SOLUTION FORMATION

The following paragraph details the approach employed by Sugihara and May, called the Simplex Projection Method. The next section then demonstrates how genetic operators fit into this simplex projection methodology.

The basic idea of the Simplex Projection Method compares forecasts (using a specific time step) of future values with actual values in a time series. When a complete series is finished, the overall accuracy of the many individual forecasts is determined by use of the correlation coefficient. Then a higher time step is made, and the complete forecasting procedure is rerun. This process continues for time steps to a value of about 12 steps.

The results are then plotted. In a chaotic time series, the accuracy of the forecast diminishes as the horizon step increases; but with noise, the forecast error remains the same statistically and thus, the accuracy is somewhat constant as the horizon step increases. In their words, “comparing the predicted and actual trajectories, we can make tentative distinctions between dynamical chaos and measurement error: for a chaotic time series, the accuracy of the nonlinear forecast falls off with increasing prediction time interval, whereas for uncorrelated noise, the forecasting accuracy is roughly independent of prediction time interval [2].”

In order to make the comparison, a geometric space is generated from the time series values. This is a common methodology used in other nonlinear time series approaches such as neural networks. After obtaining a time series, they

1.  Determine four very important parameters. These are:

a.  the time delay, tau - this is the number which is used for extracting values in the series that will become geometric components in a state space.

b.  the embedding dimension, E - is an estimate of the dimension of the underlying nonlinear structure from which the time series was obtained. These are often fractal in nature, meaning the dimension is a fraction, not a whole number. The embedding dimension is often the nearest integer of the fractal dimension.

c.  the size of the pattern library and the size of the test series. The pattern library is used to form the space, and the test series is used as an independent out-of-sample series for distinguishing noise and chaos.

d.  the time step - the number of time elements in the future that are being projected. The initial value is one and is incremeneted by 1 for a total of 12 time steps.

2.  Establish a library pattern and a test series, using the parameters from 1 above. This is accomplished by converting the time series into a lagged delay coordinate space. The space (called a state space) consists of vectors determined from the parameters as follows: x =[x_t, x _t-tau, x _{t-2 tau}, … x _{t-(E-1) tau}] where x's are individual time series elements. Each lagged coordinate vector represents a data point in an E-dimensional space.

An example of establishing data points in a state space from a time series is the following. Consider the small time series 12, 2, 8, 4, 11, 17, 19, 13, 17, 4, 9, 7, 16, 12, and 9. Assume a time lag of one, an “embedding dimension” of three, a pattern library size of six, and a test series of four. Then the following data points comprise the pattern library: (8,2, 12), (4,8,2), (11,4,8), (17,11,4), (19,17,11), and (13,19,17). The following four vectors comprise the test series: (17,13,19), (4,17,13), (9,4,17), and (7,9,4).

3.  Here is the expensive search. Take each data vector in the test series, one at a time, and compare it to each data vector in the pattern library and note the Euclidian distance between them. Obtain the E+1 closest vectors. Check to see if the four points form a simplex around the test vector. If not, obtain the next closest pattern vector and check, in ascending order of distance, if any four of the five points form a simplex around the test vector. If not, obtain the next closest pattern vector and check, in ascending order, if any four of the six points form a simplex. In short, the calculations involve an expanding (in combinations) set of data vectors that generate enormous search times as the size of the time series increases. Note if any of the data vectors form a simplex around the test vector, continue the process until the pattern library is exhausted. From the list of all simplices that were obtained, find the one with the smallest diameter.

4.  Calculate the weighted distance from the test vector to each vertex (pattern vectors) for the selected simplex.

5.  Project the simplex into the future using a selected time step, s, and then note where the points end up after's elements into the future. As previously stated, the reason for this assumption is that the projection falls off in a chaotic system, but, in the presence of noise, the projection is expected to be fairly constant. The measurement used in the projection (to indicate whether the projections fall off or not), is the correlation coefficient.

The following illustrates the projection using the example already discussed. Assume that the vector (17, 13, 19) represents the test vector and (13, 19, 17), (19, 17, 11), (17, 11, 4), and (11, 4, 8) represent the vertices of the simplex obtained in step three. Then the projection for the vertex at (13, 19, 17) is (17, 13, 19). Because the x component, i.e., the 13 projects to 17 (which is in fact the next element in the original time series), the x component, i.e., the 19, projects to 13, and the x component, a 17, projects to 19. Likewise, the vector at (7, 9,4) projects to (16, 7, 9). For a time step value of 2, the projections for the vector (13, 19, 17) is (4, 17, 13) because the x component, a 13, skips to 3 forward elements in the original time series for a value of 4. The x component, a 19, projects two time elements for a value of 17, and the x component projects two time steps for a value of 13.

6.  A new projected simplex is formed using the coordinates of the projected vertices.

7.  Calculate the projected value of the test vector by using the weights previously established; calculate it as the weighted distance from the vertices of the projected simplex.

8.  Record the difference between the actual results and the projected estimate. Then repeat the above process for each of the vectors in the test series. After all of the vectors in the test series have been used, calculate the correlation coefficient.

9.  Repeat the above process for each time step. The time step starts at one and is incremented by 1 each cycle for a total of 12 cycles.