KNN and Naïve Bayes  Data Science
1 Introduction
Machine learning is an application of artificial intelligence (AI) that provides systems the ability to automatically learn and improve from experience without being explicitly programmed. The machine learning is classified into different categories viz. supervised machine learning, unsupervised learning, semisupervised learning, and reinforcement machine learning. The supervised learning algorithm takes features as input, maps to a mapping function and approximates a result. The goal is to approximate the mapping function so well that when it gets new input that it can predict the output variables for that data. The supervised learning algorithm aims to find the pattern of the features to a particular result.
Classification, which is the task of assigning objects to one of several predefined categories, is a pervasive problem that encompasses many diverse applications. Examples include detecting spam email messages based upon the message header and content, categorizing cells as malignant or benign based upon the results of MRI scans, and classifying galaxies based upon their shapes.
The classification problems like email is spam or not, tumor is benign or malignant, etc. are binary classification as it deals with two categories in the target class. When there are more than two categories in the target class, the classification problem resides to multilabel classification and example might be like classifying cars company based on image whether it is Honda or Volkswagen or Renault.
The classification algorithm kNearest Neighbors classifier and Naïve Bayes classifier are two classifiers that better suits the classification problem. The performance metrics like Confusion matrix, Accuracy, F1 score, Precision, Recall, Heatmap, etc. gives the insight of model performance.
2 Algorithms
The convention used in the derivation includes a collection of labeled examples $$ {(x_i,yi)}_{i=1}^N $$ , where N is the size of the collection, $x_i$ is the Ddimensional feature vector of example $ i=1, 2, …, N $ , $y_i$ is a realvalued target and every feature $ x_i^{(j)} $ , $ j=1, 2, …, D $ , is also a real number.
2.1 kNearest Neighbors Classifier
kNearest Neighbors (kNN) is non parametric and instancebased learning algorithm. Contrary to other learning algorithms, it keeps all training data in memory. Once new, previously unseen example comes in, the kNN algorithm finds k training examples closest to x and returns the majority label.
The closeness of two examples is given by a distance function. For example, Euclidean distance is frequently used in practice. Euclidean distance between $ x_i $ and $ x_k $ is given as,
$$ d(\boldsymbol {x_i, x_k}) = \sqrt{(x_i^{(1)}x_k^{(1)})^2 + … + (x_i^{(N)}x_k^{(N)})^2} \tag1 $$
The Euclidean distance in summation of the vector is given as;
$$ d(\boldsymbol {x_i, x_k}) = \sqrt{\sum_{j=1}^{D}(x_i^{(j)}x_k^{(j)})^2} \tag2 $$
Another popular choice of the distance function is the negative cosine similarity. Cosine similarity defined as,
$$ s(\boldsymbol {x_i, x_k})=\frac{\sum_{j=1}^{D}x_i^{(j)}x_k^{(j)}}{\sqrt{\sum_{j=1}^{D}(x_i^{(j)})^2}\sqrt{\sum_{j=1}^{D}(x_k^{(j)})^2}} \tag3 $$
The Equation (3) gives a measure of similarity of the directions of the two vectors and $𝑠(\boldsymbol {x_i, x_k}) $ can also be denoted as $ cos(\boldsymbol{x_i, x_k})$ ). If the angle between two vectors is $ 0 $ degrees, then two vectors point to the same direction, and cosine similarity is equal to $ 1 $. If the vectors are orthogonal, the cosine similarity is $ 0 $. For vectors pointing in opposite directions, the cosine similarity is $ −1 $. If we want to use cosine similarity as a distance metric, we need to multiply it by $ −1 $. Other popular distance metrics include Minkowski distance, Chebychev distance, Mahalanobis distance, and Hamming distance. The choice of the distance metric, as well as the value for k, are the choices the analyst makes before running the algorithm. The kNN classifier starts with loading the data into memory. The value of k (number of neighbors) defines the prediction boundaries that means how much sorted distances are taken into account to find the mode of the k labels. The algorithm takes votes to classify the labels among selected kneighbors. It returns the majority class labels leaving behind minority. The flowchart of the kNN classifier is;
The selection of the hyperparameter k has a significant effect on the classifier. In general, for the lower value of k, the classifier may overfit on new unseen data. The value of k is chosen such that balances bias and variance. When k is small, we are restraining the region of a given prediction and forcing our classifier to be “more blind” to the overall distribution. A small value for K provides the most flexible fit, which will have low bias but high variance. Graphically, our decision boundary will be more jagged. On the other hand, a higher k averages more voters in each prediction and hence is more resilient to outliers. Larger values of k will have smoother decision boundaries which means lower variance but increased bias. The value of k is chosen such that the desired accuracy of kNN classifier is achieved. The simple method to calculate the value of k is plotting error versus k graph and choosing the k on which error is minimum.
2.2 Naïve Bayes Classifier
Bayes’ Rule or Bayes’ Theorem is a statistical principle for combining prior knowledge of the classes with new evidence gathered from data. The classconditional probability 𝑃(𝑋𝑌), and the evidence, P(X):The Bayes’ Rule (also known as the Bayes’ Theorem) stipulates that:
$$ P(𝑌\boldsymbol{X}) = \frac{P(\boldsymbol{X}𝑌) P(𝑌)} {P(\boldsymbol{X})} \tag4 $$
In Bayes’ rule (4), it finds the probability of event 𝑌, given that the event 𝑋 is true. Event 𝑋 is also termed as evidence. 𝑃(𝑌) is the priori of 𝑌 (the prior probability, i.e. Probability of event before evidence is seen). 𝑃(𝑌𝑿) is a posteriori probability of 𝑋, i.e. probability of event after evidence is seen. A Naïve Bayes classifier estimates the classconditional probability by assuming that the attributes are conditionally independent, given the class label 𝑦. Here, 𝑃(𝑿) is a class probability and 𝑃(𝑿𝑦) is a conditional probability. The conditional independence assumption can be formally stated as follows:
$$ 𝑃(\boldsymbol{X}𝑌 = 𝑦) = \prod_{i=1}^d𝑃(𝑋_𝑖𝑌 = 𝑦)\tag5 $$
Where each attribute set $$ \boldsymbol{X} = {𝑋_1 ,𝑋_2, … ,𝑋_𝑑} $$ consists of d attributes. The Naïve Bayes is also called Simple Bayes as it assumes that features of a measurement are independent of each other and makes equal contribution to the outcome.
3 METRICS
The classifier model doesn’t always give the accurate result. There are some parameters to measure how the classifier behave with unseen data to classify like Confusion matrix, Accuracy, F1 score, Precision, Recall, Heatmap etc. The different evaluation metrics are used for different kinds of problems. We build a model, get feedback from metrics, make improvements and continue until we achieve a desirable accuracy. Evaluation metrics explain the performance of a model. An important aspect of evaluation metrics is their capability to discriminate among model results.
3.1 Confusion Matrix
The confusion matrix is a table that summarizes how successful the classification model is at predicting examples belonging to various classes. One axis of the confusion matrix is the label that the model predicted, and the other axis is the actual label. In a binary classification problem, there are two classes. Let’s say, the model predicts two classes: “spam” and “not_spam”:
spam (predicted)  not spam (predicted)  

spam (actual)  23 (TP)  1 (FN) 
not spam (actual)  12 (FP)  556 (TN) 
The above confusion matrix shows that of the 24 examples that actually were spam, the model correctly classified 23 as spam. In this case, we say that we have 23 true positives or TP = 23. The model incorrectly classified 1 example as not_spam. In this case, we have 1 false negative, or FN = 1. Similarly, of 568 examples that actually were not spam, 556 were correctly classified (556 true negatives or TN = 556), and 12 were incorrectly classified (12 false positives, FP = 12).
3.2 Precision/Recall
The two most frequently used metrics to assess the model are precision and recall. Precision is the ratio of correct positive predictions to the overall number of positive predictions:
$$ precision = \frac{𝑇𝑃}{𝑇𝑃 + 𝐹𝑃}\tag6 $$
Recall is the ratio of correct predictions to the overall number of positive examples in the datasets:
$$ recall = \frac{𝑇𝑃}{TP+FN} \tag7 $$
In the case of the spam detection problem, we want to have high precision (we want to avoid making mistakes by detecting that a legitimate message is spam) and we are ready to tolerate lower recall (we tolerate some spam messages in our inbox). The goal of classifier model is to choose between a high precision or a high recall. It’s usually impossible to have both. The hyperparameter tuning helps to maximize precision or recall.
3.3 Accuracy
Accuracy is given by the number of correctly classified examples divided by the total number of classified examples. In terms of the confusion matrix, it is given by:
$$ accuracy = \frac{TP+TN}{TP+TN+FP+FN} \tag8 $$
Accuracy is a useful metric when errors in predicting all classes are equally important.
3.4 F1 Score
F1Score is the harmonic mean of precision and recall values for a classification problem. The formula for F1Score is as follows:
$$ F1 = \frac{recall^{1}+precision^{1}}{2} \tag9 $$
$$ F1 = 2.\frac{precision.recall}{precision + recall} \tag{10} $$
The general formula for positive real β, where β is chosen such that recall is considered β times as important as precision, is:
$$ F_{\beta}=(1+{\beta}^2) \cdot \frac{precision \cdot recall}{({\beta}^2 \cdot precision)+recall} \tag{11} $$
The equation (11) or $ F_\beta $ measures the effectiveness of a model with respect to a user who attaches $ \beta $ times as much importance to recall as precision.
3.5 Heat Map
The heat map can be elucidated as a cross table or spreadsheet which contains colors instead of numbers. The default color gradient sets the lowest value in the heat map to dark blue, the highest value to a bright red, and midrange values to light gray, with a corresponding transition (or gradient) between these extremes. Heat maps are wellsuited for visualizing large amounts of multidimensional data and can be used to identify clusters of rows with similar values, as these are displayed as areas of similar color.
4 RESULT
The value of hyperparameter like k in the kNN classifier plays a significant role to correctly classify the labels or target variables. The error versus k values plot provides a guideline to choose k and the value of k with minimum error is chosen.
Fig shows the fluctuation of error at different values of k and the graph is not continuous. We would rather prefer to calculate minimum error kvalue than maximum error kvalue as minimum error kvalue gives more accurate prediction. The minimum error of kNN classifier model for test set is at 𝑘 = 12 and the error is 0.0467 (i.e. 4.67%). Hence, 𝑘 = 12 is chosen as kvalue for kNN classifier. The performance metrics of kNN classifier with parameters metric as ‘minkowski’, neighbors as ‘12’are:
Confusion matrix: [[136 6]
$\qquad$ $\qquad$ [8 150]],
Precision for label ‘0’ prediction: 0.94,
Precision for label ‘1’ prediction: 0.96,
Recall for label ‘0’ prediction: 0.96,
Recall for label ‘1’ prediction: 0.95,
F1score for label ‘0’ prediction: 142,
F1score for label ‘1’ prediction: 158,
Accuracy: 0.95
The model has classified the labels with 𝑇𝑃 = 136, 𝐹𝑁 =
6, 𝐹𝑃 = 8, 𝑇𝑁 = 150 that means model misclassified 6 labels
as label ‘1’ which is actually label ‘0’ and misclassified 8 labels as label ‘0’ which is actually label ‘1’. Hence, the model has an accuracy of about 95%.
Figure Heat map predicted label over the true label Heat map is a graphical representation of value in the confusion matrix obtained from the predicted label and actual target name. In the above heatmap, the red square denotes the maximum value on the confusion matrix and with a decrease in value the color fades up. Diagonal elements have a higher value as shown in the heatmap which shows a higher performance of the classification model and informs predicated label matches the true label for any given input.
For the given model, “prime minister of nepal” supplied as input assign a label “talk.politics.mideast” similarly , when “joker” is supplied as input assign a label “comp.sys.ibm.pc.hardware”. Here for two different input two different label has been assigned out of which one label assigned for the input “prime minister of nepal” is correct whereas for “joker” correct label has not been assigned properly which is due to naïve base treating the input as independent values as well as lack of data being supplied.
5 Conclusion
The two popular classifiers kNN and Naïve Bayes provide good accuracy to the model. Many parameters contribute to model performance. The right choice of hyperparameter also yields a better result. There is no rule of thumb to select the right value of hyperparameter for the first trial and the hyperparameter value that works fine for one model may not yield the same result for another model. The good model is that which considers all the performance metric parameters like Accuracy, F1score, Precision, Recall, etc. Though we have so many metrics parameters to evaluate the model performance, some analytics is needed to better explain the metric that addresses classification problems in the best possible way. The contribution of all performance metrics needs to be analyzed to make the model accurate.
6 Appendix
A kNN classifier


B Naïve Bayes classifier


References
 P. Tan, M. Steinbach, V. Kumar and A. Karpatne, Introduction to Data Mining, Global Edition. Harlow, United Kingdom: Pearson Education Limited, 2019.
 A. Burkov, The hundredpage machine learning book, Global Edition. Quebec City, Canada, 2019.