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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.

If the time-domain data went from zero to infinity, the frequency plot would be that perfect line. In figure 5, the time-domain data actually went from zero to 10 seconds not all of which was plotted.

It appears that there is some confusion about this simple model. To draw that conclusion, the IPCC had to make an assumption about the global temperature series.

The assumption implies, among other things, that only the current value in a time series has a direct effect on the next value.

For example, if the last several years were extremely cold, that on its own would not affect the chance that next year will be colder than average.

Hence, the assumption made by the IPCC seems intuitively implausible. The confusion in the statement above is that mathematically the AR1 model does only rely on the last value to calculate the next value — you can see that in the formula above.

If day 2 has a relationship to day 1, and day 3 has a relationship to day 2, clearly there is a relationship between day 3 and day 1 — just not as strong as the relationship between day 3 and day 2 or between day 2 and day 1.

And it is easy to demonstrate with a lag-2 correlation of a synthetic AR1 series — the 2-day correlation is not zero.

For now we will consider the simplest model, AR1, to learn a few things about time-series data with serial correlation.

Note that the standard deviation sd of the data gets larger as the autoregressive parameter increases. DW is the Durbin-Watson statistic which we will probably come back to at a later date.

Now the frequency transformation using a new dataset to save a little programming time on my part :.

As the autoregressive parameter increases you can see that the energy shifts to lower frequencies. Here are the same models over events instead of 10, to make the time-based characteristics easier to see:.

As the autoregression parameter increases you can see that the latest value is more likely to be influenced by the previous value. The resulting time series is called red noise by analogy to visible light depleted in the shorter wavelengths, which appears reddish..

It is evident that the most erratic point to point variations in the uncorrelated series have been smoothed out, but the slower random variations are essentially preserved.

In the time domain this smoothing is expressed as positive serial correlation. Luckily, we can still use many standard hypothesis tests but we need to make allowance for the increase in the standard deviation of serially correlated data over independent data.

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Email Address:. Sign me up! Create a free website or blog at WordPress. Feeds: Posts Comments. We saw in Part Three that this particular paper ascribed a probability: We find that the latest observed year trend pattern of near-surface temperature change can be distinguished from all estimates of natural climate variability with an estimated risk of less than 2.

Subsequent articles will continue the discussion on natural variability. Knutson et al The models [CMIP5] are found to provide plausible representations of internal climate variability, although there is room for improvement..

Later The model control runs exhibit long-term drifts. We judge this as likely an artifact due to some problem with the model simulation, and we therefore chose to exclude this model from further analysis From Knutson et al From van Oldenborgh et al From Jones et al , figure S From Wittenberg Weather Forecasting The basic idea behind ensembles for weather forecasts is that we have uncertainty about: the initial conditions — because observations are not perfect parameters in our model — because our understanding of the physics of weather is not perfect So multiple simulations are run and the frequency of occurrence of, say, a severe storm tells us the probability that the severe storm will occur.

Because we only had one occurrence. This is exactly what currently happens with numerical weather prediction.

Climate Forecasting The idea behind ensembles of climate forecasts is subtly different. Item 3 is what I want to discuss in this article, around the paper by Rowlands et al.

From Rowlands et al Digression on Statistics The foundation of a lot of statistics is the idea of independent events.

Yet this work on attribution seems to be fundamentally flawed. Here was the conclusion: We find that the latest observed year trend pattern of near-surface temperature change can be distinguished from all estimates of natural climate variability with an estimated risk of less than 2.

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.

I've listed the editors first, then the other tools. I've only listed the most popular tools over the archive I have, and given our bias towards editing old levels those tend to be the classics like DEU and BSP.

Lies, Damned Lies, and Statistics When I first started Doom Underground , I knew that since I was keeping the information very organised and doing things like generating indices automatically, one really cool thing I could do was generate some statistics on the levels reviewed.

Investigating the interaction of direct and indirect relation on memory judgments and retrieval. Also, super proud - both of these are student theses turned papers: Maxwell, N.

Investigating the interaction of direct and in-direct relation on memory judgments and retrieval. I enjoyed talking to the group, meeting Twitter friends in real life!

I have happily acquired a new Mac Book yay! As I was working on reconnecting my GitHub repositories to the files, I was trying to understand why several of my repos were saying I had a bunch of file changes but nothing in the files themselves had changed.

I noticed they were mostly.

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