"Total" EH - incorporating different damage types into EH

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Re: "New EH" - incorporating different damage types into EH

Postby Serv » Thu Dec 03, 2009 5:05 am

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Re: "New EH" - incorporating different damage types into EH

Postby Rhidach » Thu Dec 03, 2009 7:56 am

Meloree wrote:You two are both totally batshit insane. And I love you for it.


Yeah, seconded. I have no idea what they're saying... but I can't wait for the tldr.
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Re: "New EH" - incorporating different damage types into EH

Postby theckhd » Thu Dec 03, 2009 8:15 am

Xenix: That was exactly the form I was planning on using in the appendix. Summation notation is much more convenient for the general form.

However, I wanted to avoid using images for the math text, despite being prettier, because they have a tendency to disappear long after the fact, making the content of posts unreadable. I was planning on including a pdf version of the derivation for those of us that like the formal math notation, which you've conveniently provided. Thanks. :)
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Re: "New EH" - incorporating different damage types into EH

Postby Xenix » Thu Dec 03, 2009 8:22 am

Heh - for the TLDR, read Theckd's conclusion at the end of his first post on the thread. All the stuff after is just further theory talk along the same lines which might be useful for other derivations later.

Continuing what I did in my previous post, Theckd, here is the differential NEH equation in terms of the post-mitigation variables:
Image
Ri is defined above and represents your percent damage intake from the i'th damage type (e.g. if a boss hits you for 100,000 raw physical damage and you take 30,000, Rp = 30%).
Phi-hat_i is the percent of mitigated damage that is from the specific damage type, Phi_i represents the percent of raw damage that is from the specific type and Q-bar is defined the same as previously (your weighted percent damage intake).

In reality, this equation is exactly the same as my previous one - three terms, each meaning the same thing. In fact, if you use the third equation to change variables back to Phi_i, it will devolve into the same formula. Also, while the second term looks much simpler in this equation, dPhi-hat_i/dU is not a simple derivative to take. The main thing you'd want to use this form of the equation for is differentiating with respect to either one of the Phi-hat_i variables or R. Anything else and the previous form would probably work better.

<Edit>: PDF of the derivation is at here.

Oh, and I just noticed with the newly defined R_i variable, the second term in the pre-mitigation version can be re-written in a form that makes the second terms look similar between the two:
Image
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Re: "New EH" - incorporating different damage types into EH

Postby Kriskringle » Thu Dec 03, 2009 11:39 am

theckhd wrote:Upon looking at it more closely, I have to revise my answer. Resistance auras will make absolutely no difference in the armor/stam graph. This should be clear by the form of equation (22):
Code: Select all
      12.54*(K+A)      1
dA = -------------*---------*dS                 (22)
           H        (1-X-Y)

Note that this depends only on (1-X-Y), or the percentage of damage intake due to "regular" physical sources. In other words, your resistances are encapsulated in X and Y, so by increasing resistance we just move along the graph to a point of larger (1-X-Y).


Just wanted to pop back in and say thanks for the answer (and boo to me for not thinking about it before I asked). Love the new Stam vs. Armor effectiveness plots.
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Re: "New EH" - incorporating different damage types into EH

Postby Wrathy » Thu Dec 03, 2009 12:47 pm

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Re: "New EH" - incorporating different damage types into EH

Postby Kihra » Thu Dec 03, 2009 1:42 pm

This came up in another thread, but just a reminder that 12.54 is not the right number. 12.54 is exactly what you get using additive stacking of stamina and not multiplicative stacking of stamina.

1.1 from kings * (1 + .08 + .06) = 1.1 * 1.14 = 12.54 (WRONG)

1.1 from kings * 1.08 * 1.06 = 12.59 (RIGHT)
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Re: "New EH" - incorporating different damage types into EH

Postby theckhd » Thu Dec 03, 2009 2:05 pm

Kihra wrote:This came up in another thread, but just a reminder that 12.54 is not the right number. 12.54 is exactly what you get using additive stacking of stamina and not multiplicative stacking of stamina.

1.1 from kings * (1 + .08 + .06) = 1.1 * 1.14 = 12.54 (WRONG)

1.1 from kings * 1.08 * 1.06 = 12.59 (RIGHT)

Now that you mention it, I think I remember that discussion. When I get some time I'll go back and fix the numbers.
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Re: "New EH" - incorporating different damage types into EH

Postby Xenix » Thu Dec 03, 2009 3:54 pm

I just had another thought - this will most likely come up eventually, so I'll post about it now, and you might consider mentioning it, Theckd. All of these equations that both Theckd and I derived assume your health to be constant over all types of damage. In reality, this is true -except- when you have a flat damage reduction ability that doesn't affect all damage types equally. The only things like that I can think of off the top of my head are:

  • Block - Only affects certain physical damage
  • Resist Potions - Only affect one element of magical damage

Before I go any further, let me remind you that we're talking about EH here, which takes into account only -guaranteed- damage reduction. If you're not block capped, you can't add block into your EH.

So, the next question is how can this be incorporated into the EH equation? Well, first we have to look at exactly what flat damage reductions do: They subtract damage off the post-mitigation damage you would have taken. Another way to look at this is they increase your health by an amount equal to the damage they absorb. If we define Hi to be your static health H plus any flat damage reductions Fi you have against damage type 'i', we can simply substitute Hi for H in the post-mitigation form of the equation and expand to get:
Image

There is no similar form for the raw damage % equation because flat damage reductions only happen after mitigation. Furthermore, with the differential form, we have to change all the H's to Hi's and keep them inside the summation:
Image

We can then plug in H + Fi for Hi and expand to get:
Image

The first row is exactly the same as before but we have a new second row that represents the extra flat damage reductions. If you have none of these, it's exactly the same equation as before.

TLDR:
If you have any guaranteed flat damage reductions (such as block or a resist potion) against a specific type of damage, you can consider them as extra health against that kind of damage only. Just add the flat reduction to your health for that damage type when calculating NEH.

As an example, 1 block value = 1 health against a single attack that is blockable (which should be obvious). If only half the damage that killed you was from a blockable attack though, 1 block value would only be worth 0.5 health, etc. You could then plug these into the health-armor equivalence equations to see how much armor is equivalent to 1 block value. Also, note that block is special as it scales with the number of blockable attacks. (1 block would be worth 5 health if it took 5 blockable attacks to kill you, etc).
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Re: "New EH" - incorporating different damage types into EH

Postby Reynardadin » Sat Dec 05, 2009 6:09 am

Hey Theck (and everyone else),

I'm the author of TankTotals, an addon which, among other things, provides the user with their total physical mitigation, per-school guaranteed and average spell mitigation, EH, expected TTL, etc. It's primarily concerned with stats that don't change in different circumstance or from one fight to the next, but with 3.3 on the horizon and considering the interest that this thread has generated, I've been thinking about how best to incorporate your formula as a feature. Gravity at pwnwear prodded me to look into it, and I was initially going to post this at his site, but I figured it'd be more productive to put it up here.

I had actually considered adding almost exactly your calculation to TankTotals back when I first implemented the EH and TTL numbers, although I didn't think to separate bleed damage from physical, and I intended to express it as an overall mitigation or "survivability" figure rather than as a health value (using your notation, it would have been 1 - [(1-Y)(1-Ma)(1-Mt) + ΣYi(1-Mgi)(1-Mri)]). Mostly, I toyed with the idea because of the attraction of having a single number to sum up an entire fight from the tank's point of view. However, there were a few issues that ultimately made me decide against it (some of which have already been touched upon, so apologies for partial repetition). Please note that the following observations are made from the perspective of calculating a single NEH value for an entire fight, as has been proposed in this thread:

1. The formula obviously depends entirely upon the relative proportions of damage types encountered in each individual fight. As such, EH ceases to be an invariant benchmark of damage absorption, and instead becomes a (broad) estimate of health on a per-fight basis. This seemed to me to be of dubious utility in general gear/spec deliberations, which is the purpose of EH in the first place.

2. The beauty of the old EH calculation for benchmarking purposes is its consistency; no matter what fight you're in, whether you're stunned or frozen or dancing with your back turned to the boss, you can always absorb at minimum (H/(1-M)) physical damage at any time. This formula, by contrast, operates by retrospectively weighting the relative contributions of different damage sources across an entire fight, as though each were happening constantly and simultaneously; it makes little sense to express this as a health value, or to draw a parallel between it and EH. What if (as is often the case) the entire contribution of magic damage is concentrated in 2-3 massive nukes, which - despite dealing far less damage than my supposed NEH - will easily kill me unless I have a cooldown active? It breaks the EH paradigm.

3. In the large majority of cases, it's more useful to decouple these numbers and show them separately. To take the exaple of Anub, it's much more beneficial for the MT to know that Freezing Slash and Leeching Swarm will be reduced by A at least and B on average, and that the melees he takes while frozen will be mitigated by C. Applying the NEH formula to encounters like this, where the instantaneous percentage of damage caused by magic changes drastically over the course of the fight, would result in a number which is at best unhelpful and at worst actively counter-productive.

It was for these reasons that I decided not to implement the feature, and why I continue to be highly skeptical of efforts to come up with a "magic number" for each fight based on plugging WoL totals into this formula. (As a side question, again from the perspective of calculating NEH across an entire fight - how does avoidance interact with your approach? If you're using WoL to sum up the percentage of total damage contributed by magic and bleeds as has been discussed througout this thread, and taking (1-X-Y) as the percentage of damage due to physical attacks, aren't you implicitly treating avoidance as mitigation? Doesn't this take another step away from the old meaning of EH?)

However, as you discussed earlier, it's very useful as a means of determining how alternate gearing could have saved you from a specific death; in other words, analysing the window between the most recent time you were at 100% and your death. At the moment, I'm thinking that this would be the best way to implement the formula as an addon feature - to start monitoring incoming damage every time you're brought to 100% health, record the percentage contribution from each damage source, and then present a summary when you die. You'd then be given the option of using those percentages as X and Y in the NEH calculation, which would show up alongside EH on the TankTotals display. You could then try different gear combinations to see how they affect NEH in this particular case - perhaps the addon could show the NEH value in red until you gear/spec in a way that would have prevented your death, at which point it would turn green. I could also implement a configuration screen with sliders allowing an arbitrary X and Yi to be specified, similar to the Mob Damage slider used in calculating TTL.

Anyway, hopefully I can come up with something useful before 3.3 :) Any input would be appreciated!
Last edited by Reynardadin on Sat Dec 05, 2009 7:24 am, edited 1 time in total.
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Re: "New EH" - incorporating different damage types into EH

Postby Awyndel » Sat Dec 05, 2009 7:24 am

Kihra wrote:This came up in another thread, but just a reminder that 12.54 is not the right number. 12.54 is exactly what you get using additive stacking of stamina and not multiplicative stacking of stamina.

1.1 from kings * (1 + .08 + .06) = 1.1 * 1.14 = 12.54 (WRONG)

1.1 from kings * 1.08 * 1.06 = 12.59 (RIGHT)


Is this proven? Buffs multiply but are we sure talents do as well?

I have to admit when I suggested that number to Satrina in the armor/stam thread and he didn't correct me I asumed it was correct.

On another note, this is the first time this year you've completely lost me lol. Little bit too much math. I asume making an estimate of how much % of the worst spikes is magic damage will still give me a good enough number? So 20% magic damage means the armor is worth 0.8 of what it normally is?
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Re: "New EH" - incorporating different damage types into EH

Postby Xenix » Sat Dec 05, 2009 11:22 am

Reynard -

IMO, what you want to do to make fight-specific numbers is use the formula based on the raw damage percentages for a specific combination of abilites instead of the one based on the post-mitigation damage percentages. While it is true, like you said, that basing it on X/Y/Z, etc. would change all the time, the worst-case raw damage you have to survive in most fights won't change. Furthermore, what you'd be pulling from WoL logs would not be the % of each damage type over the entire fight, but either the average or maximum damage that each attack you're interested in caused. You would then manually string these numbers together into whatever the worst-case spike damage scenario would be. As an example, for Gormok, you need to be able to survive 1 melee hit + 1 bleed tick + 1 impale hit at whatever % damage increase stack you get to. You could get the average or maximum post-mitigation values from WoL, and if needed, you could back out the raw damage of those to come up with a worst-case N-EH requirement.

Average damage totals over a fight don't matter though, as you said, just the worst-case scenario that will kill you, and it's entirely possible a fight will have more than one worst-case scenario. (e.g. Algalon where it might be 2 melee hits + black hole explosion + phase punch, or just Big Bang by itself). If your N-EH isn't enough to survive that hit combination, it doesn't invalidate the equation, it just says you'll probably need a cooldown, PW:Shield or some other damage reduction method to increase your N-EH for that worst-case scenario.

But no, like EH, it's not a metric for evaluating an entire fight, but for evaluating if you can survive a specific set of attacks, and there's normally only one combination you're worried about for any given fight - that might have gotten lost in all the discussion.
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Re: "New EH" - incorporating different damage types into EH

Postby jere » Sat Dec 05, 2009 2:49 pm

Awyndel wrote:Is this proven? Buffs multiply but are we sure talents do as well?

I have to admit when I suggested that number to Satrina in the armor/stam thread and he didn't correct me I asumed it was correct.


Well I can't say I remember anyone testing it for WotLK, but we did actually do tests on it in BC (for sacred duty and combat expertise stacking). We came up with scenarios where the results would be different based on additive versus multiplicative. We found them to be multiplicative. However, it is not impossible that they changed it for WotLK (though I am doubtful they would have). Someone could always check and see again.
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Re: "New EH" - incorporating different damage types into EH

Postby Xenix » Sat Dec 05, 2009 3:42 pm

Just tested on my character. Unspecced and post-specced, I got the following:

Code: Select all
             Unspecced:       Prot:      Prot w/ Kings:
Random gear set:
Stamina:           645         737                 811
Health:          13204       14124               14864

Full tanking set:
Stamina:          3208        3671                4038
Health:          39109       43739               47409


A few things to note:
Both values of (prot stam w/ kings)/(prot stam) are 1.1. This means Kings does stack multiplicatively.

Both values of (prot stam)/(unspecced stam) are 1.142+ They would be 1.140 or under from rounding if it was additive stacking.

Both values of (prot stam w/ kings)/(unspecced stam) are 1.257+, and they would be 1.254 or under if it was additive stacking. Due to rounding, it doesn't quite reach 1.259x

(Health2 prot with kings - Health1 prot with kings - 275 health from gear enchant)/(stam2 unspecced - stam1 unspecced) = 12.59, exactly as expected.

TLDR:

The talents and BoK do stack multiplicitavely at 12.59 extra health per added point of stam for a paladin.
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Re: "New EH" - incorporating different damage types into EH

Postby theckhd » Sat Dec 05, 2009 7:34 pm

Before I make any other comments, I'd like to make a point about "new" vs. "old" in regards to this calculation. I've seen a lot of people assuming that "new EH" means that this is a completely new idea. That's probably mostly my fault for the title of the thread - it was supposed to be a play on "New Math," but that's probably too outdated of a reference for many readers to get.

In any event, it's not a new idea. This is still the same EH we know and love. All I've done is demonstrate the correct formula for it, so that people can use it more accurately. There are far too many people who use it incorrectly to make decisions about gearing ("Armor trinket A gives me more EH than stamina trinket B, so I'll wear it for Jaraxxus").

The hope is that by providing both the correct definition and a better insight into what EH actually represents, it will become a more useful tool to tanks that know what they're doing.

Reynardadin wrote:1. The formula obviously depends entirely upon the relative proportions of damage types encountered in each individual fight. As such, EH ceases to be an invariant benchmark of damage absorption, and instead becomes a (broad) estimate of health on a per-fight basis. This seemed to me to be of dubious utility in general gear/spec deliberations, which is the purpose of EH in the first place.


I will agree that EH is a metric better suited to per-fight analysis. But that's true of any EH value, and it's definitely not the "purpose" of EH. There's no reason to think that the "old" all-physical definition is any more general. It's just as incorrect on a fight with any significant amount of magical damage as the "new" version is on a fight with all physical damage (and assuming you don't choose X=Y=Z=0).

Reynardadin wrote:2. The beauty of the old EH calculation for benchmarking purposes is its consistency; no matter what fight you're in, whether you're stunned or frozen or dancing with your back turned to the boss, you can always absorb at minimum (H/(1-M)) physical damage at any time. This formula, by contrast, operates by retrospectively weighting the relative contributions of different damage sources across an entire fight, as though each were happening constantly and simultaneously; it makes little sense to express this as a health value, or to draw a parallel between it and EH. What if (as is often the case) the entire contribution of magic damage is concentrated in 2-3 massive nukes, which - despite dealing far less damage than my supposed NEH - will easily kill me unless I have a cooldown active? It breaks the EH paradigm.

This EH version can still be used for dealing with pure physical burst, just set X, Y, and Z to zero.

More pertinently, we decided somewhere around page 3 or 4 that choosing X, Y, and Z based on the total damage taken in a fight isn't the right way to use the formula, as Xenix mentioned a few posts up. As you correctly noted, avoidance would affect the X/Y/Z values if you used total damage taken, and avoidance has no place in an EH calculation.

What you want to do is identify the burst situation that you're concerned about, and calculate X, Y, and Z based on that.

Reynardadin wrote:I'm thinking that this would be the best way to implement the formula as an addon feature - to start monitoring incoming damage every time you're brought to 100% health, record the percentage contribution from each damage source, and then present a summary when you die. You'd then be given the option of using those percentages as X and Y in the NEH calculation, which would show up alongside EH on the TankTotals display. You could then try different gear combinations to see how they affect NEH in this particular case - perhaps the addon could show the NEH value in red until you gear/spec in a way that would have prevented your death, at which point it would turn green. I could also implement a configuration screen with sliders allowing an arbitrary X and Yi to be specified, similar to the Mob Damage slider used in calculating TTL.

This is a neat idea. You could also add presets for common burst death scenarios (melee+Impale+ImpaleDoT, melee+Freezing Slash+melee, etc).
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