In my last post, “Fluctuating Tails I,” we explored the effects of a black swan on a single normal distribution. In this post, we will look at a the effects of a black swan on a multinormal distribution from the perspective of a linear regression.

Lets start off with the results of a linear regression of multidimensional data. These regressions give rise to ellipse containing the multidimensional data. This data also gives rise to many normal distributions summed into a multinormal distribution.

I modified the underlying figure. The thick purple line on the distribution on the p(y) plane represents the first black swan. The thin purple line projects the black swan across the ellipse resulting from the regression. The data to the right of the black swan is lost. The perpendicular brown lines help us project the impacts on to the distribution on the p(x) plane. The black swan would change the shape of the light green ellipse, and it would change the shape of the distribution, shown in orange, on the p(x) plane.

In the next figure, we draw another black swan on the p(y) plane distribution further down the tail. We use a thin black line to represent the second black swan. This black swan has fewer impacts.

In neither of these figures did I project the black swan onto the p(x) plane distribution, or draw the new x’ and y’ axes as we did in the last post. I’ll do that now.

Here we have projected the black swan and moved the x and y axes.

Notice that the black swan is asymmetrical, so the means of the new distributions would shift. This means that any hypothesis testing done with the distributions before the black swan would have to be done again. Correlation and strength tests depend on the distance between the means of the hypotheses (distributions).

**Parameter Distributions**

After drawing these figures, I went looking for Levy flight parameters. I wanted to show how a black swan would affect pumps in a Levy random walk. I settled instead on a Rice distribution.

The shades of blue in the figure are the standard deviations of sigmas from the mean. Sigma is one parameter of the Rice distribution. V is another.

Here are the PDFs and CDFs of a Rice distribution given the relevant parameter values. The blue vertical line through both of the graphs is an arbitrary black swan. Some of the distributions are hardly impacted by the black swan. A particular distribution would be selected by the value for the parameter v. The distributions would have to be redrawn after the black swan to account or the change in the ranges of the distributions. Once redrawn, the means would move if the black swan was asymmetrical. This is the case for the Rice distribution and any normal distributions involved.

If the parameters themselves were distributions, a black swan would eliminate parameter values and the distributions for those parameter values.

When we base decisions on statistical hypothesis testing, we need to deal with the impacts of black swans on those decisions.

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