What+is+a+model

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=What is a model?=


 * (Concepts:** finite state machine, unique next state function, autonomous system, (multiple) attractors, garden of eden states, domains of attraction, ergodicity)

Before defining what a model is, let us introduce the **[|Finite state machine]** (FSM), as a prototype of a model. We will then try to understand various model formalisms in terms of simplifications, or short-cuts, made relative to FSMs, as well as discuss some of the things we can learn from models.

//Finite state machines as a model prototype//
A FSM is composed of: We denote such an FSM by . As the name suggests, the number of states N must be a finite number. The **state** of the system is defined as all internal information of the system that we need to uniquely determine the next state and the next output. The **NSF** specifies how a set of inputs and current state leads to the next state. A NSF can be defined by a [|transition table] which specifies the next state (one out of the s1, s2, ..., sN possibilities) for all possible combinations of inputs and states. If an NSF is defined, the NSF should uniquely determine which state an FSM takes in the "next time" given the current state and input of the FSM. Thus, an FSM is deterministic, and has a **unique next state function**. Similarly, a NOF specifies the next output for all possible sets of input and state. This is also defined by a table, and therewith, one can infer which output an FSM emits in the next time given the current state and input of the FSM.
 * a set of inputs I = {i1, i2, i3, ... iN},
 * a set of states S = {s1, s2, s3, ..., sN},
 * a set of outputs O = {o1, o2, o3, ..., oN},
 * a next state function (NSF) and
 * a next output function (NOF).

An important requirement for a FSM is that it must be fully specified: NSF and NOF must //uniquely// specify the next state and output for //all// possible combinations of states and inputs; and the state and input of the FSM must be //completely// known at any moment in time. If this requirement is not fulfilled, one can not determine or compute the next state of FSM, and thus its temporal dynamics.

FSMs, in general, do not necessarily have all I, S, O, NSF and NOF. FSMs can be composed of I, O and NOF, which are called **input-output systems** . FSMs can also be composed of I, S, O and NOF , which may be called input-output systems **with memory**. To elaborate on this difference in terms of modeling, consider a [|pendulum] and consider an FSM as a model of it. Suppose one sets up the following experiment: A pendulum is deviated from its resting position, and is quietly freed from the deviation point. One then measures the position of the pendulum after a fixed amount of time since freeing. By considering an initial deviation as an input, and the position of pendulum after a fixed amount of time as a response to the input, one can characterize a pendulum by an input-output system . Alternatively, an experiment can be set up as follows: One gives to a pendulum a deviation from its resting point **and** a certain push, measures the position of the pendulum at an arbitrary time point (starting point) and the position of the pendulum after a fixed amount of time since the first measurement. Clearly, one now also requires angular momentum as an explanatory property for understanding of the system (the same deviation at the starting time could lead to multiple deviations at the measuring time, dependant on angular momentum). In this case, an FSM should hence be  instead, where angular momentum is the internal state of the pendulum. Therefore, what components an FSM should have are not intrinsic characteristics of the system of concern, but are determined by how one sets up experiments.

(//Interestingly, most behavioural and learning experiments are set up as input-output systems in an attempt to control for variation between individuals, i.e. the role of the state (memory) is minimized.//)

Let us now consider an **[|autonomous system]****,** which is a typical form of a model (such as autonomous [|ordinary differential equations] (ODEs)). In the FSM formalism, an autonomous system is defined as . Since the NSF uniquely determines the next state, each state has only one next state. This can be visualized as follows: Consider a [|graph] (nodes connected by arrows) in which nodes represent states, and directed edges (arrows) represent transitions between states. In this scheme,  will be such that each state has only one out-going edge, but one state can have multiple incoming edges. This simple visual consideration of  leads to several important concepts (which are generally useful in modeling): > from which it does not leave)
 * **[|Attractor]** (a subset of states where the system eventually goes and from which it does not leave)
 * **[|Garden of Eden states]** (those states where the system can start but never returns to)
 * **Multiple attractors** (there can be several distinct subsets of states where the system can go and
 * **Domain of attraction** (the union of states that lead to a certain attractor)

(Note that not all  have attractors and Garden of Eden states. If all states are visited by a system a possibly infinite amount of times, this is called [|ergodicity]. If  is ergodic, then there are no attractors nor Garden of Eden states. However, this is a special case among all possible .)

It was pointed out that FSMs must be fully specified because, otherwise, one can not compute the future state of the system. This requirement holds in general: A model must be **fully specified** (we consider only such models in this course).

When the number of states is substantially large, specifying a transition table NSF for an FSM is a formidable, if not impossible, task. Thus, in practice, most modelers make short- cuts with respect to fully specifying a unique next state function of a model. We will discuss some important short-cuts in the next section. These include ODEs, [|MAPs (difference equations)], and [|Cellular Automata (CA)].

Next: Short-cuts or Modeling formalisms

(CHANGELOG 2014-2015) - Removed non-visible figure of example of FSM. - Added figure: example of autonomous FSM. - Added definition of state.