Math.md

+# Linear Regression
+
+This document contains information regarding the Odinary Least Squares linear regression implemented in the evaluation tool of the visualtisation platform.
+
+## Model
+
+The regression used in our evaluation follows the linear model:
+
+* y = __X__β + ε
+
+where y is the vector of scores (e.g community engagement, information knowledge) and __X__ is the matrix of independent demographics (e.g gender, ethnicity) selected by the user. β is the fixed effect of parameters and ε is the vector of random errors.
+
+## Estimating the Coefficients
+
+The estimate of β using Ordinary Least Squares is given by:
+
+* &beta;<sup>_e_</sup> = (__X'X__)<sup>-1</sup> __X'__ y
+
+where __X'__ is the transpose of __X__. We assume normality meaning (&epsilon; ~ N(0, &sigma;<sup>2</sup>_I_<sub>n</sub>). Since &epsilon; is a random variable, the regressors are all deterministic and the regressor matrix containing a series of 1s (coefficient of x<sup>0</sup>), we have _E_(&epsilon;) = 0, where _E_ is the expected value. We substitute the value of y, we get:
+
+* &beta;<sup>_e_</sup> = (__X'X__)<sup>-1</sup> __X'__ y = (__X'X__)<sup>-1</sup> __X'__ (__X__&beta; + &epsilon;)
+        = &beta; + (__X'X__)<sup>-1</sup> __X'__ &epsilon;
+
+leaving us with &beta;<sup>_e_</sup> - &beta; = (__X'X__)<sup>-1</sup> __X'__ &epsilon;. As &beta; is a constant, we have Var(&beta;<sup>_e_</sup> - &beta;) = Var(&beta;<sup>_e_</sup>) 
+where the variance is the square of the standard deviation giving us
+
+* Var(&beta;<sup>_e_</sup> - &beta;) = _E_[(&beta;<sup>_e_</sup> - &beta;)(&beta;<sup>_e_</sup> - &beta;)'] - _E_[(&beta;<sup>_e_</sup> - &beta;)]_E_[(&beta;<sup>_e_</sup> - &beta;)]'
+
+(primes denote the transpose of that particular matrix)
+Since we assume __X__ to be deterministic, the expected value only applies to &epsilon;
+* Var(&beta;<sup>_e_</sup>) = (__X'X__)<sup>-1</sup>__X'__ _E_(&epsilon;&epsilon;')__X__(__X'X__)<sup>-1</sup> - (__X'X__)<sup>-1</sup>__X'__ _E_(&epsilon;)_E_(&epsilon;)'__X__(__X'X__)<sup>-1</sup>
+
+with _E_(&epsilon;) = 0, it becomes :
+* Var(&beta;<sup>_e_</sup>) = (__X'X__)<sup>-1</sup>__X'__ _E_(&epsilon;&epsilon;')__X__(__X'X__)<sup>-1</sup>
+
+and we assumed normality so Var(&epsilon;) = _E_(&epsilon;&epsilon;') = &sigma;<sup>2</sup>_I_, where &sigma;<sup>2</sup> > 0 is the common variance/error of each element in the vector of errors or the mean-squared error. This gives us:
+* Var(&beta;<sup>_e_</sup>) = (__X'X__)<sup>-1</sup>__X'__ &sigma;<sup>2</sup>_I_ __X__(__X'X__)<sup>-1</sup>
+
+simplifing this, we get the error or variance for each coefficient given by
+
+* Var(&beta;<sup>_e_</sup>) = &sigma;<sup>2</sup>(__X'X__)<sup>-1</sup>
+
+where the error of each coefficient in &beta;<sup>_e_</sup> corresponds to the diagonal elements of the resulting matrix above.
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