C.G.P.Grey lays out the problems with the U.S.A’s current “first past the post/winner take all” voting system and then explains how to fix them:
The problems:
Minority rule: You can win with less than half the vote (at least in the beginning).
Inevitable, unavoidable, two-party system evolves over time.
Third-parties become spoilers; the better a third-party does the worse it is for their voters because it takes away votes from the other party they would favor.
Here I’ll go over how to do Least Squares Regression, as simply as possibly, using Excel and its Solver*. Least Squares Regression can be used to match pretty much any type of function to any type of data. Most spreadsheet programs, like Excel, will do some curve matching for you when you add trendlines to graphs, but for more sophisticated work — in pre-Calculus and beyond for example — you need a more general approach.
Figure 1. Selected annual average carbon dioxide concentrations since 1959 (see Table 1 for data). Data from NOAA.
We’ll start with a data set that we want to match. The atmospheric CO2 data from Mauna Loa is a convenient dataset. It’s also an important data set to model because matching a function to the data will allow us to predict the change in CO2 over the next 100 years, and those predictions are the ones the IPCC uses in their estimates of the impact of global climate change around the world. Billions, if not trillions of dollars depend on those predictions (and how governments decide to respond). For ease of demonstration, I’ve picked a few of the annual average CO2 concentration measurements at random to work with:
Table 1: CO2 Data
Year (x)
CO2 concentration (ppm) (y)
1959
315.97
1964
319.62
1969
324.62
1981
340.10
1985
346.04
1996
362.59
2003
375.77
Now, looking at the data, we can see there is some sort of trend. Our first decision is about how to model it.
Straight Line Model
The simplest approach, and the one we’ll try first is to fit a straight line through the data. (The final spreadsheet is here).
The equation of a straight line is:
In this case, as we can see from the graph (Figure 1), the y axis is CO2 concentration, and the x axis is the year. m and b are our two unknown constants that set the slope of the line (m) and move it up and down (b). If you need to get a better feel for what this means, try changing the slope and intercept in the Straight Line Grapher (I used this in class to demonstrate).
To match our straight line model to the data we need to determine the values of m and b that give the best fit.
Figure 2. The formula ("=C$3*$A8+C$4") for our straight line model uses the year (column A) and the m and b coefficients (cells C3 and C4 respectively) . You'll note the $ signs in the C3 and C4 references in the formula (i.e. $C$3 and $C$4); these tell Excel to always refer to these specific cells when the formula is copied and pasted down the entire column.
So we create a spreadsheet with the data and in the adjacent column set up the straight line function by setting two cells to the values of the constants (m and b) and using those values to calculate the modeled CO2 concentration.
Figure 3. The initial values for m and b (1 and -1600 respectively) don't match the data very well.
You’ll notice that I have values of m = 1 and b = -1600 . These are just my initial estimates of these values. The initial values are not crucial, as you’ll see, but are just there for me to check that my formula is in right.
Once I have the correct formulas in, I can play around with these values until my line matches the data. However, this is where Solver comes in to save a lot of time.
Finding a match
First we need a quantitative way of telling if we have a good match or not. We can start by taking the difference between each real data point and the modeled value. We’ll call this the error.
Figure 4. Calculating the error -- the difference between the actual and modeled data.
Now we could get a single value for the total error by adding up all the individual error values, or taking the average. However, as one of my students pointed out, we could end up with a case where the modeled line crossed through the data and we’d end up with positive differences of the data points above the line canceling out the negative differences of the data points below the line. His solution was to take the absolute value of the differences (errors) instead, which should actually work just as well in the method we’re taking here.
Instead of using the absolute value, however, we’ll square the errors instead. This achieves the same effect we need because the squares of both negative and positive numbers are positive. This approach is where the “squares” in Least Squares Regression comes from. The “Least” part comes from the fact that we’re now going to try adjusting our coefficients until we get the average of the squares of the errors to be as small as possible.
Figure 5. Calculating the square of the error.
Now we take the calculate the average of the errors (the sum of the errors would work just as well) using the spreadsheet’s “AVERAGE” function.
Figure 6. Calculating the average error using the "AVERAGE" function.
Now we can adjust m and b and not just see how they compare to the data points on a graph, but know that we have the best fit if we minimize the average error.
Solver
Instead of changing m and b by hand, we can use Excel’s Solver to minimize the average error for us. The matching of a straight line can be done using algebra (e.g. here and here) but that approach won’t help us when we get to more complex functions.
We select the average error cell (E16) and tell Solver to minimize its value by changing the values of m and b.
Figure 7. Setting up Excel's Solver.
When we hit solve, Solver should converge on a solution. Because of the way the problem is set up — using the square of the error for example — makes this a non-linear problem for the solver to solve. As of this writing, Excel is the only spreadsheet program I know of that has a built-in, non-linear solver.
Figure 8. The Solver solution.
You’ll notice that Solver’s solution gives:
m = 1.360
b = -2351.9
So now we have the equation for the best fit line (our model) being:
Prediction
Using this model we can predict the atmospheric CO2 concentration for the year 2050 by setting x = 2050 in the modeled equation, which gives 436.5 ppm.
Figure 9. The straight line model matches the data very well and can be projected to predict atmospheric carbon dioxide concentrations in the future.
The final spreadsheet I used to do these calculations can be found here.
Parabolic Model
As good as the straight line model seems to be, it does not account for the slight upward curve the data seems to have (see Figure 1). So instead of a straight line, perhaps we could try a parabolic function. Just as before, we’ll need to figure out the coefficients for the parabolic function. In this case the general equation we’ll use is:
and we’ll need to find the coefficients a, b and c. To see how changing these coefficients change the curve, you can play around with the interactive parabola model.
We set up our spreadsheet in the same way as for the straight line, but with our new coefficients.
Figure 10. Setting the equation for a parabolic model.
Note that the only column that needs to change from the straight-line spreadsheet is the “Model” column (column C), to add the coefficients and to change the formula to that of a parabola.
Now we can use Solver to minimize the average of the squares of the errors, just as we did before, only having it change the three coefficients instead of two.
a = 0.00060091
b = -1.018
c = 0.9933
and a final equation:
Figure 11. Matching data using a parabolic function.
Prediction
The parabolic model predicts for the year 2050 (x = 2050) that the CO2 concentration will be 439.4 ppm.
The Excel spreadsheet for the parabolic model is here.
Which model is better?
The easiest way to compare the two models is to see which did better at minimizing the squared errors.
Table 2: Comparison of models.
Model
Average Squared Error
Straight Line
4.94
Parabola
4.47
So the parabolic model does better, but not by much.
Conclusions
With this approach, you can use any type of function — exponential, sinusoidal etc. — or combination of functions to match any data, and be able to compare among the models. A better measure of how well the models match the data is the regression coefficient or coefficient of determination, but I’ll save those for another post.
Conclusions
* UPDATE (Dec 31, 2012): Gnumeric’s Solver works.
* You’ll need to use Microsoft’s Excel, or better yet Gnumeric for this because, as of this writing, none of the other common spreadsheet options — OpenOffice, Google Docs, and Mac’s Calc — have a non-linear solver built in — and even some of the newer versions of Excel seem buggy.
Perhaps the key reason for the profound influence of Darwin’s “On the Origin of Species” is that it’s such a well written and well reasoned argument based on years of study. It is a wonderful example of how science should be done, and how it should be presented. In the past I’ve had my middle schoolers try to translate sections of Darwin’s writing into plainer, more modern English, with some very good results. They pick up a lot of vocabulary, and are introduced to longer, more complex sentences that are, however, clearly written.
Diagram and notes on the bird species P.Nanus from The Zoology of the Voyage of H.M.S. Beagle, Part 3: Birds by J. Gould and G.R. Gray (edited by C.Darwin). Image via Darwin Online.
The text of “On the Origin of Species” is available for free from the Gutenberg library. Images of the original document can be found (also for free) at the UK website, Darwin Online (which also includes the Darwin’s annotated copy). Darwin Online also hosts lot of Darwin’s other works, as well as notes of the other scientists on The Beagle, among which is included some wonderful scientific diagrams.
This year, I’m going to have the middle schoolers read the introduction, while the honors environmental science students will read selected chapters and present to the class — this will be their off-block assignment.
Diagram of the fish Cofsyphus Darwini by L. Jenyns in The Zoology of the Voyage of H.M.S. Beagle, Part 4: Fish (edited by C.Darwin). Image via Darwin Online.. .
Pepper plant responding to being watered. These images were taken over the course of two hours.
During class on Friday, I watered my Chinese five-color hot pepper plant for the first time in three days. It responded quite well, helping to illustrate one reason (to maintain their rigidity/prevent wilting) why plants need water. I did this because I was curious about how fast plants respond to water, and with the data from the images I should be able to demonstrate what a scientific report should look like.
The full plant’s response:
Notes
The original camera images were cropped for the gif-animation using Imagemagick’s convert
convert $i -crop 500x400+1550+1100 crop-$i
The image file sequences were converted to mp4 video using ffmpeg (instructions here):
Dan Ariely concludes (video by RSA) that making people think about morality increases the likelihood that they’ll act honestly.
People try to balance the benefiting they gain from cheating against being able to feel good about themselves by being honest. While very few people tend to cheat a lot, many people cheat a little and self-rationalize their dishonesty.
Our school has adopted a short honor code that we’ll ask students to write at the top of tests and other assignments that is intended to remind them of their moral obligations.
Based on one of Ariely’s other conclusions, I’m also considering having students confess their in-class transgressions — talking out of turn; improper use of technology — every month or so, since this type of thing also seems to encourage probity.
The introduction of snakes to Guam has reverberated through the ecosystem.
Accidentally introduced to the island in the 1940s, the snake decimated the island’s native bird species in one of the most infamous ecological disasters from an invasive species.
By the 1980s, 10 of 12 native bird species had been wiped out.
Since many birds consume spiders, compete with spiders for insect prey and utilize spider webs in their nests, their loss has led to a spider explosion on the island, researchers said.
Note (for the Algebra students): The scientific article includes a nice box and whisker plot showing how many more spiderwebs there are on Guam compared to other islands.
Number of spider webs on different islands. Guam is the only island shown that has had a severe reduction in birds. Image from Rogers et al., 2012.
Changing colors of universal indicator show how blowing bubbles acidifies water (light green-second beaker) from neutral pH (dark green-third beaker) standard. For comparison, the first beaker (red) is acidified while the last beaker (blue) is made alkaline.
CO2 + H2O —-> H2CO3
This useful little reaction, where carbon dioxide reacts with water to produce carbonic acid, came up in my middle school class when we talked about respiration, it’ll come up soon in environmental science with the effects of carbon dioxide on the oceans (acidification), and it offers the opportunity to discuss pH and balancing chemical reactions in chemistry.
The middle school class did the neat little experiment where students blow bubbles in water (through a straw), and the carbon dioxide in their breath reacts with the water to slightly acidify it. A little universal pH indicator in the water (or even cabbage juice indicator) shows the acidification pretty well if you make sure to keep a standard nearby so students can see the change in color.
The fact that the CO2 in your breath is enough to acidify water begs the question — which was asked — how much of the air you exhale is carbon dioxide? According to the Oak Ridge Carbon Dioxide Information Analysis Center’s FAQ page, it’s concentration is about 3.7% by volume. Which is a lot more than the 0.04% average of the atmosphere.
Of course if you really want to talk about the pH you need to get into the acid equilibrium and the dissociation of the carbonic acid to produce H+ ions; you can get the these details here.
My students are researching the organisms they collected from the creek, and I was outlining the types of information I wanted them to find. We were talking about how many animals have seasonal reproductive cycles, and I pointed out that plants flower seasonally as well. One of my students put two and two together and came up with something close to a whole number: “You mean to say that flowers are … some sort of … creepy … sexual things?”
