Many of the really interesting parts of the world are now recognized as exhibiting complex behaviour. If we use the simplest definition as described in here, that suggests that it is unpredictable. We now recognize that one of the elements of complexity is the emergence of complex behaviour within a system that is actually described by simple equations (even if we don't know what those are). As described in earlier posts, these systems may be studied in a parameter space defined either by the original data set plotted against its time derivative, or a lagged data plot.
Today I will try to justify my assertion that the stock market shows many of the properties of complex systems. In order to show the typical behaviour of such a system, let us consider the climate system.
The particular component we will look at is the deep ocean delta O-18 record, which is a proxy for global ice volume. By O-18 I mean the isotope of oxygen with a mass of 18 atomic units. The delta O-18 record is the difference between the "standard" isotopic composition of the ocean and the particular measurements, expressed as per mil (parts per thousand).
The basic idea here is that there is an isotopic fractionation that occurs as water is evaporated. Water molecules with an O-18 in them are heavier, and so are less likely to be evaporated; furthermore, if they are evaporated, they are more likely to be the first molecules to condense out of the water vapour in the tropical to subtropical areas and fall as rain. Thus the water vapour that reaches arctic areas is already very depleted in O-18, so that falling snow in arctic areas is relatively depleted in O-18.
This falling snow is what builds glaciers. Glaciers are made from water that is very depleted in O-18. When glacier volume increases, this increase in ice volume is reflected by a relative enrichment in O-18 in ocean water, as the total amount of O-18 in the world's waters is pretty much constant. The enrichment of ocean water is reflected in the oxygen isotopic content of single-celled, carbonate-shelled organisms (which are recovered in abundance in subsea cores). When these fossils are sampled by coring, the downcore variations in isotopic composition of the shells is interpreted to provide a proxy record of global ice volume.
Variations in deep ocean O-18 over the past one million years from ODP 677 (Shackleton et al., 1990). Original data available here.
At first glance, the most recent part of the ice volume record is dominated by asymmetric saw-tooth shaped cycles--marked by long periods of glacial advance and short periods of rapid glacial retreat.
In an earlier post, we saw how to construct phase space portraits in two dimensions from a time series.
The two-dimensional reconstructed phase space of this data set reveals that there are at least three "regions" of stability in phase space, each representing relatively stable volumes of ice.
Two-dimensional phase space portrait of the ice volume proxy record from 210 thousand years ago until about 8,000 years ago, showing regions of stability (marked G). The upper right of the chart represents greater ice volumes (glaciations) and the lower left represents lower ice volumes (interglacials).
This chart shows us a specific trajectory through phase space of the ice volume system over the past 210 thousand years. The line is marked at ten thousand year intervals with a dot.
At the beginning of the plot (the point labelled 210) ice volume is low. We follow the dashed curve up and to the right, and we may note that there is a lot more space between 200 and 190 than there is between 210 and 200. This implies that ice volume changed (in this case, increasing) a lot more between 200 thousand and 190 thousand years ago (or yBP) than it did between 210 thousand and 200 thousand yBP.
In the upper right of the graph, we seem numerous points plotted fairly close together. The rate of change of global ice volume was pretty slow from about 180-140 thousand yBP. This was during the next-to-last glacial maximum. There was a rapid deglaciation from 140-120 thousand yBP, followed by a long (120-70 thousand yBP) interglacial period.
During the following glaciation, we see fairly rapid growth to the loop from 60-30 thousand yBP, then more growth, leading to the most recent deglaciation.
The same curve can be followed over the past million years (but it gets a bit difficult to follow with all the line crossings). We would note certain consistencies. Firstly, the curve shows the same alternations between regions of slow movement of the curve (lots of points grouped together) and regions in which the system evolves very quickly.
Secondly, in cycle after cycle, we would note the areas of slow motion (labelled 'G' in the figure above) occur in the same regions of phase space---meaning that there are particular ice volumes (or glacial configurations) that are more stable than others. Such a system is described as having numerous metastable modes of operation.
Model of a system with feedbacks. Some portion of the output signal may react on the input, or may alter the parameters of the model.
The reasons systems behave this way is because of the presence of feedbacks. For our purposes, there are two types of feedbacks--positive and negative. Negative feedbacks tend to counter the input to the system, or rather tend to lead the system to resist changing in response to any external driving mechanism. Positive feedbacks cause the system to enhance the effects of a driving mechanism, creating the appearance that the system is careening out of control.
Schematic diagram showing the elements of a dynamic system with multiple metastable modes of operation (viz. Kauffman, 1993). Depending on the starting postion, the system will tend to evolve to a fixed solution (either a point or a limit cycle) within a region of phase space defined by a separatrix.
While the system is evolving towards one of the regions of stability (attractors in the above figure), postive feedbacks dominate, and the system evolves rapidly. While the system is within one of the regions of stability, the negative feedbacks dominate.
The climate system is subject to forcing (changes in heat received from the sun due to variations in orbital geometry, among other things) which attempt to drive the system away from the centre of attraction. As long as the system remains close to the attractor, negative feedbacks will tend to force the system back to its local equilibrium. (Arguably that is the situation we are in now with atmospheric carbon dioxide).
If the system is driven across a separatrix, it will evolve rapidly towards a new centre of attraction, and positive feedbacks will again dominate. For many systems we do not know where the next area of attraction is or what it will be like. (In our current situation, even though the Earth is resisting changes due to increases in carbon dioxide, however its capacity for continuing to do so is finite, and we 1) do not know when we will cross the separatrix nor 2) do we know what the effect of crossing the separatrix will be.)
In part 2 we will expand on this idea of attractors and separatrices and see how the concept applies to stock prices.
Reference:
Kauffman, S., 1993. The Origins of Order: Self-Organization and Selection in Evolution.
Shackleton, N.J., A. Berger, and W. R. Peltier, 1990. An alternative astronomical calibration of the lower Pleistocene timescale based on ODP Site 677. Transactions of the Royal Society of Edinburgh: Earth Sciences 81: 251.
*Well, perhaps it's the other way around.
Today I will try to justify my assertion that the stock market shows many of the properties of complex systems. In order to show the typical behaviour of such a system, let us consider the climate system.
The particular component we will look at is the deep ocean delta O-18 record, which is a proxy for global ice volume. By O-18 I mean the isotope of oxygen with a mass of 18 atomic units. The delta O-18 record is the difference between the "standard" isotopic composition of the ocean and the particular measurements, expressed as per mil (parts per thousand).
The basic idea here is that there is an isotopic fractionation that occurs as water is evaporated. Water molecules with an O-18 in them are heavier, and so are less likely to be evaporated; furthermore, if they are evaporated, they are more likely to be the first molecules to condense out of the water vapour in the tropical to subtropical areas and fall as rain. Thus the water vapour that reaches arctic areas is already very depleted in O-18, so that falling snow in arctic areas is relatively depleted in O-18.
This falling snow is what builds glaciers. Glaciers are made from water that is very depleted in O-18. When glacier volume increases, this increase in ice volume is reflected by a relative enrichment in O-18 in ocean water, as the total amount of O-18 in the world's waters is pretty much constant. The enrichment of ocean water is reflected in the oxygen isotopic content of single-celled, carbonate-shelled organisms (which are recovered in abundance in subsea cores). When these fossils are sampled by coring, the downcore variations in isotopic composition of the shells is interpreted to provide a proxy record of global ice volume.
Variations in deep ocean O-18 over the past one million years from ODP 677 (Shackleton et al., 1990). Original data available here.
At first glance, the most recent part of the ice volume record is dominated by asymmetric saw-tooth shaped cycles--marked by long periods of glacial advance and short periods of rapid glacial retreat.
In an earlier post, we saw how to construct phase space portraits in two dimensions from a time series.
The two-dimensional reconstructed phase space of this data set reveals that there are at least three "regions" of stability in phase space, each representing relatively stable volumes of ice.
Two-dimensional phase space portrait of the ice volume proxy record from 210 thousand years ago until about 8,000 years ago, showing regions of stability (marked G). The upper right of the chart represents greater ice volumes (glaciations) and the lower left represents lower ice volumes (interglacials).
This chart shows us a specific trajectory through phase space of the ice volume system over the past 210 thousand years. The line is marked at ten thousand year intervals with a dot.
At the beginning of the plot (the point labelled 210) ice volume is low. We follow the dashed curve up and to the right, and we may note that there is a lot more space between 200 and 190 than there is between 210 and 200. This implies that ice volume changed (in this case, increasing) a lot more between 200 thousand and 190 thousand years ago (or yBP) than it did between 210 thousand and 200 thousand yBP.
In the upper right of the graph, we seem numerous points plotted fairly close together. The rate of change of global ice volume was pretty slow from about 180-140 thousand yBP. This was during the next-to-last glacial maximum. There was a rapid deglaciation from 140-120 thousand yBP, followed by a long (120-70 thousand yBP) interglacial period.
During the following glaciation, we see fairly rapid growth to the loop from 60-30 thousand yBP, then more growth, leading to the most recent deglaciation.
The same curve can be followed over the past million years (but it gets a bit difficult to follow with all the line crossings). We would note certain consistencies. Firstly, the curve shows the same alternations between regions of slow movement of the curve (lots of points grouped together) and regions in which the system evolves very quickly.
Secondly, in cycle after cycle, we would note the areas of slow motion (labelled 'G' in the figure above) occur in the same regions of phase space---meaning that there are particular ice volumes (or glacial configurations) that are more stable than others. Such a system is described as having numerous metastable modes of operation.
Model of a system with feedbacks. Some portion of the output signal may react on the input, or may alter the parameters of the model.
The reasons systems behave this way is because of the presence of feedbacks. For our purposes, there are two types of feedbacks--positive and negative. Negative feedbacks tend to counter the input to the system, or rather tend to lead the system to resist changing in response to any external driving mechanism. Positive feedbacks cause the system to enhance the effects of a driving mechanism, creating the appearance that the system is careening out of control.
Schematic diagram showing the elements of a dynamic system with multiple metastable modes of operation (viz. Kauffman, 1993). Depending on the starting postion, the system will tend to evolve to a fixed solution (either a point or a limit cycle) within a region of phase space defined by a separatrix.
While the system is evolving towards one of the regions of stability (attractors in the above figure), postive feedbacks dominate, and the system evolves rapidly. While the system is within one of the regions of stability, the negative feedbacks dominate.
The climate system is subject to forcing (changes in heat received from the sun due to variations in orbital geometry, among other things) which attempt to drive the system away from the centre of attraction. As long as the system remains close to the attractor, negative feedbacks will tend to force the system back to its local equilibrium. (Arguably that is the situation we are in now with atmospheric carbon dioxide).
If the system is driven across a separatrix, it will evolve rapidly towards a new centre of attraction, and positive feedbacks will again dominate. For many systems we do not know where the next area of attraction is or what it will be like. (In our current situation, even though the Earth is resisting changes due to increases in carbon dioxide, however its capacity for continuing to do so is finite, and we 1) do not know when we will cross the separatrix nor 2) do we know what the effect of crossing the separatrix will be.)
In part 2 we will expand on this idea of attractors and separatrices and see how the concept applies to stock prices.
Reference:
Kauffman, S., 1993. The Origins of Order: Self-Organization and Selection in Evolution.
Shackleton, N.J., A. Berger, and W. R. Peltier, 1990. An alternative astronomical calibration of the lower Pleistocene timescale based on ODP Site 677. Transactions of the Royal Society of Edinburgh: Earth Sciences 81: 251.
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