README Update

Author: GeorgeSeif

README.md

@@ -39,15 +39,15 @@ Here are the descriptions of the above machine learning algorithms
 
 ### Linear and Logistic Regression
 
-Regression is a technique used to model and analyze the relationships between variables and often times how the contribute and related to producing a particular outcome together. Beginning with the simple case, a _Single Variable Linear Regression_ is a technique used to model the relationship between a single input independant variable (feature variable) and an output dependant variable using a linear model i.e a line. The more general case is _Multi Variable Linear Regression_ where a model is created for the relationship between multiple independant input variables (feature variables) and an output dependant variable. The model remains linear in that the output is a linear combination of the input variables. There is a third most general case called _Polynomial Regression_ where the model now becomes a _non-linear_ combination of the feature variables; this however requires knowledge of how the data relates to the output. For all of these cases of the regression the output variable is a real-number (rather than a class category). We can also do logistic regression where instead of predicting a real-number, we predict the class or group that the input variable represent. This can be done by modifying the regression training such that the error is computed as the probability that the current example belongs to a particular class. Regression models can be trained using Stochastic Gradient Descent (SGD)Regression is fast to model and is particularly useful when the relationship to be modelled is not extremely complex (if it is, better off).
+Regression is a technique used to model and analyze the relationships between variables and often times how they contribute and are related to producing a particular outcome together. Beginning with the simple case, _Single Variable Linear Regression_ is a technique used to model the relationship between a single input independant variable (feature variable) and an output dependant variable using a linear model i.e a line. The more general case is _Multi Variable Linear Regression_ where a model is created for the relationship between multiple independant input variables (feature variables) and an output dependant variable. The model remains linear in that the output is a linear combination of the input variables. There is a third most general case called _Polynomial Regression_ where the model now becomes a _non-linear_ combination of the feature variables; this however requires knowledge of how the data relates to the output. For all of these cases of the regression the output variable is a real-number (rather than a class category). We can also do logistic regression where instead of predicting a real-number, we predict the class or group that the input variable represent. This can be done by modifying the regression training such that the error is computed as the probability that the current example belongs to a particular class. This can be done simply by taking the sigmoid of the regular linear regression result and using a one vs. all scheme, or simple applying the Softmax function. Regression models can be trained using Stochastic Gradient Descent (SGD). Regression is fast to model and is particularly useful when the relationship to be modelled is not extremely complex (if it is complex, better off using something like a neural network).
 
 ### Neural Networks
 
-Neural Networks: A neural network consists of an interconnected group of nodes called _neurons_. The input feature variables from the data are passed to these neurons as a multi-variable linear combination, where the values multiplied by each feature variable are known as _weights_. A non-linearity is then applied to this linear combination which gives the neural network the ability to model complex non-linear relationships. A neural network can have multiple layers where the output of one layer is passed to the next one in the same way. At the output, there is generally no non-linearity applied. Neural Networks are trained using Stochastic Gradient Descent (SGD) and the backpropagation algorithm. Neural networks can be used for either classification or regression in the same way as the linear and logistic regressions' description above. Since neural networks can have many layers with non-linearities, they are most suitable for modelling complex non-linear relationships in which taking the time to train them properly is worthwhile. 
+A _Neural Network_ consists of an interconnected group of nodes called _neurons_. The input feature variables from the data are passed to these neurons as a multi-variable linear combination, where the values multiplied by each feature variable are known as _weights_. A non-linearity is then applied to this linear combination which gives the neural network the ability to model complex non-linear relationships. A neural network can have multiple layers where the output of one layer is passed to the next one in the same way. At the output, there is generally no non-linearity applied. Neural Networks are trained using Stochastic Gradient Descent (SGD) and the backpropagation algorithm. Neural networks can be used for either classification or regression in the same way as the linear and logistic regressions' description above. Since neural networks can have many layers with non-linearities, they are most suitable for modelling complex non-linear relationships in which taking the time to train them properly is worthwhile. 
 
 ### Naive Bayes
 
-Naive Bayes is a classification technique based on Bayes' Theorem with the assumption that all feature variables are independant. Thus a Naive Bayes classifier assumes that the presence of a particular feature in a class is unrelated to the presence of any other feature. Naive Bayes classifier combines 3 terms to compute the probability of a class: the class probability in the dataset, multiplied by the probability of that class given the values of the example feature variables, divided by the probability of those particular example feature variables occuring. To compute the probability pf particular feature variables occuring there are 3 main optional techniques. One can assume that the value of a particular variable is _Gaussian_ distributed which can be a common case and thus this method is useful when the variables are real numbers. _Multinomial_ division is good for feature variables that are categorical as it computes the probability based on histogram bins. The final option is to use a _Bernouli_ probability model when the data is binary. Naive Bayes is simplistic and easy to use yet can outperform other more complex classification algorithms. Is has fast computation and thus is well suited for application on large datasets.
+Naive Bayes is a classification technique based on Bayes' Theorem with the assumption that all feature variables are independant. Thus a Naive Bayes classifier assumes that the presence of a particular feature in a class is unrelated to the presence of any other feature. Naive Bayes classifier combines 3 terms to compute the probability of a class: the class probability in the dataset, multiplied by the probability of the example feature variables occuring given the current class, divided by the probability of those particular example feature variables occuring in general. To compute the probability of particular feature variables occuring there are 3 main optional techniques. One can assume that the value of a particular variable is _Gaussian_ distributed which can be a common case and thus this method is useful when the variables are real numbers. _Multinomial_ division is good for feature variables that are categorical as it computes the probability based on histogram bins. The final option is to use a _Bernouli_ probability model when the data is binary. Naive Bayes is simplistic and easy to use yet can outperform other more complex classification algorithms. Is has fast computation and thus is well suited for application on large datasets.
 
 ### Support Vector Machines
 
@@ -106,4 +106,4 @@ In addition the the main algorithm files, we have the following set of helper fu
 5. Matplotlib
 
 ## Installation
-The above packages can be installed by running the commands listed in the "install.txt" file
+The above packages can be installed by running the commands listed in the "install.txt" file