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    Statistics Of Doom

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    I have started a new github site where all the materials for courses will appear, to make it easier for you to find everything you need.

    I have provided entire courses for you to take yourself, use for your classroom, etc. If you are an instructor and want to check out the answer keys, please drop me a line by using the email icon at the bottom of the screen.

    The Year of the Thesis! Just wanted to highlight several publications from this year, which were mostly theses from some fabulous young researchers: Scofield, J.

    How the presence of others affects desirability judgments in heterosexual and homosexual participants.

    Investigating the interaction of direct and indirect relation on memory judgments and retrieval. And not chaotic. And not interesting. So we need something to keep it moving.

    The equation that results note 1 has the massive number of three variables — position, speed and now time to keep track of the driving up and down of the pivot point.

    Three variables seems to be the minimum to create a chaotic system note 2. This is typical of chaotic systems — certain parameter values or combinations of parameters can move the system between quite different states.

    As we increase the timespan of the simulation the statistics of two slightly different initial conditions become more alike.

    But if we look at the statistics of the results we might find that they are very predictable. This is typical of many but not all chaotic systems.

    The orbits of the planets in the solar system are chaotic. In fact, even 3-body systems moving under gravitational attraction have chaotic behavior.

    So how did we land a man on the moon? This raises the interesting questions of timescales and amount of variation. Therefore, in principle, the Solar system can be chaotic, but not necessarily this implies events such as collisions or escaping planets..

    Such variations are not large enough to provoke catastrophic events before extremely large time. Just to round out the picture a little, even if a system is not chaotic and is deterministic we might lack sufficient knowledge to be able to make useful predictions.

    If you take a look at figure 3 in Ensemble Forecasting you can see that with some uncertainty of the initial velocity and a key parameter the resulting velocity of an extremely simple system has quite a large uncertainty associated with it.

    This case is quantitively different of course. By obtaining more accurate values of the starting conditions and the key parameters we can reduce our uncertainty.

    Many chaotic systems have deterministic statistics. Other chaotic systems can be intransitive. That is, for a very slight change in initial conditions we can have a different set of long term statistics.

    Lorenz gives a good example. Lorenz introduces the concept of almost intransitive systems. Note 2 — This is true for continuous systems.

    Discrete systems can be chaotic with less parameters. Climate sensitivity is all about trying to discover whether the climate system has positive or negative feedback.

    A hotter planet should radiate more. Suppose the flux increased by 0. That is, the planet heated up but there was no increase in energy radiated to space.

    In this case it would indicate negative feedback within the climate system. Consider the extreme case where as the planet warms up it actually radiates less energy to space — clearly this will lead to runaway temperature increases less energy radiated means more energy absorbed, which increased temperatures, which leads to even less energy radiated..

    As a note for non-mathematicians, there is nothing inherently wrong with this, but it just makes each paper confusing especially for newcomers and probably for everyone.

    The model is a very simple 1-dimensional model of temperature deviation into the ocean mixed layer, from the first law of thermodynamics:.

    T is average surface temperature, which is measured around the planet on a frequent basis. The forcing f is, for the purposes of this exercise, defined as something added into the system which we believe we can understand and estimate or measure.

    For the purposes of this exercise it is not feedback. Feedback includes clouds and water vapor and other climate responses like changing lapse rates atmospheric temperature profiles , all of which combine to produce a change in radiative output at TOA.

    N is an important element. Effectively it describes the variations in TOA radiative flux due to the random climatic variations over many different timescales.

    This oft-cited paper reference and free link below calculates the climate sensitivity from using measured ERBE data at 2. Their result indicates positive feedback, or at least, a range of values which sit mainly in the positive feedback space.

    This equation includes a term that allows F to vary independently of surface temperature.. Some results are based on 10, days about 30 years , with , days years as a separate comparison.

    First, the variation as the number of time steps changes and as the averaging period changes from 1 no averaging through to days.

    Second, the estimate as the standard deviation of the radiative flux is increased, and the ocean depth ranges from m.

    The daily temperature and radiative flux is calculated as a monthly average before the regression calculation is carried out:. Third, the estimate as the standard deviation of the radiative flux is increased, and the ocean depth ranges from m.

    The regression calculation is carried out on the daily values:. If we consider first the changes in the standard deviation of the estimated value of climate sensitivity we can see that the spread in the results is much higher in each case when we consider 30 years of data vs years of data.

    This is to be expected. This of course is what is actually done with measurements from satellites where we have 30 years of history.

    The reason is quite simple and is explained mathematically in the next section which non-mathematically inclined readers can skip.

    We mean the random fluctuations due to the chaotic nature of weather and climate. In this case, the noise is uncorrelated to the temperature because of the model construction.

    These figures are calculated with autocorrelation for radiative flux noise. This means that past values of flux are correlated to current vales — and so once again, daily temperature will be correlated with daily flux noise.

    And we see that the regression of the line is always biased if N is correlated with T. Evaluating their arguments requires more work on my part, especially analyzing some CERES data, so I hope to pick that up in a later article.

    The relationship between global-mean radiative forcing and global-mean climate response temperature is of intrinsic interest in its own right. While we cannot necessarily dismiss the value of 1 and related interpretation out of hand, the global response, as will become apparent in section 9, is the accumulated result of complex regional responses that appear to be controlled by more local-scale processes that vary in space and time.

    If we are to assume gross time—space averages to represent the effects of these processes, then the assumptions inherent to 1 certainly require a much more careful level of justification than has been given.

    Measuring the relationship between top of atmosphere radiation and temperature is clearly very important if we want to assess the all-important climate sensitivity.

    The value called climate sensitivity might be a variable i. In the last article we saw some testing of the simplest autoregressive model AR 1.

    Before we move onto more general AR models, I did do some testing of the effectiveness of the hypothesis test for AR 1 models with different noise types.

    The Gaussian and uniform distribution produce the same results. So in essence I have found that the tests work just as well when the noise component is uniformly distributed or Gamma distributed as when it has a Gaussian distribution normal distribution.

    The next idea I was interested to try was to apply the hypothesis testing from Part Three on an AR 2 model, when we assume incorrectly that it is an AR 1 model.

    Remember that the hypothesis test is quite simple — we produce a series with a known mean, extract a sample, and then using the sample find out how many times the test rejects the hypothesis that the mean is different from its actual value:.

    This simple test is just by way of introduction. The AR 1 model is very simple. In non-technical terms , the next value in the series is made up of a random element plus a dependence on the last few values.

    There is a bewildering array of tests that can be applied, so I started simply. First of all I played around with simple AR 2 models.

    The results below are for two different sample sizes. For each sample, the Yule-Walker equations are solved each of 10, times and then the results are averaged.

    In these results I normalized the mean and standard deviation of the parameters by the original values later I decided that made it harder to see what was going on and reverted to just displaying the actual sample mean and sample standard deviation :.

    Then I played around with a more general model. With this model I send in AR parameters to create the population, but can define a higher order of AR to test against, to see how well the algorithm estimates the AR parameters from the samples.

    In the example below the population is created as AR 3 , but tested as if it might be an AR 4 model.

    The histogram of results for the first two parameters, note again the difference in values on the axes for the different sample sizes:.

    Rotating the histograms around in 3d appears to confirm a bell-curve. Something to test formally at a later stage. The MA process, of order q, can be written as:.

    This means, in non-technical terms, that the mean of the process is constant through time. Examples of the terminology used for the various processes:.

    This is unlike the simple statistical models of independent events. And in Part Two we have seen how to test whether a sample comes from a population of a stated mean value.

    The ability to run this test is important and in Part Two the test took place for a population of independent events. The theory that allows us to accept or reject hypotheses to a certain statistical significance does not work properly with serially correlated data not without modification.

    Instead, we take a sample and attempt to find out information about the population. This bottom graph is the timeseries with autocorrelation.

    When the time-series is generated with no serial correlation, the hypothesis test works just fine. As the autocorrelation increases as we move to the right of the graph , the hypothesis test starts creating more false fails.

    With AR 1 autocorrelation — the simplest model of autocorrelation — there is a simple correction that we can apply.

    We see that Type I errors start to get above our expected values at higher values of autocorrelation. So I re-ran the tests using the derived autocorrelation parameter from the sample data regressing the time-series against the same time-series with a one time step lag — and got similar, but not identical results and apparently more false fails.

    Curiosity made me continue tempered by the knowledge of the large time-wasting exercise I had previously engaged in because of a misplaced bracket in one equation , so I rewrote the Matlab program to allow me to test some ideas a little further.

    It was good to rewrite because I was also wondering whether having one long time-series generated with lots of tests against it was as good as repeatedly generating a time-series and carrying out lots of tests each time.

    So this following comparison had a time-series population of , events, samples of items for each test, repeated for tests, then the time-series regenerated — and this done times.

    So 10, tests across different populations — first with the known autoregression parameter, then with the estimated value of this parameter from the sample in question:.

    The rewritten program allows us to test for the effect of sample size. The following graph uses the known value of autogression parameter in the test, a time-series population of ,, drawing samples out times from each population, and repeating through 10 populations in total:.

    This reminded me that the equation for the variance inflation factor shown earlier is in fact an approximation.

    The correct formula for those who like to see such things :. And this is done in each case for tests per population x 10 populations..

    Fortunately, the result turns out almost identical to using the approximation the graph using the approximation is not shown :. With large samples, like , it appears to work just fine.

    In the next article I hope to cover some more complex models, as well as the results from this kind of significance test if we assume AR 1 with normally-distributed noise yet actually have a different model in operation..

    The statistical tests so far described rely upon each event being independent from every other event. Typical examples of independent events in statistics books are:.

    If we measure the max and min temperatures in Ithaca, NY today, and then measure it tomorrow, and then the day after, are these independent unrelated events?

    Now we want to investigate how values on one day are correlated with values on another day. So we look at the correlation of the temperature on each day with progressively larger lags in days.

    The correlation goes by the inspiring and memorable name of the Pearson product-moment correlation coefficient. And so on. Here are the results:.

    And by the time we get to more than 5 days, the correlation has decreased to zero. By way of comparison, here is one random normal distribution with the same mean and standard deviation as the Ithaca temperature values:.

    As you would expect, the correlation of each value with the next value is around zero. The reason it is not exactly zero is just the randomness associated with only 31 values.

    Many people will be new to the concept of how time-series values convert into frequency plots — the Fourier transform.

    For those who do understand this subject, skip forward to the next sub-heading.. Suppose we have a 50Hz sine wave.

    If we plot amplitude against time we get the first graph below. If we want to investigate the frequency components we do a fourier transform and we get the 2nd graph below.

    That simply tells us the obvious fact that a 50Hz signal is a 50Hz signal. So what is the point of the exercise? What about if we have the time-based signal shown in the next graph — what can we tell about its real source?

    When we see the frequency transform in the 2nd graph we can immediately tell that the signal is made up of two sine waves — one at 50Hz and one at Hz — along with some noise.

    This wiki. This wiki All wikis. Sign In Don't have an account? Start a Wiki. So there's no way it is random enough to be considered representative of Doom WADs in general.

    Tools I've religiously recorded the tools used to make every WAD that has been reviewed, or at least as well as I can using the information provided in the text files.

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