Mastery of Focus Skills - using nonlinear regression tool for skill breakpoints

[jimhsu 2]

I was bored, so I played around with a nice nonlinear regression tool trying to fit data for skill breakpoints.

 

Aspect Focus. (trivia section)

 

Can someone verify to see if these results are even close?

[Schot 407]

Very interesting grphical display of hard and soft point benefits jimshu.

 

I think further investigation into the nonlinear regression is needed however. For those of us that are not mathmetically inclined for it would certainly provide greater value to your contribution of you were to explain what a, b and c represent. In addition, since you are asking here if someone could verify if your results are even close it might be a good idea to confirm them before adding to the wiki. The graph is really a fantastic way to impress upon player's just how valuable mastery is. I really like it.

[Ysne58 236]

I was bored, so I played around with a nice nonlinear regression tool trying to fit data for skill breakpoints.

 

Aspect Focus. (trivia section)

 

Can someone verify to see if these results are even close?

 

I'm pretty math challenged. If you can walk me through what I would need to do to help verify this, I will be happy to try.

[jimhsu 2]

I see the edit was reverted - that's fine. This is probably a good place to discuss, and get hard values from the game for more evidence.

 

As for the regression:

 

I am working under the assumption that values generated for the tables are derived from some equation, with a unknown structure. (I.e. I don't know what parameters were used). So, we have an equation

y = f(x) where x is the Skill Level, and y is the regeneration rate.

 

By simply plotting regen rate with skill level, we see something like this (blue line). The blue line is actually under the red line, but you can't see it.

 

The sharp break at 75 suggests that the function is not differentiable at that point, and thus f(x) is actually a piecewise function defined over the ranges [1,75) and [75,+infinity]; hence one function f1(x) gives regen rates from 1-74, and another f2(x) gives the rates from 75+.

 

Taking the data and tweaking it a bit, I plugged it into a nonlinear regression tool (results here). I automatically excluded anything with three or more parameters, because of parsimony (the real equation isn't likely complicated), or anything with things like atan(x) or sin(x).

 

As to why I settled on y = a / (b + x) + c :

 

Plot

 

 

Residuals

 

 

1. The curve "looks like" a square root curve (plot y = sqrt(x) and see what I mean)

2. High correlation (r > 0.9999). This cannot be perfect (r = 1) because the display value is rounded. Likewise, the values for a, b, and c cannot be exact because the input data is rounded.

3. Residual analysis shows low correlation of residuals. This is a good thing.

 

Now is this right? I don't know. Unless a dev comes over, no one will know. But the equation does work with the data.

 

If anyone has questions, I'll try to write simpler explanations later. I will also try to find a simpler expression with the tool.

 

PS BTW this was nothing compared to what I had to go through for another game (Eve online) to analyze stacking penalties. If you're bored enough, you might find the relevant post via google.

Edited November 9, 2009 by jimhsu

[jimhsu 2]

Another interesting graph for people to ponder on:

 

 

Shown are skill levels with Mastery (x axis) and without Mastery (y axis) that give the same regen according to my calculations. The further out you are, the worse it gets (a 200 skill level with mastery is equivalent to nearly 1700 without, according to the formula). Can that possibly be right?!

[chattius 2,522]

In the The how to become a demigod thread I did some tries to find the formula which is used to calculate the damage reduction done by armour. It seems it is something like (x-1)/(x-b). So I suspect similiar easy formulas in your case too. If you have time you can perhaps translate my calculations in this thread into nice english maths and test first if they are right.

 

(x-1)/(x-b) gets close to 1 for x-> infinity but never reaches it. X would be the factor of how much higher your armour is than needed to have no seen effect against a hit.

Doing tries with different armour numbers at always the same enemies. If last kill shows 100 damage done and you see 100 damage when you have 250 armour then what I call fixpoint-armour is 250. If You have 1000 armour, the enemy does 100 damage, then x = 4.

The (x-1) gets zero for the fix point armour, so no damage reduction done.

If you are hit naked then x = 0 and you get the b in the formula.

[ka243 3]

I understand the second graph about equivalent skill levels for mastery and nonmastery. I don't get the first part at all. The link you posted : http://zunzun.com/FunctionFinderResults/2/...d=1257751651.45

 

Does not work.

Edited November 9, 2009 by ka243

[jimhsu 2]

Your hypothesis is close. I'm almost absolutely sure that the equation is some sort of rational function.

 

One hypothesis is the classic Michaelis-Menten equation (from enzyme kinetics): y = ax/(x+b), most commonly seen as rate = Vmax*/(+Km)

 

This gives something that is close, but look at the residuals:

 

 

 

Clearly there is a higher order term in there. So try the Michaelis-Menten equ with offset: y = ax / (b + x) + c

 

 

 

That looks quite good, in my opinion.

 

For the regression model:

 

y = ax / (b + x) + c

 

a = 74.364

b = 21.220

c = 5.6017

 

Since this IS the Michaelis-Menten equation basically, I'll try to use my knowledge of enzyme kinetics to try to interpret the parameters:

 

a is equivalent to Vmax; basically the "maximum possible" regen rate benefit assuming skill level goes to infinity. In other words, you can't achieve >74.36% regen bonus without mastery, no matter how many soft points you have.

b is equivalent to Km. This is the skill level at which you get "half" of the potential regen bonus (74.36/2, or 37.18%)

c is basically the regen time at skill level "zero". Of course, such a thing doesn't actually exist.

 

Results here.

 

I understand the second graph about equivalent skill levels for mastery and nonmastery. I don't get the first part at all. The link you posted : http://zunzun.com/FunctionFinderResults/2/...d=1257751651.45

 

Does not work.

 

Hm.. results page seems to be dynamic. Try a PDF.

All the top results have identical residuals and standard errors, suggesting that all those equations are basically equivalent.

 

The second graph is IMO the most useful. It's basically a transformation of the 1st graph. Imagine on the 1st graph, drawing a horizontal line at a regen time - the 2nd graph is the (x,y) coordinates of the points at which each curve in the 1st graph touches that line.

Edited November 9, 2009 by jimhsu

[jimhsu 2]

As for verification:

 

I will assume that the tooltip value is right. If not, thing may be more complicated.

 

1. Get a focus skill to 74 hard points and play around with +All Skills to get soft point skill levels. Record regen values. Something like a +1 All skills ring will help here (see step 2).

2. Get the focus skill to 75 hard points (mastery) and generate skill levels to (1). Record regen values.

 

(a local save game might help here in case you screw something up).

 

Values from (1) should fall onto the red line, while values from (2) should be on the blue line (and match the sacredwiki numbers).

 

This is all assuming the displayed value is correct...

Edited November 9, 2009 by jimhsu

[ka243 3]

What are the practical applications of this (things we should know/do as a result)? Ex: Don't put more than 1 point into a skill unless you plan to take it to mastery, attain 24 points as quickly as possible?

 

Can this analysis be applied to all skills or only focus skills?

[jimhsu 2]

I did a similar thing with lore and it shows the same pattern. The specific numbers may change however. I currently don't feel like going through all the skill data and regressing it with the same equation.

 

I guess the real point is shown in 2nd figure. I.e. If you wanted to match the regen of a char with 110 focus mastery, you would need ~190 skill level without mastery. And so on - as the level with mastery becomes higher, the # of soft skill points needed to match becomes so high that it becomes basically impossible even if you manage to get +1K all skills. Thus, mastery becomes more and more essential at high skill levels -- but you already knew that didn't you? This is just an attempt to put some real numbers onto the observation.

 

The 24 points (actually, 21) also makes sense. If you don't plan to take the skill to mastery, that is basically where you can expect to derive "half" the benefit from the skill. I think the formula and specific cutoffs will change from skill to skill, but I expect the general equation to look the same.

 

PS For damage in skill lore:

 

a = 394.04

b = 100

c = 5.895

 

These values intrinsically have standard error, so the real equation could just as likely be

 

a = 394

b = 100

c = 6

 

The quality of fit is the same in both cases.

 

 

It's the same equation. You can plot it yourself if you like.

Edited November 10, 2009 by jimhsu

[jimhsu 2]

Did some quick verification testing with an unmastered focus skill. I don't have any uber +All Skills gear (only up to +80 or so), so don't expect to see values like 200 or so. (And Ice & Blood nerfed that even more).

 

Skill level Regen (calc) Actual (game)

77 63.89968412 63.9

84 64.96851239 65

90 65.77756765 65.8

 

So far the equation appears to be correct. What I want to see though are unmastered skill stats in the extreme high end (like several hundred or so). The regression equation might be off a bit at that high of a range, but not more than 0.1%.

Edited November 10, 2009 by jimhsu

[gogoblender 3,068]

I've been following this thread for a few days now, and would like to see how this can be applied to the wiki. jimhsu, if you are confident about the math, my next question is... can this be boiled down to a thesis statement that you can write up that would be a decent "factesque" article for the wiki?

 

I'll spin this another way... has this topic here come up with a "truth" that can be used as strategy for players? I'm seeking to understand how this offers utility to players, thereby winning itself a place in wiki on the page. Are you able to put together an article detailing what trends this math points out, and why certain actions would be useful for players with proofs?

 

Cheers!

 

 

gogo

[Schot 407]

Very good points by gogo and I'm along the same thinking. We were talking about your work jim and trying to wrap our heads around the application of the data you're collecting. Finding an absolute equation to any event in Fallen Angel as well as Underworld has always been a mysterious matter.

 

Something that the wiki would greatly benefit from, as well as players, is to have a comparison of Hard Point vs. Soft Point benefits in the form of a line graph such as yours. As it is we only have hard point table data. Reason being that it's difficult to generate point by point Soft Point data over 74 but creating a curve based on 9-12 point intervals or so would be easy enough and quite useful. A graph with a max point limit of about 500 would demonstrate the difference wonderfully.

[chattius 2,522]

A similiar question can be done for alchemy. What will Alchemy mastery give in trophy effects or potion duration. The full table in the wiki was done by a test characters and then slowly putting one point after the other in alchemy? So all values above 75 are mastery ones?

 

I am quite sure currently that my armour formel is correct. I fight same beast with two different (not too close) armour values and I can predict the damage I will take for other armour values quite exactly. So (x-1)/(x-b) seems to work for that. We tried to easy our formula by inventing a fix point armour (FPA), which depends on difficulty, level and damage of enemy. If you have FPA fitting for the received damage then the damage you received equals the damage shown in last kill. So no damage reduction from armour will be done. So x = your armour divided by FPA. For x = 1 the formula should be zero, which it is. For no armour we could easily test how much bigger our received damage was compared to the shown damage, and we could calculate the b.

 

If you get hit in niob without any armour to the received damage type the damage can easily more than tripled. So knowing this I know always socket at least one artamark to prevent one hits. Damn I am talking about damage formula again. Well I will do the maths for alchemy if noone else will do

[jimhsu 2]

I will see when I have time to do a full write-up. Hm...

[Sarcolemming 8]

From what I've gathered reading all this it boils down to basically the following...

 

a is equivalent to Vmax; basically the "maximum possible" regen rate benefit assuming skill level goes to infinity. In other words, you can't achieve >74.36% regen bonus without mastery, no matter how many soft points you have.

b is equivalent to Km. This is the skill level at which you get "half" of the potential regen bonus (74.36/2, or 37.18%)

c is basically the regen time at skill level "zero". Of course, such a thing doesn't actually exist.

 

Now given that without mastery our regen times cannot ever exceed ~74% the only reason to bother boosting any skill via 'soft points', is with +all skills rings, which would benefit the other skills we do need. As far as ingame player usage I can see this as being a different more mathy sort of way of saying after level xx, there is no real benefit in adding 'soft points' because we will never achieve greater than a 74% regen bonus. No matter how monstrous of a bonus we actually have.

 

That said it's really quite rare in my experience that a skill in any particular build is not mastered, unless of course it's either A. to unlock a secondary skill we need or B. to add a specific and notable benefit at level 1, such as concentration or combat discipline. Aside from those reasons the actual formula itself could be used to find a particular benefit far beyond what is already listed in wiki, however very few players especially on console(ps3 has only a single level 200 atm) even have the ability to push skills well into the 3 or even 4 hundred levels.

 

Perhaps a more useful formula would be penalty associated with eating too many runes? if my max level is 98 and I have read 150 runes, what would be the penalty? is there a cap to that penalty? how do we know at what level above max will our penalty be completely maxed out? I personally am unsure of such things, perhaps there is already a formula in use.

 

To me at least, this formula above while useful.... isn't terribly so unless we are referring to a much grander scale 200+ where it may come in handy finding potential benefits, as opposed to smaller already known and well listed benefits(200 and below). Now assuming a sacred 3 does ever come out, it may well be that a similar sort of formula will be used and in that event, having the same formula we can then proceed to calculate out the values much more quickly for the particular skills it will involve. Assuming of course it even uses this formula of ax/(b+x)+c. Great work fellas but I sadly can't see much use for this other then giggles

[Numerii 1]

One thing that I've gotten from this, and this is something that some people have simply been doing without really knowing the math behind it, is that when you first start a character it would be wiser to invest 21 hard points(assuming a character with limited to no access to +skills) than to leave all skills at 1 point until you are ready to master them.

 

This would provide more "bang for the buck" in those first 100 levels or so.

 

It's actually something that I did for my main Shadow Warrior since 6/10 skills are defensive skills. I brought each one up to 20 hard points then started mastering them individually from that point on. My intent was simply to provide more balance to his defense. Now I have a reference showing that 20 points(21 actually) is a very good number for that purpose.

[ka243 3]

One thing that I've gotten from this, and this is something that some people have simply been doing without really knowing the math behind it, is that when you first start a character it would be wiser to invest 21 hard points(assuming a character with limited to no access to +skills) than to leave all skills at 1 point until you are ready to master them.

 

This would provide more "bang for the buck" in those first 100 levels or so.

 

It's actually something that I did for my main Shadow Warrior since 6/10 skills are defensive skills. I brought each one up to 20 hard points then started mastering them individually from that point on. My intent was simply to provide more balance to his defense. Now I have a reference showing that 20 points(21 actually) is a very good number for that purpose.

 

Or just get one point and make sure to have at least +20 all skills. Because of the big benefits of mastery, I never put more than one point in a skill unless I'm trying to master that skill next or I need mod points or I need points in that skill to unlock another skill.

Edited November 18, 2009 by ka243

[Numerii 1]

That said it's really quite rare in my experience that a skill in any particular build is not mastered, unless of course it's either A. to unlock a secondary skill we need or B. to add a specific and notable benefit at level 1, such as concentration or combat discipline. Aside from those reasons the actual formula itself could be used to find a particular benefit far beyond what is already listed in wiki, however very few players especially on console(ps3 has only a single level 200 atm) even have the ability to push skills well into the 3 or even 4 hundred levels.

 

Perhaps a more useful formula would be penalty associated with eating too many runes? if my max level is 98 and I have read 150 runes, what would be the penalty? is there a cap to that penalty? how do we know at what level above max will our penalty be completely maxed out? I personally am unsure of such things, perhaps there is already a formula in use.

 

To me at least, this formula above while useful.... isn't terribly so unless we are referring to a much grander scale 200+ where it may come in handy finding potential benefits, as opposed to smaller already known and well listed benefits(200 and below). Now assuming a sacred 3 does ever come out, it may well be that a similar sort of formula will be used and in that event, having the same formula we can then proceed to calculate out the values much more quickly for the particular skills it will involve. Assuming of course it even uses this formula of ax/(b+x)+c. Great work fellas but I sadly can't see much use for this other then giggles

 

1. Don't forget there are builds that pick an aspect focus for the sole purpose of modifying a single Combat Art(think High Elf and Grand Invigoration) and therefore they would only invest 9 points to do so.

 

2. While the formula isn't that useful to the majority of us, the same could be said about the smallest piece of a jigsaw puzzle. Afterall, when a community is making a concerted effort to "crack the code" of a game, a tiny piece of information might just help flesh out that puzzle just enough to easily figure out 3 more pieces that were previously unnoticed.

[Numerii 1]

Or just get one point and make sure to have at least +20 all skills. Because of the big benefits of mastery, I never put more than one point in a skill unless I'm trying to master that skill next or I need mod points or I need points in that skill to unlock another skill.

(assuming a character with limited to no access to +skills)

[Antitrust 32]

This is easily verifiable by modding the +skills bonus in single player. Depends on what exactly you are trying to verify of course, but access to +skills is hardly an issue.

[nokka 0]

I've also done a small analysis of the skill progressions. Only afterwards did I notice that this subject had already been dealt with to an extent.

 

Because the threads are so similar in content, I decided to try and link them to eachother ^^

(not to mention the fact that my thread is missing the nice graphs that this one can boast...)

 

Also, posting will drag the thread to the top and I'd really like to know how or if people want this info put up onto the Wiki.

[SolomonGrundy 1]

numbers that make sense for a focus skill

 

1 - Assumes high +all skills/+focus

5 - Unlocks other skills

9 - Focus onl,y complete modding for 1

12 - Assumes Lore, 75+12= 87 mod point

14 - Focus only, mod point

22 - Focus only, mod point

31 - Focus only OR Focus and Lore, mod point. I believe 31 is a an excellent place to leave a focus

42 - mod point

55 - mod point

75 - MASTERY

 

Investment beyond mastery is not a good option until level 125 or higher. Too many other skills to master, or bring to 31