Three days ago, out on Twitter, a peep tweeted a graph that was supposed to show how a market event amounted to nothing. The line graph dropped the baseline, rose above the baseline, and dropped again to the baseline. It was a quick thing that had me spending the rest of the day, and parts of the following three days hammering on it.

The peeps point was that nothing happened. Grab a hammer and join me in building a case showing just how much did happen.

This was their graph. If you’re in a hurry, you won’t notice the net loss.

I rotated the minima so I could see if the loss was completely recovered. It was not. The vertical symmetry is asymmetric. Rotating the minima reveals a gap, labeled A, shows that the upside did not completely recover the value lost in during the first downside.

The second downside loss stops at the line labeled B, the new baseline. There is a gap between the initial baseline and the final baseline. The gap between the baselines is larger than the gap between the peaks. I coped the gap between the peeks and put it below the initial baseline to demonstrate that loss at A did not account for all the loss between the baselines. Subtracting the loss A from the loss between the baselines gives us the gap labeled B.

Notice that the baseline at B moves up slightly. I just saw this after drawing many diagrams. I annotate my error. We will ignore this slight upside. Just one more thing that the peep and I overlooked. I will remove it from subsequent diagrams.

Going back to the first diagram, we had a downside, an upside, and another downside. The first downside (A) and the second downside (B) account for the difference between the initial and final baselines.

In the figure on the right, I explored the symmetries. The vertical red lines represent the events embedded in the signal. The notation for the symmetry for an event n, span the interval from n-1 to n+1. These spans are shown in gray.

Since I rotated the minima, the symmetry above the signal is actually a vertical (y-axis) symmetry around the origin. I drew purple lines from the vertex at the top to the vertexes at the baseline. Then, I moved the purple lines to the top of the figure. They looked symmetric, but are slightly asymmetric. The left side was three units wide; the right, four units wide.

Both of the horizontal (x-axis) symmetries are asymmetric. The gray box notation demonstrates that these signal components are very asymmetric.

Asymmetries indicate locations where something was learned or forgotten. The Glass-Steagall Act often gets cited as one of the causes of the housing crisis. It was a forgetting. In Stewart Brand’s “*How Buildings Learn*,” they learned by accretion. We accret synapses as we learn. When we put a picture on a wall, the wall learns about our preferences. The next resident may not pull that nail out, so such remodeling artifacts accret. Our house becomes our home, because we teach our house, and our house learns. So it is with evolution.

Before I created the box notation, I was drawing the upside and downside lines and rotating them to see how much area was involved in each of the asymmetries. I’m using the rotation approach in the figure to the left. I’ve annotated the three asymmetries, The white areas are cores, and the orange areas are tails. The asymmetry annotated at the top of the figure is, again, horizontal. The tail is just a line as the asymmetry is slight. The cores are symmetric about vertical lines, not shown, that represent the events encoded into the signal.

In an earlier figure, I just estimated the area of the tail. When I highlighted that area, because I use MS Paint to draw these things and it dithers, I got a line of green areas, rather than a single area. I numbered them in order. They are labeled as Area Discontinuities. In a sense, they would be Poisson distributions in individual Poisson games. In area 8, those Poisson distributions become a single normal distribution. That normal has more than 32 data points. With 20 data points, that normal can be estimated. In a sense, there is a line through those Poissons and the normal. This is what happens in the technology adoption lifecycle as we move from early adopters each with their own Poisson game and sum towards the vertical/domain specific market f which the early adopter is a member. This line is one lane of Moore’s bowling alley.

Where the figure mentions “Slower,” that is just about the slope of that last diagonal, the second loss. The red numbers refer to the earlier unrefined gaps we are now calling A and B.

When there are tails, the normal distribution involved will exhibit kurtosis. I built a histogram of the data in the area that I highlighted in green and then, looked at the underlying distribution along the line through those areas. There seemed to be two tails: one thicker and one thinner. Of course, all of this is meaningless, as it results from the dithering. With a vector rendering, there would only be one more consistent area.

The tiny thumbnail in the middle of the thumbnails at the bottom right of the figure shows a negatively skewed normal, but in another interpretation, the distribution is four separate normals. Where I mentioned theta, the associated angle quantifies the kurtosis.

One more thing is happening where a Poisson distribution finally becomes a normal distribution, the geometry shifts from hyperbolic to Euclidean.

In the next figure, I look at the black swan view of the signal. A black swan is usually drawn as a vertical line cutting off the tail of the normal distribution, labeled Original and highlighted with yellow and light green. Here we are talking generally. The next figure we will use this to show how the three black swans generate the signal that we’ve been discussing. The negative black swan throws away the portion of the distribution remaining beyond the event driving the black swan, then the remaining data is used to renormalize the remaining subset of the original data. The lifetime of the category is reduced. The convergence with the x-axis contracts, aka moves towards the y-axis. The positive black swan moves the distribution down. The normal becomes enlarged, so it sits on the new x-axis below the original baseline. The new distribution includes the light green and green areas in the figure. The lifetime of the category is lengthened. The convergence moves out into the future, aka moves further away from the y-axis.

In the continuous innovation case, the positive black swan will stay aligned with the driving event. The normal distribution is enlarged just enough to converge with the new x-axis below the prior x-axis. In the discontinuous innovation case, the positive black swan would begin at the B2B early adopter phase of the technology adoption lifecycle. In the discontinuous case, the size of the addressable market would drive the size of the normal, and it is not correlated with the prior distribution.

Now we go back to the example we’ve worked on throughout this post. We will apply the black swan concepts to the signal using the diagram below. There are three black swans. A negative black swan that generates the first loss. A positive black swan follows with a recovery that does not fully recover the value lost in that first loss. This recovery is followed by another negative black swan that contributes to the net loss summed up by the signal. The normals are numbered 0 through 3. The numbers are to the right of the events, and they are on the baseline of the associated normal. The original distribution (0) is located at the event driving the first black swan. The new distribution (1) associated with the first loss, the first negative black swan. The x-axis of this black swan is raised above the original x-axis. This distribution lost the projected data to the right of the event, data expected from the future. Renormalizing the distribution makes it higher from peak to the new baseline, and the distribution contracts horizontally. The rightmost convergence of the normal with the x-axis is where the category ends. The leftmost convergence is fixed. The x-axis represents time. The end of the category will arrive sooner unless some other means to generate revenues is found, aka a continuous innovation is found. The first gain, aka the positive black swan, generates a larger distribution (3). The x-axis is lower than that of the immediately prior x-axis. The convergence moves into the future relative to the immediately prior distribution. This is followed by another loss, the second loss, the second negative black swan. Here the x-axis rises above the previous x-axis. The distribution (3) is renormalized and is smaller than the immediately previous distribution (2).

From a signal perspective, the original signal input was above the output. The black swans move the signal to the line labeled “Restatement.” The shape of the original and restatement generate and output the same signal.

Next, we look at the logic underlying the signal. I’ll use the triangle model. In that model, every line is generated by a decision tree represented by a triangle. The x-axis has decisions trees, aka triangles associated with it. Each interval on the x-axis has its own decision tree. The y-axis has its own intervals and decision trees. The events that drove the black swan model drive the intervals and associated decision trees.

The pink triangles represent the y-axis decision trees involved in the losses. The green triangle represents the y-axis decision tree for the gain. The green triangle is higher than the gain, because it does not recover the entire loss from the first loss. I annotated the shortfall. The asymmetry in the vertical axis, that we discussed earlier, appears on the upper right side of the triangle is thicker. This thickness is not constant. The colors and the numbers show the patterns involved on that side of the triangle. The axis of symmetry associated with the green triangle is an average between the baseline of the input signal and the baseline of the output signal. Putting this symmetry axis would increase the asymmetry of the representation.

The erosion would be shown more accurately as subtrees, rather than a single subtree starting at the vertex, like a slice of pie.

On the x-axis, each triangle is shown in blue. The leftmost triangle consists of a blue triangle and yellow triangle. The blue triangle represents the construction of the infrastructure that generates that interval of the signal. The yellow triangle represents the erosion that infrastructure. The black sway, the first lost resulted from that erosion.

Keep in mind that the negative black swan reduces the probability, so they move their baselines up vertically. Positive black swans increase the probability, so they move their baselines down vertically.

In the very first figure, I annotated the asymmetries and symmetries. Asymmetries are very important because they inform us that learning is necessary. Asymmetries in the normal distribution show up as kurtosis due to samples being too small to achieve kurtosis-free normality or symmetry.

The vertical orientation of those pink triangles is new to me as I wrote this. They represent the infrastructure to stop loss, a reactive action. The results may appear positive, but in the long run, represents exposure. These actions will be instanced for the situation being faced. Given that a black swan can happen at any moment, you don’t want to have to invent a response. You want to move from reactive, predictive, proactive time orientations as quickly as possible. Many people see OODA loops as a reactive mechanism. The military trains on the stuff, on the infrastructure–decision trees being part of that infrastructure. Know before you go. Eliminate or reduce those asymmetries before you get into the field, before the black swan shows up.

The events in the original signal view ties to the black swan/distribution view and the logical view are tied together by the red lines representing the events.

I drew another figure that is a bit cleaner about the signal view. The

Even if the signal looks like nothing, a net zero, take a closer look, there was much to be seen, much learning got done to produce the result. Know before you go.

Enjoy.