Euclidean Distance

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Draft pseudocode for an algorithm

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Pseudocode for K-Nearest Neighbors (KNN) Algorithm

Input:

  • Training dataset D={(x1,y1),(x2,y2),…,(xn,yn)}, where xi are feature vectors and yi are the labels.
  • Test data point xtest.
  • Number of neighbors k.

Output:

  • Predicted label for xtest.

Steps:

  1. Initialize:
    • Set k (number of neighbors).
  2. Compute Distance:
    • For each point xi in the training dataset D:
      • Compute the distance distance(xtest,xi), usually using Euclidean distance:distance(xtest,xi)=∑j=1m(xtest,j−xi,j)2
      • Where m is the number of features.
  3. Sort Neighbors:
    • Sort all points (x1,x2,…,xn) in the training dataset based on their computed distance to xtest.
  4. Select Top k Neighbors:
    • Select the top k closest points xi1,xi2,…,xik with the smallest distances.
  5. Vote for the Label:
    • For classification:
      • Let {yi1,yi2,…,yik} be the labels of the top k neighbors.
      • Predict the label y^test of xtest as the most frequent label in {yi1,yi2,…,yik}.
    • For regression:
      • Predict the label y^test as the average of the k nearest labels:y^test=1k∑i=1kyi
  6. Return the Prediction:
    • Return the predicted label y^test.

Pseudocode Example:

php
function KNN(D, x_test, k):
distances = []
for each point (x_i, y_i) in D:
distance = EuclideanDistance(x_test, x_i)
distances.append((distance, y_i))

# Sort by distance
distances.sort() # Sort based on the first element (distance)

# Select top k neighbors
nearest_neighbors = distances[:k]

# For classification, vote for the most frequent label
labels = [label for _, label in nearest_neighbors]
predicted_label = MostFrequentLabel(labels)

return predicted_label

function EuclideanDistance(x1, x2):
distance = 0
for i in range(len(x1)):
distance += (x1[i] - x2[i])^2
return sqrt(distance)

function MostFrequentLabel(labels):
# Return the most common label
return mode(labels)

Explanation:

  • Input: The training dataset D and the test data point xtest are provided as inputs. Additionally, the number of neighbors k is a critical parameter.
  • Distance Calculation: For each training point, the distance to the test point is calculated using a distance metric (commonly Euclidean distance).
  • Sorting: The dataset is sorted based on these distances, and the top k closest points are selected.
  • Label Prediction: The final step involves predicting the label of the test point either through a majority vote (for classification) or by averaging (for regression).
  • Efficiency: The complexity of KNN is O(n⋅m), where n is the number of training samples, and m is the number of features. Efficient data structures like KD-trees or Ball-trees can be used to speed up the nearest neighbor search for larger datasets.
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