Wednesday, March 04, 2009

The Idea of Mathematical Delay

The Idea of Mathematical Delay

Delay mechanisms are the essential devices by which dynamic phenomena are understood. These fundamental dynamic sub-assemblies define specific arrangements of rates-of-change with the state variables that they control. Such structures are best visualized in 'signal path diagrams', the animation of which requires application of numerical methods. While signal path analysis is a rather daunting topic, we do hope to provide some sense of comfort (however false) with SFEcon's use of this discipline.

Exponential Decay and the Signal Path Diagram

The most primitive, naturally-occurring pattern of dynamic adjustment is exponential decay. This pattern is also familiar: consider a the matter of a pizza delivered to a room such that the temperature differential between the pizza and the room equals some amount A; obviously A will decay away to zero with the passage of time T. According to Newton's curve of cooling, the value of A will vary with T according to the exponential pattern shown below.

Exponential functions are characterized by a unique property: striking a tangent at any point, and extending the tangent to cross the ordinate, will always define the same distance 1/V along the time axis. The functional relation between A and T is given by V and A's initial value A0.

Comparatively few dynamic systems are simple enough to yield closed-form expressions of their behavior such as equation for exponential decay above. More complicated systems can be made to disclose their dynamics if they can be perceived in terms of a 'signal path diagram'. The signal path for exponential decay, shown below, makes use of the three elements by which all signal paths are composed.

The current level of temperature differential is depicted by a square. Dissipation of the potential embodied in A is visualized by flow along a solid line leading away from the square. The rate of decay is depicted by a valve symbol attached to the line of flow away from A. Broken lines indicate flows of information, which are considered instantaneous. Thus the instantaneous rate of decay E is always computed by the current level of A, multiplied by the parameter V, which is depicted by a circle. This mechanism indicates that time's passages always has potential A flowing out in proportion to the amount of A that is left. Thus A depletes at the ever decreasing rates characteristic of exponential decay.

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First-order Delays

First-order delays are to dynamics what exponentially-weighted averages are to economics, viz. filtering mechanisms. A delay is constructed by attaching an exogenous driving function R to the signal path depicting exponential decay:

Whatever pattern is evident in the time-series of R will emerge in attenuated and 'spread out' form at 1/V time units later in the time-series of E. E is a delay or lag on R; and the current value of E is an exponentially-weighted average on all past values of R. If R is held constant, A will attain a level such that E = VA = R.

First-order delays have familiar uses in economics and operations analysis. If, for example, R is a series of sales rates for some product, E might disclose appropriate production rates by smoothing the R series. Economists have reference to a certain delay model of consumption and income whereby consumption E is thought to be determined by a lag on income R. This model is considered at some length in SFEcon's treatment of Keynesian economic theory, which we commend as another source of familiarity with the notion of delay.

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Higher-order Delays

The emulation of certain phenomena require that the dynamics of smoothing and delay be arrayed in more complex ways. Higher-ordered delays are constructed by arranging first-order delays in series, such that the effluent of one level cascades into the next. This arrangement is shown below for a third-order delay in which three levels A, B, and C are used to filter and smooth an exogenous input rate R.

Here the delay on R would be given by E = V (A + B + C), and equations for continuously re-computing the levels would follow the pattern shown in the diagram. The functions for updating the levels are in the form required for emulation by numerical methods: levels depart from their prior values by allowing their controlling rates' influence to persist over a differential element of time d.

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Sigmoid Decay

Higher-ordered delays are used for the emulation of behaviors having more complexity than can be rendered by simple, first-order, or exponential delays. The particulars behind such a statement can be as numerous as there are possible driving functions R. But differences in behavior between first-order and higher-ordered delays are best exposed for the case of an established steady-state, in which R = E, that is set on a course of dynamic adjustment by suddenly making the input rate equal 0.

When this is done to the first-order delay, we observe the exponential pattern of decay introduced at the top of this page.

When this is done to a third-order delay, we observe the sigmoid pattern shown below.

The sigmoid pattern of decay shows very little initial loss in the collected levels A + B + C. This follow by a brief period in the vicinity of T = 1/V wherein the collection of levels depletes quite rapidly. The pattern is completed in a third phase characterized levels that are now so low that they have comparatively little force with which to propel their final exhaustion.

A sigmoid pattern of decay might be appropriate for mapping the useful life of a capital asset, such as a family automobile purchased with an intention to be used for, say, 1/V = 8 years. For much of the automobile's useful life it functions as if new. As its age begins to approach the point for which its obsolescence has been designed, the automobile's functionality is only maintained by increasingly frequent and costly repairs. Finally, maintenance expenses are so high that the automobile must be re-sold (i.e. re-valued) at a low enough price for it to be economically useful to its new owner.

A sigmoid pattern's exact shape is controlled by the order of the higher-ordered delays generating it: the higher the order, the sharper the transition between initial and final states. Different orders of delay are therefore likely to be indicated for tracing out the useful economic lives of commodities composing an economic model. SFEcon's generic prototypes use third-order delays to emulate the processes by which economic goods exhaust themselves in producing the next generation of goods. But this is only a formality intended to 'hold place' for a more incisive analysis.