Steve Eager just lately wrote “I’m not Discreet, and Neither is Time” during which he discusses the alleged defects of discrete time fashions versus steady time ones.
(In discrete time, the mannequin state is outlined on a time axis that may be labelled as integers: step 1, step 2, and so forth. In a steady time, a mannequin’s time axis is the true axis. A discrete time mannequin might be regarded as being outlined by distinction equations, whereas steady time is generally outlined by differential equations.)
Steve Eager has completely different modelling priorities than I do, so I can’t try to reply to him point-by-point. He desires his fashions to do various things than I would like mine, which is a extra philosophical conflict than a query of discrete time versus steady. I’ll as an alternative simply cowl extra generic points, which overlap his factors. I already mentioned this within the appendix to my ebook An Introduction to SFC Fashions Utilizing Python — the place the fashions are discrete time.
Even when the Actual World is Steady, The Greatest Fashions Might Not Be
My undergraduate diploma was in Electrical Engineering, and programs coated each steady time and discrete evaluation and design, in addition to the tough problems with integrating the 2 kinds of time axis.
The explanation why discrete time is required is that digital circuits are greatest modelled because the variables having quantised values on a discrete time axis. (Most often, the truth that values are quantised is ignored, and we use an actual quantity to precise values.) That is even though all circuits are finally analog. (The Brits write “analogue,” however I persist with the American spelling for circuitry, I write “analogue” for the phrase’s different makes use of.)
For a single wire, it would have a voltage that varies been the bottom voltage (nominally 0) and the facility provide voltage, which I’ll label V. Voltages which are inside some tolerance of 0 are handled as a binary 0, whereas voltages close to V are handled as a binary 1. Values outdoors a tolerance of 0 or V are undefined, and presupposed to be prevented.
The circuitry is analog, and so if we have a look at a steady time view of the voltage on an oscilloscope, we’ll see the voltage transfer pretty easily between 0 and V (assuming the circuit is working correctly). Since that leaves the binary interpretation undefined generally, this seems dangerous. That is handled by working the circuit with a clock frequency — what issues for the circuit is what the worth is when the clock ticks. The design is meant to be such that the voltage settles appropriately near 0 or V earlier than the following clock tick. If we push the clock fee to the tolerance of the circuitry, it will likely be simply settling in time earlier than the tick — which poses dangers of errors as a result of actuality that not all chips behave identically.
One of the best ways to analyse a pure digital circuit is to not fear concerning the intermediate time values, and simply mannequin the system on the clock ticks. You solely have a look at the continual time axis when the digital circuit is interacting with an analog one — which has gotten rather more uncommon than once I graduated in 1990.
There’s a direct analogy to economics. All financial transactions happen at distinct deadlines, and accounting invariably makes use of end-of-period closing values. The closest to steady time variables are quotes from digital markets — however even there, the transactions occur in discrete jumps. Though my information of fastened revenue is more and more antiquated, I’ve solely seen fastened revenue money flows as being modelled as (end-of-day) flows.
All Financial Knowledge Are Discrete Time
All noticed financial information re discrete time. There’ll by no means be steady time financial information revealed, notably once we take into account the truth that even one second of steady time comprises an infinite variety of information factors.
If all information are at all times discrete time, any relationship between them by definition is a discrete time mannequin.
Conversions Between Steady Time and Discrete Time Is Difficult
We lose a spectacular quantity of data once we transfer from steady time to discrete time. Conversions should not one-to-one; an infinite variety of steady time fashions may give rise to the identical discrete time mannequin. This makes discussing relationships between them tough for individuals who don’t have any been skilled within the matter — which is basically electrical engineers.
For instance, I sadly as soon as encountered a physicist who positioned himself as an knowledgeable on communication techniques. Nonetheless, this knowledgeable was unable to know the problems concerned in changing a linear first order steady time mannequin to a discrete time one.
On condition that noticed information won’t ever be steady, it makes little sense to say {that a} steady time mannequin is the “right” one given the paradox in conversions.
Steady Time Arithmetic Is Onerous
Doing proofs with steady time fashions is difficult. Fashions are allegedly outlined by equations utilizing differentials (derivatives), whereas we very often features that should not have derivatives outlined in all places — and even wherever, if the perform is the results of a random course of. And if we wish fashions that embed optimising behaviour, we most likely need random processes to point out up someplace — if the mannequin is deterministic, all digital mannequin inhabitants are fully conscious of what is going to occur forever.
For instance of the complexity, we want the Dirac step “perform” to generate jumps in time collection. The one strategy to correctly take care of these objects is through measure concept; undergraduate educating includes some hand-waving that may increase points as quickly as we try and do the evaluation in a extra complicated system.
Undergraduate educating skates over the mathematical complexity by the expedient tactic of utilizing handwaving (or bullshit, utilizing the technical math time period). Practitioners miss this by counting on digital simulations (that are successfully discrete time except the equations are solved symbolically).
The issue is analysis. Individuals who use undergraduate arithmetic to do utilized math solely have an opportunity in the event that they keep of their lane and keep away from something with hidden complexity. Management techniques engineers are ready to do that by sticking to matrix algebra of their proofs, and never worrying concerning the particulars of the differential equations that drove using matrices. Stepping out of that consolation zone ends in a literature that’s stuffed with “proofs” that don’t actually show something, as a result of they skip key points. That is each an issue for engineers in management concept, in addition to just about the whole revealed DSGE literature.
In management techniques, this led to a quiet underground tutorial warfare once I was an inmate of instructional establishments. The issue is easy: does an accurate theorem assertion with an incorrect proof deserve precedence over a later paper that does the arithmetic correctly? In the meantime, it was doable to search out whole sub-literatures that relied on theorems that have been outright incorrect. Though that sounds insane, it’s the completely predictable consequence of the publish-or-perish atmosphere: everyone is so busy publishing papers, that no one has the time (nor desires to take the tutorial political dangers) of shredding current papers that most likely no one will learn anyway.
This creates the state of affairs {that a} competent researcher must quietly fake massive parts of the literature don’t exist, although anyone studying the abstracts would suppose the papers are pertinent. As one would possibly count on, this will generate a tough assessment course of. One might be aware that that is precisely the type of behaviour post-Keynesians (together with myself!) complain about of their dealings with neoclassicals. (The important thing distinction is that I truly learn and understood the papers concerned, and will quietly clarify my exclusions by mentioning factual issues with the work.)
Agent-Primarily based Hybrid Fashions
If we wished a really correct mannequin, we might use an agent-based mannequin the place transactions happen in a sequence of some kind inside an accounting interval, then as soon as the transactions cease firing, the mannequin does end-of-period accounting. The top-of-period accounting is what would correspond to measured financial information.
Concluding Remarks
For what Steve Eager desires to do, steady time fashions could also be one of the simplest ways ahead. But when we need to argue fundamentals, since all financial information are discrete time, discrete time fashions are fully pure.