Complexity and Scientific Modelling--论文代写范文精选

噪声可以被视为是不可预知的，不同的方法几乎区分噪声，造成不同的复杂性之间的权衡。复杂性是区分开来的,总体上是独立的。下面的essay代写范文进行详述。

Abstract
There have been many attempts at formulating measures of complexity1 of physical processes (or more usually of the data models composed of sequences of measurements made on them). Here we reject this direct approach and attribute complexity only to models of these processes in a given language, to reflect its “difficulty”. This means that it is a highly abstract construct relative to the language of representation and the type of difficulty that concerns one. A framework for modelling is outlined which includes the language of modelling, the complexity of models in that language, the error in the model's predictions and the specificity of the model (which roughly corresponds to its refutability or to the information it gives about the process).

Such a framework makes sense of a number of aspects of scientific modelling. 
1. As a result complexity is not situated between order and disorder , as several authors have assumed, but rather such judgements arise given certain natural assumptions about the language of modelling and the desirable trade-offs between the model complexity, its specificity and its error rate. 
2. Noise can be seen as that which is unpredictable given the available resources of the modeller. In this way noise is distinguished from randomness. Dif ferent ways of practically distinguishing noise can thus be seen as resulting from different trade-offs between complexity, error, specificity and the choice of modelling language. 
3. Complexity is distinguished from concerns of specificity such as refutability , entropy and information. Complexity is thus seen to have context-dependent relations with such measures but in general is independent from them.
4. Less complex models are not a priori likely to be more accurate, but rather that given the typical structure of expressive modelling languages and our limitations in searching through such languages, choosing the simpler model can be a useful heuristic.

Complexity
There is an understandable wish to measure the complexity of real systems rather than just of models of systems, but if natural systems have inherent levels of complexity they are beyond us. In practice there is no practical upper bound on their complexity as one has only to consider them in more detail or by including more aspects. On the other hand, the ef fective complexity of systems does depend on our models - the exact motions of the planets may be puzzling when you have to describe them in terms of epi-cycles but much simpler in terms of ellipses. It may be objected that even if such “real complexity” is so intractable, one can still make comparative judgements; i.e. 

it is natural to judge some natural systems as more complex than others. An example of this is the claim that a cell is simpler than a whole or ganism. However in defending such judgements one is always forced into relating that which is compared within a common model. It is only once you have abstracted away what you consider to be irrelevant details, that such judgements of relative complexity become evident. It may be argued that some such models and frameworks are privileged but this pre-judges decisions about relevance and so is not helpful to an analysis of complexity and its place in scientific modelling. In any case complexity is more critically dependent upon the model rather than what is modelled. So we will approach it from this point of view and leave the reader to judge whether this distinction is fruitful in understanding the processes involved.

A Framework for Analysing Modelling
Firstly it is important to distinguish between the form and predictive meaning of such models. The models themselves are always held in some form. The set of such possible forms can be considered as a language in its broadest sense - frequently it may correspond closely to an actual language, either natural or formal. I will call this the modelling language. Such models are amenable to some form of inference, in that they can be used to predict some property given some other information (even if sometimes some of the necessary information is only available after the predicted event). At least some of the information comes from what is modelled in the form of measurements. 

The models correspond, loosely, to what they model via these predictions. Thus we distinguish two aspects of a model: its form and the correspondence between possible information and the predictions that one could infer from it. This set of information along with the respective predictions can be thought of as defining a subspace of the space of all relevant possibilities. I will call this subspace the model's semantics, because one can draw an analogy between a logic’s syntax and its semantics in terms of the set of logical models a statement is true for. The primary way in which these models can be judged is by the degree of correspondence between what is modelled and the predictions of the models - its error . 

This however does not rule out the default model, that “anything can happen”. Such a model is always trivially correct (and thus is typically chosen as a starting point). Thus we also need an additional goal of preferring the more specific (or refutable) model. I will call this the model's specificity. A modeller with infinite resources and time need only use these two measures as guides in its choice of model. In some cases, of course these dual aims might be in conflict. In a given modelling language one might be forced to choose between a vague but accurate model and a specific but more erroneous one. In some accounts the specificity of models are sometimes left out of analyses of modelling because the types of modelling languages considered are inherently precise. For us more limited beings, with very distinct practical considerations, the complexity of our models become important. As we shall see below we have to balance the complexity , the error and the specificity of out models. Note that here, complexity is a property of the form of the models while the error and specificity are properties of its corresponding model semantics.

Other Formulations of Complexity 
There is not room to do anything but mention but a few of the other formulations of complexity here (see [4] for a reasonably complete listing). I will only look at three here (I am distinguishing here between complexity and simplicity as it is traditionally used in philosophy , for this see section 8, below). The Algorithmic Information of a pattern can be considered as the difficulty of storage of a program to generate the pattern (or alternatively the dif ficulty in finding such a program when working though possible programs in order of length). Grassberger [6] defines the Ef fective Measure Complexity (EMC) of a pattern as the asymptotic behaviour of the amount of information required to predict the next symbol to the level of granularity . 

This captures an aspect of the scaling behaviour of the information required for successful prediction by a markov process model. This thus captures the asymptotic behaviour of a special case of my definition. A similar approach is taken by Badii and Politti [1]. The topological complexity described by Crutchfield [3], is a measure of the size of the minimal computational model (typically a finite automaton of some variety) in the minimal formal language in which it has a finite model. Thus the complexity of the model is both ‘objectivised’ by considering only minimal models but also related to the fixed hierarchy of formal languages. 

This has a number of disadvantages. Firstly this does not give a unique complexity for any pattern, as there is not necessarily such a “minimal” formal language, secondly in some formal languages the minimal model is uncomputable and thirdly in stochastic languages the minimal model will frequently be a completely random one, so one is forced to trade specificity with complexity to get a reasonable result. He also defines a measure of specificity similar to EMC above, as complementary to the topological complexity. In each case the desire to attribute complexity purely objectively to a physical process seems to force a relativisation to either some framework for which privilege is claimed (e.g. a Turing Machine), to some aspect of the problem (e.g. granularity of representation) or by considering only the minimal size. This, of course, does not completely eliminate the inherent subjective ef fects in the process of modelling (principally the language of modelling), and obscures the interplay of complexity , specificity and the error involved.

To see this possibility consider the following situation. A modeller has an infinite and precise symbolic language with a limited number of symbols and some fixed grammar such that it includes some small expressions, but expressions of increasing size can be constructed. Suppose this language describes members of a class of data strings of any length of any sequence of symbols taken from a fixed alphabet. A simple counting ar gument shows that most such patterns are disordered (as defined either by something like Shannon information or algorithmic information measures), but a similar counting argument shows that only a few of these patterns can correspond to models with relatively small minimal representations. 

That is, most of the disordered patterns will correspond with the models with the relatively large minimal representations. Whatever the ordering in terms of ease of search, in general the bigger forms will be more difficult to find, i.e. more complex. Thus, in this case, far from complexity and disorder being antithetical, one would be hard pushed to arrange things so that any of the most complex models would correspond to even slightly ordered patterns. So if complexity does not necessarily lie between order and disorder , where has our intuition gone wrong? Without any prior knowledge about the process that produces the data we have no reliable way of distinguishing what is mer ely very complex behaviour and what is irr elevant noise. The diagrams above mislead us because our experience about the patterns we typically encounter , has led us to recognise the noise, and separate it out from the relevant pattern. That this is not necessarily so, see figure 3, where we show each pattern as a magnification of a section of the one to its right.（essay代写)

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