Regression analysis for kg/hp vs. Ring laptimes

[lucid]

After reading the previous discussion on the rumored 8:02 M3 time, and the recent discussion on the 7:38 GT-R time, I did a regression analysis for weight/power vs. ring laptimes. You'll see the cars that were included in the spreadsheet. The redline in the graph is the best fit line, which is:

Time=422.15 + 15.95 * Weight/Power

Adjusted R^2: 0.78
P<0.001
These numbers basically mean that a significant linear relationship, as described by the above formula, exists.

The crank power and weight figures were taken from supercars.net. I don't have a position on their accuracy; just plotting some of what's on that page.

http://www.supercars.net/PitLane?vie...ID=0&tID=10073

If we can build a database for known wheel dyno numbers, I can re-run the regression.

It would be interesting to discuss the cars that don't fit the trend.

[13eastie]

Very interesting.

It's notable that the E46 CSL is the only BMW whose performance exceeds that indicated by the trendline, and by some margin....

Using lucid's formula with power 420hp and mass 1655kg, we get a predicted E92 M3 time of 8:05.

Also the weight/power ratio is 3.94 (almost exactly the same as for the E46 M3 CSL).

The previous M3 was about 7s "softer" than the trendline. How will the new one differ in this regard? And to what extent is the CSL's awesome performance due to differing tyres?

[swamp2]

Nice work lucid, you have quantified nicely exactly the basic physics that we all (many of us at least...) understand quite well.

Those asking about the CSL the answer is indeed obvious - tires make a huge difference and that car comes basically with street legal race tires. That being said the CSL may be below the curve fit without those tires. Cars below the line are in a way the best designed and engineered cars for going fast, those above less so.

This also makes my point perfectly about the Nissan GT-R. I have sloppily added the car to you graph (man that was harder than Excel - I should have just asked you to do it...). The GT-R is an massive outlier in the data. This means one (or more) of a couple things to me.

1. They have really reinvented the sports car and bested everyone by a significant margin. Maybe it could be their AWD system.
2. The car is under rated in term of power
3. The car is under rated in terms of weight (claimed>actual)
4. The car had street legal race tires

What is the most likely scenario. I'd vote in order 4,3,2,1.

Lastly, bringing it back OT... I have added the new M3 at 7:57 and 8:03 (with US hp figure - 414). Again 8:03 is a provisional time and 7:57 is a good rumor directly from Richter and both times are still up in the air as to which tires are invlolved.

Nissan, BMW, whom ever, you simply can't beat physics.

[Sedan_Clan]

Nice work guys!

[paulyd]

this is awesome. Great idea to run a regression lucid and swamp.

I noticed R square = 0.78 (proportion of the time directly explained by weight/power ratio)

This leaves 22% of the time to be explained by some other variables.

Tires???

If the E46 CSL and the GTR were on race tires, then their outlying points could be the result of superior tires.

AKA give them comparable tires, and their times would fall closer to the line of best fit (reality).

EDIT:
Tires supposedly reduce time by around ~20 sec.

Look at graph, move E46 CSL time(dot) up two grid lines (20 sec). It falls just above the best fit line (just as the standard E46 does - which makes sense given that it is 95% the same car)

Move the GTR point up 20 sec (two grid lines) and it falls almost exactly on the line of best fit!

I smell tires

[jaiman]

this thread is wothless without error statistics.

[lucid]

OK guys, I did some modifications to help sort things out.

1. I sorted the sheet by kg/hp, and the then by time.
2. I added 2 columns: one for the projected time according to the regression formula, and another for the difference between the projected and actual times. This quantifies the divergence from the regression line.
3. I added a row for the E92 M3 with 8:02 time, and 2 rows for the GT-R with the 7:38 (one is for the base 480 hp version and the other for the speculated 530hp version by Motortrend). Since there is ambiguity associated with these runs, I did NOT include them in the regression analysis, so the regression line formula has not changed.
4. I redid the plot two show the speculated M3 and GT-R kg/hp vs. time datapoints.

[paulyd]

Originally Posted by jaiman

this thread is wothless without error statistics.

You can guess by looking at the grid lines but here are the exact error stats:

I added the E92 M3 and the Nissan GTR into the regression, they are at the bottom.

The GTR is the clear outlier by 21 seconds below expectations

The E46 M3 CSL (near the top) is 13 seconds below expectations

The 997 GT2 is 16 seconds below expectations

[lucid]

Originally Posted by swamp2

This also makes my point perfectly about the Nissan GT-R. I have sloppily added the car to you graph (man that was harder than Excel - I should have just asked you to do it...). The GT-R is an massive outlier in the data. This means one (or more) of a couple things to me.

1. They have really reinvented the sports car and bested everyone by a significant margin. Maybe it could be their AWD system.
2. The car is under rated in term of power
3. The car is under rated in terms of weight (claimed>actual)
4. The car had street legal race tires

What is the most likely scenario. I'd vote in order 4,3,2,1.

It makes some sense if the car tested was the speculated 530 hp version. Otherwise, I'd go with: 4, 2, 3, 1.

[paulyd]

Originally Posted by lucid

It makes some sense if the car tested was the speculated 530 hp version. Otherwise, I'd go with: 4, 2, 3, 1.

If you do 530 hp instead of 473 for the GTR

It is still an outlier by over 15 seconds below expectations...

Still doesnt make sense?

[lucid]

Originally Posted by chitown08

You can guess by looking at the grid lines but here are the exact error stats:

I think he might have been referring to these:

[lucid]

Originally Posted by chitown08

If you do 530 hp instead of 473 for the GTR

It is still an outlier by over 15 seconds below expectations...

Still doesnt make sense?

Look at the GT2.

[paulyd]

Originally Posted by lucid

Look at the GT2.

Yeah I guess 16 sec below for GT2 so 15 below for the GTR isnt impossible

Do you know if any of these tests were done on race tires? (we obviously don't know abou the M3 or the GTR, but the GT2 or anyone else)

[lucid]

Originally Posted by chitown08

Do you know if any of these tests were done on race tires? (we obviously don't know abou the M3 or the GTR, but the GT2 or anyone else)

For the GTR, http://www.supercars.net/PitLane?vie...ID=0&tID=10073 now says:

"company test driver Suzuki, slick cut tyres, track partially wet"

I don't know about the rest.

[dechoong]

Power to weight ratio though an important indicator of performance, doesn't reveal everything especially how effectively a car puts its power down on the road.

I'm taking an example of the R8's track time at the Bedford Autodrome circuit from the july issue of EVO:

1. Porsche Carrera GT 1:19.7
2. Caterham CSR 260 1:21.0
3. McLaren F1 1:21.2
4. Ferrari Enzo 1:21.3
5. Audi R8 1:22.1
6. Koenigsegg CCX 1:22.3
7. Lamborghini Gallardo 1:22.8
8. Mosler MT900 1:23.0
9. Ascari KZ1 1:23.2
10. Porsche 997 GT3 1:23.4

It's pretty obvious all of the 5 cars below the R8 has a higher power to weight ratio yet they are slower. The thing similar between the R8 and GT-R is both utilizes trick 4WD drivetrain which I believe is well developed to the point where it provides superior traction which shows in their lap time. Remember the Audi Quattro in the BTCC (or was it WTCC) years ago? They were so successful on track that their Quattro 4WD was banned. Likewise the C4S is faster over a lap of the Nurburgring than the lighter C2S.

[enigma]

Until we can control for tires charts like this are of limited value. On an 8 min track tires alone can cause a 16s or so swing in results. Its why the CSL does much better than its projected time while the base car is worse than its projection.

The general rule with tires is going from a street tire like the Pilot Sport 2 to a DOT-R like the Pilot Sport Cup or Hoosier R6 is worth about 2s per min of track.

If you go to a full slick those are worth another 1s or more. In the video the noise you hear from the tires makes it unlikely that they are running a tire like a full slick or an R6. However, they could still be running a moderate dot-r.

Basically we don't have enough information to make a good comparison.

One thing that is possible. Nissan may have pulled a elise/z06/s2000 like move and put a good suspension in a stock car. The reason BMWs don't usually put up good times stock is the horrid front camber curve combined with the limited room for alignments adjustments and narrow tires for the weight of the car.

The reason the Z06/Nissan/Elise put up very, very good times for a stock car is the run an aggressive double wishbone suspension, a lot of tire for the weights of the car, and often have good alignment possiblites without needing to replace struts.

I have an E46 M3 and an Elise. Both make about the same HP/LB. Yet the elise is much faster at a track stock than the M was stock. The difference is almost 5s/min. 2 of those are from the fact the elise comes with tires similar to the CSL, still that leaves 3s thats mostly due to the suspension.

[sdiver68]

Don't forget track conditions.

At Road America in the early (colder) Spring, I was lucky to run 2:35. In the heat of the summer, I ran 2:28. That's 7 seconds on a track about 1/4th the size of the Nordshleife. And I ran "softs" in the Spring and "mediums" in the Summer.

At GIR (my times were about 3 seconds off in the colder Spring versus summer times), track about 1.6 miles.

[lucid]

Originally Posted by enigma

Until we can control for tires charts like this are of limited value.

Not true because this is a statistical approach (the power of n).

Even if we control for tires, we run into other uncontrolled variables such as weather, driver skill, driver mood, etc. That issue has been debated to death on various threads here. I am not saying we shouldn't control for what we can control for, but the point of pooling together a large database is using statistics to even things out. One can calculate a 95% confidence interval as a performance envelope for instance. That obviously doesn't mean much for making sense of an individual data point, and 29 cars do not make up a large database either. However, if we were to add 200 data points to the chart, then the outcome would make me have more faith in the regression line as the confidence interval would most likely get tighter.

That said, even with 29 cars, you can arrive at some observations. We know for a fact that the CSL was on slicks. Draw a straight line between the CSL point and the 997 GT2 point, which ends up being pretty much parallel to the regression line, and some std deviations away from it. Any car below and to the right of that line, regardless of what type of tires its on, is either breaking some exciting new ground in engineering or is bogus. 480hp GTR falls into that category. However, we don't have enough information to say that is indeed new territory or just bogus. Time will tell.

[swamp2]

Originally Posted by enigma

Until we can control for tires charts like this are of limited value. On an 8 min track tires alone can cause a 16s or so swing in results. Its why the CSL does much better than its projected time while the base car is worse than its projection.

The general rule with tires is going from a street tire like the Pilot Sport 2 to a DOT-R like the Pilot Sport Cup or Hoosier R6 is worth about 2s per min of track.

If you go to a full slick those are worth another 1s or more. In the video the noise you hear from the tires makes it unlikely that they are running a tire like a full slick or an R6. However, they could still be running a moderate dot-r.

Basically we don't have enough information to make a good comparison.

One thing that is possible. Nissan may have pulled a elise/z06/s2000 like move and put a good suspension in a stock car. The reason BMWs don't usually put up good times stock is the horrid front camber curve combined with the limited room for alignments adjustments and narrow tires for the weight of the car.

The reason the Z06/Nissan/Elise put up very, very good times for a stock car is the run an aggressive double wishbone suspension, a lot of tire for the weights of the car, and often have good alignment possiblites without needing to replace struts.

I have an E46 M3 and an Elise. Both make about the same HP/LB. Yet the elise is much faster at a track stock than the M was stock. The difference is almost 5s/min. 2 of those are from the fact the elise comes with tires similar to the CSL, still that leaves 3s thats mostly due to the suspension.

What happened to my ally in science? Well that is not so fair of me... If folks always agreed on everything life would be boring. Perhaps lucid has already properly defended the use of simple statistics to understand the performance cars obtain. But I will continue.

Like he said the chart covers tons of random variables the idea is the see what parts are random and what parts are due to the physics. I certainly agree that having two complete regressions, one with street tires and one with race tires would be more valuable, but for the most part that data are unavailable. Hence my comment about cars below the line and those above the line. Those like the Elise should (didn't check it myself) be below the line due to their inherent design strengths specifically to make them perform better than other cars of a similar power to weight ratio. For example their suspension (as you mentioned) low moments of inertia, low cg, etc., etc. All such factors engineered into the car will simply push it below the line. Tires are the same, they just happen to give a "big push".

This leads back to the GT-R. I do not doubt that Nissan has put some real high speed "trickery" into this car. By "trickery" I don't mean magic, I simply mean engineering and design effort focused on making it a real track monster. The trick AWD system, traction control system and DCT as well will push it below the line (or at least in that direction). This is then THE question: how far below the curve is reasonable/possible with a totally stock vehicle? lucid could translate this to a percentage/# of std. deviations out etc. or other more concrete numbers, but my gut tells me it is simply too far out to be running street tires and have the power and weight it claims to. My first observation of this was simply the fact that almost all cars <7:40 have >500 hp and <1500kg. Now that immediate observation is even more strongly shown as how far of an outlier it is - the furthest - even given cars known to be running (street legal) race tires. Heck we even now have the direct admission from the Nissan fellow about "shaved" tires or whatever. Even if I did not now know this extra piece of informaton I would still stick to my guns - the car is simply too far below the curve.

Give it just a bit more thought .

My conclusion and standing bet (if anyone is interested) is that at least one of my previous points 1-4 are the case for the GT-R and this 7:38 lap.

[enigma]

Originally Posted by swamp2

This is then THE question: how far below the curve is reasonable/possible with a totally stock vehicle?

One we don't know its stock. You see in the video they are putting some diffrent tires on it. The term used is "cut slick" which may mean groved slick. Note: that is important because that would be a large upgrade over the tires on the M3 CSL. On a track as long as the ring that could mean 10-15s over a DOT-R whcih is already 10-15s over a street tire. So put street tires on the car and it might regress all the way to 8:00 or so.

If they did run a "slick" that also has other implications. That the stock suspension was able to run the tire without being overloaded means it must have a very, very good camber curve and a lot of roll stiffness, or the car has been upgraded from stock in some way. Possibly as an unanounced option package al-la the Exige Cup 240.

Statistics are great, but not when unknown factors can influence the times by far more than the accuracy of the statistics you are trying to use.

All we know is a GT-R like car did a specific time. We don't know any of the specs of the car in question. They clearly put special tires on it for the lap, were other things changed? Either as a straight mod or a performance option on the car, unkown.

[coaster]

Originally Posted by swamp2

My conclusion and standing bet (if anyone is interested) is that at least one of my previous points 1-4 are the case for the GT-R and this 7:38 lap.

Well swamp it's also possible that none of your four previous points is the case for the GTR. If you hadn't seen the GTR's ring time before you looked at this linear regression chart how would you have explained the GT2's "outlier" status? Probably with the same arguments I bet.

In my eyes and looking at this chart the GTR's time is not that much of an outlier (and neither is the GT2). Now if either was 2X further outside the envelope of all the other cars I would squint hard and be concerned about trickery on Nissan's part.

By the way, regression has nothing to do with "physics". Let's not confuse a least squares fit to some data with the physical laws governing vehicle motion.

[swamp2]

Originally Posted by enigma

One we don't know its stock. You see in the video they are putting some diffrent tires on it. The term used is "cut slick" which may mean groved slick. Note: that is important because that would be a large upgrade over the tires on the M3 CSL. On a track as long as the ring that could mean 10-15s over a DOT-R whcih is already 10-15s over a street tire. So put street tires on the car and it might regress all the way to 8:00 or so.

If they did run a "slick" that also has other implications. That the stock suspension was able to run the tire without being overloaded means it must have a very, very good camber curve and a lot of roll stiffness, or the car has been upgraded from stock in some way. Possibly as an unanounced option package al-la the Exige Cup 240.

Statistics are great, but not when unknown factors can influence the times by far more than the accuracy of the statistics you are trying to use.

All we know is a GT-R like car did a specific time. We don't know any of the specs of the car in question. They clearly put special tires on it for the lap, were other things changed? Either as a straight mod or a performance option on the car, unkown.

OK some disagreement but I have much more simply thinking out loud to do. Bear with me. I think this reply get better as it goes...

Indeed a fully modified race car can place you some significant distance below the regression line - absolutely. This regression line is for street cars (and some exotics - but still steet cars). With the implication you make about the the vehicle being very far from stock, tires and suspension, maybe more, the entire point and impact of the video is called into question. The purpose of the video is marketing and despite the use of non street tires the ENTIRE POINT of the video is that Nissan has made a street car that is also a Porshe Turbo "killer". I trust we can agree on that? Surely if the vehicle is in a state of "tune" anywhere close to what you imply the entire exercise is a giant case of "apples to oranges" and all of the marketing advantages would be lost in a flurry of calls of fraud upon Nissan (and I'm assuming they would get found out somehow). So I contend that the car is likely stock except for two large variables: engine tune and or tires. One of these is easy to cover and to explain around ("it was just tires"...). Engine tune would certainly be a huge cheat as well but who knows. Again the most I can speculate on at this point is back to my points 1-4.

Not to split hairs on the tires but I will modify my possibilty #4. You clearly know more about tires than I do and I may have used incorrect terminology in that statement. I will revise my statement to:

4. The car had a tire of the grade track/competition and DOT legal or higher, potentially all the way to full race tire.

Considering the innovation of the car, as well as the marketing effort of Nissan focusing on the Ring time, I do think that the sole change of a tire could be enough to explain the distance from the regresson line. In other words tires could easily be worth 10+ seconds (this we know) and if the car was 10 seconds slower I would not consider it an outlier and consistent with the range of possibility based on the other data points. This is the way I am strongly leaning at this point.

Look how much more sense the GT-R point makes if you simply move it vertically up 10-15 seconds which is easily explainable by tires. However, to move it horizontally to imagine improving its power to weight ratio you have to move it by around 25% to get it to even reasonably cluster with other cars which are still well below the mean. Looking at things this way is simply another very valuable feature of looking at the data in this fashion. It shows very cleary how important tires are and how much power or weight it takes to match the effect of tires. I like!

In conclusion my guess is dead stock except tires, a great performing car well above almost all at this power to weight ratio.