## Field goal kicking—like putting in 3D with oblong balls

Putting

Andrew Gelman (the author of most posts on this blog, but not this one), recently published a Stan case study on golf putting [link fixed] that uses a bit of geometry to build a regression-type model based on angles and force.

Field-goal kicking

In American football, there’s also a play called a “field goal.” In the American football version, a kicker (often a player migrating from the sport everyone else in the world calls “football”) tries to kick an oblong-ish “ball” between 10 and 70 meters between a pair of vertical posts and above a post at a certain height. If you’re not from the U.S. or other metrically-challenged country still using (British) imperial measures, it’ll help to know that a meter is roughly 1.1 yards.

Sounds kind of like putting, only in 3D and with no penalty for kicking too hard or far and wind effects instead of terrain features. This modeling problem came to my attention from the following blog post:

Unlike Gelman’s golf-putting example, Long’s model combines a kick-by-kick accuracy model with a career-trajectory model for kickers, another popular contemporary sports statistics adjustment. Long used brms, a Bayesian non-linear multilevel modeling package built on top of Stan, to fit his model of field-goal-kicking accuracy. (For what it’s worth, more people use brms and rstanarm now than use Stan directly in R, at least judging from CRAN downloads through RStudio.)

Model expansion

The focus of Gelman’s case study is model expansion—start with a simple model, look at the residuals (errors), figure out what’s going wrong, then refine the model. Like Gelman, Long starts with a logistic regression model for distance; unlike Gelman, he expands the model with career trajectories and situational effects (like “icing” the kicker) rather than geometry. An interesting exercise would be to do what Gelman did and replace Long’s logistic model of distance with one based on geometry. I’m pretty sure this could be done with brms by transforming the data, but someone would need to verify that.

Similarly, Gelman’s model still has plenty of room for expansion if anyone wants to deal with the condition of the greens (how they’re cut, moisture, etc.), topography, putter career trajectories, situational effects, etc. My father was a scratch golfer in his heyday on local public courses, but he said he’d never be able to sink a single putt if the greens were maintained the way they were for PGA tournaments. He likes to quote Lee Trevino, who said pro greens were like putting on the hood of a car; Trevino’s quotes are legendary. My dad’s own favorite golf quote is “drive for show, putt for dough”—he was obsessive about his short game—his own career was ended by knee and rotator cuff surgery—hockey wasn’t good to his body, either, despite playing in a “non-contact” league as an adult.

It would be fun to try to expand both Long’s and Gelman’s models further. This would also be a natural discussion for the Stan forums, which have a different readership than this blog. I like Gelman’s and Long’s post because they’re of the hello-world variety and thus easy to understand. Of course, neither’s ready to go into production for bookmaking yet. It’d be great to see references to some state-of-the-art modeling of these things.

Other field goals

Field goals in basketball (shots into the basket from the floor as opposed to free throws) would be another good target for a model like Gelman’s or Long’s. Like the American football case and unlike golf, there’s a defense. Free throws wouldn’t be a good target as they’re all from the same distance (give or take a bit based on where they position themeselves side to side).

Are there things like field goals in rugby or Australian-rules football? I love that the actual name of the sport has “rules” in the title—it’s the kind of pedantry near and dear to this semanticist’s heart.

Editorial

I thought twice about writing about American football. I boycott contact sports like football and ice hockey due to their intentionally violent nature. I’ve not watched American football in over a decade.

For me, this is personal now. I have a good friend of my age (mid-50s) who’s a former hockey player who was recently diagnosed with CTE. He can no longer function independently and has been given 1–2 years to live. His condition resulted from multiple concussions that started in school and continued through college hockey into adult hockey. He had a full hockey scholarship and would’ve been a pro (the second best player after him on our state-champion high-school team in Michigan played for the NY Rangers). My friend’s pro hopes ended when an opponent broke both his knees with a stick during a fight in a college game. He continued playing semi-pro hockey as an adult and accumulating concussions. Hockey was the first sport I boycotted, well over 30 years ago when my friend and my father were still playing, because it was clear to me the players were trying to hurt each other.

I’m now worried about baseball. I saw too many catchers and umpires rocked by foul tips to the face mask this season. I feel less bad watching baseball because at least nobody’s trying to hurt the catchers or umpires as part of the sport. The intent is what originally drove me out of watching hockey and football before the prevalence of CTE among former athletes was widely known. I simply have no interest in watching people trying to hurt each other. Nevertheless, it’s disturbing to watch an umpire get led off the field who can no longer see straight or walk on his own or see a catcher don the gear again after multiple concussions. As we know, that doesn’t end well.

1. D Kane says:

> I boycott contact sports like football and ice hockey due to their intentionally violent

Any concerns about soccer? Repeated heading of the ball is unlikely to be good for you. And, certainly, the game is designed (and the players intend) to engage in this behavior.

• I’m neither a neurologist nor a soccer fan, so I don’t have a pre-formed opinion. The worst injury I’ve ever seen in the few World Cup games I’ve watched is a twisted ankle. It sounds more like baseball than American football or ice hockey in that I don’t think the players are intentionally trying to hurt each other. I could easily be wrong about that because I have no idea why anyone’s doing anything in soccer.

• I think with a few exceptions it’s rare for intentional injuries to happen in soccer. There are some players who sort of specialize in heading the ball in from corner kicks and things, but I think the average soccer player retires with their health in general, and brain in particular largely intact.

That being said, I am discouraging my kids from heading the ball at all. At their age they’re not allowed to, but later when they’re allowed I’ll still be recommending against it.

• > I could easily be wrong about that because I have no idea why anyone’s doing anything in soccer.

Also, Bob, it’s not really that complicated of a game: try to take the ball down the field, and put it in the other team’s goal without the other team blocking the shot or taking the ball away from you, and without using your hands, unless you’re the goalkeeper inside your own penalty box.

If the ball leaves the field of play on the sides, the team that touched it last gives it up to the other team to throw it in… if the ball leaves the field in the back of the field, the team that touched it last gives it up to the other team to either corner kick it in and try to score a goal (defending team touched last), or goal kick it from near the defending goal and try to carry the ball to the opposing side (attacking team touched last).

All the rest is just details.

• D Kane says:

> It sounds more like baseball

In baseball, it is not uncommon for pitchers to purposely through the ball at opposing batters. If that is not “intentionally trying to hurt each other,” then I am not sure what would be.

• Is this really “not uncommon”? I mean, out of 1000 pitches how many times does that happen?

• Phil says:

Pitching to intentionally hit an opposing player is rather rare, as most of the readers of this blog probably know. When it is done, it’s usually not done with an intent to do serious damage. Most times a batter is hit by a pitch it is unintentional…although that is partly a matter of definition: if a pitcher is trying to pitch way inside to move a batter off the plate or ‘stop him from getting too comfortable’, and the inside pitch hits the batter, do we call that intentional?

Getting hit with a baseball going 90 miles an hour must hurt a lot, but empirically it rarely causes permanent injury and indeed the player who has been hit rarely leaves the game. That said, there is the risk of hitting (and breaking) a wrist, elbow, knee, or the face or head, and in pro baseball history one player has been killed by a pitch and at least a dozen others have had their careers ended or greatly harmed by head injuries.

Pitching way inside is less common now than it used to be, presumably because umpires are quicker to throw out pitchers who are pitching dangerously but I think there has also been a cultural shift based on the knowledge that the pitcher could end the batter’s career. To give an example of the old days (meaning the’50s and 60s), Don Drysdale was famous for throwing at, or very close to, batters. He had some great quotes about the practice, and other players had great quotes about him. A few favorites are:

“The pitcher has to find out if the hitter is timid, and if he is timid, he has to remind the hitter that he’s timid.” – Don Drysdale

“My own little rule was two for one. If one of my teammates got knocked down, then I knocked down two on the other team.” – Don Drysdale

“Don Drysdale would consider an intentional walk a waste of three pitches.” – Mike Shannon.

“I hated to bat against Drysdale. After he hit you he’d come around, look at the bruise, and say ‘Do you want me to sign it?'” – Mickey Mantle

• D Kane says:

We began this side conversation with Bob writing:

> I boycott contact sports like football and ice hockey due to their intentionally violent

This struck me as virtue signalling. Lots of sports are violent, by design. Hit batsmen in baseball and slide tackle recipients in soccer are two obvious examples. Phil replies with:

> Pitching to intentionally hit an opposing player is rather rare

And that makes it OK? Bob seemed to be saying that some sports were “intentionally violent” and others were not. He boycotted the former. This is a very different from claiming that sports exist on a spectrum of violence and that there was some point along that spectrum beyond which Bob boycotted.

Daniel notes, correctly, that:

> it’s rare for intentional injuries to happen in soccer

I find the notion of intentionality to be interesting in this context. No pitcher who throws a beanball wants to permanently maim the batter. No football player tackling an opponent wants to permanently maim an opponent. And so on. (Of course, there are sociopaths in every sport.)

Bob seems to be implying the opposite, that, say, hockey players are intentionally trying to permanently injure their opponents. That seems an unfair slur. Does Bob know many hockey players?

• Bob does apparently know many hockey players, including one who “an opponent broke both his knees with a stick during a fight in a college game” (quoted from the original post). Also at one point Bob mentioned his father was a serious hockey player. So I assume Bob knows quite a few serious hockey players either directly or indirectly as friends of friends etc.

Fights in hockey are so common that the old adage goes “Last night I went to a boxing match and a hockey game broke out…”

https://en.wikipedia.org/wiki/Fighting_in_ice_hockey

Boxing, yep, intentionally injurious.

MMA, yep, trying to hurt people.

Football, absolutely players are sometimes trying to hit other players hard enough to take them out of the game.

Soccer, Baseball, Basketball, not so much.

• AllanC says:

Daniel,

The existence of fighting in hockey does not presuppose intent to injure. Although it is without doubt that some fights, just as some instances in all sports, involve participants who carry malice this is not the necessarily the norm.

Fighting in hockey has in the past existed for three primary reasons: show fighting (i.e. staged fights between heavyweights), energy fighting (e.g. to get the crowd / team into the game), and disciplinary fighting (e.g. teaching someone else a lesson).

The vast majority of fights are either staged fighting, which is mostly phased out of the game, or energy fighting. These fights usually do not involve participants who intend to injure one another. In fact, their goal is quite auxiliary to that; their actions do carry the possibility of injury of course but this is also true of the nature of the sport in general.

As far as disciplinary fighting is concerned, it usually ends before it begins. The norm is a skirmish ensues after a particular on-ice event and it ends before fighting ever begins and at worst a few roughing penalties are assessed. The escalation to fighting (where arguably the participants are intending to hurt one another) is actually rather rare.

There is also quite a big difference between intending to temporarily subdue / hurt and to permanently injure, which I think is an important distinction. In my experience, there are very few circumstances where a player in hockey would outright want to permanently injure someone else. To the extent that a player does intend to permanently injure on a consistent basis, I think that person just happened to be playing hockey and would otherwise have the same intent if they played other sports.

• I’m fine with saying there are different degrees of intent. A person fighting in hockey is unlikely to be actually trying to permanently disable or kill an opponent, whereas a couple of drug dealers in a shootout are… But I do think fighting in hockey sometimes intends to harm sufficiently to remove the person from the game, and such fights dramatically escalate the risk of more serious injuries, whether the more serious injuries are intended or not.

MMA is not the same thing as marines in combat, but it’s a damn sight more injurious than trying to outswim your opponent in your own lane…

In soccer contact is generally allowed, but a yellow card is given when contact is “without regard for the safety of an opponent” and a red card is given when “without regard for the safety of the opponent and with excessive force or brutality”. Yellow cards can be given for “dangerous play” where no contact occurred such as high kicking too close to other players. That’s dramatically different from hockey.

Tolerance of some level of excessive force or brutality is *built in* to hockey, american football, boxing, MMA, kickboxing, etc.

I’m not going to pass judgement on other sports, but I will say that if you are looking for a contact sport where intentional injury is generally not tolerated and penalized sufficiently to usually prevent it from occurring… soccer is a good choice.

• AllanC says:

We are largely in agreement then. But this brings us back to the point made by D Kane where sports exist on a spectrum of violence.

• Andrew says:

AllanC:

Sports do exist on a spectrum of violence. Based on his personal experience and his further reflection, Bob is not comfortable at the more violent end of this spectrum. The existence of a spectrum should not stop people from drawing the line somewhere!

• Terry says:

I can’t believe boxing is still legal. I don’t have any coherent moral argument, I just look at it and think “how can this be legal?”

• AllanC says:

Andrew,

See D Kane’s comment above which precipitated this side discussion between Daniel and myself

“And that makes it OK? Bob seemed to be saying that some sports were “intentionally violent” and others were not. He boycotted the former. This is a very different from claiming that sports exist on a spectrum of violence and that there was some point along that spectrum beyond which Bob boycotted.”

• Andrew says:

Allan:

I’ll defer to Bob here, but very quickly it’s my impression that there’s nothing in soccer comparable to a hockey player breaking both of someone’s knees with a stick. That said, we each have our own experiences and perspectives. This also doesn’t mean that the sport of hockey is inherently violent—maybe it’s just the pro or near-pro level that Bob’s friend experienced, and maybe things are better than they used to be. I don’t think that anything Bob said contradicts the idea that sports, as they are played, exist along a continuum of violence, intentional and otherwise.

• Phil says:

I said “Pitching to intentionally hit an opposing player is rather rare” and D Kane says ‘And that makes it OK?’

Different people feel differently, but yeah, I’m OK with pitching to hit but not maim an opposing player in circumstances in which that is considered acceptable under the current culture of baseball. In my opinion it used to be much too common but now it’s calibrated about right. I’d be fine with it going away altogether, too.

What does it mean to pitch to hit but not to maim? You don’t throw at or behind the batter’s head, you don’t throw at his knee, and you don’t throw it as hard as you can.

• Manoel Galdino says:

In Soccer, in general they don’t try to hurt other people on purpose. However, there are so many injuries that several places have problems late in life. Former player Ronaldo (Brazilian Ronaldo, who played for Real Madrid) said he can’t really climb (walk?) stairs

• Yeah, to the extent they have problems, I think it’s almost always in the knees, a problematic joint.

It seems like in current play, when a really serious injury occurs though it’s usually splashed all over the news, because it’s relatively rare and therefore sensational. For example Mohammed Salah dislocated a shoulder before the world cup… Some people speculated it was intentional on the part of Sergio Ramos… but Mo still played in world cup.

And recently Andre Gomes broken an ankle during a play involving Son Heung-min. Son was apparently inconsolable with regret for having tripped Gomes (I think Gomes broke his ankle when he planted the foot after Son tripped him up so it wasn’t even directly that Son collided and broke the ankle). If you are trying to hurt someone, it seems unlikely you’d be inconsolable with regret for having done it.

So I think it’s a very different game from Hockey or Football or Boxing.

2. mpledger says:

Like most things in rugby – it’s complicated.

There are three types of kicks for goal (penalties, field goals, conversions). Penalties are the most interesting. The referee awards a penalty at a particular position on the field and the kicker can choose to 1) shoot for goal, 2) field positions or 3) tip and run/pass. If shooting for goal, the player can move the ball directly backwards so that the player gets to weigh up the angle of kick and distance from goal. I would guess that a weaker team would attempt more penalty goals and from further away, so like golf, the average kick/putt success rate is not the same as the average success rate of kickers/putters.

3. Neil D says:

There has been a lot of studies of this in Australian Football. When taking a set shot the statistics for the player involved for the particular distance and angle are shown by the broadcaster-it might be say 5 to the left, 22 for a goal, and 7 to the right. A previous broadcaster showed the probability of a goal-I’m not sure of the methodology they used. A recent study, not using Bayesian methods, is by Anderson, Breed, Spittle, and Larkin (2018) “Factors affecting set shot goal-kicking performance in the Australian Football League” published in the journal Perceptual and Motor Skills and can be downloaded at https://www.researchgate.net/profile/Paul_Larkin2/publication/325662079_Factors_Affecting_Set_Shot_Goal-kicking_Performance_in_the_Australian_Football_League/links/5ca6a97892851c64bd50b414/Factors-Affecting-Set-Shot-Goal-kicking-Performance-in-the-Australian-Football-League.pdf

• Neil D says:

Just to add to my original comment, it was very surprising to see from the Anderson et al. paper how many more set shots were taken on the right side of the oval compared to the left (see their Figure 3), despite it being easier for a right footer to kick a goal on the left side of the oval. Of course, as in other sports, left footers are over-represented (about 20% in the AFL in 2015), and these days most AFL footballers can kick on both feet. I’m not sure of the explanation for the discrepancy, but maybe footballers are more confident of playing on and passing the ball accurately on the left side of the oval where they do not have to kick across their body.

4. Andrew says:

Bob:

Yeah, it seems like a mistake to fit the model using the logistic transform. Based on my golf modeling experience, I think a geometry-based model should work much better. The funny thing is, he begins his post with, “When you have a hammer, everything looks like a nail, right?” I’d recommend he program something directly in Stan rather than using brms or rstanarm, as this would give him the flexibility to fit a geometry-based model. To me, the natural way to set things up would be for there to be a single parameter, sigma, representing the standard deviation of the angle of the shot. Assuming the angular error has a normal(0, sigma) distribution, this directly gives Pr(success|distance,sigma). Then you can just let sigma vary by player, weather, etc. It would be simplest to use a linear model for log(sigma), I think.

Long considers “using a clog-log link rather than the logistic link, showing that it has better predictive accuracy under some conditions. I am going to ignore that advice for a few reasons, most importantly because the advantage is small and also because the clog-log link is computationally intractable with the software I’m using.” I think it would be better to start with a geometry-based model. Also, it’s really easy to write these models in Stan (you can use the golf case study as a template); really, software should not be driving your model choice here. I suspect a geometry-based model will do a lot better than polynomials!

• Andrew says:

Thinking about it more, there’s a big difference between golf putting and field goal kicking, and that is that there’s a maximum distance for a field goal. There’s only so far that someone can kick the ball with the required accuracy. So the model should reflect this: the difficulty of kicking the ball really far, and the decrease in the angular accuracy if you kick it really hard.

• Alex says:

Is this not also true for golf? That there’s only so far someone can putt with the required accuracy? If nothing else, at some point you would be putting from the rough or a green (or something else; I’m not a big golf fan) and the characteristics of the grass would change the conditions, I assume.

Plus, NFL kickers are getting more accurate from larger distances over time; there’s a link in Long’s blog post showing that kickers now are as accurate on 50 yard field goals as kickers in the 60s were on 20-30 yard field goals. I’m not going to say that there is no maximum distance for kicking accuracy, but it’s at least a moving target.

• I actually think putting is pretty similar, but it’s much more rare for the physical maximum quantity of kinetic energy you can put into the ball to be relevant for putting. I could see how some field goals you might be unable to kick it hard enough to reach the target.

• Jacob Long says:

I’d never seen the golf case study, but it’s fascinating to say the least. I hadn’t really thought about this general approach to modeling this kind of problem — of course my geometry wouldn’t have been up to snuff to innovate it anyway.

I decided to try to do a very simple FG model using the golf case study — I binned the kicks by distance much like the golf data and just plugged it into the same model, other than changing the values for the “radius” of the ball and field goal posts. It’s always possible I made some kind of error, but the simple one-parameter model does not fit the data well at all. I suspect it’s because there’s two other important aspects in kicking field goals that are absent from the putting problem:

1. Sometimes the challenge with a kick is not just trying to kick straight, but also kicking it far enough. This is maybe related to the aspect of putting that regards hitting the ball the right speed, except in kicking you don’t ever hit it too hard.

2. You also have to launch the ball at the proper angle, setting aside the matter of striking the ball hard enough. Hitting a long kick too high will make it fall short even if the raw leg power was there. Although I don’t know how significant it would be, on long kicks hitting the ball higher may also introduce more air resistance due to the added hangtime.

At any rate, the final “just for fun” model in the case study in which the overshot and distance_tolerance parameters are estimated rather than plugged in (I have no idea how to translate those concepts to this problem to come up with numbers myself) performs better. This model fits the data well for about 50 yards and then greatly overestimates everything longer than that.

I’m sure there’s a good, principled way to handle the extra complexities of kicking in this framework that ultimately outperforms a multilevel logit with many fewer parameters but it’s not as simple as I hoped.

• Andrew says:

Jacob:

Yeah, that makes sense. You’d want a model with some upper limit on how hard the kicker can kick the ball. I’m not quite sure what would be a good way to do this, but the effect on the model would be for Pr(success) to go all the way down to zero at some point (and to drop quickly right before that point), rather than seeing the sort of gentle asymptote that we see with logistic regression or with the golf model.

• Put a parameter for maximum initial velocity. Then for that max there is a velocity and vertical angle combination which leads to a trajectory. Then based on the trajectory it can either make it or miss. The larger the successful area in velocity,angle space the higher the prior probability to make it.

what you’d want to do is estimate this area in a pre computation, and then use that function as your prior probability to make it and further refine it based on data using a function with parameters that multiplies the base function.

• You could probably start with drag free, which you could do algebraically, but to get good results you’d probably need to do an ODE with drag as a function of Reynolds number. The Reynolds number depends on the ball speed mostly but also air properties that change with temperature and humidity I’m not sure if that has a noticeable effect but it might between say Green Bay vs Florida

5. Samuel says:

Just FYI – the first link to the Stan putting case study is non-functional (looks like hyperlink but clicking it does nothing).

6. Andre says:

These are both great examples, and I love how the golf putting case study builds off of geometric intuition than a stock statistical model. But what if I am given something, like, a pesticide for a certain species of tree. I have absolutely no intuition for how this pesticide works. There’s no first principals intuition”. It would make sense to have some spatial correlation, anyway. That is, trees nearby those pesticides applied are less likely to have canopy loss, which can be modeled with GP. As a first move, I tried hierarchies on species, but the model was horribly identified, indicated by a bunch of divergences. Just partial pooling as in the radon example didn’t help at all. I guess white ash trees and blue ash trees aren’t likely to vote similarly? The best information I have is the spatial component.

• Andrew says:

Andre:

But you actually should have a lot of information in this example! For one thing, if you think this pesticide works, you might have some biological model for it, or else some evidence that it works for similar trees. Conversely, if you have no prior reason at all to think the pesticide works, then you could consider this pesticide-tree combination to be one of many in a population obtained by trying some pesticide on some species of tree to see what happens—in which case this lack of information can be expressed as weak but not empty prior information. A hierarchical model is fine, and you can make the model more effective by including prior information on group-level mean and variance parameters. Spatial information is fine to use too: one advantage here is that if you have a lot of trees, then this represents internal variation which can help you fit the spatial model.

7. Rok Č. says:

Pro Football Focus (that work with most of the NFL teams) has also recently made a blog post on bayesian evaluation of kickers..They did only use the kicking distance, but it’s still a nice insight: https://twitter.com/PFF_Moo/status/1202294875774951424

It is behind the paywall, but most of the interesting charts can be seen in the tweet and the free part of the blog post. If anyone wants to read the entire blog post I am also happy to share a PDF to a few :)

8. jd says:

In the Stan golf putting case study, can someone explain why the Pr(|angle|<threshold angle)=2*Phi(threshold angle / sigma) – 1 ?

• I think Phi is the CDF of the normal distribution. So what’s the probability for a normal random variable to be in the range [-a,a]?

It’s Phi(a)-Phi(-a)

but due to symmetry:

Phi(-a) = 1-(Phi(a))

so it’s Phi(a) -(1-Phi(a)) = 2*Phi(a) – 1

9. Terry says:

In climate science, there is a similar application involving tree rings. Maybe skilled statisticians could improve the current state of the art there.

Width of tree rings have been used as a “proxy” for temperature because tree ring widths depend on temperature as well as other things.

Many papers have been written on tree ring proxies, but there is one part of the analyses that seems understudied. Tree ring widths decrease over the life of the tree due to the aging process and this non-temperature-related trend needs to be removed to isolate the temperature-related fluctuations. As I recall (its been a while) they model this aging trend mathematically (IIRC the age-related trend is modeled with a fairly simple equation). Since a lot of this work was done decades ago, the modeling was pretty crude.

So, is there a better way to remove this aging trend using more modern techniques?

Further, there is a deeper question of how much error the de-ageing analysis introduces into subsequent analysis of temperature trends. I have never seen this question addressed. If the de-ageing analysis is off, wouldn’t that leave a trend in the data that would be mistaken for a temperature-related trend? Indeed, how can you have confidence in computed temperature trends once you have de-trended the data for ageing? And if the ageing process is more complicated than the assumed trend, wouldn’t it introduce errors over at least portions of the time period?

It has been a long time since I looked at this, so I don’t know the current state of the art.

• Andrew says:

Terry:

Tree-ring analysis is really hard! I don’t know the current state of the art either. My colleagues and I published this paper back in 2015 laying out some of our struggles. I like where we were going in that paper, but we were not able to include enough data in our analysis to make strong conclusions.

• Terry says:

From Andrew’s paper:

However, in subsequent use the chronologies are treated as data and the standardization process does not feature in the climate reconstruction model. One consequence is that uncertainties from Steps (i) and (ii) above do not propagate through to the final predictions. Any incorrectly modeled variation from the standardization of the proxy measurements will be carried forward to the climate reconstruction

TS [Traditional Standardization] has two major problems. First, the uncertainty about parameters in the standardization is ignored in all further modeling. Second, incorrectly modeled variation in the standardization output can distort modeling of the climate signal, inducing bias in the reconstructed climate values. The problem is that TS effectively detrends each tree-ring series in an attempt to remove the nonclimatic growth. This can easily lead to the unintentional removal of climatic influences on growth from the tree-ring chronology (Cook et al. 1995), that is, some of the low-frequency climate “baby” can get thrown out with the nonclimate “bathwater.”

This second problem has been named the “segment length curse” (Cook et al. 1995): one can only expect to recover climate signals that have a higher frequency (in relation to climate cycles) than the inverse of segment length of individual tree-ring measurements. As low-frequency climate signal (and possibly some medium-frequency signal) is removed during standardization, the examination of changes in climate over hundreds, or thousands of years, becomes problematic. We use simulated temperature and tree-ring series to demonstrate the problem in S3.

Bingo!

I’m just a schmo, but I never understood how you could get reliable information about temperature trends after detrending the series, and I didn’t understood how you could get a reliable model of tree-ring growth for an individual tree.

Tree-ring chronologies were huge around 2000 in climate science. The IPCC featured them prominently. But nobody seemed to care very much about these issues.

• Terry,

If you took all your tree ring cores from trees that sprouted at approximately the same time, then age of the tree and time trends in CO2 would be confounded… But if you take cores from trees of different ages over a spread of several hundred years, then trees that are young and trees that are old are at different phases of their aging trends for the same calendar year. So the trend in aging should be identifiable separately from the CO2 trend in calendar years.

That being said, I think Bayesian models using uncertainty in aging related behavior and CO2 related behavior would be highly preferable to point estimates.

10. Hi Andrew,

For Australian Rules Football check out.

https://jamesblandecon.github.io/AFLAngleSetShots.html