Hi,
I try to make a distinction between linear and nonliner models and between static and dynamic ones. Let my model contain one equation:
T = A/D + B + C; T = plus infinity for D = 0, and D = [0, 1].
It is a linear model. Here T is not a rate equation, but just a variable that I want to predict its value when the values for the rest variables are known. Strogatz suggests a system is said to be linear if all the independent variables, on the right hand side, appear to the first power only (Nonlinear Dynamics and Chaos, p. 6). So, I my model is perfectly linear.
Can the same criterion be used to classify SD models? Long time ago, Forrester said that external effects on a linear system are purely additive (Industrial Dynamics, p. 50). Recently, Sterman put it clearly: in a linear system, the rate equations are always a weighted sum of the state variables (Bussiness Dynamics, p. 264). They don't mention the power, but does Strogatz.
Now consider static models. The output of any static model cannot be plotted against time to create a time series. If plotted, the output is no more than a single point (x,t), called 'event' by SD authors. The model is used for point prediction, not for behavior pattern projection. Moreover, static models have no feedback loop. That is, the independent variables are not self-related, but they all affect the same dependent variable which never affects (gives a feedback to) the independent variables on the right hand side.
Below are some simple examples of the classified models.
Linear model:
T = A+B+C
Nonlinear model:
T = A*B*C
Static model:
T = A+B+C
Dynamic model
T(t) = A(t)+B(t)+C; t=[0, tn]
Feedback model
???
(please give me some simple examples)
Classification of Models
Static versus dynamic
Hi Monte.
Your first equation is not linear on D but linear on the other variables.
The best definition of linearity is the analytical one:
A variable Y dependant on n variables Xn: Y = f(Xn) is linearly dependant on Xi if the
Partial derivative dY/ dXi is constant whatever the constant values of the other variables.
With this definition you do not need to analyze the power of the polynomial expression of Y
Or anything else, which will be difficult anyway to express if you have a model expressed with many equations.
So a model is not linear or not linear, but can be totally linear or totally non linear or
Partially linear.
You have to consider too that a model may have different outputs. Some being linear and
the other not.
For a SD model, an output being a function of the input, the definition is valid too.
A characteristic of linear functions is that if f and g is linear then f°g is linear too.
If y = f(x) is linear and x = g(z) is linear than y = f ( g (z)) is linear too .
Linearity is too conserved with addition of two linear functions and multiplication of a linear function with a constant.
Than if all the equations of a SD model are linear, than logically all the output should be
linear too. It should be too true for the levels that accumulate (addition of two linear functions).
Example: if a rate is equal to 4 * A and linearly dependant from A, and accumulate in a stock.
After 5 time periods it will be equal to 20 A which is always linearly dependant from A.
But if the rate is equal to A * the value of the level, if the level has an initial value of 1,
then the values of the stock will be equal to 1, 1 + A, 1 + A +(A * (1 + A)) etc… which is no more linearly dependant from A. In fact it is because the rate is no more linearly dependant from A but from A and the stock something else that is too dependant from A.
About the output of a static model not creating any time series, why?
For me the definition of a static model is a model or a function where the time is not
represented explicitly.
But it is not because the time is not represented explicitly that you cannot use a static model
to deliver the same kind of results that would deliver a dynamic one.
Example: a hotel manager has noticed that his weekly turnover is linearly dependant from the number of phone calls per week. He wants to predict his turnover for each next 4 weeks.
He has noticed the last year’s number of phone calls for the same periods.
So to predict the turnover he can predict the number of phone calls equal to those from last year and then the turnover.
Now is the turnover dependant from the time or not?
One can say that it is not dependant from the time, and strictly dependant from the number of
phone calls, but the number of phone calls being in that case dependant from the time, one can say that the turnover is too dependant from the time.
Of course there is no behaviour or understanding in these predictions, but it can be sufficient for the business man in question.
I have thought about the definition of static and dynamic models.
It is not so simple.
In a field that I am interested in, revenue management, models are mainly dynamics, but the problems are not studied, trying to understand the causality of the behaviour, but trying to
optimize the outcomes with models that are not dynamic from an SD point of view, but which are termed dynamical and in fact are. These models never mention feed back loops, or very rarely.
In fact in life everything is dynamical, and nothing is really static, and if the SD point of view was the only way to study dynamic problems, SD would be much more used.
Regards.
JJ.
Your first equation is not linear on D but linear on the other variables.
The best definition of linearity is the analytical one:
A variable Y dependant on n variables Xn: Y = f(Xn) is linearly dependant on Xi if the
Partial derivative dY/ dXi is constant whatever the constant values of the other variables.
With this definition you do not need to analyze the power of the polynomial expression of Y
Or anything else, which will be difficult anyway to express if you have a model expressed with many equations.
So a model is not linear or not linear, but can be totally linear or totally non linear or
Partially linear.
You have to consider too that a model may have different outputs. Some being linear and
the other not.
For a SD model, an output being a function of the input, the definition is valid too.
A characteristic of linear functions is that if f and g is linear then f°g is linear too.
If y = f(x) is linear and x = g(z) is linear than y = f ( g (z)) is linear too .
Linearity is too conserved with addition of two linear functions and multiplication of a linear function with a constant.
Than if all the equations of a SD model are linear, than logically all the output should be
linear too. It should be too true for the levels that accumulate (addition of two linear functions).
Example: if a rate is equal to 4 * A and linearly dependant from A, and accumulate in a stock.
After 5 time periods it will be equal to 20 A which is always linearly dependant from A.
But if the rate is equal to A * the value of the level, if the level has an initial value of 1,
then the values of the stock will be equal to 1, 1 + A, 1 + A +(A * (1 + A)) etc… which is no more linearly dependant from A. In fact it is because the rate is no more linearly dependant from A but from A and the stock something else that is too dependant from A.
About the output of a static model not creating any time series, why?
For me the definition of a static model is a model or a function where the time is not
represented explicitly.
But it is not because the time is not represented explicitly that you cannot use a static model
to deliver the same kind of results that would deliver a dynamic one.
Example: a hotel manager has noticed that his weekly turnover is linearly dependant from the number of phone calls per week. He wants to predict his turnover for each next 4 weeks.
He has noticed the last year’s number of phone calls for the same periods.
So to predict the turnover he can predict the number of phone calls equal to those from last year and then the turnover.
Now is the turnover dependant from the time or not?
One can say that it is not dependant from the time, and strictly dependant from the number of
phone calls, but the number of phone calls being in that case dependant from the time, one can say that the turnover is too dependant from the time.
Of course there is no behaviour or understanding in these predictions, but it can be sufficient for the business man in question.
I have thought about the definition of static and dynamic models.
It is not so simple.
In a field that I am interested in, revenue management, models are mainly dynamics, but the problems are not studied, trying to understand the causality of the behaviour, but trying to
optimize the outcomes with models that are not dynamic from an SD point of view, but which are termed dynamical and in fact are. These models never mention feed back loops, or very rarely.
In fact in life everything is dynamical, and nothing is really static, and if the SD point of view was the only way to study dynamic problems, SD would be much more used.
Regards.
JJ.
Thank you very much for your very helpful note. It is now clear that a linear equation, Y = mX + c, is said to be linear because the slope (dY/dx or m) is constant. If m is variable, then the behavior of Y is no longer linearly changing.
In this linear model, it seems that Y does not depend on time, but on X. Y is purely determined by X, which in turn can exhibit nonlinear behavior.
Static model contains variables that are related in mass dimension, not in temporal dimension. For example,
Y = 0.02X
where Y my money measured in US dollars; X is my money in Thai Baht. Here X is another level which vary depending on revenue and expenditure. There is no rate in a static model. There are levels only. So, my definitions of static and dynamic model are as follows:
Static model is a model without rate variable.
Dynamic model is a model with level and rate variables.
Are there definitions acceptable?
In this linear model, it seems that Y does not depend on time, but on X. Y is purely determined by X, which in turn can exhibit nonlinear behavior.
Static model contains variables that are related in mass dimension, not in temporal dimension. For example,
Y = 0.02X
where Y my money measured in US dollars; X is my money in Thai Baht. Here X is another level which vary depending on revenue and expenditure. There is no rate in a static model. There are levels only. So, my definitions of static and dynamic model are as follows:
Static model is a model without rate variable.
Dynamic model is a model with level and rate variables.
Are there definitions acceptable?
static and linear model
Hi
In the equation Y = mX +c, m can be a variable and Y sitll depend linearly from X as long as m is independant from X. But m can vary on its own. What one considers is a partial derivative.
About the right definition of static and dynamic, one can refer to the mechanical definition of static and dynamic, where in static the time is not included and in dynamic time is included.
But the definition by level and rates might be an SD definition, but I am sure that people from other fields will not aggree with it.
An interesting point of view about static and dynamic is Magnet Myrveit's one.
You can see it on the site dynaplan.com blogs and articles the two last articles on SD and spreadsheets. It compares static complexity with dynamic one.
Regards.
JJ
In the equation Y = mX +c, m can be a variable and Y sitll depend linearly from X as long as m is independant from X. But m can vary on its own. What one considers is a partial derivative.
About the right definition of static and dynamic, one can refer to the mechanical definition of static and dynamic, where in static the time is not included and in dynamic time is included.
But the definition by level and rates might be an SD definition, but I am sure that people from other fields will not aggree with it.
An interesting point of view about static and dynamic is Magnet Myrveit's one.
You can see it on the site dynaplan.com blogs and articles the two last articles on SD and spreadsheets. It compares static complexity with dynamic one.
Regards.
JJ