Multiple Linear Regression, Linear Programming, Decision Theory - Assignment Example

Introduction Multiple linear regression involves the use of several predictors to determine an outcome (the response). Roundy and Frank (2004) intended to apply a multiple linear regression model in the investigation of the relationships between interacting wave modes usually characterized by different frequencies. The model so developed had nonlinear power terms (implying that the form of the relationship was not exactly like shown in the example equation y = β0 + β1x + β2x + β3x + ε but more like y = β0 + β1x1 + β2x22 + β3x33 + ε). Understandably, this model is linear in the sense that every predictor variable is either a constant or the product of a parameter (β’s) and a predictor variable (x’s). The researchers further investigated whether the multiple linear regression model provided a better description of the relationship between the wave modes than would a linear regression model with only a linear predictor. Analysis In the model, y (the response) is the ISOw (westward moving intraseasonal modes) and x (the predictor variable) is the ISOe (eastward moving intraseasonal modes). ISOe is further broken down to into more variables by applying power functions of the predictor variable to create a polynomial. Higher power terms are included in the model in order to seek evidence of any improvements in how they increase the accuracy of how wave modes are displayed. This selection is arbitrary and purely based on the assumption that it may lead to development of a better model for depicting the relationship between the independent and dependent variables. Each of the introduced independent variables is then evaluated for significance (at the 5% level of significance) in order to establish its relevance to the entire model. Each item with a coefficient whose p-value falls below the 0.05 (5%) threshold is considered as being statistically significant. Such variables are retained in the model. The test of significance was repeated several times using the bootstrapping technique. The actual equation utilized is: Y^subs,t+ T^ = X^sub t^a^sub s,T^ and is further modified to: A^sub s,T^ = (X^sup T^^sub t^X^sub t^)^sup -1^X^sup T^^sub t^Y^sub s,t+T^ by solving for a specified lag for the regression coefficients. In this equation, “T” is the matrix transpose, “a” the coefficients, and s the grid points (more easily interpreted as the lags). The regression equation involving the nonlinear terms is then tested for suitability against the ordinary linear regression. The model that appears to explain more variance in the response is deemed better. Conclusion The analyses confirmed that the multiple linear regression model applied was able to reveal processes that help ascertain the relationships between interacting wave modes. The researchers noted that the inclusion of the non-linear terms in the models helped in improving the resolution of wave interaction than when purely linear versions of the models were applied. In investigating the characteristics of the regression coefficients, the authors observed that these coefficients were statistically significant, signifying the importance of each of the predictors in the prediction model. The model is therefore more powerful than the ordinary linear regression model in studying the characteristics of wave modes which reveal the importance of inclusion of nonlinear terms in the multiple regression equation. The results presented the initial insight into how the use of multiple linear regression with nonlinear power terms could help in the study of atmospheric waves. Linear Programming Introduction The aim of linear programming techniques is to seek the optimal solutions among a set of alternatives available at a time. Lev and Kowalski (2009) attempted to develop an alternative optimization method that would simplify the solution seeking process. The new technique that the authors present involves the introduction of bounded variables that eliminate the need for additional variables for preserving non-negativity. This implies that the new variables introduced take a possible range of values, unlike in the Simplex method where a single value for an introduced slack variable is usually introduced. Furthermore, the need for substituting variables is eliminated and operations remain on the original equations. Perpetuation of errors through the process is easily eliminated when the proposed method is employed. This is because a similar variable evaluation process is employed afresh whenever another variable is being investigated. Analysis The proposed method dwells on the principle of linearity. The procedure begins by seeking an initial feasible solution using established methods. The analyst then moves to select m+1 loose variables for analysis. Out of these variables, “m” are computed to comply with the provided constraints while the extra one is adjusted by one unit (either increased or decreased) from its original value. The author recommends that the first iterations involve the slack variables. Rates of change for the “m” variables are recorded simultaneously. Accordingly, the “z” value is tracked to determine which of the “m” variables leads in reaching the bound. The procedure is repeated until the analyst obtains “m-n” tight variables. A further step of ascertaining optimality is undertaken. Optimality is ascertained through perturbation analysis, carried out on each of the tight variables against the set of “m” loose variables (loose variables have values unequal to the bounds). The desired change in the value of “z” results in bound violation for every tight variable (a variable whose value is equal to its bounds) in case the solution is optimal. In case a tight variable and the “z” move further away from the set bound then it will be exchanged for a loose variable identified through the rate of change analysis. Further violation of the above step will imply that the process needs to be repeated once more until optimality is attained. The steps outlined above are given a fresh point-by-point explanation with calculations showing the step-by-step calculations on real data. Conclusion One big step in the formulation of the new method is the removal of the requirement for variable non-negativity. This bold step eliminates the need for introduction of additional variables. This is seen as a boost for reduced consumption of computer memory usually associated with linear programming applications/ programs. The new method is deemed simpler than the Simplex method, which is normally considered difficult, especially for learners in the lower grades. However, the simplicity of the new method allows its introduction in the lower grades of learning since it both allows the use of hand calculations and entails understandable criteria for selection of variables, thus making operations research even more applicable further down the system. Besides increased efficiency, the method eliminates perpetuation of errors that would render answers obtained through the Simplex method wrong without the option for partial auto-correction. Forecasting Introduction Forecasting employs several techniques to provide insights into how future operations could be; mainly on the basis of past figures collected through time. Reilly (1994) explored upcoming forecasting techniques that were taking shape at the time of the publication. The article was largely motivated by the upcoming influence of computers at the time as a tool for forecasting, presenting a deviation from the traditional manual calculations system. In essence, the author was upbeat that adoption of the computerized forecasting system would invigorate forecasting to the extent larger data volumes would replace minimal sampling. The author began by listing some of the major shortcomings presented by the outdated manual system, including difficulties in detecting problems with data, such as autocorrelation and heteroscedasticity. Furthermore, the system posed extreme challenges with detection of errors, sometimes at the advanced stages of analysis, and requirement for large file storage space. Analysis The author starts by noting the massive improvements in processing speed for mainframe computers up to the early 1990s when personal computers were beginning to become the mode of ownership. Improvements include the reduction of computation time from 7.5 hours in 1967 to 15 minutes in 1970, and 10 minutes in 1994. Another improvement highlighted was in data storage and transfer from one point to another. The improvement in communication technology is noted as a basis for revolutionized exchange of ideas in judgemental forecasting. The significance of such advanced communication is envisaged as a means to improve global interaction of forecasting experts; which is indeed the case today. The author further argues for the introduction of a system that enables uniform product codes for various products. This system, as argued by the author, would enable experts to access data relating to specific products more easily and analyse greater and more reliable volumes of data as opposed to sampling which is prone to errors in sample selection. The rise of the field of artificial intelligence was seen as a background for providing conditional instructions for computational basis. The use of more complex and newer methods of forecasting such as newer forms of regression analysis with a basis in theory was also associated with improved interaction among experts in forecasting. Beyond economics, the author noted that econometrics was gaining greater use in business. The main roles of the technique were outlined as gaining insight into cause and effect interrelationships, play “what and if” games, and predict turning points. Conclusion Advanced forecasting techniques are shown as an important step in the development of more reliable forecasts. As a result, the author was upbeat that future forecasting roles would result in more reliable results, which would be the result of the use of larger and more comprehensive datasets. More reliable forecasting models (such as more improved econometric techniques) would necessitate multidimensional view of data. Furthermore, the various forecasting techniques and concurrent computerization of operations would help the analysts undertake massive analysis of more variables than the 1990s technology would allow. Where all these variables are relevant to the response (predicted) variable the results would be more accurate and of greater importance to policy makers. The author further noted that reliance on the use of computer-generated output must be guided by proper understanding of the concepts behind the analysis since that is the only way analysts can make informed judgements. Decision Theory Introduction Decision making often involves selecting options from a set of possible ones when the choice maker (presumably at the policy or management level) is unaware of the actual outcomes that could accompany the choice. Yager (1999) discussed the problem of making decisions when the decision maker is unaware of some of the values of variables associated with the solution function. The authors discuss the importance of adopting an attitude (either optimistic or pessimistic) when making a choice. The article attempts to show the association between making decisions under uncertain terms and competitive decision making (game theory). The authors then rely on the association of a selected probability with every alternative instead of decisive selection of alternatives to make their decisions under uncertainty. Analysis For every procedure in which we are required to make a decision from a point of “not knowing”, we face a few alternatives, each of which has a potential payoff scheme. This necessitates selection of the alternative that resonates with the amount of resources at the disposal of the decision maker. For instance, the highest payoff (the main motivator) may be accompanied by higher investment in alternatives that require more expensive resources. The authors demonstrate this as the point at which a pessimist and an optimist deviate. For the optimist, there is no problem putting in more resource in the hope that the returns will be equally high. However, the pessimist may opt for the solution that involves minimal investment capital despite sometimes appearing to offer lesser payoff. These concepts are demonstrated through model formulation for pessimistic decision making and optimistic decision making through the discussed frameworks. By assigning various choices respective weights, the author underscored the fact that such weights translate into probabilistic chances of occurrence of each event, thereby implying creating an attitudinal vector (function) with its respective payoff. Using these attitudinal weights, it was determined that the degree of optimism can be assigned to the expected probability of an occurrence happening as an analyst’s subjective expectation for differing degrees of success. Therefore, the merger between the weight-assigned and the probability-assigned model is treated as just a redefinition of the subjective valuation function into the expected value. By extension, the subjective nature of assigning expected values to the decision options open up the payoff to bias. Conclusion The results indicate that there is a strong relationship between decision making under uncertainty and competitive decision making (game theory). The procedure offers insight into the justification of decisions under the optimistic and pessimistic lenses. Drawing from purely theoretical options, the analysis provides proof for certain decisions being overall more suitable than others, and why some are unsuitable under similar circumstances. Further, the analysis showed that the idea of assigning probabilities to selected alternatives (as used in game theory) is helpful though it involves additional levels of uncertainty to also assign various variables respective probabilities. Nonetheless, this serves to simplify the optimization problem through assigning of best-existing estimates to be associated with each variable. The idea of turning an objective function into a linear objective function appears to provide the added challenge of ascertaining suitability of variable association. References Lev, B. & Kowalski, K. (2009). An alternative to the simplex method for solving linear programming problems: A managerial perspective. International Journal of Management. 27(2): 326-331. Reilly, T. (1994). Business forecasting: Today and tomorrow. The Journal of Business Forecasting Methods & Systems. 13(3): 7-9. Roundy, P. E. & Frank, W. M. (2004). Applications of a multiple linear regression model to the analysis of relationships between eastward- and westward-moving intraseasonal modes. Journal of the Atmospheric Sciences. 61(24): 3041-3048. Yager, R. R. (1999). A game-theoretic approach to decision making under uncertainty. International Journal of Intelligent Systems in Accounting, Finance and Management. 8(2): 131-143. Read More