Creation
The first thing I want you to do is consider an answer to a question. Mathematical scientists: what is the most interesting or elegant solution, derivation, or proof you have seen or executed and the problem it solved? Engineers: what is the most interesting or elegant thing you ever created and the problem it solved? I’ll give you a few seconds to quietly think of your answer.
Those on the end of each row, check the color of your sign. If it’s blue, raise your hand and point backward so everyone in your row can know your direction. If your sign is fuchsia, raise your hand and point forward so everyone in your row can see your direction. Again, blue backward, fuchsia forward. Those in the center, note your direction.
This might create a little chaos, but when I say “go,” turn around or lean forward and share your answer with a person or two in your vicinity. I’ll give you 1 minute total, so be quick. Go!
Thanks for sharing with one another!
Here’s my answer:
When I worked at NASA JPL in Pasadena, California, I was in a group tasked to solve the problem of collecting data at the surface of Venus. For those who don’t know, Venus is a very interesting, but inhospitable, place. Interesting, because it’s very similar to Earth in both size and location in the solar system. Inhospitable, because of the
- very dense CO2 atmosphere,
- surface temperature of 460 °C and
- surface pressure of 92 Earth atmospheres.
Oh, and the sulfuric acid droplets in the “air.” If the temperature and pressure don’t kill you, the sulfuric acid will!
**** Venus slide here. ****
It’s difficult for any machine to survive at the surface of Venus, although the Soviet Union’s Venera 9 did for a grand total of 53 minutes in 1975, sending back the first image taken on the surface of another planet. The first color images were taken by Venera 13.
**** Venera slide here. ****
**** Venus surface slide here. ****
The solution we designed was a gold-plated zylon ballon system that would fly about 50 km above the surface of Venus, descending periodically to take photos and collect in-situ data at or near the surface.
The machine used a mixture of helium and water whose boiling and condensation would change system buoyancy and, therefore, altitude. We called it a “phase-change fluid” (or PCF) balloon system.
**** PCF balloon system slide here. ****
We tested a series of prototype machines in Earth’s atmosphere, launching from Pasadena and chasing the balloons eastward.
**** Photos slide here. ****
Engineers call the process of creating new things to solve problems design: deciding sizes, shapes, and materials for pieces in a system or machine. We engineers design with materials and energy from the world around us, combining them in original and novel ways. We at JPL didn’t invent polymers, gold, helium, or water, but we combined them in new ways to be the balloon envelope, the buoyant gas, and the phase-change fluid of our system to create a new machine to solve the problem of obtaining data in the inhospitable environment at the surface of Venus. In the process, our Earth-based prototype became the first, and to date only, machine of this type to fly successfully.
Where mathematics and engineering collide
The next thing I want to discuss is how engineers go about this designing. To do so, I’ll share another story. Randy Pruim is the keynote speaker tomorrow, and, in case you’re worried, he knows I’m about to do this.
More than 10 years ago, we began collaborating on this paper published in the journal Energies. The paper contains this figure.
**** Paper and figure slide. ****
I’ll zoom in so you can see more clearly. Historical observations of GDP are indexed to 1 in 1960 and shown as a dashed line. Fitted model data are shown as a solid line. During our work together, we disagreed about the right way to calculate the sum of squared errors (SSE) between observations and a model or between a model and observations.
Without yet saying who held which view, I’ll show you the two options we debated. And by the way, although I remember this disagreement in vivid detail more than a decade later, I’m not bitter. And, as David Letterman used to say, this only an exhibition; this is not a competition. Please, no wagering on who was right.
Option A is this: the sum of squared errors is the sum over all historical observations (i) of the square of historical GDP less fitted GDP.
**** Add option A equation here. ****
Option B is the sum over all historical observations of the square of fitted GDP less historical GDP.
**** Add option B equation here. ****
I also include versions of the equations with the typical nomenclature found in textbooks: y and y-hat.
**** Add additional equations here. ****
The question to you is this: which option is right, Option A or Option B, and why? And no fair sitting on the fence. You can’t say “it doesn’t matter,” because it does! You must choose one or the other, A or B!
I’ll give you a few quiet seconds to decide.
Now comes the second thing I want you to do today. Not yet, but when I say “go,” turn around or lean forward and share your answers. If you find you disagree, make the case that your answer is the right answer. If you agree, develop arguments for your position and against the other option. Take 1 minute to discuss with the same conversation partners as before. Go!
Thanks for having this discussion! Perhaps you can use this question as a conversation starter for those awkward times when you’re early to a session and find yourself next to someone you don’t know.
Before I share more about the disagreement between Randy and me, I’ll share why I think it matters.
The subtrahend (the term after the minus sign) is the thing you consider real or correct or true. In contrast, the minuend (the term before the minus sign) is the thing you are testing.
Option A implies that the true GDP is represented by the fitted model. The historical observations are an imperfect and imprecise measurement of true GDP. When our observations are too high, we have positive measurement error. When our observations are too low, we have negative measurement error. If we had better observations, they would conform to the truth of the model. For Option A, the model is the thing that’s real.
In contrast, Option B suggests that the only thing we know is our observations in the concrete and tangible world. Abstract fitted models are meant to describe our concrete observations, not the other way around. When the fitted model is greater than an historical observation, our model is overpredicting GDP, and we have positive model error. If the model prediction is less than an observation in any year, our model is underpredicting GDP, and we have negative model error. For Option B, the observations are the thing that’s real.