Model Data

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Journal of Applied Sciences Research, *(*): *91-894, 2011

ISSN 1819-544X

This is a refereed journal and all articles are professionally screened and reviewed

ORIGINAL ARTICLES

Logical Comments on Goal Programming Approach Based on Median

Ramin Seyedi

Islamic AZAD University, Maragheh Branch, Maragheh, Iran.

ABSTRACT

In this paper we see some logical comments for Models in paper An experimental comparison of the new

goal programming and the linear programming approaches in the two-group discriminant problems, also discuss

some logical notes about the modelling and other primal subjects.

Key words: Goal programming; Linear programming; Classification, Model selection.

Introduction

Hasan Bal, Hasan orkcu and Salih Celebioglu in the paper (Bal, 2006) consider a newlinear programming

and two goal programming models for two-group classification problems. When these approaches are applied

to the data of real life or ofsimulation, proposed new models perform well both in separating the groups andthe

group membership predictions of new objects. In discriminant analysis somelinear programming models

determine the attribute weights and the cut-off valuein two steps, but suggested models determine simultaneously

all of these valuesin one step. Moreover, the results of simulation experiments show that suggested proposed

models outperform significantly than existing linear programming and statistical approaches in attaining higher

average hit ratios. Deductive logic explicates the notion of a valid argument and develops a formalism how to

discern valid inferences that preserve the truth in passing from the premises to the conclusions. Then the

premises logically entail the conclusion .Hence, deductive logic studies principles and criteria of truth-preserving

inference.It is a formal science in the sense that the meaning of the symbols does not affect soundness or

validity of the conclusions. Inductive logic tries to generalize the idea of logical entailment to inferences where

the truth of the premises doesnot guarantee the truth of the conclusions. Still, the truth of the premises might

indicate the truth of the conclusion, and it is the point of inductive logic to make the vague and informal notion

of truth-indication more precise. The central concepts become confirmation and evidential support: it is not

asked whether the premises logically entail the conclusion but whether they give good reasons toassert the

conclusion and to which degree they support it. In particular, inductive logic is supposed to quantify the effects

of observation and measurement on the epistemic status of general hypotheses and theories. Most empirical

science sinfer from data to general hypotheses, and as deductive relations between the oryand evidence seldom

hold, the degree of support is of particular interest. There inductive logic comes into play, figuring out which

hypotheses are best confirmed by the data. Usually, inductive reasoning in science proceeds along the lines of

the mathematical theory of probability. A probabilistic entailment has the general form

where and i denote sentences of a given language and y and xi denote the corresponding probabilities. In

particular, y denotes the posterior probability which the premises sentences with a given probability impose on

the conclusion .Many frequentist techniques are highly sensitive to underlying assumptions so that human

expertise and scientific understanding are required for a sensible implementation. Consequently, I conclude that

statistics mainly addresses practical worries about using data in making decisions, predicting events or describing

the mechanisms of a system. More precisely, statistics contains a patchwork of different approaches.

Corresponding Author: RAMIN SEYEDI, Islamic AZAD University, Maragheh Branch, Maragheh, Iran.

E-mail: *******@***-********.**.**

J. Appl. Sci. Res., 7(6): 891-894, 2011 892

Choosing one of them is highly sensitive to modelling assumptions, specification of goals, error tolerance

etc., and there are no conclusive arguments for a particular method. Hence, comparison of different methods

is only possible relative to far-reaching assumptions, blurring the prospects for conceptual unification of

statistics. The way how genuinely scientific insights enter the statistical model analysis suggests that statistics

resembles an empirical science more than a sophisticated inductive logic. This claim can be substantiated by

the numerical turn in statistics: computer-based design of statistical methods and their simulation-based

evaluation become more and more important.(Gilani 2002; Fitelson, 2005; Akaike,)

2. LCM Model and LPMED Models:

From paper (Burnham, 1998), we can see that Linear programming models and many of the others

determine the attribute weights and cut-off value. And divide the process of their model into two steps: the first

constitutes the determination of attribute weights, and the second determines the cut-off value for the

classification. In its first step their model makes use of an objective function minimizing the sum of deviations

from the group mean classification scores. The LCM model can be formulated as follows:

s.t.

Where ), are unrestricted variables.

By this model, wj (j = 1,2 k ), the attribute weights are found, and then the object scores are obtained.

In this model, it is reached to the weights by making object scores close to their group mean scores. And then

the object scores are used in the following model, and the classification is made:

s.t.

where hi $ 0, (i = 1,2 n) and c is an unrestricted variable. LCM model, minimizes the sum of individual

deviations of the classification scores from their group mean classification scores. Instead of the mean, we can

use the median in their model (Burnham, 1998), because the median is the point that minimizes the total

l1!norm distance from all points to it. And this model based on median is called as the LPMED. This LPMED

model minimizes the deviations of individual classification scores from their group median classification scores

in a two-group classification problem. Similar to the LCM model, the LPMED model formulated as follows:

s.t.

J. Appl. Sci. Res., 7(6): 891-894, 2011 893

Where ) are unrestricted variables

and med1,j is the median of the jth variable in G1 and med2,j is the median of the jth variable in G2. In this

model, in the first step the weights wj are found after the solution to the LPMED1. Here the weights are found

by making the object classification scores close to their group median scores. Using these weights the

classification scores for each object are evaluated and then the assignment of objects to groups are made by the

following the LPMED2 model:

s.t.

Where hi $ 0, (i = 1,2 n) and c is an unrestricted variable. Like in LCM models, the classification is made

in two independent steps. (Bal, 2006; Bajgier, 1982; Sueyoshi, 1999).

3. Logical Comments on Model Selection:

A lot of the modern debate in statistics and applied sciences focuses on the issue of model selection how

to filter a set of candidate models as to obtain a predictively successful and explanatorily helpful model. To

select a model which can be used in further study of the phenomena is such an important decision that we ought

to treat it as an integral part of statistical inference. Model selection thus involves the fitting of models to

empirical data as well as decisions on the complexity of the model and finding the causally relevant factors.

A suitable selection strategy has to evaluate the model selection uncertainty, i.e. to account for the problem that

the same data which are used for selecting a model family are also used for fitting the model and estimating

the parameters which leads to undue optimism towards the selected model. This problem, sometimes also called

selection bias, is a serious problem for statistical inference Statisticians admit this privately, but they(we)

continue to ignore the difficulties because it is not clear what else could or should be done To base statistical

inference on several sensible candidate models is a natural attempt to mitigate the problem. This is the rationale

of model averaging: Instead of using a single fitted model as the basis of statistical inference, the inference is

based on an average of all candidate models. The subsequent discussion of model selection methods show that

satisfactory inference methods are highly sensitive to prior assumptions, goals of inference and substantial

scientific insights into the underlying process. Statistical methods are optimal only relative to a variety of

external, pragmatic factors: Which types of error do we want to address? What are the practical consequences

of a fallacious inference? What is the structure of the random error? Do we have nested or non-nested, linear

or non-linear models? And so forth. It turns out to be impossible to make a neat separation between the logical

and the decision-theoretic part in statistical inference. Statistics must not be described as a branch of

mathematics that miraculously transforms messy data and vague assumptions into a trustworthy posterior

distribution. This would neglect the many uncertainties in the process. Instead, statistics seems to be much closer

to empirical work and scientific modelling: The most interesting and fruitful questions about models in science

deal with the interplay of scientific inquiry and mathematical modelling. Being able to address such questions

with the help of statistical tools has yielded an incredible progress, making statistics an indispensable part of

empirical science.

J. Appl. Sci. Res., 7(6): 891-894, 2011 894

These questions are beyond the realms of formal theories of inference as inductive logic.(Sprenger, 2009)

The theoretical properties which we can deduce about a model like LCM and LPMED, do not decide alone over

its adequacy. There are a lot of different error types, and none of the available model selection criteria takes

care of all of them. Instead, simulations that resemble typical applications are used in order to study the

properties of the proposed criterion. They help us to see whether the criterion is sufficiently robust and

applicable in a variety of circumstances. In particular, it is important to check whether the constraints given by

the intended application are adequately transformed to the parameters of the simulation (e.g. number of candidate

models, linearity, etc.). This evaluation of the model selection criteria has a quasi-empirical character, and due

to the increasing computer power, such approaches become more and more popular. In particular, the results

of the simulation analysis can lead to the introduction of adhoc information criteria that are adapted to model

selection under specific circumstances.

It should be clear by now that a solution of the model selection problem is more than the solution of an

intricate mathematical problem. Human expertise is required to decide which form of modelling is most

appropriate. It is clear that these priorities must be set by scientists, not by mathematicians. Only they

understand the objects of mathematical modelling sufficiently well to assess the adequacy of a particular

discrepancy function or the importance of model parsimony in the relevant context.

The results thus suggest a close collaboration between mathematically minded statisticians and working

scientists in order to find the most adequate model selection method in a particular problem. Indeed, this is the

route statistics has taken in the last decade, with a lot of statistical literature stemming from researchers that are

not located in a mathematics or statistics department. The increased interest in statistical methods among

researchers whose primary interests are outside mathematics and statistics shows that a crucial point has been

realized: In order to design efficient and helpful statistical methods, scientific understanding and mathematical

sophistication have to go hand in hand. It turns out to be impossible to make a neat separation between the

logical and the decision-theoretic part in statistical inference. Statistics must not be described as a branch of

mathematics that miraculously transforms messy data and vague assumptions into a trustworthy posterior

distribution. This would neglect the many uncertainties in the process. Instead, statistics seems to be much closer

to empirical work and scientific modelling: The most interesting and fruitful questions about models in science

deal with the interplay of scientific inquiry and mathematical modelling. Being able to address such questions

with the help of statistical tools has yielded an incredible progress, making statistics an indispensable part of

empirical science. These questions are beyond the realms of formal theories of inference as inductive logic.

(Sprenger, 2009; Zucchini, 2000)

References

Akaike, H., 1995. Information theory as an extension of the maximum likelihood principle, B.N. F. Petrov,

Csaki (Eds.), Second International Symposium on Information Theory, Akademiai Kiado, Budapest.

Bal, H., H. Orkcu, S. Calabioglu., 2006. An experimental comparison of the new goal programming and the

linear programming approaches in the two-group discriminant problems, Computers, Industrial Engineering.

Bozdogan, H., 1988. ICOMP. A new model selection criterion, in: H.H. Bock (Ed.), Classification and Related

Methods of Data Analysis, Elsevier Science, Amsterdam.

Burnham, P., D.R. K. Anderson, 1998. Model Selection and Inference: A Practical Information-Theoretic

Approach, Springer, New York.

Bajgier, S.M. and A.V. Hill, 1982. An experimental comparison of statistical and linear programming

approaches to the discriminant problem, Decision Sciences.

Fitelson, B., 2005. Inductive logic, Philosophy of Science, An Encyclopedia, Routledge, London.

Gilani, A. and M. Padberg, 2002. Alternative methods of linear regression. Mathematical and Computer

Modelling.

Sprenger, J., 2009. Statistics between inductive logic and empirical science, Journal of Applied Logic 7.

Sueyoshi, T., 1999. DEA Discriminant Analysis view of goal programming. European Journal of Operational

Research.

Zucchini, W., 2000. An introduction to model selection, Journal of Mathematical Psychology.

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