Time Delay Building Blocks:
Eureqa provides the delay(x, c) building block to represent an arbitrary time-delay, where x could be any expression. The expression delay(x, c) returns the value of x at c time units in the past. When used as a building-block, Eureqa can automatically optimize expressions or variables to be delayed and the time-delay amount c.
The figure above plots an arbitrary variable x and a delayed value delay(x, 1.0), where the values are ordered by some time variable t. The delayed version is equal to x at 1.0 time units into the past.
To use time-delay building-blocks, your data must have some notion of time or ordering. You also need to tell Eureqa which variable in your data represents the time or ordering value:
If you don’t specify a time variable, Eureqa will use the row number in the spreadsheet as the time value of each data point.
If a particular delayed time value falls between two points in the data set, the value is linearly interpolated between the two data points using the time value.
Eureqa also provides the delay_var(x, c) building-block which is identical to delay(x, c), except that it only accepts a variable as input. It’s provided as a special case of the delay(x, c) building-block to allow you to constrain the types delays used in the solutions. But in the end they are effectively identical.
Control the Fraction of Data Used for History
Notice that the delayed output plotted above does not have values on the left side of the graph for the first few time points. This is because these points request previous values of x that lie before the first point in our data set. Eureqa will automatically ignore these data points when calculating errors.
However, there is a way to control how much of the data set Eureqa is allowed to ignore – or effectively, specify a maximum delay offset. You can limit the fraction of data used for time-delay history values in the Advanced Solutions Options menu:
The default maximum fraction is 50% of the data. If you find that Eureqa is identifying solutions with very large time delays, perhaps just to avoid modeling difficult features in the first half of the data set, you may want reduce this fraction
Additionally, you can control the number of delayed values per variable (including a zero delay of an ordinary variable use) in this dialog.
Another way to model a value as a function of its previous values is with fixed delays. You can enter in fixed time-delays, or “lags” of the variable, directly into the Search Relationship option. For example:
x = f( delay(x, 2.1), delay(x, 5.6) )
This search relationship tells Eureqa to find an equation to model the value of x as a function of it’s value at 2.1 and 5.6 time units in the past.
You may also want to specify a minimum time-delay offset. If you entered a search relationship such as x = f(x), Eureqa would find a trivial answer f(x) = x. More likely, you wanted to find a model of x, but as a function of x at least some amount of time in the past. The way to do this is to again use a fixed delay, such as:
x = f(delay(x, 3.21))
Here, 3.21 is the minimum time-delay. Now, if the time-delay building-blocks are enabled, Eureqa can delay this delayed input further if necessary.
Delay Differential Equations:
Another common use for time-delays in for modeling using Delay Differential Equations. Finding delay differential equations is just like searching for ordinary differential equations. For example, entering a search relationship like:
D(y,t) = f(y)
but also enabling time-delay building blocks. This relationship has a trivial solution however: Eureqa will return the slope formula such as
f(y) = ( y – delay(y, 0.1) )/0.1
Therefore, you most-likely want to limit the total number of delays per variable to one (which includes the zero delay of the normal variable use). You can set this in the Advanced Solution Settings menu. The default is unlimited.
Implementing Delays Outside of Eureqa
In Matlab, you can implement a time delay using the interp1 function. For example, the expression delay(x, 1.23) would be implemented as:
interp1(t, x, t – 1.23, ‘linear’)
Implementing delays in Excel is a littler harder. You need to download an Excel add-on that adds an interpolate function. For example, the package XlXtrFun adds a function “Interpolate” that is just like Matlab’s interp1. There are also other guides for Linear Interpolation with Excel.