Multiple Linear Regression

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1.1 Background

Multiple linear regression efforts to model the relationship between two or more explanatory variables and a response variable by fitting a linear equation to observed data. Every value of the independent variable x is associated with a value of the dependent variable y. The population regression line for p explanatory variables x1, x2… xp is defined to be μy=β°+β1x1+β2x2+ … + βpxp

This line describes how the mean response f changes with the explanatory variables. The observed values for y vary about their means μyand are assumed to have the same standard deviation σ. The fitted values b0, b1... bp estimate the parameters β1,β2,…,βp of the population regression line. Formally, the model for multiple linear regressions, given n observations, is yi= β0 + β1xip + β2xip + ... βpxip + εi for i= 1,2, ... n. In the least-squares model, the best-fitting line for the observed data is calculated by minimizing the sum of the squares of the vertical deviations from each data point to the line (if a point lies on the fitted line exactly, then its vertical deviation is 0). Because the deviations are first squared, then summed, there are no cancellations between positive and negative values. The least-squares estimates b0, b1, ... bp are usually computed by statistical software. The values fit by the equation b0 + b1xi1 + ... + bpxip are denoted yl, and the residuals ei are equal to yi - yi, the difference between the observed and fitted values. The sum of the residuals is equal to zero. The variance σ2 may be estimated by s2= ei2n-p-1 , also known as the mean-squared error (or MSE). The estimate of the standard error s is the square root of the MSE. Nowadays, there are many computer languages that can be used for computer programming. Some of them are C and C++. C++ is a type of computer programming language. Created in 1983 by Bjarne Stroustrup, C++ was designed to serve as an enhanced version of the C programming language. C++ is object oriented and is considered a high level language. However, it features low level facilities. C++ is one of the most commonly used programming languages. The development of C++ actually began four years before its release, in 1979. It did not start out with the name C++; its first name was C with Classes. In the late part of 1983, C with Classes was first used for AT&T’s internal programming needs. Its name was changed to C++ later in the same year. C++ was not released commercially until the late part of 1985. Developed at Bell Labs, C++ enhanced the C programming language in a variety of ways. Among the features of C++ are classes, virtual functions, templates, and operator overloading. The C++ language also counts multiple inheritance and exception handling among its many features. C++ introduced the use of declarations as statements and includes more type checking than is available with the C programming language. Considered a superset of the C programming language, C++ maintains a variety of features that are included within its predecessor. As such, C programs are generally able to run successfully in C++ compilers. However, there are some issues that may cause C code to perform differently in C++ compilers. In fact, it is possible for some C code to be incompatible in C++. Library creation is cleaner in C++. The C++ programming language is considered portable and does not require the use of a specific piece of hardware or just one operating system. Another important feature of C++ is the use of classes. Classes help programmers with the organization of their code. They can also be beneficial in helping programmers to avoid mistakes. However, there are times when mistakes do slip through. When this happens, classes can be instrumental in finding bugs and correcting them. The original C++ compiler, called Cfront, was written in the C++ programming language. C++ compilation is considered efficient and fast. Its speed can be...
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