Mathematics - 2017-01-19 (page 2 of 5)

the at least one hole part is important

it's a statement of existence, was thinking something else. derp

Socrates

you are not the first one who derped today

thought for food

or the last :-)

@Socrates It's expected. I'm a professional derper, as well as a professional procrastinator, a pro-procrastinator.

A prodigious pro-procrastinator, hoping to becoming a most prolific pro-procrastinator some day.

thought for food

3:26 AM

you'll "flourish" in here

:-D

hi can anyone here help me with a stochastic calculus problem?

Socrates

@thoughtforfood could you imagine a big monitor sitting in the pose of the seemingly philosoph, looking on him^^

3:50 AM

Define $X_t^S$, where $S$ is a stopping time as
$X_t^S = X_{min\{t,S\}}\mathbb{1}_{S>0}$
Prove that if $X_t^S$ is a martingale with respect to the filtration $\mathcal{F}_{min\{t,S\}}$ then it is also a filtration with respect to the filtration $\mathcal{F}_t$

Oh no, who starred that mess?

I've been stuck on this problem for quite a while

4:40 AM

hi guys..

I'm a newbie in stats. I was wondering how does Explained Sum of Squares (ESS) actually explain accuracy of a model

because if the model / function was highly inaccurate, the ESS will be large

yet, wikipedia says "In general, the greater the ESS, the better the estimated model performs."

4:59 AM

Need some guidance with basic function stuff. First, the image of the rationals under $x\;\mapsto\;2 x$ is still the rationals, right?

yes

I'm guessing this is true for any nonzero multiple (with the mapping over $\Bbb R$ oc)

@Ming Why do you say that?

@Brody consider $x\mapsto \sqrt{2}x$

hah right. nonzero *rational multiple

thanks

I'm investigating when $f(A_1\cap A_2)= f(A_1)\cap f(A_2)$. (no outright spoilers please)

5:34 AM

I have an answer that has received two down votes; but I am clueless to what mistake I made. Can someone please point out the mistake if any?
http://math.stackexchange.com/questions/2103963/translate-the-statement-into-english-and-something-about-the-order/2104063#2104063

@Brody because ESS formula is difference between sum of predicted value and mean squared. So, if the predicted value is far off the mean, the value will be large. EG the average test score for students is 60%. My model predicts 15%. the difference is huge because my model is clearly inaccurate. Therefore, ESS will be large

@Ming mm now I'm confused, because my text uses different terms and notation, and I also don't understand the relationship in question. One sec...

Any logicians in the room?

@Brody i used wikipedia

5:53 AM

@Ming Wikipedia seems right. The larger this particular sum of squares, the closer our data points are to the regression line, so it indicates a stronger correlation

Balarka Sen

Hi @Brody, @Akiva, @Kaj.

Certainly we have $\subset$ containment

It looks like lack of injectivity might mess up $\supset$ containment

Hey @BalarkaSen

the model analyzed should be a least-squares regression @Ming

Hello @Balarka, @Kaj

@KajHansen you mean generally and over all of $f$'s domain?

Is it possible to ask a question on Stack Exchange about an answer that was wrong?

If so; how would one even begin if the answer did not have any comment as to why it is wrong?

@Brody if the least-squares regression did a poor job in predicting the values eg: it was very inaccurate, will ESS be large or small?

5:58 AM

Not necessarily in general, but I can think of examples where $\supset$ containment messes up, and it happens when $f$ isn't injective on the $A_1 \cup A_2$ piece of the domain

EG: if the ESS was small, does it mean that the average of the predictions were close to the mean? is that considered an inaccurate model ? @Brody

I think. I'm playing chess, so this doesn't have my full attention

Equality definitely holds when $f$ is injective

@Ming accuracy means your predictions are close to your observations. the mean is just the average of your observations

@Brody but ESS is sum of predicted values minus mean. So if ESS is large, means the sum of differences was large. Correct?

if sum of differences was large, means that the predicted values were far away from the mean. correct?

yes, overall those deviations from the mean would be large @Ming

but that's just how much your predictions deviate from the center line

6:09 AM

therefore, wikipedia days the larger the deviation of predictions from the center line, the better the model does in explaining

what is it explaining? if my mean is 10. and my model predicts 100, yes the difference is large. but what does that explain?

@Brody

@Ming you have to take every value into consideration. your regression would typically involve many data pairs

@Brody assume i take all values into consideration. They all predict very far off values. 100, 96, 78, -100, -50 etc...

the ESS will be very large.

according to wikipedia, the larger the values, the better the model does in explaining. what is it explaining?

@Ming as said earlier, greater ESS means your least-squared predictions are closer to your observations. Rather, the points in the data distribution (scatter plot) are tight against your regression line

Is it possible to ask a question on Stack Exchange about an answer that was wrong?
If so; how would one even begin if the answer did not have any comment as to why it is wrong?

I only understand this mathematically (and not all that well), and not in a direct intuitive way, sorry @Ming

6:23 AM

@Brody i think i get it. Wikipedia says "generally". their statement is true only when ESS < TSS. If ESS is greater than TSS, then their statement is not valid. Thats my thoughts. this video helped me youtube.com/watch?v=I8cRj0wefi8

@user400188 did you try posting a comment under the answer? did you try asking others in the same thread? did you try asking about it elsewhere, e.g. in chat? anyway, yes you can ask questions about answers of previous questions.

I asked in chat and in a comment under the answer but no one has responded so far. The original thread was: math.stackexchange.com/questions/2103963/…

oh, so it was your answer. you failed to mention that.

sorry; I thought I would try a different tack on asking the question because no one answered the first few times i tried

also it looks like your answer has only been there 2 hours

6:27 AM

@Ming I don't think ESS > TSS should ever occur

Mike Miller

morning

@Brody true

Isn't it like 10:30 PM where you're at @MikeMiller ?

i guess i was assuming ESS > TSS

@Ming The interpretation I understand is by looking at the identity ESS=TSS-RSS

6:29 AM

@arctictern I suppose I should wait longer. I spent this last hour however checking for a mistake in it, which has made me impatient I guess.

nah

Good morning

Morning

TSS is how much the observations vary about the mean line, RSS is how much they vary about the least-squares line. since ESS=TSS-RSS, the explained sum measures how much this variance reduces using a least-squares prediction rather than a mean prediction @Ming

@user400188 A couple things. First, the thing you wrote after "You are correct in writing..." is not actually what OP wrote, and secondly just because you can eliminate an extra C(x) doesn't mean you should (the task isn't to find the most compact equivalent statement after all, it's to translate the statement directly into symbols).

6:32 AM

so, a big ESS means your least-squares is really advantageous. a large proportion of the variance can be explained/resolved by the least-squares model @Ming

although in the future keep in mind downvotes don't necessarily mean logical errors. they could also mean a voter doesn't understand your answer, doesn't like its style, doesn't think it's useful, or else finds it misleading

@arctictern Why is it not what the op wrote? Its the first part of the expression. I also copied and pasted it into my answer.

@arctictern, sometimes I have a tendency to give away too much information to a potential homework problem, and I think that's the reason for a majority of my downs

@user400188 you mean the second part of the expression?

the fact that you wrote only part of it means you are not faithfully copying OP's attempt at translation. thus, you should mention that you're only referring to part of OP's expression.

The first part of ∀x(C(x))∨(∀x∃y(C(y)∧(F(x,y)))

6:36 AM

the first part of that is ∀x(C(x) and the second part is (∀x∃y(C(y)∧(F(x,y))) no?

what you wrote after "You are correct in writing..." is only the second part (∀x∃y(C(y)∧(F(x,y)))

@arctictern What I wrote after that is ∀x(C(x)... Is it just not showing up on some people computers? I did use the $\$ stuff to produce it

@KajHansen I laughed a bit at this because "downs" is interwebz slang for down syndrome. no offense intended toward anyone, it just read funny at first

I didn't**

hahaha @Brody

Didn't use the $\$ stuff to produce the ∀x(C(x)**

6:38 AM

@user400188 yes, you wrote ∀x(C(x) after it. not sure of your point.

sorry, I kinda forgot about that image problem @Kaj

@arctictern did you just say "what you wrote after "You are correct in writing..." is only the second part (∀x∃y(C(y)∧(F(x,y)))"

doesn't matter either way. like I said, the exercise isn't to find the most compact logically equivalent form, it's to translate the statement as faithfully as possible. calling a part of the correct answer "not necessary" might be misleading if it is necessary for OP to get a correct answer or to successfully translate the statement as faithfully as possible

@user400188 yes

I don't mind @Brody

mmm, the notation for the subset relations kinda bothers me

6:39 AM

You mean $\subset$, $\supset$ ? Why?

Dunno, maybe I'd prefer $\subseteq$ to mean "proper subset of or equal to"

Oh yeah I see

It actually is kinda strange now that I think about it. We don't say $x<y$ might imply equality

@arctictern I think I understand the mistake now. I did not write out the full expression and thus it was not what the op originally wrote. Then I did not answer the question correctly because I wrote a simpler expression than the sentence the op asked us to write in logic.

right, it's not analogous. not that one should expect notation 'n stuff to be logical

but still... :P

I make a point to use $\subsetneq$ if necessary

6:43 AM

like when we say the subset direction $\subset$ obviously holds, now what of the superset direction $\supset$?

seems a bit awkward given that the latter stands only as equality for this problem, i.e. when the sets are the same. so it's not even really a matter of superset containment. sorry, it's hard to explain, lol

haha, not sure I follow, but I do see the general inconsistency with notation