kNN Classifier from Scratch (numpy only)
k-Nearest Neighbors is a supervised machine learning algorithm for regression, classification and is also commonly used for empty-value imputation. This technique "groups" data according to the similarity of its features.
KNN has only one hyper-parameter which is the size of the neighborhood (k):
- k represents the amount of neighbors to compare data with. Most of the times, at least in classification and imputation, k is odd just in case there is a tie between different neighbors.
- the bigger the k, the less 'defined' the classification areas.
Distance is a key factor in order to determine who is the closest. Distance impacts the size and characteristics of the neighborhoods. The most commonly used is Euclidean distance since it gives the closest distance between 2 points but there are some others.
Most Common Distances
- Euclidean: the shortest distance between to points that might not be the best option when features are normalized. Typically used in face recognition.
- Taxicab or Manhattan: is the sum of the absolute differences of the Cartesian coordinates of 2 points. It works the same way as when a car needs to move around 'blocks' to get to the destination. So, it is basically as adding the horizontal and vertical distances in a 2 dimensional setting.
- Minkowski: is a mix of both Euclidean and Minkowski.
The amount of features impacts kNN significantly because the more points we have, the more 'unique' each neighborhood becomes. It also affects speed because we need to measure each distance first in order to determine who are the closest k neighbors.
The kNN Algorithm
For downloading the code or testing it with the classic iris dataset, please see the GitHub repository.
The most efficient way to calculate the algorithm is in a vectorized form, so instead of calculating the points one by one is better to vectorize the final table and then sort the elements with shortest distances.
1.- Create a matrix with all the distances. The size of the matrix is i x j where i = rows in training set and j = rows in testing set.
[code language='python'] import pandas as pd import numpy as np def knn(xTrain, xTest, k): """ Finds the k nearest neighbors of xTest in xTrain. Input: xTrain = n x d matrix. n=rows and d=features xTest = m x d matrix. m=rows and d=features (same amount of features as xTrain) k = number of nearest neighbors to be found Output: dists = distances between xTrain/xTest points. Size of n x m indices = kxm matrix with indices of yTrain labels """ #the following formula calculates the Euclidean distances. distances = -2 * [email protected] + np.sum(xTest**2,axis=1) + np.sum(xTrain**2,axis=1)[:, np.newaxis] #because of numpy precision, some really small numbers might #become negatives. So, the following is required. distances[distances < 0] = 0 #for speed you can avoid the square root since it won't affect #the result, but apply it for exact distances. distances = distances**.5 indices = np.argsort(distances, 0) #get indices of sorted items distances = np.sort(distances,0) #distances sorted in axis 0 #returning the top-k closest distances. return indices[0:k, : ], distances[0:k, : ] [/code]
2.- Sort the matrix by columns. Since all testing point distances to each training points is now in a matrix, we can sort the indexes for each testing point to find the closest k-neighbors.
3.- Get the y-label that repeats more (classification) or the average of the y-labels (regression). Find the points in the training set that are closer to the testing set points. Use mean for regression or mode for classification.
4- Create a new array with the projected label of the testing set. The size of the array is the same size as the y of the testing set.
[code language='python'] def knn_predictions(xTrain,yTrain,xTest,k=3): """ Input: xTrain = n x d matrix. n=rows and d=features yTrain = n x 1 array. n=rows with label value xTest = m x d matrix. m=rows and d=features k = number of nearest neighbors to be found Output: predictions = predicted labels, ie preds(i) is the predicted label of xTest(i,:) """ indices, distances = knn(xTrain,xTest,k) yTrain = yTrain.flatten() rows, columns = indices.shape predictions = list() for j in range(columns): temp = list() for i in range(rows): cell = indices[i][j] temp.append(yTrain[cell]) predictions.append(max(temp,key=temp.count)) #this is the key function, brings the mode value predictions=np.array(predictions) return predictions [/code]
5- Calculate accuracy of the projected labels. Evaluate the differences between the projected label of y in the testing set with the actual y of the testing set. If accuracy is low, we can change it by modifying k.
#from sklearn.neighbors import KNeighborsClassifier from sklearn.metrics import accuracy_score from sklearn import metrics import matplotlib.pyplot as plt #will first check which is the best k Ks = 15 mean_acc = np.zeros((Ks-1)) std_acc = np.zeros((Ks-1)) #ConfustionMx = ; for n in range(1,Ks): #Train Model and Predict #neigh = KNeighborsClassifier(n_neighbors = n).fit(xTrain,yTrain) #yhat=neigh.predict(xTest) yhat=knn_predictions(xTrain,yTrain,xTest,n) mean_acc[n-1] = metrics.accuracy_score(yTest, yhat) std_acc[n-1]=np.std(yhat==yTest)/np.sqrt(yhat.shape) print( "The best accuracy was:", np.round(mean_acc.max()*100,2), "% with k=", mean_acc.argmax()+1) plt.plot(range(1,Ks),mean_acc,'g') plt.fill_between(range(1,Ks),mean_acc - 1 * std_acc,mean_acc + 1 * std_acc, alpha=0.05) plt.legend(('Accuracy ', '+/- 3xstd')) plt.ylabel('Accuracy ') plt.xlabel('Number of Neighbors (k)') plt.tight_layout() plt.show()