Prismatic Prison of Pride: Haste Breakpoints

If you’ve had any contact with finding haste breakpoints then you’ve probably seen the healer breakpoint cards at Totemspot which help you find what haste rating to get when going for extra ticks of healing spells.  Eternal Flame functions at a base level of one tick per 3 seconds for 30 seconds.  With haste, the time between ticks decreases as the total duration of the HoT stays the same.  Therefore haste increases the number of ticks.  Sacred Shield functions similarly however the time between ticks is 6 seconds.

The Prismatic Prison of Pride (drops off Sha of Pride) is Siege of Orgrimmar healer specific trinket which increases our haste and mastery rating, critical strike healing and damage, and spirit by a certain percentage depending on the ilvl.  The tooltip displayed is a rounded version of the true multiplier.  Since many reforge programs don’t consider PPP in to their calculations, what you must do is to remove the trinket and calculate the haste needed before the multiplier and reforge to that.

I’m gonna try to make your life more simple and give you a table of the necessary breakpoints with any of PPP version out there.  The massive table below is composed of 6 mini-tables for each un-upgraded version of the trinket (LFR, Flex, Normal, Normal WF, Heroic, Heroic WF).  In each table, there are 3 major columns for un-upgraded (0), upgraded (1) and upgraded (2) versions.

hastebkpt_PPP

Thanks to Totemspot and Keldion‘s work in making this!

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5.4 Siege of Orgrimmar Healing Trinkets Overview

Few words on PTR changes since there hasn’t really been any news…

5.4 PTR Release – August 27, 2013 (Source)

Confirmed Patch Release – September 10th, 2013

The period of mechanics changes to Holy Paladins seem to be over as Blizzard gets into more of the scaling changes period.  You can check out all of the 5.4 Holy Paladin changes here.  On the whole, holy paladins are looking fairly decent in 10-man situations but weaker in 25-man.  We’ll see what kind of scaling changes Blizzard will roll out to balance the healing classes in that scenario.  It seems like the patch is just on the horizon.  With the Arena season coming to a close on August 27th, the release of the new patch should be some time close to this, either on the same day or a week or two afterwards.

I’m pretty excited about this new raid involving Garrosh and this monster or hatred and fear that has brewed up inside him.  It’ll be really interesting to see how the storyline plays out in relation to the old world and some of his loyal soldiers, namely Nazgrim who is near and dear to our own hearts as the first NPC we interact with around almost all of level 85 in the Jade Forest, as well as the friendly and chubby pandas we’ve come to know well in Pandaria.  The first few raids MSV, HoF, and ToES were visually to me mediocre and the storyline behind them was not as strong considering our brand new introduction to the land of Pandaria.  The Throne of Thunder raid was in my opinion a lot better visually with some more interesting fights but the story line is still a bit removed from the standard World of Warcraft storyline.  Now that we’ve become more and more familiar with the mantid and Mogu story lines along with the Sha creatures, it’ll be really nice to see them meld in with characters we’ve known for many expansions.

I’m less impressed with the healing changes to Holy Paladins.  Understandably the class needed adjustments but in my opinion, it would’ve been better to put the changes off till an expansion when there’s more time to talk over and test changes and just roll the coefficient back on Mastery for 5.4.  No one in the holy paladin community debated the fact that EF was a crutch since it was the more efficient spell in 99.9% of situations and at least for me I’m not arguing to get IH back because I think it’s OK to have.  But apart from the developers I hardly see any holy paladin happy with the way things currently are.  We need to be left with something that is still engaging to play.  It’s fine to say that HoTs or absorbs weren’t a design consideration for holy paladins.  It’s not OK to leave the class in the dark about what their role is in a raid because we need to know our role, hell the whole community needs to know our strength as a class to bring us over other healing classes which have clear defined roles. Regardless of my disappointment, it won’t affect my decision to play a Holy Paladin and I’m hoping to explore some of the new things the class offers which I know will be there.

On the state of lvl 45 talents, EF still appears to be the go-to talent (even after weeks and weeks of PTR changes which I find amusing) for 25-man and possibly 10-man.  In 10-man SS may still be a viable talent as the absorb covers more of the raid.  People are still dancing around SH since the nerf to the mana efficiency of the talent build.  With tuning though, this priority could change around a great deal.

Stop. Trinket Time.

One thing I’ve been updating in the 5.4 PTR Patch Notes page is the difference trinkets available as well as the legendary healing cloak.  My goal here is to give you an overview of what the 4 new trinkets of the SoO raid are going to provide for you and some predictions as to what will be the best choice for Holy Paladins as well as get you off on a good start once the patch hits.

Legendary Cloak

For those of you unlucky individuals only able to get Titan Runestones off bosses with 100% chance to drop them, better hurry up and get yours soon because very quickly after 5.4 hits, you will be able to obtain an upgraded version of the Special Cloak from the quest line you’re currently able to obtain (ilvl 600) called Jina-Kang, Kindness of Chi-Ji. This cloak provides no additional stats compared to the one currently obtainable but comes with an additional Equip effect.  For healers it is the chance of a periodic buff called “Spirit of Chi-ji“.

  • Spirit of Chi-ji: Your healing spells have a chance to grant you Spirit of Chi-Ji, increasing all healing done by 5% and causing all overhealing to be redistributed to 5 nearby injured friends, for 10 sec. (0.54 PPM)

How to get the legendary cloak:

  1. You must have completed up to the quest to get your ilvl 600 cloak.
  2. Go meet up with your buddy ol’ pal Wrathion at the Tavern of the Mists in the Veiled Stair (his usual hang out place with his friend) (Quest)
  3. Wrathion’s like meet me at the Timeless Isle yo. Now on the PTR I was able to access this place by going to the Mogu’shan Palace entrance and talking to an NPC there but you can optionally also fly to it.  It’s located to the southest of the Jade Forest. (Quest)
  4. At the Timeless Isle, you will pick up two quests Secrets of the Timeless Isle and The Emperor’s Way.
    • Secrets of the Timeless Island is a quest to obtain 5000 Timeless Coins.  Timeless Coins are obtained through several things including doing dailies, killing mobs, treasure chests, the typical.  There are many articles out there about obtaining these.  The general consensus is that these are not hard to obtain so getting 5000 of them should be doable within 1 day.  (Quest)
    • The Emperor’s Way is a quest to defeat the 4 celestial legends in combat. This requires a full 25-man raid group to do.  Here is Wowhead’s amazing in-depth article about the Timeless Isle and includes a section describing these 4 bosses. (Quest)
  5. After turning these two in, you will meet Wrathion at the Seat of Knowledge which is basically the step in front of Mogu’shan Palace in the Vale.  You will then receive a cool item to use on your cloak.  Depending on what your cloak is, the item will upgrade that version of it.  (Quest)
  6. Finally, he will tell you to kill someone.

The cloak prioritizes healing people who are more injured.  I hear currently that the cloak is extremely overpowered for restoration shamans.  In Testing, the trinket contributes around 5-10% depending on the overhealing and the type of fight for holy paladins from the information I’ve thus far seen so it’s a fairly nice buff.

Siege of Orgrimmar Healing Trinkets

Four healing trinkets drop from the SoO raid, much like ToT.  These drop from Sha of Pride (4), General Nazgrim (8), Thok the Blood Thirsty (11), and Siegecrafter Blackfuse (12). Remember there are 14 bosses in SoO.  For all the trinkets, there will be 6 versions.

  • LFR (528)
  • Flexible Raid (540)
  • Normal (553)
  • Normal Warforged (559)
  • Heroic (566)
  • Heroic Warforged (572)

Item upgrades I believe will still be available so your highest possible ilvl gear will be 580.  Warforged is the same thing as Thunderforged.  Higher chance to drop in 25-man than 10-man and 6 levels above either the normal or heroic equivalent.  I’ll go in to the trinkets in a bit more detail.  It’s important to remember that these might change any time between now and August 27th and also what kind of healing spells the trinkets proc off of (direct healing or any type of HoT + direct healing) will affect their usefulness.

Contemplation of Chi-ji (Timeless Isle, Vendor Purchase): Increases your Spirit by 8281 for 15 sec. (1 Min, 30 Sec Cooldown)

  • 14020/1.5 Minutes (535), 14552/1.5 Minutes (539), 15104/1.5 Minutes ( 543)

This trinket is obtained through Timeless Isle items.  Just wanted to mention it so people were aware of it in case they wanted to go for a catch-up item.

Thok’s Acid-Grooved Tooth (SoO Drop, Thok the Bloodthirsty)
Equip: Your heals have a chance to Cleave, dealing the same healing to 5 nearby targets. Chance to do cleave healing based on item level.
Equip: Each time your spells heal you have a chance to gain Intellect for 20 sec. 115s internal cooldown. 15% chance for intellect proc regardless of item level, intellect proc value dependent on item level.

Item Level Chance to Cleave Intellect Buff
LFR (528) 2.46 9317
Flexible Raid (540) 2.75 10420
Normal (553) 3.11 11761
Normal Warforged (559) 3.29 12436
Heroic (566) 3.51 13274
Heroic Warforged (572) 3.71 14039

There’s not too much to say about this trinket in terms of its complexity.  Essentially, the spell will cause a certain percentage of your heals to heal nearby 5 targets for the same amount of healing. I’m not sure about the range on the trinket so if it is something standard like 10 or 8 yards, then it may not be as good for non-stack fights or for 10-man raids.  I believe this spell can only be activated by beneficial casts and so it won’t proc from HoTs.  I believe that currently the trinket is really buggy so it’s hard to test.  I’ve got my eye on this one though until I can confirm more details about it.  It seems like a decent trinket however due to its RNG nature, I’m not sure how much of it will go to overheal.  Granted overhealing with it would be beneficial with the legendary cloak. (Have I mentioned how ridiculous I think it is to have overhealing actually be more beneficial?)

Prismatic Prison of Pride (SoO Drop, Sha of Pride)
Amplifies your Critical Strike damage and healing, Haste, Mastery, and Spirit by a percentage.
Each time your spells heal you have a chance to gain Intellect for 20 sec. 115s internal cooldown. 15% chance for intellect proc regardless of item level. Effective for healer specializations only.

Item Level Amplify Percentage Intellect Buff
LFR (528) 6% 9317
Flexible Raid (540) 6% 10420
Normal (553) 7% 11761
Normal Warforged (559) 7% 12436
Heroic (566) 8% 13274
Heroic Warforged (572) 8% 14039

Similar intellect proc to some of the other trinkets with a base increase in stats.  I think this trinket is decent but it remains to be determined whether it’s better than the others.  an 8% increase on the heroic version for example would only translate to around a 5.6% increase in mastery for example or for with 15,000 spirit around 1,200 spirit.  My gut feeling is that this type of trinket is much better for DPS classes who have extremely high secondary stat priorities.  It could also be better for high item levels as the percentage increase will amplify a large value.  I especially don’t see much use in the trinket if spirit is not an issue for the fight as the spirit part of the trinket was just recently added on.

Dysmorphic Samophlange of Discontinuity (SoO Drop, Siegecrafter Blackfuse):
+ Static Intellect
Your heals have a chance to grant you Spirit for 10 sec. Every 0.5 sec, this effect is reduced by a portion. 0.92 PPM regardless of ilvl.

Item Level Static Intellect Beginning Spirit Spirit Decrement Every 0.5s Spirit to Mana Conversion Per Proc
LFR (528) 1552 18640 932 11022
Flexible Raid (540) 1735 20840 1042 12325
Normal (553) 1959 23520 1176 13925
Normal Warforged (559) 2072 24880 1244 14719
Heroic (566) 2211 26540 1327 15715
Heroic Warforged (572) 2339 28080 1404 16616

The last column in the table above shows the total amount of mana gained back for each proc.  The PPM of this trinket dictates that it procs approximately once every minute.  This is effectively a mana return trinket with a fancy mechanic.  It also benefits classes which stack more haste which Paladins are not turning out to do very much despite the sanctity of battle change.  However, even for a once per minute proc, on just a 8 minute fight this trinket will return 80,000 mana.  It may be that using this trinket would allow us to go more of our secondary stats (such as haste) and be beneficial that way.

Nazgrim’s Burnished Insignia (SoO Drop, Nazgrim)
Your heals have a chance to trigger Multistrike, which causes instant additional healing to your target equal to 33% of the original healing done.
Your helpful spells have a chance to grant intellect for 10 sec. 0.92 PPM for Intellect proc regardless of item level.  Chance to trigger changes based on item level.

Item Level Multistrike Chance
LFR (528) 11.1%
Flexible Raid (540) 12.4%
Normal (553) 14.0%
Normal Warforged (559) 14.8%
Heroic (566) 15.8%
Heroic Warforged (572) 16.7%

Apart from the intellect buff which we’ve already discussed, this trinket provides a chance to trigger an additional 33% heal off any heal you perform.  I’m wary of trinkets like this because a large part of our heals are already going to overhealing so I suspect that this trinket may result in mostly overhealing.  The chance to multistrike however is quite large so the sheer amount of procs may cause this trinket to pull ahead of the others.  Also, the legendary cloak mechanic will help with the overhealing.

What 5.4 trinkets are Holy Paladins using?

Based on my guesses which may be as wrong as they are right, I think the interplay with the trinkets and the overhealing buff will make either the cleave or multistrike trinket at least one of our best.  My gut feeling is the multistrike trinket and the cleave trinket are pretty close for 10-man groups with perhaps the cleave trinket pulling ahead a bit more on stacked fights and much farther ahead for 25-man.  I think the spirit trinket is a must have for progression with its static intellect buff and the spirit proc returning quite a lot of spirit.  It should be a decent competitor if not better than Horridon’s Last Gasp due to its reliability.

Revisiting the Golden Days: RPPM Mechanics Changes

Four months ago I started this blog at the onset of 5.2 with the introduction of the RPPM mechanic and the new healing trinkets.  It was a whole lot of fun going through and digesting each of the new trinkets and the in-depth analysis of RPPM mechanics that I found and I plan to do something similar when the new trinkets for 5.4 are finalized.

Just yesterday, Lore posted about changing the on proc percentages at the start of fights.  If you recall, Blizzard implemented an alteration improving on how the trinkets would proc after not proccing for a long time.  In short, your chance to proc on each attack would increase the longer you went without a proc.  This made sense.  But in short what happened was between pulls you would go with a 2 or 3 minute break in between and sometimes more and almost every single RPPM item in the game had at least a 0.5 RPPM (roughly 1 proc in 2 minutes) and right at the first hit, all of your trinkets would proc leading to frenzied DPS chaos at the beginning of fights.  This made people who perhaps didn’t have RPPM mechanic items or who capitalized on specific procs by changing their whole gearing strategy unhappy.  Raids were essentially forced to plan Bloodlust and strategies around it wait a certain time before pulling to insure trinket procs, etc.  All in all it was affecting how happy people were.

So they changed it. The new change to the trinket is that at the start of the pull (I’m assuming upon entering combat) the “time since last proc” variable is automatically set to 90 seconds no matter how long you’ve actually waiting.  This of course will affect trinkets differently depending on their RPPM.  Following is a list of items affected by RPPM proc mechanics, and their chance to proc on pull now compared to how long you would have to wait currently to get a 100% proc change on pull.

rppm_5.4changes

In effect, the new change to trinket procs did not actually buff the proc change of any classes who were complaining that they didn’t get an on pull proc but rather nerfed only a few of the classes who were more likely to get a proc (yellow times above).  For example, classes who use Rune of Reorigination with a wait time of about 3 minutes for a 100% chance to proc on pull now will be left with only a 1/5 of a chance.  These are the classes with a mediocre waiting time before 100% pull now have their proc on pull %s drastically reduced.  As you can imagine from numbers, the classes impacted most were the classes wearing items with a 100% proc on pull time around 1-2 minutes from 90 seconds.  Where previously, they could wait around for 2-3 minutes or how ever long it takes your raid to wipe and pull again and you can pretty much guarentee a proc on pull, now for these classes this is no longer the class.  An example of this above would be someone with Gaze of the Twins and 30% haste for example.  Your chance to proc on pull is quite low.

I realize all of this doesn’t affect healers so much but it’s still interesting to see what kind of effect the changes brought about and is a nice change from the endless whining that dominated my last few post.

And since no post can be complete with some whining, I’d just like to say the recent comments by all the Devs have left me in the dark.  I just said a few posts ago how I believed that they knew what the problem was and addressing the changes with the right solution was the problem but now I don’t even know if they realize what the problem is.  The whole PTR has felt like I’ve been testing out and analyzing a class completely different from what the devs are testing.  I’m not completely disheartened by the changes nor would I change classes or quit the game just because my class does poorly, but when someone comes out and says that the class is fine, we’re happy with it, it feels like I’ve been slapped in the temples.  There are so many examples of what kind of reduction in healing the nerf to EF will be.  I made a whole post on it with a farm log from real data a while ago.  No simulation bullshit.  No random fight mechanics skewing the data.  Primordius is a fairly typical tank and spank fight with increasing raid and tank damage.  5% reduction?

A few works on the judgement nerf.  The increase of 5% to 12% base mana for Judgement is a change of 4200 mana.  Considering a typical 6 minute fight and assuming a 5 second Judgement and 80% Judgement usage efficiency, you’ll use approximately 58 judgements.  The nerf brings a difference of 241,920 mana.  It’s no laughing matter and with more efficient Judgement usage, the nerf is actually more damaging.  Now the issue is that with the SH build, a heavy haste/mastery gearing strategy is practically necessary to keep it up and competitive with the playstyle.  With too much haste, we don’t heal enough with HR and our other general spells. With too much mastery we don’t get as many uses out of Judgement and that equates to less HPS from less buffed HR.  Spirit was not necessary because generating 3 HP required much lower mana due to the cost of Judgement.  The SH build as has been stated is less a HP generation increase and more a mana regen buff as I’ve stated before.  The nerfed Judgement will now even remove that aspect from the SH build.  Please don’t let this change go live.

BUT, bright side is at least they seem to be willing to see things differently so lets not get discouraged and keep trying to bring the right issues to light.  I’m sure if things were to go live today, we would find a way to make things work or be in such a terrible place that we’ll have to get a hotfix sometime in to the patch.  🙂

Lightning-Imbued Chalice, Revisited

On March 29th, Blizzard decided to buff the proc chance of the Lightning-Imbued Chalice, a healing trinket that drops from Lei Shen. Just as a reminder, this is what version 1 of the trinket did.

  • Static intellect: 1218/1467/1552/1657/1728*
  • Smart heal proc: 46561/56071/59207*/63294/66816*
  • 3 RPPM (real procs per minute) (here’s an explanation of what this means) (http://us.battle.net/wow/en/forum/topic/6893549789)
  • Procs off direct or periodic healing spell
  • Healing buff called “Restoration of the Infinite”, accumulated charges called “Infinite Power”

Recall earlier in the month around March 13th, they buffed the proc chance on all of the trinkets together once. Therefore this trinket has been essentially buffed twice since we looked at it previously.  Lets take a look at what the changes resulted in.

Here are some basic statistics about the logs I gathered:

table1

Using the WoL import mentioned in one of my previous posts, I imported all of the information.  All logs are in English and are full kill logs.  I realized that including wipe logs would incorporate much more information, but in wipes, weird things happen especially towards the end and I can not quickly distinguish when the raid has given up so I left those out.  A caveat of the analysis is that all of these logs are from around 5-6 individuals therefore there may be some bias in play style.

Our first step is to verify that our data makes sense.  What I mean by this is longer fights for example, we would expect on average more healing done, more direct heals, etc. Lets check some of these points first.

1

The plot above shows the comparison between healing done and fight length.  We would expect this plot to have roughly a positive slope which you can see here is true.  This plot is interesting on several levels. Notice that the points are clustered together roughly according to group for some fights (ex. Horridon (H)). For every heroic fight, if you’re a DPS you can calculate on average how much DPS you will need to do to kill the boss before enrage.  It is kind of interesting that you can tell from this boss around what healing you will need to do in terms of HPS to keep people alive through a boss.  This number is of course extremely variable due to healing composition but the here is undeniable.

Here is a type of image called a dendrogram.  This type of analysis which is a simple unsupervised hierarchical clustering is designed to identify similarities between two things.  Imagine if you had a group of balls, some of which are red and some of which are blue.  It would be very easy to distinguish these into two groups and in fact since you have no other information about them such as their size or any patterns they would separate into exactly two groups.  But now suppose you had other information such as size and perhaps some balls have different patterns; to group these into similar groups would be a much more difficult task.  What hierarchical clustering does is takes input, in our example it would be ball size, color, and pattern and identifies grouping based on these multiple input categories numerically.  You can read more about the math behind it here but basically in a sense, the same action I’ve just describe to you can be achieved mathematically by the use of matrices.  For example, we could denote different colors and patterns by values such as 1 for red and 2 for blue.  Ball size is already a numerical value.  Mathematically we can find the similarity between observations for one ball compared to another by metrics such as the ones stated on the Euclidean page.  By doing this recursively we can find which balls are more similar to each other numerically and end up with a robust system of dividing the groups.

3

So what is this picture saying?  This tree denotes a similarity structure where the longer the arm of the tree, the more different the two fights are from each other.  Remember this is based on two factors, the healing done and the fight length. If we for example split this picture in to four groups, we notice the top group contains a lot of Dark Animus fights.  This indicates that a majority of the Dark Animus fights have very similar fight lengths and healing done.  We also notice in the second group, there is a majority of the Horridon (H) fights in the top half of the group.  These are two groups where the two input variables were very similar among different logs.  This could indicate many things.  Perhaps this fight is relatively easy and everyone doesn’t need to heal a lot and therefore, difference in play style or raid comp don’t make a huge difference on healing.  It’s also possible that these fights have very strict healing requirements and thus most successes in killing the boss have similar values.

Now lets look at a fight which doesn’t cluster well such as the Tortos fight.  In my experience, Tortos is a very different fight depending on healing composition. Some ToT fights are very suited towards certain classes such as those primarily for AoE healing or single target healing.  A paldin healing with a Monk and Shaman might not do as well as a paladin healing with a Disc priest.

Lets move on to looking at direct heals + HoTs since these are the spells which have the chance to trigger charges of Infinite Power.  We would expect that a plot of this value versus fight length would be roughly similar to the one above.

2

The plot above shows the fight length by the Direct Heals + HoT ticks value.  Recall that the LIC procs either on direct or HoT ticks.  The purpose of this plot was to show that as the fight length went on longer, you would naturally have more direct heals and HoT values, regardless of how much healing was done.  We would expect this plot to have a roughly positive slope and we do see that shape here.  Once again notice how the fights are roughly grouped together indicating possibly that play style for these fights are all very similar.

We can estimate from each person’s logs their spell power very roughly.  I was recently lucky enough to become the owner of the Thunderforged Hydra-spawn trinket.  I am positive the absorb from this trinket is not affected by any holy paladin base talents such a Seal of Insight (1.05) or Holy Insight (1.25).  Therefore, I can make an educated assumption that it does not affect the healing from the lightning-imbued chalice as well.  This means that fights such as Jin’rokh (H) which give a healing bonus will not affect our calculation of spell power.

The LIC’s heal component is 56071 + 125% of SP for the 522 normal version.  This value changes very little between the normal and normal thunderforged version which gives a base healing amoutn of 56286 and maintains the same SP coefficient but goes up drastically in the heroic and heroic thunderforged versions. Using this knowledge, and the crit coefficient of 2x because the heal from this trinket can crit, we can estimate the average SP.

4

Looking at the plot, you can see that a majority of the SP values are around 40,000 which I think is pretty much impossible even with Jade Spirit, Intellect Flask up continually.  If you look at the outlying values which I plotted the fights specifically on to because the coloring was too close, they are both related to the Jin’rokh heroic fight.  Thus we know that the healing from this trinket is affected by external fight mechanics.  Is it possible that I was incorrect in assuming this trinket is not affected by holy paladin buffs like the Hydra-spawn trinket?

I replotted the values including both the innate holy paladin buffs and the values are a lot more realistic.  The important thing from this plot we see is that the SP does not scale based on fight length which is what we expect.

I therefore have reasonable evidence to believe that this trinket is affected by both the Seal of Insight and Holy Insight buffs as well as any external boss mechanic healing buffs. I can also say that the general trend of our data from looking at these points is roughly similar to what we would expect in the typical raid situation.

Buffed PPM Proc Rate ~6.4

The first time I looked at this trinket, I made a mistake and took the proc chance from the number of procs of the actual heal and forgetting that the heal takes 6 charges to build up.  I calculated the average proc for the heal to be around 105 seconds meaning that there was on average a proc ever 17.5 seconds.  The original PPM value was therefore very close to the true value of 3 PPM.  Let us not make the same mistake and look at the proc rate in the new data.

5

The plot above shows the total procs by fight length.  The red points represent the buffed trinket and the blue points represent the original version.  I have a fitted a linear model to the points because I expect the number of procs to go up linearly with fight length.  By using R’s lm function which essentially takes a set of points and tries to find a linear line which will best describe the set of points.  It does this by plotting a large number of lines with different plots and finding out how much error, or how far away the actual points are to the line.  The sum of these errors which is the least is the line of best fit and R will return the slope of it.

The slope of the original trinket is 0.058208.  The units of this value is procs per second.  Therefore this trinket comes out to be around 3.49 RPPM.  The slope of the new trinket is 0.107103 procs/second or ~6.42 RPPM.  Essentially they buffed this trinket by almost two times so that in one minute it is very likely for the actually heal to go off once.

Notice the two red points close to the blue group.  I suspected that these were actually non-buffed trinkets.  Since the hotfix came in relation to the American time zone, I’m not sure whether the trinkets were buffed even though the log was recorded on the 29th.  I went back and checked the log dates and the dates for these two were clearly after the 29th.  However I noticed that in one log, the person survived only 70% through the fight and the other was around 56%.  I suspect this is why these values are outliers.

Haste

I originally had tried to pull out information from the expression browser however that page is written with a javascript instead of HTML.  Since I didn’t record the haste, I had wanted to calculate it from looking at Holy Light cases and deducing an approximate haste value from the time it took for the cast to finish.  However I’m not familiar enough with HTML to extract a javascript source code so I went through and did it manually after making the URL which would find the page.  I was interested in haste because from our previous analysis, it appeared to scale the best with PPM.  However, the logs also have some lag which is different between different logs so I realized that this is probably not a viable method of calculating haste as there would be a lot of error in the value.

I feel like the only way to make a reasonable analysis of haste is for me to test it out on the trinket I came into possession of so we’ll hold off on this discussion for now.

Direct Heals + HoTs scales with number of Procs

6

Here we see a comparison of procs (of Infinite Power) and direct heals and HoTs.  From the description of the trinket, we expect this relationship to be positie.  Here we can see that relatively this is true.  Up to around 4000 DH+HoT, the value of Procs is consistently increase.  After a certain number however, the number of procs become extremely varied.  If you look at the number of DH+HoT for people who received 60 procs, the number rangse between 2000 and 6000.  Perhaps this haas something to do with the fight?

7

We see a slight grouping of the type of fight.  For example, Horridon (H) points are once again grouped together.  The relationship is however not extremely clear.  We do notice that Megera (H) fights have a lot of DH+HoT but on average, less procs than other fights where paladins used the same number of DH+HoT.

8

I therefore conjectured that the fight might have ore to do with fight length after a certain point. First off, I want to point out something quite striking, we see essentially the same rainbow pattern consecutively from left to right indicating number of procs is very closely associated with fight length.  From a vertical point of view however, it is almost like the number of procs you get is unrelated to the number of heals you do.  Therefore, any class who has a higher active time from HoTs or something such as paladins or druids will not benefit from this trinket over classes which have lower active times such as priests. This is of course expected since the healing from this trinket is unaffected by how many times you cast.  The equation could easily have told you that.

Healing Done from Trinket is ~1% Greater than Unbuffed Version

table2

When reading the next second, remember the number of data points in each set and just keep this in mind.

9

This is a plot of the percent of total healing taken by LIC by item level and version.  As you can see the previous version of the trinket was healing under 1% for the normal version and around 1.5% for the thunderforged version.  The difference between the normal and normal thunderforged version is actually quite large.  While this the difference between normal and normal thunderforged decreases, we can see that in the last two for the buffed version, t he average healing is around the 2-2.5% range.  This is approximately 1% better than the older normal version.  The thunderforged version of the buffed trinket has a shorter bottom quantile meaning while the average of this trinket versus the normal version of the buffed trinket is similar, the chance for a higher % of overall healing is a lot higher with the thunderforged version whereas the normal buffed version even reaches down below 1%.  Even though the buffed version of the normal trinket has the most datapoints, it has the longest tails.  This means that the variability of this trinket is still extremely large.  While the old normal version only had 7 samples, you see that the range of values is surprisingly small.

Summary

The buff to this trinket is quite noticeable.  The trinket is proccing a lot more and thus healing a lot more than the old version.  There is no denying it’s an improvement, but the source of argument is whether the buff was enough.  Let’s put things in perspective.  Our spell daybreak heals approximated 1-4% each fight.  This is a paladin spell.  We are talking about a trinket which provides on-par healing with a paladin spell.  I don’t think there should be any argument about how good this is if it on average can provide an addition 2.5% healing.

However…this trinket still has two problems.  The first is the overhealing. As expected we see an extremely positive trend in the data when comparing healing vs procs.

11

Look however at this plot for overhealing vs healing.

10

This plot shows the effective healing from the trinket by the over healing percentage.  This kind of negative trend strongly indicates that the decrease in healing from this trinket for some people was mainly due to overhealing.  Of course we expect to see a trend because the more effective healing you do, the more overhealing you do.  What would be ideal would be if the overhealing % was relatively independent of effective healing.

The second problem I saw was that for the new version of the trinket, the number of procs is highly variable.  Some people using the trinket reported numbers around 1.5% which from our data is perfectly viable.  But this value is not much different from what the old version was giving.  Of course people are going to complain that the new version didn’t buff anything.

Therefore, what I think will solve this (and I’ve said this before) is if we decrease the SP scaling or the base heal amount AND make the number of procs more consistent.  I realize the model for RPPM is fairly dynamic and dependent on play style but I would argue for a method to make the value less variable or dependent on so many variables.  With more consistent procs and less healing per proc, I think the overall amount of healing would increase.

Ultimately I would use this trinket if it dropped for me.  Despite my frusturations at it, I feel like the trinket is better than say Relic of Chi-ji.  As people gear up, we are already noticing the trend towards using less spirit simply because the fights are not as mana intensive.  I personally think sitting around 15-16K spirit unbuffed will last you through most of the heroic fights.  The main problem now, is not that we’re going OOM but during damage heavy damage phases, we won’t be able to keep people up because our heals are not strong enough.  Therefore this trinket with its static INT would be the go to choice with Horridon’s Last Gasp as the preferential spirit trinket.  I suspect in the following days, we will see some buff to the Stolen Relic.  That one is currently and was before the worst performing trinket.  We’ll have to se what they do to it because unlike this trinket and the Hydra-spawn bag, the trinket is an on use item and if changed properly could end up being stronger than both LIC and the IHS.

In-depth RPPM Analysis Part 4 (and last!)

Beginning from where we left off last time:

Single Proc Trinket, no ICD
Recall this equation for the chance of n events in time t:

10

Recall that the lambda term as derived in the paper comes from this term representing the total number of events over a time period [a,b]:

3_3

By plugging in a with 0 and b with t, we will get the number of events in a period of time [0,t].  Dividing this value by the time period will get the rate of events, the lambda-t term in the first equation.  This is because rate is defined as the number of occurrences over time. This term can then be plugged into the lambda terms in the first equation.

To find the uptime of a trinket, first we need find the average time to proc by multiplying the above formula by t to find the average number of procs and integrating the time from 0 to infinity to find the overall average time to proc. Very little explanation is given for the reasoning of this equation. In a sense, the equation above representing the chance of n events in timeframe [0,t] is multiplied by t.

4_1

In the formula above, to represent the chance of 1 event, n is taken to be 1.  Thus we multiply the chance of one event in time t, by t.   This is approximated as the average time between procs given a certain time period, t.  By integrating the expression over infinity, we sum up all possibilities for different time periods.  I’m not quite sure where the average part comes in to all of this as I don’t see a normalizing term to number of events or something similar to that.

Assumptions (which are not mentioned) are made to arrive at the following equation after plugging in the equation P above into the equation for t.  I do not own Mathematica, but if you do, you can plug the code given into it and evaluate the result to see exactly what assumptions were made.

4_2

The uptime given the duration of the proc will be defined then as followed:

4_3

It is clear where uptime is calculated this way as the bottom term represents the average time between single procs and the term on the top represents the duration of the proc.  In terms of healing trinkets however for example for Horridon’s Last Gasp with a stacking debuff the uptime will be calculated differently.

 Single Proc Trinket, with ICD
This equation is exactly the same as above, the only difference is the times are adjusted for the duration of the ICD and changing any probabilities for procs in the ICD duration to be zero.  For example, the first two equations (which are actually the same with one generalized so the time factor is not negative which is impossible) adds a case where if the time period is under ICD, the probability of a proc is zero.

The second equation which is the average time to proc rather than integrating from 0 to infinity, is now integrated from the ICD to infinity. This returns a very complicated variation of the above equation for the time between procs.  The t(ICD) term is the same equation for t with the integral above, taking the ICD to be the time value at the beginning.

Further variations of this formula can be changed around to find the average time between procs for different situations.  I won’t go into them here but the following are calculated:

  • Time to first proc give a time delay factor (time since last proc) less than limit of the time in the max equation required for the largest term to go above 1
    2
  • Time to first proc give a time delay factor greater than the limit, note that the time must go from the beginning of the time delay factor to infinity in calculating the time between procs

The last equation seeks to find the time to first proc which makes the equation below equal to 1 (one event must occur).

3_3

To do this we simply change the time limits, a to the time delay (last time since proc) and b to time at first proc).

4_4

The term in the integral comes from the second point of this old equation:

3

By solving for the time at first proc the following equation is derived:

4_5

By plugging in the time delay (typically the time since last wipe) we can calculated to an approximation how much time into the fight until your first proc will happen.

I hope these analyses have at least been an enjoyable read for everyone and not too confusing.  Essentially all distributions in mathematics can be approximated with certain functions, in this case exponential functions.  What I hope to do after this is to use the derive equations and put in known values for the healing trinkets to see how close the approximation is to the real values.  Speaking of real values, I’ve been gathering a database of world of logs posts for people who have used the healing trinkets in fights.  I plan to summarize what hopefully will be a larger data set than the initial analysis and see where they are at.  I learned a lot about what to look for in analyzing the trinket logs and my new analysis should be a lot cleaner and answer exactly the questions proposed.

Also, I hope to talk about and simulate some common problems or questions that always go unanswered, including topics such as EF blanketing and when to switch tier 14 to tier 15.

In-depth RPPM Analysis Part 3

I’ll skip describing the proc on pull portion because it’s not extremely relevant but in a gist, if you recall the equation for the chance to proc equation where t is the last time since a proc:

6_3

By setting this equation to 1 (definite chance to proc) and knowing your RPPM and Haste, you can calculate t so that you are guaranteed a proc on pull.

The next equation approximates the maximum time before a proc during combat. This is different from the above approximation because the change in time term is no longer 10 seconds (simulating no attacks) and you are not scaling the approximation by any kind of attack factor.  The 10 seconds is the maximum time which was provided by Blizzard to count between attacks meaning even if you were not in combat, the time between attacks would max at 10 seconds in their calculations.

In this equation, we assume similarly that tmax, the maximum possible theoretical time before the RPPM mechanics has 100% chance to proc is greater than 3/2 * lambda.  This value was not pulled out of a hat.  Recall the first part of this analysis where I talked about what time would be necessary for the penalty factor, the one which accounts for the last time the mechanic procced, would be above 1.

2

For this value to be above 1, t must be greater than the value 3/2*lambda.  This is a valid assumption since otherwise, the maximum value in the above max() expression would be 1 and nothing would change with the proc chance.  We’re assuming there’s a reason for the penalty factor to be implemented in that the maximum time before a proc is going to cause the second penalty term above to increase greater than 1.

With the same reasoning as the proc on pull mechanic, the following equation is derived:

1_3

I was confused about the reasoning behind this equation.  Since they want the chance to proc to be 100%, I thought the second part of the equation above would already be one.  Furthermore, I don’t see any kind of summarization term which combines all attacks.  To make an educated guess at the reasoning behind this equation, based on the attacks per second, you want to find the probability of the proc occurring with 100% certainty.  You must include the ‘attacks per second’ term because your 100% certainty must include this scaling factor based on attack speed.

By plugging our equations into the above expression (we have everything in these equations from parts 1 and 2) we find that the maximum time before a proc is proportional to:

2_3

The author demonstrates that the RPPM constant has limits (can not go below a certain amount) for there to be a 100% chance to proc.  In other words, if the RPPM constant was under 0.019, you might never get a single proc. This is calculated by setting t (time since last proc) to the Blizzard stated limit of 1000 seconds and finding the proc rate lambda (Blizzard’s equation).

Average Proc Rates (Single Proc)

The author here defines some initial terms.

3_3

This term represents the total number of events over a period a-b.  This essentially integrates the function for proc rate based on the time since last proc.  Integrating the function allows you to find the area under the curve, the sum of a number of different proc chances.  It is entirely possible that this value will not add to 1, but if it does it means that at least one event will occur in this time period a to b. To average this over a time interval [0,t) we divide the above equation substituting a by 0 and b by t and divide it over the time range t, which is exactly how you would calculate a rate (number of events over time).  If we identify an equation for the average proc rate at time t over interval (0,t) we get:

4_3

This is essentially a modified form of one of the first equations we derived and the one we were referring to in the previous second:

6_3

We have substituted the integral above into the first term in this equation.  We have replaced a chance to proc (based on last time since proc value) which is essentially a rate value with an average proc rate value.  With some math, this turns out to be”

5_3

This part is confusing to me because we are replacing one kind of term with a different kind of term.  The first term in the new equation is a proc rate, and the first term in the original equation is a probability term.  We can look at the two symbols and thing well they’re both lambda but they actually represent what is in my opinion different things.  In fact, the term in the new equation is actually the integral of the term in the second equation which makes the reasoning behind the substitution even more dodgy.  Lets just give the author the benefit of the doubt though.

Let me say a few sentence about the graph.  First, we notice the protection doesn’t kick in until the time between procs reaches above 80 seconds. After a certain point, the protection system will actually return less proc chance than without the protection system.  It is likely however, that the time between procs will never reach this period but that still needs to be determined.

Overall, this part of the analysis is at times confusing to me, not in the mathematical sense but in the logical reasoning sense.  Approximations can only be made if the base considerations follow what we would expect in real life.  For example, you can not approximate the speed of a car if you only take its mileage for example and how long you’ve owned the car.  These are not related to the speed.  Regardless, perhaps the reasoning for these is beyond by level of understanding and you can in fact use these equations to calculate timings.  These calculations might be a little more complicated since you will be forced to take the integrals yourself.  If you have access to Mathematica or even some online websites such as Wolfram Alpha, you can use those to calculate you integrals. In this section, the authors have included a graph which allows the analogy behind all of this math and the real word to be discerned. You can easily use a plot like this to identify your proc rate after knowing a period of no procs and see for example, if your trinkets (by either looking at world of logs or other things) is a following this approximation.

Remember the words of our good friend George E. P. Box, “All models are false but some models are useful.”

In-depth RPPM Analysis Part 2

The next approximation is for the probability that one proc happens at a particular time t.  This event is mirrored by a mathematical process which is known as a probability density function.In probability theory, the chance of anything occurring or the addition of the probability of a number of different states always sums up to 1.  For example, you pick a ball from a bag with a blue, red and yellow ball.  The chance that you pick a ball (disregarding the color) is 1 because we’ve already established that you’re picking a ball.  The chance you pick a ball of a particular color is 1 out of 3 because there are 3 different colored balls.

The integral in the equation shown is a fancy way of summing up all possibilities to 1 as I’ve stated.  P(t) is a probability distribution function which is simply what the probability is over a range of t values. At one single t, there will be one unique probability (chance to proc).

Recall the equations from last time:

1

The paper has reasoned the equation for a proc at time t, by multiplying the chance for it not to proc between time [0,t) by the chance for it to occur between [t, t+dt].  ‘dt’ or adding a ‘d’ in front of a variable such as t, is a fancy way to say instantly or some like to say an infinitesimally short period of time.  It’s used to describe an event occurring at time t. It’s a little more complicated than this but this is the essential gist of it.

From the equations above, the chance of a proc occurring at time t-t+dt means it could not have procced before time t, and must proc in that range.  Thus we multiply the two equations from above:

2

By integrating this equation from 0 to infinity with respect to t, we get the probability that a proc occurs at any time t.  Note in the above equation, that the interval t-t+dt is symbolized by a triangle which is what mathematicians like to use to describe change.  Here it represents a change in time.  With some math, this equation becomes:

3

The paper states this is in the form of a basic exponential distribution.  Basically, there are some distributions or functions which commonly occur so they are recognizable.  This equation, when plotted out, symbolizes one of these known as the exponential distribution (http://en.wikipedia.org/wiki/Exponential_distribution). The paper also mentions that the inter-proc time is distributed according to P(t) which is essentially saying in real English what we’ve just calculated.  Between any two procs, there is a time where it does not occur, here we’ve denoted it as the period [0,t) and a time which it does occur [t,t+dt].  We are calculating the probability of a proc at this time [t,t+dt], therefore over the scale of times, which will be see from plotting the above function P(t) over a range of t, the probability to proc at certain times.  From this plot, we might see a significant peak.  This is what the average inter-proc time will be.

The next approximation is the general case of n events occurring in a time interval.

4

The author mentions it can happen in two cases.

1. All n events occur before time t, no events occur in time t – t+dt, approximated by the equation:

5

2. n-1 events happen before time t, 1 event in time t—t+dt

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Note that the variables are used here so that the same two cases can occur for any time t. Combining these equations we arrive at the final equation:

7

We also know, from properties of differential functions that this holds true:

8

As I mentioned before, approximations like this are ways of calculating the value of a function like this without using difficult or complicated mathematics.

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Solving this equation for Pn(t) yields:

10

(! is the mathematical symbol for factorial.  3! = 3 * 2 *1)

This is as mentioned, the standard form of another common distribution called the Poisson distribution, poisson being the name of a famous mathematician. The author mentions something important here.  This chance of n procs of interval [0, t] is independent of your attack speed.

I think I’ll stop at this point.  This section had a lot mathematical terms in it but I hope I’ve made it a little bit clearer to you.  Similar to last time, I want to remind everyone that these equations are actually usable in real life.  Often mathematicians forget about the purpose of their approximations and equations.  Suppose you wanted to see how many procs you would get in an 8 minute fight.  Given the constants for the trinket and your haste, you can plot the function out above and see what the distribution looks like.  Then you can say with reasonable certainty, the number of procs you will get falls somewhere along the distribution you plotted.  This is what makes math cool.

Look out for the last part of the explanation coming soon 🙂