e., medial ball representation) is worked out and its implications on the whole segmentation pipeline are studied. Such algorithms have wide applications for industrial processes and medical imaging, when data are recorded under varying illumination conditions, are corrupted with high noise or are occluded.
2013, 7(1): 27-46 doi: 10.3934/ipi.2013.7.27 +[Abstract](2738) +[PDF](389.3KB)
Abstract:
It has been shown in [10] that on a simple, compact Riemannian 2-manifold the attenuated geodesic ray transform, with attenuation given by a connection and Higgs field, is injective on functions and 1-forms modulo the natural obstruction. Furthermore, the scattering relation determines the connection and Higgs field modulo a gauge transformation. We extend the results obtained therein to the case of magnetic geodesics. In addition, we provide an application to tensor tomography in the magnetic setting, along the lines of [11].
2013, 7(1): 47-79 doi: 10.3934/ipi.2013.7.47 +[Abstract](2942) +[PDF](4753.9KB)
Abstract:
In order to prevent influx of highly enriched nuclear material throu-gh border checkpoints, new advanced detection schemes need to be developed. Typical issues faced in this context are sources with very low emission against a dominating natural background radiation. Sources are expected to be small and shielded and hence cannot be detected from measurements of radiation levels alone. We consider collimated and Compton-type measurements and propose a detection method that relies on the geometric singularity of small sources to distinguish them from the more uniform background. The method is characterized by high sensitivity and specificity and allows for assigning confidence probabilities of detection. The validity of our approach can be justified using properties of related techniques from medical imaging. Results of numerical simulations are presented for collimated and Compton-type measurements. The 2D case is considered in detail.
2013, 7(1): 81-105 doi: 10.3934/ipi.2013.7.81 +[Abstract](3631) +[PDF](939.5KB)
Abstract:
The full application of Bayesian inference to inverse problems requires exploration of a posterior distribution that typically does not possess a standard form. In this context, Markov chain Monte Carlo (MCMC) methods are often used. These methods require many evaluations of a computationally intensive forward model to produce the equivalent of one independent sample from the posterior. We consider applications in which approximate forward models at multiple resolution levels are available, each endowed with a probabilistic error estimate. These situations occur, for example, when the forward model involves Monte Carlo integration. We present a novel MCMC method called $MC^3$ that uses low-resolution forward models to approximate draws from a posterior distribution built with the high-resolution forward model. The acceptance ratio is estimated with some statistical error; then a confidence interval for the true acceptance ratio is found, and acceptance is performed correctly with some confidence. The high-resolution models are rarely run and a significant speed up is achieved.
Our multiple-resolution forward models themselves are built around a new importance sampling scheme that allows Monte Carlo forward models to be used efficiently in inverse problems. The method is used to solve an inverse transport problem that finds applications in atmospheric remote sensing. We present a path-recycling methodology to efficiently vary parameters in the transport equation. The forward transport equation is solved by a Monte Carlo method that is amenable to the use of $MC^3$ to solve the inverse transport problem using a Bayesian formalism.
2013, 7(1): 107-122 doi: 10.3934/ipi.2013.7.107 +[Abstract](3248) +[PDF](332.6KB)
Abstract:
This article is concerned with the representation of curves by means of integral invariants. In contrast to the classical differential invariants they have the advantage of being less sensitive with respect to noise. The integral invariant most common in use is the circular integral invariant. A major drawback of this curve descriptor, however, is the absence of any uniqueness result for this representation. This article serves as a contribution towards closing this gap by showing that the circular integral invariant is injective in a neighbourhood of the circle. In addition, we provide a stability estimate valid on this neighbourhood. The proof is an application of Riesz--Schauder theory and the implicit function theorem in a Banach space setting.
2013, 7(1): 123-157 doi: 10.3934/ipi.2013.7.123 +[Abstract](2538) +[PDF](1415.4KB)
Abstract:
The aim of our work is to reconstruct an inclusion $\omega$ immersed in a fluid flowing in a larger bounded domain $\Omega$ via a boundary measurement on $\partial\Omega$. Here the fluid motion is assumed to be governed by the Stokes equations. We study the inverse problem of reconstructing $\omega$ thanks to the tools of shape optimization by minimizing a Kohn-Vogelius type cost functional. We first characterize the gradient of this cost functional in order to make a numerical resolution. Then, in order to study the stability of this problem, we give the expression of the shape Hessian. We show the compactness of the Riesz operator corresponding to this shape Hessian at a critical point which explains why the inverse problem is ill-posed. Therefore we need some regularization methods to solve numerically this problem. We illustrate those general results by some explicit calculus of the shape Hessian in some particular geometries. In particular, we solve explicitly the Stokes equations in a concentric annulus. Finally, we present some numerical simulations using a parametric method.
2013, 7(1): 159-182 doi: 10.3934/ipi.2013.7.159 +[Abstract](2679) +[PDF](479.0KB)
Abstract:
We study the inverse problem of the simultaneous identification of two discontinuous diffusion coefficients for a one-dimensional coupled parabolic system with the observation of only one component. The stability result for the diffusion coefficients is obtained by a Carleman-type estimate. Results from numerical experiments in the one-dimensional case are reported, suggesting that the method makes possible to recover discontinuous diffusion coefficients.
2013, 7(1): 183-197 doi: 10.3934/ipi.2013.7.183 +[Abstract](2746) +[PDF](373.1KB)
Abstract:
We investigate two inverse scattering problems for the nonlinear Schrödinger equation $$-\Delta u(x) + h(x,|u(x)|)u(x) = k^{2}u(x), \quad x \in \mathbb{R}^2,$$ where $h$ is a very general and possibly singular combination of potentials. The method of Born approximation is applied for the recovery of local singularities and jumps from fixed angle scattering and backscattering data.
2013, 7(1): 199-216 doi: 10.3934/ipi.2013.7.199 +[Abstract](2954) +[PDF](522.8KB)
Abstract:
In this paper we integrate the SART (Simultaneous Algebraic Reconstruction Technique) algorithm into a general iterative method, introduced in [8]. This general method offers us the possibility of achieving a new convergence proof of the SART method and prove the convergence of the constrained version of SART. Systematic numerical experiments, comparing SART and Kaczmarz-like algorithms, are made on two phantoms widely used in image reconstruction literature.
2013, 7(1): 217-242 doi: 10.3934/ipi.2013.7.217 +[Abstract](3433) +[PDF](1839.1KB)
Abstract:
Electrical impedance tomography (EIT) aims to reconstruct the electric conductivity inside a physical body from current-to-voltage measurements at the boundary of the body. In practical EIT one often lacks exact knowledge of the domain boundary, and inaccurate modeling of the boundary causes artifacts in the reconstructions. A novel method is presented for recovering the boundary shape and an isotropic conductivity from EIT data. The first step is to determine the minimally anisotropic conductivity in a model domain reproducing the measured EIT data. Second, a Beltrami equation is solved, providing shape-deforming reconstruction. The algorithm is applied to simulated noisy data from a realistic electrode model, demonstrating that approximate recovery of the boundary shape and conductivity is feasible.
2013, 7(1): 243-252 doi: 10.3934/ipi.2013.7.243 +[Abstract](2525) +[PDF](325.4KB)
Abstract:
We study the spherical mean transform on $\mathbb{R}^n$. The transform is characterized by the Euler-Poisson-Darboux equation. By looking at the spherical harmonic expansions, we obtain a system of $1+1$-dimension hyperbolic equations. Using these equations, we discuss two known problems. The first one is a local uniqueness problem investigated by M. Agranovsky and P. Kuchment, [ Memoirs on Differential Equations and Mathematical Physics, 52 (2011), 1--16]. We present a proof which only involves simple energy arguments. The second problem is to characterize the kernel of spherical mean transform on annular regions, which was studied by C. Epstein and B. Kleiner [ Comm. Pure Appl. Math., 46(3) (1993), 441--451]. We present a short proof that simultaneously provides the necessity and sufficiency for the characterization. As a consequence, we derive a reconstruction procedure for the transform with additional interior (or exterior) information.
We also discuss how the approach works for the hyperbolic and spherical spaces.
2013, 7(1): 253-265 doi: 10.3934/ipi.2013.7.253 +[Abstract](2358) +[PDF](325.2KB)
Abstract:
Photoacoustic tomography is a rapidly developing medical imaging technique that combines optical and ultrasound imaging to exploit the high contrast and high resolution of the respective individual modalities. Mathematically, photoacoustic tomography is divided into two steps. In the first step, one solves an inverse problem for the wave equation to determine how tissue absorbs light as a result of a boundary illumination. The second step is generally modeled by either diffusion or transport equations, and involves recovering the optical properties of the region being imaged.
In this paper we, address the second step of photoacoustics, and in particular, we show that the absorption coefficient in the stationary transport equation can be recovered given certain internal information about the solution. We will consider the variable index of refraction case, which will correspond to an inverse transport problem on a Riemannian manifold with internal data and a known metric. We will prove a stability estimate for a functional of the absorption coefficient of the medium by finding a singular decomposition for the distribution kernel of the measurement operator. Finally, we will use this estimate to recover the desired absorption properties.
2013, 7(1): 267-282 doi: 10.3934/ipi.2013.7.267 +[Abstract](2932) +[PDF](1846.2KB)
Abstract:
The X-ray phase contrast imaging technique relies on the measurement of the Fresnel diffraction intensity patterns associated to a phase shift induced by the object. The simultaneous recovery of the phase and of the absorption is an ill-posed nonlinear inverse problem. In this work, we investigate the resolution of this problem with nonlinear Tikhonov regularization and with a joint sparsity constraint regularization. The regularization functionals are minimized with a Gauss-Newton method and with a fixed point iterative method based on a surrogate functional. The algorithms are evalutated using simulated noisy data. The joint sparsity regularization gives better reconstructions for high noise levels.
2013, 7(1): 283-290 doi: 10.3934/ipi.2013.7.283 +[Abstract](2375) +[PDF](757.5KB)
Abstract:
Source extraction in audio is an important problem in the study of blind source separation (BSS) with many practical applications. It is a challenging problem when the foreground sources to be extracted are weak compared to the background sources. Traditional techniques often do not work in this setting. In this paper we propose a novel technique for extracting foreground sources. This is achieved by an interval of silence for the foreground sources. Using this silence interval one can learn the background information, allowing the removal or suppression of background sources. Very effective optimization schemes are proposed for the case of two sources and two mixtures.
2013, 7(1): 291-303 doi: 10.3934/ipi.2013.7.291 +[Abstract](3670) +[PDF](330.1KB)
Abstract:
We consider an interior inverse scattering problem of reconstructing the shape of a cavity. The measurements are the scattered fields on a curve inside the cavity due to one point source. We employ the decomposition method to reconstruct the cavity and present some convergence results. Numerical examples are provided to show the viability of the method.
2013, 7(1): 305-306 doi: 10.3934/ipi.2013.7.305 +[Abstract](2640) +[PDF](217.6KB)
Abstract:
In paper "Strongly Convex Programming for Exact Matrix Completion and Robust Principal Component Analysis", an explicit lower bound of $\tau$ is strongly based on Theorem 3.4. However, a coefficient is missing in the proof of Theorem 3.4, which leads to improper result. In this paper, we correct this error and provide the right bound of $\tau$.

2020 Impact Factor: 1.639
5 Year Impact Factor: 1.720
2020 CiteScore: 2.6


text:
## Estimation in a Markov chain regression model with missing covariates

Dorota M. Dabrowska1, Robert M. Elashoff1 and Donald L. Morton2

1 Department of Biostatistics, University of California, Los Angeles, CA 90095-1772

2 John Wayne Cancer Institute, Santa Monica, CA 90404

Summary. Markov chain proportional hazard regression model provides a powerful tool for analysis of multiple event times. We discuss estimation in absorbing Markov chains with missing covariates . We consider a MAR model assuming that the missing data mechanism depends on the observed covariates, as well as the number of events observed in a given time period, their types and times of their occurrence. For estimation purposes we use a piecewise constant intensity regression model.

### 1 Introduction

Missing covariate measurements arise frequently in regression analyses of survival time data. The most common approach to handling such measurements corresponds to the case deletion method. It consists of exclusion of subjects with missing covariates and analysis of the data based on information collected on the remaining subjects. This method can be highly inefficient, and can lead to biased estimates, if complete cases do not form a random sample of the original data (Little and Rubin, 1987).

Several authors have proposed methods for analysis of the proportional hazard model with missing covariates. In particular, Zhou and Pepe (1995) and Lin and Ying (1993) suggested methods for regression analysis in the case of covariates missing completely at random (MCAR). This model assumes that the distribution of the missing data mechanism does not depend on the outcome variables. The approach taken by Zhou and Pepe and Lin and Ying corresponds to estimation of regression coefficients based on modified partial likelihoods obtained by approximating the conditional expectation of a covariate Z(t) given the risk process based on subjects who have complete measurements and remain at risk for failure at time t. Martinussen (1999) and Chen and Little (1999) considered the more parsimonious model assuming that covariates are missing at random (MAR). Under assumptions of this model, the missing data mechanism may depend the observed data, but not on the values of the missing covariates. Several methods for handling missing covariates in both proportional hazard model and parametric survival analysis models were also proposed by Lipsitz and Ibrahim (1996, 1998), Chen and Ibrahim (2001).

In this paper we consider estimation in a multivariate counting process corresponding to a finite state proportional hazard Markov chain model (Andersen, et al. 1993, Andersen, Hansen and Keiding, 1992). As opposed to single endpoint models, here inferences refer to a stochastic process {J(t),t e [0,r]} such that at time t, J(t) takes on values in a finite set E = {1,..., k} representing possible events in the evolution of a disease. Along with a possibly censored realization of the process J(t), we also observe a vector of time independent covariates Z. The model assumes that conditionally on Z, the process J(t) forms an inhomogeneous Markov chain and intensities of transitions among adjacent states have proportional hazard form. In Section 2 we allow some components of the vector Z to be missing. We define a MAR model, assuming that the missing data mechanism depends on the observed covariates as well as the number of events observed during a specified period of time, their types and times of the occurrence. We consider censoring corresponding to the termination of the study at a fixed time point t, and random censoring representing an absorbing state of the observed model. For purposes of estimation of the parameters of the Markov chain we use a modification of Freedman's (1982) approach to analysis of the proportional hazard model with piecewise constant hazard rates. The method uses histogram approximation to the hazard rates and a piecewise linear approximation to cumulative intensities.

In their analysis of MCAR and MAR models, Little and Rubin (1987, p. 90) showed that the likelihoods for estimation of the parameters of interest are the same under assumptions of both models. More precisely, the likelihoods differ only in the proportionality factors depending on the parameters describing the missing data mechanism at hand. In the present setting, the MCAR model allows for estimation of the unknown parameters as well as estimation of a modified matrix of transition probabilities (Section 2.1). On the other hand, the MAR condition depends on the sequence of states visited and times of entrances into these states. Similarly to Little and Rubin (1987, p. 90), the likelihood for estimation of the regression coefficients is the same under the MAR and the MCAR model (up to proportionality factors), however, we show that the MAR model does not allow in general for estimation of the transition probabilities among different states of the model.

For illustrative purposes in Section 3 we use data on 4141 patients diagnosed with malignant melanoma and treated at the John Wayne Cancer Institute, Santa Monica (JWCI) and UCLA. The JWCI/UCLA database was initiated over 30 years ago. This clinio-pathological demographic database has been established in late 1970's and identified as a national resource for melanoma studies. The database has expanded in the number and type of variables included as well as addition of clinical trials data developed by JWCI. Data analyses of this database encountered the not uncommon sit uation where observations on some important variables are missing. Thus for example, depth of invasion of the primary tumor has not been observed in a moderate number of cases.

### 2 The model and estimation 2.1 The model

Throughout we consider estimation in a finite state Markov chain process. We assume that the chain {J(t) : t G [0,t]} is observed over a finite time period (t < and its state space E = {1,...,k} can be partitioned into two disjoint sets TU A = E, T nA = 0, representing transient (T) and absorbing (A) states. A pair of distinct states (i,j) G E x E,i = j is called adjacent if transition from state i to state j is possible in one step. The collection of all such adjacent pairs is denoted by E0, E0 C E x E.

A Markov chain regression model can be specified in terms of two parameters. They correspond to (i) the joint marginal distribution of the initial state Jo and the covariates Z = (Zi,..., Zd); and (ii) the conditional cumulative intensity matrix A(t; z) = [Aij(t; z)]i,jeE. The entries of the matrix A(t; z) are given by

For i = j, the functions aij (u, z) represent conditional hazard rates of one-step transitions among adjacent states of the model. The negative on-diagonal entries form cumulative hazard functions accounting for the sojourn time in each state of the model. The (i,j) entry of the conditional transition probability matrix

P (s,t; z) = [Pij (s,t; z)]i,j=i,,.,k = [Pr(J (t) = j\J (s) = i,Z = z)]i,j=i,,.,k provides the conditional probability that the process occupies state j at time t, J(t) = j, given Z = z and given that that at time s, s <t the process is in state i. The matrix P(s,t; z) forms solution to Kolmogorov equations

P(s,t; z) = I + P(s,u-; z)A(du; z) = I + A(du; z)P(u,t; z) , ss where I is the identity matrix. Methods for its computation are discussed in Chiang (1968), Aalen and Johansen (1978) and Andersen et al (1993), among others.

Associated with the pair (Z, {J(t),t G [0,r]}) is a marked point process {Z, (Tm, Jm)m>o}, where 0 = To <Ti < ... < Tm < ... are times of consecutive entrances into the possible states of the model, and Jo, Ji,..., Jm,... are states visited at these times. Let Wm = (T£, J£ : I = 0,...,m),m > 0, be the first m pairs in the sequence (Tm, Jm)m>o. The assumption that, conditionally on the covariates, the process J(t) forms a Markov chain entails that

Pr(Tm < t,Jm = jm\Wm-i ,Z) = f (u,jm\Tm-i, Jm-i, Z) ,

where f (u,jm\t m— 1, jm-i, Z) = 1( u > tm — 1)F (u\tm-i ,jm — U z)ajm-i,jm (u; z) and

The function F(■\tm-i,jm-i,z) represents the conditional survival function in state jm-i, that is

Finally, the probability of one-step transition into state j at time Tm is given by

From this it also follows that lim - Pr(Tm G [t,t + s],Jm = j\Z, Tm > t, Wm-i) =

If we denote by Nh(t) and Yh(t),h G E0, the processes

Nh(t) =Y Nhm(t) , Yh(t) =Y, Yhm(t) , m> i m> i where

Nhm(t) — l(Tm-i < Tm < t, Jm-1 — i,Jm — j) , Yhm(t) — l(Tm > t > Tm-1, Jm-\ — i) , then N(t) — {Nh(t) : t G [0,r],h — (i,j) G Eq} is a multivariate counting process whose components record transitions among adjacent states occurring during the time interval [0,t]. Assuming that the process N(t) is defined on a complete probability space (f, F, Pr), its compensator A(t) — {Ah(t) : t G [0,r],h G Eq}, relative to the self-exciting filtration {Q ® Ft}t<T, Q — a(Z), Ft — a{Jo, Nh(s),Yh(s+) : s < t,h G E0} satisfies

Ah(dt) — E[Nh(dt)| Q x Ft-] — Yh(t)ah(t; Z)dt .

The assumption that the process forms a proportional hazard Markov chain corresponds to the choice

Ah(dt)— Yh(t)epTZhah(t)dt , where a — [ah : h G Eq] are unknown baseline hazards, [Zh : h G Eq] is a vector of transition specific covariates, and f3 — [fih : h G Eq] is a conformal vector of regression coefficients. For any pair of adjacent states, h G Eq, the vector Zh is either equal to the covariate Z, or else it represents a function Zh — @h(Z) derived from the covariate Z.

We assume now that the covariate Z can be partitioned into two nonempty blocks, Z — (Zq,Zi) such that Zq — (Zoi,...,Zoq) and Z\ — (Zn,..., Zi,d-q). We shall use the following regularity conditions. Condition 2.1

(i) The conditional distribution of Zq given (Z\, Jq) has density gg(zq| zi,jo) with respect to a product dominating measure and dependent on a parameter 0 G O C Rq.

(ii) Conditionally on (Zq, Z\, Jq), the sequence m>0 forms a proportional hazard Markov chain model with parameters a — [ah : h G Eq] and ¡3 —[Ph : h G Eq].

(ii) The parameter 0 is noninformative on (a,3).

We denote by ^ — (0, a, ¡3) the unknown parameters. In Appendix 2, we give a recurrent formula for the conditional density of the the covariate Zq given the vector V, V — [N.(r), (J^Te)1^:^ ),Z1]. We also show that the "marginal" model, obtained by omitting the covariate Zq forms a non-Markovian counting process.

Here we assume that some components of this vector may be missing. Let R — (Ri,..., Rq) be a binary vector defined by

Rj — 1 if ZQj is observed Rj — 0 if ZQj is unobserved .

Then for a subject whose missing data indicator R is equal to r — (ri,..., rq), we observe the vector (V, ZQ(r)), where V — [N (t), (Te, J^^f ^, Zi] and

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How to put a markov into a regression model?
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