Adv Quant: Compelling Topics

Compelling topics summary/definitions

• Supervised machine learning algorithms: is a model that needs training and testing data set. However it does need to validate its model on some predetermined output value (Ahlemeyer-Stubbe & Coleman, 2014, Conolly & Begg, 2014).
• Unsupervised machine learning algorithms: is a model that needs training and testing data set, but unlike supervised learning, it doesn’t need to validate its model on some predetermined output value (Ahlemeyer-Stubbe & Coleman, 2014, Conolly & Begg, 2014). Therefore, unsupervised learning tries to find the natural relationships in the input data (Ahlemeyer-Stubbe & Coleman, 2014).
• General Least Squares Model (GLM): is the line of best fit, for linear regressions modeling along with its corresponding correlations (Smith, 2015). There are five assumptions to a linear regression model: additivity, linearity, independent errors, homoscedasticity, and normally distributed errors.
• Overfitting: is stuffing a regression model with so many variables that have little contributional weight to help predict the dependent variable (Field, 2013; Vandekerckhove, Matzke, & Wagenmakers, 2014). Thus, to avoid the over-fitting problem, the use of parsimony is important in big data analytics.
• Parsimony: is describing a dependent variable with the fewest independent variables as possible (Field, 2013; Huck, 2013; Smith, 2015). The concept of parsimony could be attributed to Occam’s Razor, which states “plurality out never be posited without necessity” (Duignan, 2015).  Vandekerckhove et al. (2014) describe parsimony as a way of removing the noise from the signal to create better predictive regression models.
• Hierarchical Regression: When the researcher builds a multivariate regression model, they build it in stages, as they tend to add known independent variables first, and add newer independent variables in order to avoid overfitting in a technique called hierarchical regression (Austin, Goel & van Walraven, 2001; Field, 2013; Huck 2013).
• Logistic Regression: multi-variable regression, where one or more independent variables are continuous or categorical which are used to predict a dichotomous/ binary/ categorical dependent variable (Ahlemeyer-Stubbe, & Coleman, 2014; Field, 2013; Gall, Gall, & Borg, 2006; Huck, 2011).
• Nearest Neighbor Methods: K-nearest neighbor (i.e. K =5) is when a data point is clustered into a group, by having 5 of the nearest neighbors vote on that data point, and it is particularly useful if the data is a binary or categorical (Berson, Smith, & Thearling, 1999).
• Classification Trees: aid in data abstraction and finding patterns in an intuitive way (Ahlemeyer-Stubbe & Coleman, 2014; Brookshear & Brylow, 2014; Conolly & Begg, 2014) and aid the decision-making process by mapping out all the paths, solutions, or options available to the decision maker to decide upon.
• Bayesian Analysis: can be reduced to a conditional probability that aims to take into account prior knowledge, but updates itself when new data becomes available (Hubbard, 2010; Smith, 2015; Spiegelhalter & Rice, 2009; Yudkowsky, 2003).
• Discriminate Analysis: how should data be best separated into several groups based on several independent variables that create the largest separation of the prediction (Ahlemeyer-Stubbe, & Coleman, 2014; Field, 2013).
• Ensemble Models: can perform better than a single classifier, since they are created as a combination of classifiers that have a weight attached to them to properly classify new data points (Bauer & Kohavi, 1999; Dietterich, 2000), through techniques like Bagging and Boosting. Boosting procedures help reduce both bias and variance of the different methods, and bagging procedures reduce just the variance of the different methods (Bauer & Kohavi, 1999; Liaw & Wiener, 2002).

 

References

• Ahlemeyer-Stubbe, Andrea, Shirley Coleman. (2014). A Practical Guide to Data Mining for Business and Industry, 1st Edition. [VitalSource Bookshelf Online].
• Austin, P. C., Goel, V., & van Walraven, C. (2001). An introduction to multilevel regression models. Canadian Journal of Public Health, 92(2), 150.
• Bauer, E., & Kohavi, R. (1999). An empirical comparison of voting classification algorithms: Bagging, boosting, and variants. Machine learning,36(1-2), 105-139.
• Berson, A. Smith, S. & Thearling K. (1999). Building Data Mining Applications for CRM. McGraw-Hill. Retrieved from http://www.thearling.com/text/dmtechniques/dmtechniques.htm
• Brookshear, G., & Brylow, D. (2014). Computer Science: An Overview, 12th Edition. [VitalSource Bookshelf Online].
• Connolly, T., & Begg, C. (2014). Database Systems: A Practical Approach to Design, Implementation, and Management, 6th Edition. [VitalSource Bookshelf Online].
• Dietterich, T. G. (2000). Ensemble methods in machine learning. International workshop on multiple classifier systems (pp. 1-15). Springer Berlin Heidelberg.
• Duignan, B. (2015). Occam’s razor. Encyclopaedia Britannica. Retrieved from https://www.britannica.com/topic/Occams-razor
• Field, Andy. (2013). Discovering Statistics Using IBM SPSS Statistics, 4th Edition. [VitalSource Bookshelf Online].
• Gall, M. D., Gall, J. P., Borg, W. R. (2006). Educational Research: An Introduction, 8th Edition. [VitalSource Bookshelf Online].
• Hubbard, D. W. (2010). How to measure anything: Finding the values of “intangibles” in business. (2^nd e.d.) New Jersey, John Wiley & Sons, Inc.
• Huck, Schuyler W. (2011). Reading Statistics and Research, 6th Edition. [VitalSource Bookshelf Online].
• Liaw, A., & Wiener, M. (2002). Classification and regression by randomForest. R news, 2(3), 18-22.
• Smith, M. (2015). Statistical analysis handbook. Retrieved from http://www.statsref.com/HTML/index.html?introduction.html
• Spiegelhalter, D. & Rice, K. (2009) Bayesian statistics. Retrieved from http://www.scholarpedia.org/article/Bayesian_statistics
• Vandekerckhove, J., Matzke, D., & Wagenmakers, E. J. (2014). Model comparison and the principle of parsimony.
• Yudkowsky, E.S. (2003). An intuitive explanation of Bayesian reasoning. Retrieved from http://yudkowsky.net/rational/bayes

Adv Quant: General Least Squares Model

Regression formulas are useful for summarizing the relationship between the variables in question (Huck, 2011). There are multiple types of regression all of them are tests of prediction (Huck, 2011; Schumacker, 2014).  The least squares (linear) regression is the most well-known because it uses basic algebra, a straight line, and the correlation coefficient to aid in stating the regression’s prediction strength (Huck, 2011; Schumacker, 2014).  The linear regression model is:

y = (a + bx) + e                                                                   (1)

Where y is the dependent variable, x is the independent variable, a (the intercept) and b (the regression weight, also known as the slope) are a constants that are to be defined through the regression analysis, and e is the regression prediction error (Field, 2013; Schumacker, 2014).  The sum of the squared errors should be minimized per the least squares criterion, and that is reflected in the b term in equation 1 (Schumacker, 2014).

Correlation coefficients help define the strength of the regression formula in defining the relationships between the variables, and can vary in value from -1 to +1.  The closer the correlation coefficient is to -1 or +1; it informs the researcher that the regression formula is a good predictor of the variance between the variables.  The closer the correlation coefficient is to zero, indicates that there is hardly any relationship between the variable (Field, 2013; Huck, 2011; Schumacker, 2014).  Correlations never imply causation, but they can help determine the percentage of the variances between the variables by the regression formula result when the correlation value is squared (r2) (Field, 2013).

Assumptions for the General Least Square Model (GLM) modeling for regression and correlations

The General Least Squares Model (GLM) is the line of best fit, for linear regressions modeling along with its corresponding correlations (Smith, 2015).  There are five assumptions to a linear regression model: additivity, linearity, independent errors, homoscedasticity, and normally distributed errors.  Variables should be linearly related the independent variables(s), and the combined effects of multiple independent variables should be additive. A residual is the difference between the predicted value from the observed value: (1) no two residuals should be correlated, which can be numerically tested by using the Durbin-Watson test; (2) the variance of these residuals should be constant for each independent variable; and (3) the residuals should be random and normally distributed with a mean of 0 (Field, 2013; Schumacker, 2014).

Covering the issues with transforming variables to make them linear

When viewing the data through scatter plots, if the linearity and additivity assumptions could not be met, then transformations to the variables could be made to make the relationship linear. The above is an iterative trial and error process.  Transformation must occur to every point of the data set to correct for the linearity and addititvity issues since it changes the difference between the variables due to the change of units in the variables (Field, 2013).

Table 1: Types of data transformations and their uses (adapted from Field (2013) Table 5.1).

Data Transformation	Can Correct for
Log [independent variable(s)]	Positive skew, positive kurtosis, unequal variances, lack of linearity
Square root [independent variable(s)]	Positive skew, positive kurtosis, unequal variances, lack of linearity
Reciprocal [independent variable(s)]	Positive skew, positive kurtosis, unequal variances
Reverse score [independent variable(s)]: subtracting the highest value in the variable for each data set	Negative skew

Describe the R procedures for linear regression

lm( ) is a function for running linear regression, glm( ) is a function for running logistic regression (should not be confused for GLM), and loglm( ) is a function for running log-linear regression in R (Schumacker, 2014; Smith, 2015). The summary( ) function is used to output the results of the linear regression. Dependent variables are represented with a tilde “~” and independent variables are represented with a “+” (Schumacker, 2014). Thus, the R procedures for linear regression are (Marin, 2013):

> cor (x, y) # correlation coefficient

> myRegression = lm (y ~ x, data = dataSet ) # conduct a linear regression on x and y

> summary(myRegression) # produces the outputs of the lm( ) function calculations

> attributes(myRegression) # lists the attributes of the lm( ) function

> myRegression$coefficients # gives you the slope and intercept coefficients

> plot (x, y, main=“Title to graph”) # scatter plot

> abline(myRegression) # regression line

> confint(myRegression, level= 0.99) # 99% level of confidence intervals for the regression coefficients

> anova(myRegression) # anova analysis on the regression analysis

References

• Field, A. (2013) Discovering Statistics Using IBM SPSS Statistics (4^th ed.). UK: Sage Publications Ltd. VitalBook file.
• Huck, S. W. (2011) Reading Statistics and Research (6^th ed.). Pearson Learning Solutions. VitalBook file.
• Marin, M. (2013) Linear regression in R (R tutorial 5.1). Retrieved from https://www.youtube.com/watch?v=66z_MRwtFJM
• Schumacker, R. E. (2014) Learning statistics using R. California, SAGE Publications, Inc, VitalBook file.
• Smith, M. (2015). Statistical analysis handbook. Retrieved from http://www.statsref.com/HTML/index.html?introduction.html