Order of a system

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Bill Braun bbraun hlthsys.com
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Order of a system

Post by Bill Braun bbraun hlthsys.com »

Posted by Bill Braun <bbraun@hlthsys.com>
In Business Dynamics, p. 264, Sterman notes that, ""the order of a dynamic system or loop is the number of state variables, or stocks, it contains"". I've always understood this to exclude stocks that were simply sinks that accumulate activity over time but which did not make any dynamic contribution to the behavior of the model. (For example, in the classic first-order positive feedback loop model of a bank account, the interest that flows into the account balance stock could be summed into a separate stock that has no independent dynamic effect on the account balance.)

Is that understanding accurate?

In the SI model (pps. 301 & 302) he notes that, ""though the system has two stocks, it is actually a first order system because one of the stocks is completely determined by the other.""

Regarding the SIR model (p. 305) he notes, ""unlike the model considered thus far, the system is now second-order (there are three stocks, but since they sum to a constant, only two are independent).""

Could someone unpack the distinctions between these references? Thank you.

Bill Braun
Posted by Bill Braun <bbraun@hlthsys.com>
posting date Fri, 16 Sep 2005 07:39:24 -0500
John Sterman jsterman MIT.EDU
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Order of a system

Post by John Sterman jsterman MIT.EDU »

Posted by John Sterman <jsterman@MIT.EDU>
The order of a model is the number of *independent* stocks, and technically that includes the stocks that do not participate in the feedback structure. The order of the ""active structure"" is the number of stocks that do participate in the feedback structure. The order of a loop is the number of stocks in that loop.

In the SI epidemic model *with constant population (no births, in/out migration, or deaths* the order of the system is one because you can rewrite the model so that there is only one stock. Equivalently, the model is completely specified by a first-order differential equation. The model is

S = INTEG(-Infection Rate)
I = INTEG (Infection Rate)
Infection Rate = f(S, I, constants) = ciS(I/N)

where c = contact frequency, i = infectivity, and N = total population,

which appears to have 2 stocks. However, S + I = N, because we assume that all persons are either S or I (that is, the states S and I are mutually exclusive and exhaustive). Since there are no flows into or out of the S and I states other than the Infection Rate (no births, deaths, migration, or alien abductions), the total population N is constant, and we can eliminate either S or I. So, for example, eliminate S by replacing it with S = N - I, yielding

I = INTEG (Infection Rate)
Infection Rate = f(I, constants) = ci(N - I)*(I/N)

which is a first-order system (indeed, it is the familiar logistic growth model).

The same analysis applies to the SIR model: As long as N is constant, only two of the states are independent, so the order of the differential equation corresponding to the model is two.

John Sterman
Posted by John Sterman <jsterman@MIT.EDU>
posting date Sat, 17 Sep 2005 11:14:59 -0400
Nathaniel Osgood nosgood MIT.EDU
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Order of a system

Post by Nathaniel Osgood nosgood MIT.EDU »

Posted by ""Nathaniel Osgood"" <nosgood@MIT.EDU>

Hi Bill,
My colleagues may disagree, I've always found it more intuitive to consider the dimensionality of the model's trajectory through its state space -- largely because it offers us a certain degree of representational invariance. (Informally speaking, ""state space"" here refers to the multidimensional space formed by the model's state variables, associating one axis with each. This space is ""timeless"" since there is no representation of time -- the trajectory in state space just captures the possible tuples of state variables that can obtain during the operation of the model)

While we often have a great degree of choice in selecting the state variables for characterizing a dynamical system, the dimensionality of the trajectory in state space should remain constant regardless of our representational choices. Even in cases where we have many dimensions in our state space (state variables), the trajectory is may be a thin lower dimensional surface (manifold) within the state space. This reflects the fact that there are fewer degrees of freedom in the mathematical model than variables in use. (In other words, we could represent the model with fewer state variables -- just calculating the other quantities from the value of the minimal set of state variables.) Sometimes HOW we would calculate the other variables based on some minimum set is clear and sometimes it's not so clear. But depicting the trajectory in state space at least lets us know that this is indeed possible.

If you consider the SI system, we have two stocks and therefore a 2D state space. But if you look at the system's trajectory within that state space (say, envisioned in a phase plane with ""susceptible"" on the horizontal axis and ""infected"" on the vertical axis), you'll find that the trajectory is a simple 1D curve (something that has length but no area). This reflects the fact that the size of the ""infected"" population is a simple function -- in this case, a linear function -- of the ""susceptible"" population. No matter how we adjust the parameters of the model (e.g. the likelihood of infection), this trajectory will remain fixed.

If we were to envision the 3D state space associated with the SIR model, the dimensionality of the trajectory in that space is more complicated. A given run of the model will trace out a 1D curve in the space, but by adjusting the parameters (likelihoods of infection and
recovery) we can actually create a thin 2D surface that has zero volume but does have area. However, try as we might, we'll never get anything with volume, since the model only has two degrees of freedom. (All points on the trajectory manifold share the the constraint that the coordinates in the ""susceptible"", ""infected"" and ""recovered"" dimensions must total up to the total population size) There are many possible ways we could represent this mathematical system using 2 stocks, but no matter how we do it, we know that we do need at least 2 stocks.

The dimensionality of the a model is not always obvious from its mathematical characterization, but a state space depiction can often help gain insight into the underlying dimensionality. (For higher dimensional state spaces, humans don't do so well with envisioning the state space, but there are quantitative mechanisms to measure state space dimensionality).

One thing that is rather nice about such state-space approaches is that they can be used provide insights into the dimensionality of real-life systems for which mathematical models have not been developed. (A twist that isn't often appreciated is the fact that state space characteristics of coupled dynamic systems manifest themselves in the time domain and that depictions a variable and its lag can also yield insights into state space
structure.)

With respect to your particular question, a variable that is purely accumulating some information about the history of the system can be thought of as adding a dimension to state space. Now the question is whether it adds to the dimensionality of the trajectory through the state space. Even though the variable does not influence other state variables the answer can be yes. In contrast to the cases we saw in the SI/SIR models, the current value of this accumulative state variable value cannot be expressed as a function of the other state variables, but of their history.

To use your example of a bank account, imagine that in addition to the state variable representing the size of the bank account (call it ""A"") that there was another state variable that totalled up the number of dollar-months of all past bank account balances.

(""B"" can be expressed as the integral of the bank account over time). While the state space for A alone is purely 1D (we can vary the interest rate all we want but the trajectory will always trace out the same path through ""timeless"" state space), if we consider the state space for the A-B system, the trajectory is 2D (the curve in A-B space traced by runs of the system will depend on the interest rate, and the full set of possible trajectories can in fact fill the plane) If ""B"" just accumulated the interest from ""A"".

By contrast, if ""B"" captured the INTEREST from ""A"", while ""A"" remained UNCHANGED, the state space trajectory would only have 1 dimension (here ""A"" never changes and only ""B"" changes, and thus the trajectory would be 1D). This is comforting, because we indeed know that we can represent such a system using 1 stock!

If ""B"" totalled the interest generated from ""A"" and ""A"" (also) captured its own interest thereby growing the size of the bank account, the trajectory in state space would still be 1D -- no matter what the interest rate, the trajectory etched in state space would be the same. This reflects the fact that we can always determine the value of ""A"" from ""B"" and vice-versa.

I hope this different perspective helps a bit!

Nate Osgood
osgood@cs.usask.ca
Posted by ""Nathaniel Osgood"" <nosgood@MIT.EDU>
posting date Sat, 17 Sep 2005 12:17:42 -0600
Joel Rahn jrahn sympatico.ca
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Order of a system

Post by Joel Rahn jrahn sympatico.ca »

Posted by Joel Rahn <jrahn@sympatico.ca>
'Order' is a more-or-less useful concept: interesting when talking about first or second order, occasionally when talking about third-order but hardly ever useful past that. Nevertheless, here are my two-cents' worth...

The definition should read: ""the order...is the number of independent state variables..."". 'independent' means, roughly, that the value of the stock can, in principle, be changed without reference to or without being constrained by the values of any of the other stocks. Thus in the 'cumulative interest' case, the system is first-order because the 'cumulative interest' stock is equal to the 'bank balance at time t' minus the 'bank balance at time 0'. In the SIR model, the system is second-order because of the constraint that the sum of the three stocks is a constant. In this sort of case, the choice of which two stocks are deemed 'independent' is arbitrary in principle but some pairings may be more useful than others for purposes of discussion.

'Independence' in dynamic models can be related to the structure of the linearized version of the model about a chosen time. The linearized differential equation form that results can be represented using a matrix, conventionally called A in many texts, which shows the strengths of the links between the state variables. The order of the model is equal to the rank of the matrix which is the number of rows that are 'independent', that is, 'in'-(not) -'dependent' (linear combinations) of other rows . 'Dependent' variables can be represented as supplementary variables.

'Order' or the equivalent 'independence' are evidently 'brittle' concepts-the linear combinations for example, a sum of values of stocks, must be exact. Yet we know that linearization is an approximation, so in linear models with time-varying parameters or non-linear models in general, 'order' is at the whim of numerical accuracy. Thus the 'count-the-stocks' definition (subtracting the stocks that are obviously
dependent) is probably good enough. Once you get past the low order systems, there is little that helps distinguish between higher order systems in terms of possible behaviour modes in general terms. Of course, the details will be different and maybe even fascinating to some and useful to a smaller number but rarely will a model's credibility rise or fall depending on whether its order is 20 versus 21 much less 100 versus 101 (and don't get me started on what happens when you get to 1000 or so...). Posted by Joel Rahn <jrahn@sympatico.ca> posting date Sat, 17 Sep 2005 12:32:10 -0400
Mabel Fong may_belle_66 yahoo.co
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Order of a system

Post by Mabel Fong may_belle_66 yahoo.co »

Posted by Mabel Fong <may_belle_66@yahoo.com>

I play with an economic model that controls the stock
of every commodity as it exists in the inventory of
every sector. This organization of state variables can
be visualized in terms of the familiar Input/Output
table. So, by Professor Sterman’s example, the model’s
order would be at least N2, where N is the order of
the I/O matrix. This brings me to my first question:
can we clean up the usage of ‘order’ as it applies to
dynamics versus matrix algebra versus polynomials
versus analytic geometry? Would not the word ‘degree’
have its uses here? Or are we going to run out of
words anyway?

My main question though concerns whether or not the
order of my models is increased by the presence of
savings and investment. Each of the N sectors is
associated with a financial state variable in which
historical cash flows accumulate. These stocks can be
either positive or negative, with their sense
indicating either saving or investment: saving is what
a household sector has earned but not yet spent;
investment is what an industrial sector has spent but
not yet earned.

Taking the simplest case of an economy isolated from
foreign trade, the sum of these financial state
variables will always be exactly zero because the
physical transaction occurring at any cell in the
matrix will enrich one sector by exactly as much as it
will un-enrich another sector. Though the amounts
saved are always off-set by the amounts invested, the respective magnitudes of saving and investment are free to change because 1) the physical rates controlling the physical state variables are free to go wherever they need to go so as to equate marginal revenues with marginal costs at each cell in the matrix; and 2) the prices at which these physical exchanges occur are also free to change.

Each commodity price, as well as the interest rate, is
a different complicated non-linear function of all the
physical and financial state variables I have
described. Each price also references all the system’s parameters. The system’s dynamics are further complicated in that the interest rate continuously effects transfers from investment stocks to saving
stocks: any un-serviced financial position will
increase exponentially at the rate of interest.

So my second question then is whether the order of
this system should be considered N2 or N2 + N? I
want to say N2 + N because the N financial state
variables are unknowable unless it is through the
process of integration; and I believe this much is
consistent with Professor Sterman’s example. But I
also note that the financial side of this model merely abstracts the physical side, and that the sum of the financial state is always null - which would seem to be something of a disqualification for true state variables in Professor Sterman’s definition.

If I might also wedge-in a third question, what
significance are we to associate with the small fact
that an Input/Output table presumes to express the
singular economic reality of the moment irrespective
of how fine a mesh we impose on our observations. The
model I use allows for any specification of N, and automatically combines or elaborates the model’s interrelationships as N is run up or down according to ones analytic needs. This allows comparisons of behavior modes among more or less aggregated models,
e.g.: when N is at its irreducible minimum of 2, the
model’s behavior shows little dampening of the
business cycle; proper dampening shows up as N
increases from 3 on up, and quickly gets us to
pleasing and consistent example of economic causality.
How does this react on our notion of a system’s
dynamic order?

Best,

Mabel
Posted by Mabel Fong <may_belle_66@yahoo.com>
posting date Sun, 18 Sep 2005 07:46:48 -0700 (PDT)
geoff coyle geoff.coyle btintern
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Order of a system

Post by geoff coyle geoff.coyle btintern »

Posted by ""geoff coyle"" <geoff.coyle@btinternet.com>
Bill's first paragraph is correct. The order of a model is the number of stocks it contains. That includes, however, the stocks internal to delays so, a model with two explicit stocks and a third order delay is of fifth order. Information smoothing also creates a stock, though the original level is a better term. Stocks which are external to the dynamics are not part of the model's order and are probably superfluous.

No, I'm afraid that I don't understand the two quotes from Jon Sterman's book. A stock is surely fed/depleted by flows and hence cannot be directly determined by other stocks. Perhaps these are with used to be called auxiliary variables.

Hope that helps.

Regards,

Geoff
Posted by ""geoff coyle"" <geoff.coyle@btinternet.com>
posting date Sun, 18 Sep 2005 18:06:02 +0100
George Richardson gpr albany.edu
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Order of a system

Post by George Richardson gpr albany.edu »

Posted by George Richardson <gpr@albany.edu>
On Sep 19, 2005, at 6:14 AM, geoff coyle geoff.coyle btinternet.com
wrote:

>> Bill's first paragraph is correct. The order of a model is the number
>> of stocks it contains.


Not quite. Should be ""...the number of *independent* stocks it contains.""

If one has an epidemic model, say, that has a stock of susceptibles, a stock of sick, and a stock of recovered, and there are no inflows to the susceptibles and no outflows from the recovered, it is a second-order system as the third stock could be obtained algbraically by subtracting two of the stocks from the total population.

..GPR
Posted by George Richardson <gpr@albany.edu>
posting date Mon, 19 Sep 2005 16:44:16 -0400
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