On the Height of a Field

This is a short story about belief and evidence, and it starts with the GPS watch I use when I go for a run. Here’s the plot of my elevation today:

runElevation

It looks a little odd until I show you this map of the run:

runMap

Each bump on the elevation plot is one lap of the field. In the middle, I changed directions, giving the elevation chart an approximate mirror-image symmetry. (I don’t know what causes the aberrant spikes, but my friend reports seeing the same thing on his watch.)

According to the GPS data, the field is sloped, with a max height of 260 feet near the center field wall and 245 feet near home plate. It’s insistent on this point, reiterating these numbers each time I do the run (except once when the tracking data was clearly off, showing me running across parking lots and through nearby buildings.) I disagreed, though. The field looked flat, not sloped at 3 degrees. I was disappointed to have found a systematic bias in the GPS data.

But I occasionally thought of some minor consideration that impacted my belief. I remembered that when I went biking, I often found that roads that look flat are actually uphill, as can be verified by changing directions and feeling how much easier it becomes to go a given pace. I Googled for the accuracy of GPS elevation data, and found that it’s only good to about 10 meters. But I didn’t care about absolute elevation, only change across the field, and I couldn’t find any answers on the accuracy of that. (Quora failed me.) I checked Google Earth, and it corroborated the GPS, saying the ground was 241 ft behind home plate and 259 in deep center field. But then I read that the GPS calibrated its elevation reading by comparing latitude/longitude coordinates with a database, and so may have been drawing from the same source as Google Earth.

People wouldn’t make a sloped baseball field, would they? That would dramatically change the way it plays, since with a 15-foot gain, what was once a solid home run becomes a catch on the warning track. Googling some more, I found that baseball fields can be pretty sloped; the requirements are fairly lax, and in fact they are typically sloped to allow drainage.

I was starting to doubt my initial judgment, and with this in mind, when I looked at the field, it made more and more sense that it’s sloped. Along the right field fence, there’s a short, steep hill leading up to the street. It’s about five feet high and at least a 30-degree slope. It’s completely unnatural, as if it exists because the field as a whole used to be considerably more sloped, but was dug out and flattened. The high edge of the field was then below street level, so there’s that short, steep hill leading up. And if the field was dug out and flattened, maybe they didn’t flatten it all the way. The entire campus is certainly sloped the same general direction as the GPS claimed for the field. It drops about 70 feet from north to south, and it’s frequently noticeable as you walk or bike around. There’s another field I run on with essentially the same deal, and I found that when I knew what to look for, I could indeed see the slope there.

Eventually, the speculation built up enough to warrant a little effort to make a measurement. I asked a wise man what to do, and he suggested I find a protractor, hang a string down to detect gravity, and site from one side of the field to the other. I did so, expecting to feel the boldness of an impartial, truth-seeking scientific investigator as I strode across the grass. That wasn’t what I got at all.

First, I felt continuous fluctuations in my confidence. “I’m 60% confident I’ll find the field is sloped,” I told myself, then immediately changed it to 75, not wanting to be timid, then felt afraid of being wrong, and went back to 50. I’ve played The Calibration Game and learned what beliefs mean, and mostly what it’s done is give me the ability to not only be uncertain about things, but to be meta-uncertain as well – not sure just how uncertain I am, since I don’t want to be wrong about that!

Second, I felt conflicting desires. I couldn’t decide what I wanted the result to be. I wanted the field to be flat to validate my initial intuition, not the stupid GPS, but I also wanted the field to be sloped so I could prove to myself my ability to change my beliefs when the evidence comes in, even if it goes against my ego. (A strange side-effect of wanting to believe true things is that you find yourself wanting to do things not because they help you believe the truth, but because you perceive them to be the sort of things that truth-seekers would do.) I recalled a video I had seen years ago about Gravity Probe B, and the main thing I remembered from it was a scientist with long, gray hair and huge unblinking eyeballs explaining in perfect monotone that he didn’t have a desire for the experiment to confirm or refute general relativity; he only wanted it to show what reality was like.

On top of all this, there was the sense of irony at so much mental gymnastics over a triviality like the slope of a baseball field, and the self-consciousness at the absurdity of standing around in the cold pointing jerry-rigged protractors at things. So at last I crossed the field and lined up my protractor for the moment of truth

It didn’t work. I had placed my shoes down on the grass as a target to site, but from center field they were hidden behind the pitcher’s mound. I recrossed the field and adjusted them, and went back. I still couldn’t see the shoes; they were too small and hidden in the grass. I could see my backpack, though, so I sited off that. But it still didn’t really work. I didn’t have a protractor on hand, so I had printed out the image of one from Wikipedia and stapled it to a piece of cardboard, but the cardboard wasn’t very flat, making siting along it to good accuracy essentially impossible.

I scrapped that, and after a few days went to Walgreens and found a cheap plastic protractor and some twine that I used to tie in my water bottle as a plumb bob. Returning to the field, I finally found the device to be, well, marginal. Holding it up to my eye, it was impossible to focus along the entire top of the protractor at once, and difficult to establish unambiguous criteria for when the protractor was accurately aimed. I was also holding the entire thing up with my hands, and trying to keep the string in place between siting along the protractor and moving my head around to get the reading.

Nonetheless, my reading came to 87 degrees from center field to home plate and 90 degrees from home plate back to center field. This three-degree difference seemed pretty good confirmation of the GPS data. In a final attempt to confirm my readings, I repeated the experiment in a hallway outside my office, which I hope is essentially flat. It’s 90 strides long, (and I’m about two strides tall) and I found 88 degrees from each side, roughly confirming that the protractor readings matched my expectations. (I’d have used the swimming pool, which I know is flat, but it’s closed at the moment.)

I’m now strongly confident that the baseball field is sloped – something around 95% after considering all the points in this post. That’s enough that I don’t care to keep investigating further with better devices, unless maybe someone I know turns out to have one sitting around.

Still, there is some doubt. Couldn’t I have subconsciously adjusted my protractor to find what I expected? There were plenty of ways to mess it up. What if I had found no slope with the protractor? Would I have accepted it as settling the issue, or would I have been more likely to doubt my readings? It’s perfectly rational to doubt an instrument more when it gives results you don’t expect – you certainly shouldn’t trust a thermometer that says your temperature is 130 degrees – but it still feels intuitively a bit wrong to say the protractor is more likely to be a good tool when it confirms what I already suspected.

The story of how belief is supposed to work is that for each bit of evidence, you consider its likelihood under all the various hypotheses, then multiplying these likelihoods, you find your final result, and it tells you exactly how confident you should be. If I can estimate how likely it is for Google Maps and my GPS to corroborate each other given that they are wrong, and how likely it is given that they are right, and then answer the same question for every other bit of evidence available to me, I don’t need to estimate my final beliefs – I calculate them. But even in this simple testbed of the matter of a sloped baseball field, I could feel my biases coming to bear on what evidence I considered, and how strong and relevant that evidence seemed to me.  The more I believed the baseball field was sloped, the more relevant (higher likelihood ratio) it seemed that there was that short steep hill on the side, and the less relevant that my intuition claimed the field was flat. The field even began looking more sloped to me as time went on, and I sometimes thought I could feel the slope as I ran, even though I never had before.

That’s what I was interested in here. I wanted to know more about the way my feelings and beliefs interacted with the evidence and with my methods of collecting it. It is common knowledge that people are likely to find what they’re looking for whatever the facts, but what does it feel like when you’re in the middle of doing this, and can recognizing that feeling lead you to stop?

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12 Responses to “On the Height of a Field”

  1. Paul Murray Says:

    So, how do surveyors determine flatness?

  2. badger Says:

    Linked to on Less Wrong discussion.

  3. Mark Eichenlaub Says:

    Thanks for the link!

  4. blu28 Says:

    GPS is generally much more accurate for relative positioning then for absolute positioning, just because a whole class of error sources are made irrelevant. However, your GPS unit will generally arrive at a different absolute location when it changes satellite constellations, so if there are different set of satellites visible, when the change occurs it will negate the accuracy of the relative positions as well.

    But it should be obvious that if you get very consistent results like this that are invariant over time, there is no way that this could be accounted for by any sort of error. The field would have to be sloped. The only question is how much of a slope. So you could take several measurements at the bases and calculate the reported distance between them and then use a string to measure the actual distance. That would give you the calibration you need to determine the actual slope. My guess is that the calibration factor will turn out to be one. That is, that the device is accurate.

  5. Mark Eichenlaub Says:

    Thanks for the info, blu. I thought it was possible the GPS was consistently in error because it was taking its data from the same database over and over.

  6. gwern Says:

    Paul, AFAIK they do it basically the same – measuring angles – but visually: http://en.wikipedia.org/wiki/Theodolite (wouldn’t surprise me if there were lasers involved these days).

    > People wouldn’t make a sloped baseball field, would they? That would dramatically change the way it plays, since with a 15-foot gain, what was once a solid home run becomes a catch on the warning track. Googling some more, I found that baseball fields can be pretty sloped; the requirements are fairly lax, and in fact they are typically sloped to allow drainage.

    Sure they would. Nate Silver remarks somewhere in his book, IIRC, that it is a vital correction in baseball statistics to account for where a particular home run was hit, because some stadiums are much smaller and hence much easier to hit home runs out of. And if that is true for Big Baseball, how much more so for low-level or amateur fields?

  7. Mark Eichenlaub Says:

    Thanks, Gwern. I knew that different ballparks are different sizes (the distances are written in big numbers on the fences), but I hadn’t heard of different ballparks having different elevation gains before.

  8. Paul Crowley Says:

    Do modern GPS systems augment the position they get from the satellites with some sort of database-based guess, the same way they sometimes try to “snap” to streets? If so, you’re right that that could cause problems with this measurement. Otherwise the only way the GPS could be in error about this is if its idea of where sea level is is wrong, but if it were *that* wrong over such a short distance, the error over longer distances would be ridiculously large.

  9. Mark Eichenlaub Says:

    Yes, from the website for the GPS, it looks to me like they do compare to a database when recording heights.

  10. Matt Springer Says:

    At least one hiking-style GPS I’ve used also has a little pressure altimeter which it uses to supplement the actual GPS height data.

    Offhand I don’t know what the relative accuracy is for unassisted GPS, but it should be an easy and interesting exercise to find a known change in height measured with a tape measure and systematically use the GPS to make the same measurement. Then you can redo this on different days, and see how consistent it is.

  11. Anonymous Says:

    Somthing that would be easier is to borrow a laser level, then measure the distance to ground.

  12. Maxwellfire Says:

    How about taking a 5 gallon bucket of water, and dumping it on the field, then seeing where the water goes and how fast. That should pretty much tell you if it is angled.

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