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Statistical Modeling

Statistical Modeling is the process of using data to construct a mathematical or algorithmic device to measure the probability of some observation.

A statistical model is a mathematical model that embodies a set of statistical assumptions concerning the generation of sample data (and similar data from a larger population). It is usually specified as a mathematical relationship between one or more random variables and other non-random variables. Statistical models include issues such as statistical characterization 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.

Classical statistical inference are based on the frequency interpretation of probability, that is probability can be defined as long-run relative frequency. Reference is made to repeated random samples from the same population and the probabilities cited in significance tests or confidence intervals indicate the relative frequency of certain outcomes (‘significant result’ or the inclusion of a parameter value in a confidence interval) in a long series of repetitions. The frequency-based interpretation of probability can be meaningless for events that cannot be repeated (e.g. probability of raining tomorrow). In classical statistical inference, Prior information other than that in the study being analysed is only informally used in design. The parameter of interest is a fixed unknown. The basic question is "How likely is the data, given a particular value of the parameter?" and results are usually presented as P values and confidence intervals.

An alternative view, Bayesian statistical inference, interprets probability as the degree to which one believes in the occurrence of an event or hypothesis. Degree-of-belief (or subjective) probabilities change as information accumulates. The degree of beliefs interpretation of probability leads to the Bayesian approach to statistical inference. Prior information other than that in the study being analysed is used formally by specifying a prior probability distribution. The parameter of interest is an unknown quantity which can have a probability distribution. The basic question is "How likely is a particular value of the parameter given the data?" Results are presented as plots of posterior distributions of the parameter and calculation of specific posterior probabilities of interest.

Statistical models or basic statistics can be used:

• To characterize numerical data to help one 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 behaviours 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 behaviours 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.

• 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.

Mathematical Modeling

A mathematical model is a formulation or equation that expresses the essential features of a physical system or process in mathematical terms. Mathematical models are usually composed of relationships and variables. Relationships can be described by operators, such as algebraic operators, functions, differential operators, etc. Variables are abstractions of system parameters of interest, that can be quantified. Mathematical models grow out of equations that determine how a system changes from one state to the next (differential equations) and/or how one variable depends on the value or state of other variables (state equations)

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 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 behaviours of the system.

• Quantitative results from mathematical models can easily be compared with observational data to identify a model's strengths and weaknesses. ü 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.

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