statistical and mathematical models

8/18/2019 Statistical and Mathematical Models

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An Assignment

On

Statist ical and Mathematical Models

Submitted By- Aman Arora

Registration No. 20158911

Department of Geography

Faculty of Natural Sciences

Jamia Millia Islamia, New Delhi-


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Statistical and Mathematical Models Aman Arora: 20158911

What are Mathematical and Statistical Models?

These types of models are obviously related, but there are also real differences

between them.

Mathematical Models

A mathematical model is a set of descriptions of methods using mathematical tools

and techniques. The process of developing a mathematical model is termed

mathematical modelling. These Models grow out of equations that determine how a

system changes from one state to the next and/or how one variable depends on the

value or state of other variables. These can also be divided into either numerical

models or analytical models.

Numerical Models

Numerical models are mathematical models that use some sort of numerical time-

stepping procedure to obtain the models behaviour over time. The mathematical

solution is represented by a generated table and/or graph.

Analytical Models

Analytical models are mathematical models that have a closed form solution, i.e. the

solution to the equations used to describe changes in a system can be expressed as a

mathematical analytic function.

Classification of Mathematical Models

Mathematical models are usually composed of relationships and variables.

Relationships can be described by operators, such as algebraic operators, functions,

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differential operators, etc. Several classification criteria can be used for

mathematical models according to their structure:

Linear vs. nonlinear: If all the operators in a mathematical model exhibit

linearity, the result1ing mathematical model is defined as linear. Nonlinearity,

even in fairly simple systems, is often associated with phenomena such as chaos

and irreversibility.

Static vs. dynamic: A dynamic model accounts for time-dependent changes in

the state of the system, while a static model calculates the system in equilibrium,

and thus is time-invariant. Dynamic models typically are represented by

differential equations.

Explicit vs. implicit: If all of the input parameters of the overall model are

known, and the output parameters can be calculated by a finite series of

computations, the model is said to be explicit. But sometimes it is the output

parameters which are known, and the corresponding inputs must be solved for by

an iterative procedure, can be calculated through the remaining option only.

Discrete vs. continuous: A discrete model treats objects as discrete, such as the

particles in a molecular model or the states in a statistical model; while a

continuous model represents the objects in a continuous manner, such as the

velocity field of fluid in pipe flows, temperatures and stresses in a solid, and

electric field that applies continuously over the entire model due to a point

charge.

Deterministic vs. probabilistic (stochastic): A deterministic model is one in

which every set of variable states is uniquely determined by parameters in the

model and by sets of previous states of these variables; therefore, a deterministic

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model always performs the same way for a given set of initial conditions.

Conversely, in a stochastic model—usually called a "statistical model"—

randomness is present, and variable states are not described by unique values, but

rather by probability distributions.

Deductive, inductive, or floating: A deductive model is a logical structure based

on a theory. An inductive model arises from empirical findings and

generalization from them. The floating model rests on neither theory nor

observation, but is merely the invocation of expected structure. Application of

mathematics in social sciences outside of economics has been criticized for

unfounded models. Application of catastrophe theory in science has been

characterized as a floating model.

Usage of Mathematical Models

There are several situations in which mathematical models can be used very

effectively in introductory education.

Mathematical models can help students understand and explore the meaning

of equations or functional relationships.

Mathematical modelling software such as Excel, Stella II , or on-line JAVA

/Macromedia type programs make it relatively easy to create a learning

environment in which introductory students can be interactively engaged in

guided inquiry, heads-on and hands-on activities.

After developing a conceptual model of a physical system it is natural to

develop a mathematical model that will allow one to estimate the quantitative

behaviour of the system.

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Quantitative results from mathematical models can easily be compared with

observational data to identify a model's strengths and weaknesses.

Mathematical models are an important component of the final "complete

model" of a system which is actually a collection of conceptual, physical,

mathematical, visualization, and possibly statistical sub-models.

Statistical Models

A statistical model is a special type of mathematical model. What distinguishes a

statistical model from other mathematical models is that a statistical model is non-

deterministic. Thus, in a statistical model specified via mathematical equations,

some of the variables do not have specific values, but instead have probability

distributions; i.e. some of the variables are random.

A model is usually specified by mathematical equations that relate one or more

random variables and possibly other non-random variables. As such, "a model is a

formal representation of a theory" (Herman Adèr quoting Kenneth Bollen).

In mathematical terms, a statistical model is usually thought of as a pair (S, P),

where S is the set of possible observations, i.e. the sample space, and P is a set of

probability distributions on S.

Statistical Models include techniques such as statistical classification of numerical

data, estimating the probabilistic future behaviour of a system based on past

behaviour, extrapolation or interpolation of data based on some best-fit, error

estimates of observations, or spectral analysis of data or model generated output. It

embodies a set of assumptions concerning the generation of the targeted data, and

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similar data from a larger population. A model represents, often in considerably

idealized form, the data-generating process and its assumptions describe a set of

probability distributions, some of which are assumed to adequately approximate the

distribution from which a particular data set is sampled.

Statistical models are often used even when the physical process being modeled is

deterministic. For instance, coin tossing is, in principle, a deterministic process; yet

it is commonly modeled as stochastic (via a Bernoulli process).

There are three purposes for a statistical model, according to Konishi & Kitagawa.

Predictions

Extraction of information

Description of stochastic structures

Degree of Models

Experts distinguish between three levels of modelling assumption;

Fully parametric: The probability distributions describing the data-generation

process are assumed to be fully described by a family of probability distributions

involving only a finite number of unknown parameters.

Non-parametric: The assumptions made about the process generating the data are

much less than in parametric statistics and may be minimal.

Semi-parametric: This term typically implies assumptions ‘between’ fully and

non-parametric approaches.

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Usage of Statistical Model

Statistical models or basic statistics can be used:

To characterize numerical data to help one to concisely describe the

measurements and to help in the development of conceptual models of a

system or process;

To help estimate uncertainties in observational data and uncertainties in

calculation based on observational data;

To characterize numerical output from mathematical models to help

understand the model behaviour and to assess the model's ability to simulate

important features of the natural system (model validation). Feeding this

information back into the model development process will enhance model

performance;

To estimate probabilistic future behaviour of a system based on past

statistical information, a statistical prediction model. This is often a method

use in climate prediction. A statement like 'Southern California will be wet

this winter because of a strong El Nino' is based on a statistical prediction

model.

To extrapolation or interpolation of data based on a linear fit (or some other

mathematical fit) are also good examples of statistical prediction models.

To estimate input parameters for more complex mathematical models.

To obtain frequency spectra of observations and model output.

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References

• Adèr, H.J. (2008), "Modelling", in Adèr, H.J.; Mellenbergh, G.J., Advising

on Research Methods: a consultant's companion, Huizen, The Netherlands:

Johannes van Kessel Publishing, pp. 271–304.

• Burnham, K. P.; Anderson, D. R. (2002), Model Selection and Multimodel

Inference (2nd ed.), Springer-Verlag, ISBN 0-387-95364-7 .

• Cox, D.R. (2006), Principles of Statistical Inference, Cambridge University

Press.x`

•

Konishi, S.; Kitagawa, G. (2008), Information Criteria and Statistical Modeling, Springer.

• McCullagh, P. (2002), "What is a statistical model?", Annals of Statistics 30:

1225–1310, doi:10.1214/aos/1035844977.

• Andreski, Stanislav (1972). Social Sciences as Sorcery. St. Martin’s Press.

ISBN 0-14-021816-5.

• Truesdell, Clifford (1984). An Idiot’s Fugitive Essays on Science. Springer.

pp. 121–7. ISBN 3-540-90703-3.

• Billings S.A. (2013), Nonlinear System Identification: NARMAX Methods in

the Time, Frequency, and Spatio-Temporal Domains, Wiley.

• Pyke, G. H. (1984). "Optimal Foraging Theory: A Critical Review". Annual

Review of Ecology and Systematics 15: 523–575.

doi:10.1146/annurev.es.15.110184.002515.

•

Whishaw, I. Q.; Hines, D. J.; Wallace, D. G. (2001). "Dead reckoning (path

integration) requires the hippocampal formation: Evidence from

spontaneous exploration and spatial learning tasks in light (allothetic) and

dark (idiothetic) tests". Behavioural Brain Research 127 (1–2): 49–69.

doi:10.1016/S0166-4328(01)00359-X.PMID 11718884.
https://en.wikipedia.org/wiki/Stanislav_Andreskihttps://en.wikipedia.org/wiki/St._Martin%E2%80%99s_Presshttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/0-14-021816-5https://en.wikipedia.org/wiki/Clifford_Truesdellhttps://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/3-540-90703-3https://en.wikipedia.org/wiki/Digital_object_identifierhttps://dx.doi.org/10.1146%2Fannurev.es.15.110184.002515https://en.wikipedia.org/wiki/PubMed_Identifierhttps://www.ncbi.nlm.nih.gov/pubmed/11718884https://www.ncbi.nlm.nih.gov/pubmed/11718884https://en.wikipedia.org/wiki/PubMed_Identifierhttps://dx.doi.org/10.1146%2Fannurev.es.15.110184.002515https://en.wikipedia.org/wiki/Digital_object_identifierhttps://en.wikipedia.org/wiki/Special:BookSources/3-540-90703-3https://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Clifford_Truesdellhttps://en.wikipedia.org/wiki/Special:BookSources/0-14-021816-5https://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/St._Martin%E2%80%99s_Presshttps://en.wikipedia.org/wiki/Stanislav_Andreski

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