Hey Mithr,


Ignite is fire damage and wont gain any threat reduction from Frost Channeling, this is the tool tip at max rank.

*Reduces the mana cost of all spells by 10% and reduces the threat caused by your Frost spells by 10%.*

I left FoF out of the other post simply to ignore an effect that wasn't going to impact the scaling of Brain Freeze (much).

Something to bear in mind is that a cast gains the 50% crit from FoF if either of the last two casts procced FoF. So, for Frostbolt spam...



Where .15 + .85*.15 = .2775 is the chance that at least one of the last two casts procced FoF, and N_1 is the average non-crit dps.



Again, b = 1.



Where .2775 * .5 = .13875 is a constant, not dependent on b. Thus, FoF damps crit scaling slightly by adding a little more to the denominator. This is not a surprise: after all, you're critting much more often, so as a result, additional crit does less to increase your dps (by ratio).

Due to this result, we can say that the equivalent crit chance increase from FoF is...13.875%? I think. I need to give that one some thought. And, again, simplified model, as always.

And my thanks to Roywyn for catching the mistaken proc chance.

I hate to break it to you, but you used a 10% proc chance for FoF in your calculation.
There is also the bug where a reproc on cast 2 will disappear, wouldn't want to make things too easy, would we!


On a totally different note - keep in mind that Shatter creates a crit soft cap for Frost specs.
Crit above 50% becomes less valuable as is cuts into Shatter.

You get a lot of crit before gear. 10% WC, 5% MKA, 3% ToW, 4% Emp. FrB, 5% Molten, 3%+3% Focus Magic couple.
That means your soft cap is 17% crit with just AB/GotW/Kings. 20% if you can cut through the 3% crit depression vs. bosses.


You can probably reach th 17% soft cap in T7 gear if you don't dodge crit hard enough, and 20% seems like it'll be very very hard to avoid once you climb up another tier or two.

Just to clarify - crit doesn't become useless, it just becomes less good. It's not a great stat to start with anyway, haste is better for frost.
It becomes less useful because it does nothing during Shatter as you are crit capped by then.

I've been talking in PMs with Kavan about doing some peer review between the TCoM and Rawr.Mage, I'd be very interested to also compare against your modeling (once I catch up to the rest of you with non-basic stuff, that is).

If we create a basic set of test cases to compare between, we should (assuming we all have our mechanics straight) come up with similar (not equal) answers.

On the note of more precise theorycraft, last night I was investigating the possibility of treating abilities' damage as a probability distribution and performing repeated convolutions of this probability distribution to look at the distribution after some number of iterations (for example, a number of casts).

The basic idea is so: if you have a random variable X, then the distribution of X + X is the auto-convolution of X's probability density function. That is, for a probability density function f(x), the pdf for X+X is...



In signal processing, this is like taking a reversed waveform and overlaying it over the original function and sliding it back and forth.

For a spell with hit chance h, crit chance c, crit bonus b, average base damage m with maximum base damage m + ∆m and minimum base damage m - ∆m (such that 2∆m is the width of the base damage range), then we might have a probability distribution function that looks like...



Where is the Dirac delta function and is the rectangle function.

Now, taking the convolution of this...rather sucks. We could do this directly and take advantage of the distributivity of the convolution operator, or we could use...(drumroll, anyone?)

...the Fourier transform! The Fourier transform has the property that convolution in the original domain corresponds to pointwise multiplication in the image domain.

As it turns out, the Fourier transform of our pdf is fairly straightforward. Indeed, probability theorists have already collected most of these transforms as "characteristic functions" of a distribution, using the Fourier transform convention...



And thus...



So the convolution theorem says...



This could help us immensely if the transform is nice enough. Indeed, under this convention, we have...



Which, in my mind, is remarkably pretty, considering.

This is when it all falls apart. What we need to do here to consider a distribution of 2 spellcasts is multiply this function by itself and then take the inverse Fourier transform. This...sucks. Indeed, I suspect beyond this point the closed-forms are no longer...all that closed (the inverse transform of sin(x)/x is given in tables only up to (sin(x)/x)^2, which tells me something right there).

At any rate, I present this information the hope that someone smarter than me can make use of it somehow that benefits the community. If we could indeed find a pdf for N spellcasts (or N seconds of time), it would help enormously to quantify the DPS spread of a given spec--how much variation there is between best-case and worst-case performance due to procs or crits, for example. Thus, I will continue working on this problem, even though I'm not sure the results will be amenable.

Muphrid, I can't help but feel that you're going after a fly with a sledgehammer. As satisfying as determining an analytical solution for this may be, my feeling is that the most effective way to go after this problem is by simulation, with Monte Carlo methods. My impression is that the main thing holding back accurate predictions at the moment is uncertainty about the precise nature of game mechanics, including both bugs and features. But even matters that are hard to predict, like ignite-munching say, could be accounted for of a large number of simulations was checked against a significant number of parses and a fudge factor introduced.

Can you tell me the hit cap at level 80 please?



I just had to, it was too tempting
In all honesty, I think you're barking up the wrong tree there, for a two of reasons.


* We can calculate the mean and variance of a single spell cast. We always calculate the mean and never bother calculating the variance though.
That means that we can calculate mean and variance of N spellcasts exactly.

With that, we don't know the exact distrubution of N spellcasts. But we know that it's roughly Gaussian, and we know mean/variance.
What more do we want to know? I don't know really, I really really don't.

I mean, it would be really cool to find the closed form of a sample fight. I'd love that, just for the hell of it, just one sample.
But I fear that when looking at it, the only thing I'd think would be "yay, Gaussian with mean N*E and variance N*V".


* The main issue for increasing randomness is Hot Streak.
That's where your "N independent casts" model breaks down. Convolution uses multiplication of probabilities, which assumes indepence, unless I'm missing something.

The standard model gives a (c*c/(1+c)) chance to proc Hot Streak on any given cast.
That means you get one HS proc after every (1/c*c+1/c) casts on average, or every cast has a 1/(1/c*c+1/c+1) chance to be a HS Pyro.

But that's where we cheated and assumed averages. Also, HS Pyros are faster, so we move from "N casts" to "a variable amount of casts".

For exactly that reason, our "Gaussian with mean N*E and variance N*V" from my first point is pretty useless here as well.
I can't think of anything at all really that can even remotely properly model what we have there.


Long story short - I don't really see how convolutions help us with our main issue of randomness, Hot Streaks.
It becomes even more headache with Combustion.

How about crit trinkets? A trinket with clicky crit buff, and a trinket with a chance on crit proc that gives a crit buff as well?
And how about popping Combustion right at that moment, and fully incorporating Ignite bugs there?


Oh oh oh! That does really really sound like a lot of fun, but I can't see it even remotely modeled.



Oh, we can certainly deal with bugs and with features!

It becomes difficult when bugs are the features though!
Shatter combo? Combustion Combo? Finger Combo? Potency Combo?

What is bug, what is feature, and why? Where should we objectively draw the line?
The problem is not finding out how things work.

It's finding out how things are supposed to work, estimating the time it takes to actually make things behave like intended and gauging whether it's worth to interpret it a creative use of game mechanics for now and make the most of it for now.

And trying to make sense of Torment.
The vegetarian salad with boiled ham I've had definately made more sense than this.


As for Pyro DoT uptime, I use an Excel sheet, like for everything.
It consists of a handful of columns and pretends that it spams Fireballs until HS procs, then ends.
Every line contains which cast it is (this starts at 1 in the first line and increased by 1 every line), the chance that HS hasn't procced yet and the chance that the last cast was a crit and didn't proc HS.

Not sure about the details, but something like that. In every line, the values are calculated from the values of the last 2-3 lines.
For each cast, we can then calculate the chance that the first HS procs at exactly that cast.

That's copy-pasted for 500 lines/casts I think, all further calculations then are pretty elementary.
The average time until the first proc for example.

In each line, I also calculate how many Pyro ticks there were since the start.
Uses average ( there goes our randomness ) cast time, division by 3, capping at 4 and truncating.
Assuming you had a Pyro just before the first cast, which is what that is all about.


For what it's worth, my combustion estimater worked like that too, more or less.

My other calculations are a bit better.
They are basically Excel sheets with a few hundred columns, one for each spell in every relevant spec.
The first column are base stats from gear/buffs/etc. for spell power, crit, haste and such.

Each column then adjust those numbers to talents and calculates average damage, cast time + lag, mana cost and some more stuff.

For Arcane, that means that I have 15 colums of spells. AM, MBAM, ABar, AB with different specs/debuffs, other fillers.
And then some dozen columns more that take these spells to stitch together rotation.
Like ((ABar-FB)*2-ABar-MBAM) with the most favourable procs at 20% chance for example, and others.

Fire/Pyro/Bomb cycles work the same way, I'm stitching together 5 Fireballs, 1 Bomb, and X Pyros.
X being a fractional number depending on the crit rate of Fireball.


I don't think I ever thought much about Fire DoT clipping in Deep Frost. There are other issues with more impact in that spec *cough*
Hope that helps a bit

This is something that I have kind of ignored for now in Rawr.Mage. I only operate at the granularity of spammed/not spammed for dots.

What you would have to do to formulate this accurately is to include remaining dot time as part of your state in formulation of markov chain. To do it exactly I'm afraid this would bloat the state space too much to be able to solve for symbolic solution. If you're willing to sacrifice some granularity I think you could keep it at a manageable size, but I haven't attempted to do it so far. For brain freeze it should be easier since the current state space isn't that big, for the more complex Frostfire Bolt+Scorch+Living Bomb+Pyroblast I'm more sceptical.

I disagree. There are four states a given mechanic has:

1) Tooltip State (I add 10% to fire spell damage!)
2) Researched State (I actually add 10% to everything but Living Bombs DoT)
3) Intended State (???)
4) What we would like

Based on the first two in most cases we can usually divine the intent (Derr, Firepower isn't working correctly), and when we cant (Torment, WTF?), it's not difficult to model for the potential states (We assume that Torment works always, when you have slow available, or never).

Knowing how it actually works is very very important to the process in general, since tooltips alone are not necessarily indicative of what is actually going on, and knowing what bugs/issues exist are very important, otherwise we're theorycrafting the ideal, not the actual. And while in some cases it is worth it to ignore the actual (or too complex not to, IE ignite), in general I feel it is poor decision making not to allow for things to work the way they really do, and the way they appear to be intended to.

In short, I guess, as long as we broadcast the assumptions we're making, and broadcast what problems we're actively ignoring or taking into account, then we've got our bases covered. But the basis of all this is knowing what really is going on.




For the Hotstreak Pyro DoT uptimes...

Markov Chain, gotcha, that's a name I can look up in my Oreilly books.

For Roywyns solution, I see what you're doing, I like it. My issue is that I like things to be dynamic, so I can't assume spamming of fireball. Frost gets a little easier (even though the fireball DoT getting stomped for a frost spec is definitely a trivial matter), until Ice Lance on FoF procs is modeled in.

I really care about this for Hotstreak (Brain Freeze its silly obsession), since as we approach 100% crit rate we approach a base rotation of 3 fireballs to 1 pyroblast, which depending on haste can stomp a good chunk of your DoT, which is non-trivial when speaking about Pyroblast ( Theory Craft-o-Matic , around 5-6% damage difference between spamming and letting the DoT run, so 3/4 of the DoT makes up 5-6% of its overall damage)

Rawr.Mage has reached a point now where its probably the best tool out there for comparing Mage A to Mage B (in terms of spec, gear, etc), but for actual boss fights it still doesn't show anything near what really happens. This is natural and expected of all simulators. Would it be possible in a later build to be able to plug boss fights into rawr that modify its output based on fixed and predictable factors? An easy example is grull - silence every X seconds, shatter every Y second, an upper and lower limit for how often cave-in hits you based on odds of it hitting you. Its easy to model the impact of all of these on a given spec/rotation.

There are still inaccuracies but it would be a great step from comparing gear/specs to showing actual potential performance for a given encounter.

In retrospect, this isn't a mage-specific post.

Rules and Interpretation

Living Bomb DoT damage is an obvious bug. So is the Blizzard-Finger-Frostbite requirement.

I was more talking about things where we can't read the tooltip without interpreting it.
Where "written" "read" "supposed" "intended" and "actual behaviour" are 5 different things.

* Torment. The big elephant in the room. Add that Brutallus and maybe some other bosses are completely immune to Slow.

* Non-damage spells consume Arcane Potency. This does not actually violate the in-game tooltip Arcane Potency - Spell - World of Warcraft of the buff.
It is however completely silly, and it the talent tooltip states that it is for damaging spells only.
Imagine for a moment that the talent tooltip would be the same as the buff tooltip.
Then the current behaviour could be interpreted as working as written, even though it's likely not intended.

* Shatter Combo. Triple Shatter Combo. Finger Combo. Combust Combo. Potency Combo.
You honestly cannot translate those tooltips into intended behaviour without interpreting them or splitting hairs.

For me, those Combos are a real grey area of mechanics.
The only thing that counts for me right now is what you can do and what you can't do.

* Molten Armour procs eating up Finger charges. That solely depends on whether you consider Molten Armour procs as spell casts or not.

* I think I read that Arcane Potency is pretty erratic with Arcane Missiles and Blizzard, just applying the buff to one wave.
It doesn't actually matter how it works, but the tooltips are very much open to interpretation.
In particular, they should take into account that that Clearcasting only procs at the start of those two spells.
Which everyone accepts as working fine, but also is interpretation.

* Drinking 2 potions in one fight by using a speed potion a millisecond before the pull.
Exploit? Creative use of game mechanics? Not exactly game breaking, but it's about as much as you get for 3 points in World in Flames.


Long story short. I consider most of the obvious bugs to be bugs. Including Potency comsumption on non-damage spells, Finger consumption on Molten Armour and Fingers immediately fading if they proc on a cast that happens to be the second cast with Fingers up.

But I consider Torment and most or actually all the Crit buff Combos a grey zone.
There are some more things, but they don't really affect raid DPS much.


Pyroblast DoT

For the Pyroblast DoT and other things: My general approach there is - if can't calculate it, I'll just add another Excel sheet.
Which is why the file contains a lot of sheets now, I don't even want to count them *cough*

For 40% crit and 20% haste, you get 3.4 DoT ticks on average.
For 50% crit and 23% haste you get 3.1 DoT ticks, for 60% crit and 25% haste you get 2.8 DoT ticks.

At 100% crit (Loatheb) you get 2 ticks below 28% haste, 1 tick above that. Just to give you a rough idea.
You're actually on a 2:1 rotation there, not sure why you say that you approach 3:1.


Those calculations aren't linked to the actual DPS calculations though.
If I increase crit by 5%, it considers the increase in HS Procs but not the drop in Pyro ticks.
It just wasn't worth the hassle of finding out how to connect things there, and it's nearly static when you don't vary crit much.

Definately and many many thanks.

Just a little note - Demonic Pact is bugged in Rawr. Currently, it doesn't add spell power at all, but it simply increase all spell damage done by exactly 10%.

How it actually works is like that:
It takes the warlock's spell power, fully buffed with everything except the spell power buff (IDS/FTT/ToW/DP).
That includes, talents, flask, food, other buffs, probably even trinket buffs.
The game then takes 10% of this spell power and applies it as a raid-wide buff for 12s when the warlock's pet crits.

That means you'd need a text box to enter your buffing warlock's spell power or something.
It's not an issue until your Demolocks reach 3080 spell power with ToW though


Level 80 specs in Rawr
I just quickly threw some gear at Rawr, whatever was on top. Made sure to have a CSD meta.
Standard buffs/debuffs like before, 5 minute fight.

1) Mana was plenty. Frost uses no regen, Fire uses 3 gems, Frostfire uses less than 1 gem.
No mage armour, no Evocation, no Mana Potions. They currently use Destruction Potions, because Haste potions are not in yet.
Quick links to the new potions that make sense for mages:
TBC: WotLK: [Potion of Speed] [Potion of Nightmares] [Runic Mana Potion]


2) DPS ranking is pretty much like predicted and like in Simcraft.

- 5761 DPS 20/51/0 Fire Torment*
- 5478 DPS 53/18/0 Arcane Torment* (picks Fireball-Barrage-MBAM cycles for all 53/18/0)
- 5446 DPS 0/53/18 Frostfire
- 5338 DPS 20/51/0 Fire
- 5012 DPS 53/18/0 Arcane Torment, casting Slow
- 4891 DPS 53/18/0 Arcane
- 4698 DPS 18/0/53 Frost Torment*
- 4222 DPS 14/0/53 Frost

Torment* means that the spec assumes Torment working but does not cast Slow.

Rawr does not have the 5% frost damage debuff from Rune of Razorice yet.
That debuff would push Frostfire from 2% to 7% above Fire. And Frost would still be scraping the bottom of the barrel by a long shot.


3) 4/5 Frostfire Garb
Rawr doesn't register when you get 4/5 of that set. Which is good because it uses a wrong formula.
The 4/5 set bonus is worth ~50 DPS, or 1%, or 30 Spell Power. Check the OP for how it works, it's 5% like in Burnout.

I can't remember this being covered before, does FoF trigger the Torment the Weak damage bonus since it makes targets count as Frozen?

Why would it? It's not a "snare".

Skewness and Excess Kurtosis. No really, those two numbers are what else you might actually want to know, and the four of those are the sum absolutely maximal total it is theoretically possibly to give a crap about.

There's a minor problem with modeling damage as a Guassian, which is that the bell curve is symmetrically distributed around its mean and damage is not. A spell either does X damage or crits and does 2X damage (or if you're Murphid). This is assymetric for two reasons: first, lower damage ranges are more common; second, it's not possibly to anti-crit for 0 damage or some such nonsense.
That's why god invented Skewness.
It's some measure of how assymetric a distribution is around its mean. I don't quite understand everything, but like the variance, there are a few related values (true skewness, sample skewness, third moment, and third cumulant) that lets you see how everything is scaling. If you really care, give me a week and I can synthesize a synopsis for you.

Excess Kurtosis is a little more esoteric, but relates to reliability. It's some measure of how close a variable is likely to be to its variance, which gives some notion of how evenly distributed the variable is. Note that this is relative to the variance: the variance denotes how spread out the function is, the kurtosis measures how evenly the function is distributed over that spread. It's hard to understand, but it's not without use.

It turns out mean, variance, skewness, and excess kurtosis completely characterize most types of probability distribution. There's a big-ass famility of functions that have those four values as parameters, and most common distributions (including the Gaussian) are just special cases of those with trivial or default skewness/kurtosis.



This is, of course, mostly pointless since the entire problem comes from Murphid knowing how to do math but not how to use it (no offense). If your continuous probability function has rectangle functions in it, you're doing something wrong. To whit, you are using a continuous function where your random variable is obviously discrete. Crit chance is a discrete function, and crit chance over a fight is exactly modeled by a Binomial distribution. Damage range on a nuke is a uniform distribution and is probably easier to model as continuous. Luckily, the damage range and the crit factor are independent events, and therefore we can calculate them separately and multiply, and the damage ranges behave nicely.

Damage Range:
The damage range on one nuke is uniformly distributed. The damage range on several nukes is the sum of these various random variables. Computing this exactly is a bit onerous, as you get piecewise-defined n-th degree polynomials. Fortunately the sum approaches a gaussian *very* quickly. In fact, many modern computers use the sum of 12 uniform variables as their internal model for the Gaussian. Being a pure mathematician, the suggestion of using an approximation usually makes me shit a brick and beat you with it, so I want you to realize how good of an approximation this must be for me to be gung-ho about it. N casts with a damage range of X gives a Gaussian with variance X^2/12N.

Crit Chance:
A Binomial distribution models the number of successes of N consecutive independent trials with success probability P. Take N as your number of nukes, P as your crit chance, and your crit chance is now a binomial random variable. Mean is NP as expected, variance is NP(1-P). Once you have your average damage range D from above, you roll your Binomial number B and take total damage = D(N+B).

Now, it turns out the binomial variable is also well-suited for approximating with a Gaussian. In fact, the Gaussian bell curve was originally discovered as an approximation for the binomial distribution (special case of the central limit theorem). It doesn't converge quite as fast as the uniform distribution and approximating like that assumes away the skewness so I tend to avoid it, but if you want you can approximate it as such. This means that your total damage is just a product of two gaussians, which is a gaussian.

So for average noncrit nuke damage D, damage range X, crit chance C, and a fight length of N casts, you have a normal distribution G(D, X^2/12N) times a normal distribution G(N+NC, NC(1-C)). Statistics can be run on this, and various derivatives can be computed. D is a function of spell damage, N is a function of haste. Hit can be modeled as another binomial out front, giving a third Gaussian G(H, H(1-H)) (this is implicitly a two-roll system).

If you want me to think about it I can try my hand at stuff like Fingers of Frost. That's more complicated since it's, you know, dependent variables and crap. Functions of variables and whatnot. One random variable serves as the parameter for another, in the naive implimentation.

Well, skewness and kurtosis are just the third and fourth moments of the distribution. There are higher moments as well, are there not (though I understand that most of them beyond the fourth moment are not generally used)? I could imagine that a distribution is only completely characterized by the complete set of moments.

The rectangle functions I used are simply the continuous uniform distribution from the original damage range. Instead of treating hit, crit, and the damage range as three separate independent variables, I condensed them into a single independent variable and treated the cast as a single event, rather than three events.

That said, i can see some value to retaining all the separate probabilities and keeping them intact. It would be easier to consider all the various procs and effects we're dealing with as happening in a multi-dimensional probability space, would it not?

Torment of the Weak does not affect Pyroblast. That said, I would think that a build like this could provide some huge numbers. Scorch a target a few times, maybe stun them, while building a scorch stack...then AP/POM/Pyro.

Massive crit/dam to taste. 51/20/0 Pyro Spec

On the subject of TTW... According to the FAQ Slow is the only snare that will (may) work on Bosses. So is the reason FoF doesn't work is because the Boss is considered Frozen (which is not a snare)?

I keep wondering if the theorycrafting is missing something in regards to Frost Raid DPS. I know it's tough to believe the devs after BC but Ghostcrawler has said many times that Frost Raid DPS (in their testing) is very close to Fire and Frostfire which seems counter to the Sim results (unless by "close" he means within 20%)

It was excessively tested during beta.
My personal thoughts are that Blizzards and our ipinons on what a small (and acceptable) difference in dps is, is vastly different...

Frost has the same issue as Elemental: Bad scaling with spell power, crit, and haste isn't great either.

So in heroic gear in 5-mans it's all fine and dandy.
But in completely top-end maxed-out gear with epic gems (which aren't out yet) and full raid buffs, these scaling issues become apparent.

Play around with it in Rawr a bit, should give you some ideas on it.


Temporary Thread Reopening

I've had this thread reopened for a short amount of time. A few days tops.
One reason is to finalise the OP and make it a proper guideline.
The other reason is to make it easier to quote/copy/salvage things that have been said here into new threads.

The decision is to close mega-threads like these.

Think about what you want transfered into a new threads and how, and collect, quote and copy whatever you need from here.


I would love to see Garak's little synopsis about skewness and see what we can do with Murphrid's modelling as well.
The gains would be to get a better idea how volatile specs are towards randomness and in which way.
I'm expecting a positive skew due to how crits flatten the top end on the distrubutions.

I've searched through the thread for a while now and I have been unable to come to a "decent" answer to the following question....

Q: How valuable is crit rating compared to spell damage when looking at the massive crit multiplier seen when using FFB with ignite, ice shards and burnout? Most people seem to think that spell damage should still be a mage's main focus after +hit, but it just seems like having so many crit talents would put crit at or even slightly over spell dmg. I mean....Ignite + burnout alone made crit rating much more valuable, but once you throw in ice shards, you're seeing an enormous boost to crits and the following ignite damage......not to mention hot streak.

Is spell damage still the better choice for the 0/53/18 build? and if so, how much better point for point?

I played around with Rawr for a bit, and it keeps giving me super crit heavy items as my best potential upgrade for gear slots. Though I'm not sure if it's really accurate or not.

Considering you claimed to have search the whole thread, I'm wondering how you missed something in the FIRST POST!