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Some comments:
I definitely want to see paragraphs in your solutions; you need to present a description, an argument, a suggestion, or something which shows that you've thought about the results (which most of you obtained and presented). So most of you did the "Carry out" part, but not as many of you really did the "Evaluate" part.
It's a moment to criticize; to gripe; to propose; to scratch your head; to be amazed, stunned, saddened, delighted, horrified, puzzled, ....
But it's definitely not a moment for silence.
Sometimes it seemed like I was hearing the announcer telling me where the ball is, and who has the ball -- without telling why the pitcher released it, why the batter hit it with the wooden stick, why the fielder chased it, why the fielder threw it to second base, why the second baseman whacked the runner with it, and so on.
I wanted to hear it from the manager's perspective -- and you're the manager!
One purpose of models (e.g. the enormous climate models that are used to guide policy on climate change) is to simulate conditions given certain inputs -- to predict the climate in 2050, or 2100, or however many years down the line. Often changes to parameters are made, then a simulation is run -- so we change the impact of cloud cover, or we introduce a new process (e.g. ocean circulation). Then we see how the predicted climate responds.
We can simulate Wood County weather, once we have a model for it, and then see if the distribution of extreme years matches up well with Fletcher's distribution. If we use a model with a linear trend over time, then we're saying "climate change"; if we use a model with no linear trend, then we're saying "no climate change".
I also describe the process of randomization, which is essential for generating weather. We want to generate 1000 "realizations" of Wood County weather, then see whether Fletcher's extreme year distribution is outlandish (or not), given an assumption of no climate change.
MinMins are really in line with our expectation; maxmins look like they're making a move in our direction; min maxes seem confused, and maxmax years have never gotten over the 1930s....
Meanwhile, in max land, increasing cloud cover over land may be supressing max temperatures, keeping them down.
So we simulate the weather, and see if the Fletcher results fall within a "95% confidence" envelope provided by the model.
If you want to simulate a normal distribution, you need to know the mean $\mu$ and the standard deviation $\sigma$, and away you go. Simulations of normals are based on chosing random numbers from a uniform distribution (which computers can do pretty well).
Computers are good at choosing uniform random numbers, but even you can do a good job at this, with just a (fair) coin. Here it is, in four easy steps:
I threw "1001101000001101001110110110001010000110100000100100111010011111010011010101011110011100011110000000" (okay, to be honest I let the computer throw that...), which is 762827007763806961434403391360 in base 10 (I let the computer do that, too -- I'm feeling a little sheepish about this whole thing now).
So my random number is 0.6017644038715926 (I even expected the computer to do the division -- you'd think that I could do something here, and division is something that I learned in elementary school....).
Once we have our sample size of uniform random numbers, we'll use those and the cumulative distribution for a normal (which runs from 0 to 1), and work backwards:
(an inverse problem; you can solve that with Newton's method, for example: find $z$ such that $f(z)=u-normalCDF(0,1,z)=0$).
We now have a mathematical model for a sample from a random distribution.
So here's one approach:
Empirical PDF -- Probability Density Function -- for August 15th Minima
with a suggested normal-distribution overlay. Maybe the data isn't normal, but that's just to give you the idea. |
We might want to create a theoretical distribution from the empirical data -- otherwise we'd never be able to exceed the extremes of the data (to "create new records").
Empirical CDF -- Cumulative Distribution Function -- for August 15th Minima
with a normal cdf overlay (I just used the mean and the standard deviation of the empirical data to estimate the normal). Maybe the data isn't normal, but this is just to give you the idea. By the way, for the randomin
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But one way or another we'd need a distribution to sample from.
Then we address (and hopefully answer) the question: is Fletcher's result possible, given an assumption -- a null hypothesis -- of no climate change?
And all along I've been assuming the alternative hypothesis: "Nope!"
Well, actually, there's a problem with it. Weather is correlated, day by day:
Variance of mins for days a given number of "lags"
(days) apart (up to 10). This graph is really just the
"nose" on the graph at right -- we zoom in on the first
few lags, to see how the variance dives down for mins
for days that are within a few days of each
other.
We think of a "lag" as a delay -- it means we're comparing days lagging by 1, by 2, by 3 days from each other, and so on (up to 10 days apart in this graph).
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Variance for "lags" up to 365 days apart --
i.e. one year.
You may all have heard that you can't trust a weather prediction more than about a week out, and what you're seeing here is a picture of why that is: you can't trust a day 7 days out more than you can a day a year later -- i.e., "what's it usually doing on April 24th?". |
What this tells us is that tomorrow's temps are more similar (variance about 1 for minima one day apart) than those 10 days apart (variance 2.7 or so). | The graph tells us the obvious: there's a lot of variance between temperatures a half year apart (you may have a winter min compared to a summer min); and that, since temperatures are essentially periodic, temperatures about a year later are also very similar. |
So what we notice is that, after about a week, the correlation between two mins a week apart is about the same as the correlation between two mins exactly a year apart. A year apart is really "seasonal correlation" (more "climatic", if you will); the correlation for mins just a few days apart is some weather system moving through.... Which has an effect of no more than about a week. |
Also, be on the look-out for any outliers. Take a look at the data, to see if there are any funny looking data that we should investigate, too....