=Paper=
{{Paper
|id=Vol-3176/paper5
|storemode=property
|title=Effect of Distance Metric and Feature Scaling on KNN Algorithm while Classifying X-rays
|pdfUrl=https://ceur-ws.org/Vol-3176/paper5.pdf
|volume=Vol-3176
|authors=Ishan Arora,Namit Khanduja,Mayank Bansal
|dblpUrl=https://dblp.org/rec/conf/rif/AroraKB22
}}
==Effect of Distance Metric and Feature Scaling on KNN Algorithm while Classifying X-rays ==
https://ceur-ws.org/Vol-3176/paper5.pdf
Effect of Distance Metric and Feature Scaling on KNN
Algorithm while Classifying X-rays
Ishan Arora 1, Namit Khanduja 1, Mayank Bansal 1
1
Gurukul Kangri University, Haridwar, Uttrakhand. India
Abstract
In this world today medical imaging is prominently one in every of the foremost advancing
fields. Medical imaging refers to those techniques and processes that are accustomed create
images of assorted parts of the soma for diagnostic and treatment purposes [1]. These include
various radiological imaging like X-ray radiography, Magnetic resonance imaging (MRI), etc.
Many Machine learning and Deep learning algorithms are being employed in medical imaging to
spot diseases. There are many algorithms like Convolution Neural Network, Multilayer
Perceptron, Logistic Regression, etc. among which one among the algorithms is k-NN (k-Nearest
Neighbor) the foremost drawbacks related to the k-NN algorithm are (1) its low efficiency, due to
which it takes a lot of time in calculating the distance between query point and the dataset
containing many features, (2) its dependency on various distances metrics. During this paper, we
propose a k-NN type classifier that is overcoming these shortcomings. Our approach constructs a
k-NN model that performs classification by considering various distance metrics and feature
scaling so to enhance the classification accuracy. Our Experiments were administrated on some
public medical datasets collected from the Kaggle Machine Learning repository to check our
approach. Experiment results show that the k-NN model works well with Canberra distance metric
and Robust feature scaling, but is more efficient than widely used Euclidean distance and standard
feature scaling.
Keywords 1
Medical Imaging, k-NN, Robust Feature Scaling, Canberra Distance Metric
1. Introduction
The k-Nearest-Neighbors (k-NN) may be a non-parametric classification method, which is
straightforward but effective in many cases [2]. It is a Machine learning algorithm that supported the
Supervised Learning approach. Its problem domain mainly consists of a classification problem
instead of a regression problem. For a Data point P to be classified, Its k-nearest neighbors are
retrieved by measuring the distance between a given point and the associated data points, among
which the minimum distance is further mapped to classification output. However, to use the k-NN
algorithm we have got to settle on an appropriate value of k and distance metric along with
appropriate feature scaling so to reduce the computational time required for computing distance
between test point and dataset, and also the classification accuracy is extremely much addicted to
this value. There are many distance metrics and feature scaling methods , available to settle on but
the goal is to decide on that distance metric and feature scaling which may take classification
accuracy far better. There are many ways to settle on distance metrics and feature scaling, but the
easy way is to run the algorithm again and again with different distance metrics and feature scaling
and choose the one with the simplest performance.
RIF’21: The 10th Seminary of Computer Science Research at Feminine, March 08h, 2021, Constantine 2-Abdelhamid Mehri University,
Algeria
EMAIL: aroraishan51@gmail.com, (I. Arora); namit.khanduja@gmail.com (N. Khanduja ); mayankbansal231@gmail.com, (M. Bansal)
©️ 2022 Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR Workshop Proceedings (CEUR-WS.org)
Nowadays, Medical Imaging is one in every of the foremost advancing fields. One requires a
classifier that will easily process an X-ray image or another radiological image and gave a decent
accuracy.
There are many advanced algorithms like Convolutional Neural Network (CNN), Region
proposal Convolutional Network (R-CNN) that may easily provide far better accuracy on the
medical domain dataset but with more computational resources. On the other hand Linear
classification algorithms such as SVM (Supported Vector Machine) and Logistic Regression still
uses parametric classification approach and use some optimization algorithms such as stochastic
gradient descent or batch gradient descent to train there parameters before performing any
classification on test point. However, K-NN being a non-parametric classifier does not require any
high-level computational resources and prior training as linear classifiers, although it might take
time complexity of O(n2) to classify a knowledge point. Deep learning algorithms are data-hungry,
k-NN can easily outperform them on a little amount of information. k-NN encompasses a very high
cost of classifying new instances this is often thanks to all the computations happen at the time of
classification instead of all the training data encountered first. There are techniques [2, 3] that will
still reduce the computation required at query time, like indexing training examples, but it is out of
our concerns during this paper. Hence we purpose a k-NN type model during this paper to search out
whether the performance of k-NN is improved by choosing the suitable feature scaling and Distance
metric. Finally, the performance of the model is measured with ROC, AUC, Precision, etc.
2. Related Works
Several studies have been conducted to analyze the performance of KNN classifier using different
distance measures and feature scaling. Each study was applied on various kinds of datasets with
different distributions, types of data and using different numbers of distance and similarity measures.
Chomboon et al [3] analyzed the performance of KNN classifier using 11 distance measures and
standard feature scaling. These include Euclidean, Mahalanobis, Manhattan, Minkowski, Chebyshev,
Cosine, Correlation, Hamming, Jaccard, Standardized Euclidean and Spearman distances. Their
experiment had been applied on eight binary synthetic datasets with various kinds of distributions that
were generated using MATLAB. They divided each dataset into 70% for training set and 30% for the
testing set. The results showed that the Manhattan, Minkowski, Chebyshev, Euclidean, Mahalanobis,
and Standardized Euclidean distance measures achieved similar accuracy results and outperformed
other tested distances.
Hu et al [4] analyzed the effect of distance measures on KNN classifier for medical domain
datasets. Their experiments were based on three different types of medical datasets containing
categorical, numerical, and mixed types of data, which were chosen from the UCI machine learning
repository, and four distance metrics including Euclidean, Cosine, Chi square, and Minkowski
distances. They divided each dataset into 90% of data as training and 10% as testing set, with K
values from ranging from 1 to 15. The experimental results showed that Chi square distance function
was the best choice for the three different types of datasets. However, using the Cosine, Euclidean and
Minkowski distance metrics performed the ‘worst’ over the mixed type of datasets. The ‘worst’
performance means the method with the lowest accuracy.
Todeschini et al [5, 6] analyzed the effect of eighteen different distance measures on the
performance of KNN classifier using eight benchmark datasets. The investigated distance measures
included Manhattan, Euclidean, Soergel, Lance–Williams, contracted Jaccard–Tanimoto, Jaccard–
Tanimoto, Bhattacharyya, Lagrange, Mahalanobis, Canberra, Wave-Edge, Clark, Cosine, Correlation
and four Locally centered Mahalanobis distances. For evaluating the performance of these distances,
the non-error rate and average rank were calculated for each distance. The result indicated that the
‘best’ performance were the Manhattan, Euclidean, Soergel, Contracted Jaccard–Tanimoto and
Lance–Williams distance measures. The ‘best’ performance means the method with the highest
accuracy.
Lopes and Ribeiro [7] analyzed the impact of five distance metrics, namely Euclidean, Manhattan,
Canberra, Chebyshev and Minkowski in instance-based learning algorithms. Particularly, 1-NN
Classifier and the Incremental Hyper sphere Classifier (IHC) Classifier, they reported the results of
their empirical evaluation on fifteen datasets with different sizes showing that the Euclidean and
Manhattan metrics significantly yield good results comparing to the other tested distances.
Alkasassbeh et al [8] investigated the effect of Euclidean, Manhattan and a non-convex distance
due to Hassanat [9] distance metrics on the performance of the KNN classifier, with K ranging from 1
to the square root of the size of the training set, considering only the odd K’s. In addition to
experimenting on other classifiers such as the Ensemble Nearest Neighbor classifier (ENN) [10], and
the Inverted Indexes of Neighbors Classifier (IINC) [11]. Their experiments were conducted on 28
datasets taken from the UCI machine learning repository, the reported results show that Hassanat
distance [12] outperformed both of Manhattan and Euclidean distances in most of the tested datasets
using the three investigated classifiers.
Table 1
Comparison between previous studies for distance metric and Feature scaling for k-NN along with
‘best’ performing metric and Feature scaling. Comparatively our work compares the use different
variety of feature scaling and distance measures on variety of datasets.
Reference Distances Datasets Best Distance Feature Best Feature Scaling
Used Scaling
[3] 11 8 Manhattan, Standard None
Chebyshev,
Euclidean,
Minkowski
[4] 4 37 Chi-square Standard Standard
Scaling
[7] 5 15 Euclidean, Standard None
Manhattan
[5] 18 8 Manhattan, Standard None
Euclidean,Soergel
Contracted
Jaccard–
Tanimoto
Lance–Williams
[8] 3 28 Hassanat Standard None
[13] 3 2 Hassanat Standard Standard Scaling
Ours 8 3 Canberra Standard, Robust
Robust,
Min-Max
Lindi [13] investigated three distance metrics to use the best performer among them with the KNN
classifier, which was employed as a matcher for their face recognition system that was proposed for
the NAO robot. The tested distances were Chi-square, Euclidean and Hassanat distances. Their
experiments showed that Hassanat distance outperformed the other two distances in terms of
precision, but was slower than both of the other distances. Above Table provides a summary of these
previous works on evaluating various distances and feature scaling within k-NN classifier, along with
the best distance assessed by each of them. As can be seen from the above literature review of most
related works, that all of the previous works have investigated either a small number of feature scaler
or different distance measures or both.
3. k-NN with Distance metric and Feature Scaling
3.1 Basic Overview of k-NN algorithm
The k-NN classifier works by computing the distance between the unlabeled datum point and also the
majority of comparable data points among k-nearest neighbors that are closed thereto query point. The
distance between the test sample and the training data samples is calculated by a specific distance
metric.
Figure 1: AN example of k-NN type classifier classifying the query point (red dot) between two
classes
Since the above figure contains two features they can be represented in two-dimensional space
with one dimension for every feature. To classify the test sample that belongs to the class ‘yellow
dot’ or class ‘purple dot’ k-NN adopts a distance function to seek out the k nearest neighbors to
the test sample as shown in fig. Here when k = 3 test sample is classed to the class ‘purple dot’
because there are two ‘purple dot’ and one ‘yellow dot’ inside the inner circle, but when k = 6
test sample is assessed to the class ‘yellow dot’ because there are three ‘yellow dot’ and no
‘purple dot’ inside the outer circle.
The Basic k-NN classifier steps may be described as follows:
Algorithm :
INPUT: Training Samples X, Test Sample P, Value of K, Distance metric
OUTPUT: Class Label of Test Sample
• Compute the Distance between the Test Sample P and Training samples X.
• Choose the value of k (nearest neighbors for consideration).
• Assign the Test Label P the bulk class.
3.2 Distance Metrics
In math, the term distance is defined as “ how far the two objects are from one another ”. Euclid, one
among the foremost important mathematicians of ancient history, used the word distance only in his
third postulate of Principia [14]: “ Every circle can be described by a center ad distance”. In data
mining, distance means how elements in some space are close or isolated from each other. The
distance function between vectors x and y is function d( x, y ) that defines the distance between two
vectors as a non-negative real number. This function is said to be a metric if it satisfies some
properties [15] that include the following:
1. Non-negativity: It means the distance function between the vectors is always greater
than zero.
d( x , y ) ≥ 0
2. Identity of indiscernible: The distance between x and y is zero if and only if x is
equal to y.
d( x , y ) = 0 if x == y
3. Symmetry: The distance between x and y is equal to the distance between y and x.
d( x , y ) == d( y , x )
4. Triangle Inequality: Considering a third point p, the distance between x and y is
always less than the sum of the distance between x and p and the distance
between y and p.
d( x , y ) ≤ d( x , p ) + d( y ,p )
Here in our experiment, we are considering the eight types of distance metrics. We gave the
mathematical definition of distances to measure the closeness between two vectors x and y, where x
= ( x1 , x2 , x3 ,- - - - - - xn ) and y = ( y1 , y2 , y3 ,----- yn) having numeric attributes. Theoretical
analysis of these distances is beyond the scope of this work.
5. Euclidean (ED): Also known as l2 norm or Euler distance, which is an
extension to the Pythagorean Theorem. This distance represents the root of the
sum of the square of differences between the opposite value in vectors.
Euclidean distance (x, y) = √∑𝑚
𝑖=1(𝑥𝑖 − 𝑦𝑖 )
2
6. Manhattan (MD): It is also known as l1 norm or City block distance. This distance
is considered by Hermann Minkowski which represents the sum of absolute
differences between the opposite values in vectors.
Manhattan distance (x, y) = ∑𝑚 |𝑥𝑖 − 𝑦𝑖|
𝑖=1
7. Chebyshev (CD): Also known as maximum value distance [5]. It is a metric
defined on vector space where the distance between the two vectors is the
greatest of their difference in any coordinate dimension.
Chebyshev distance (x, y) = max(|𝑥𝑖 − 𝑦𝑖|)
8. Canberra (CnD): Canberra distance, which is introduced by [6] and modified in
[7]. It is an expanded version of the Manhattan distance, where the absolute
difference between the feature values of vectors is divided by the sum of absolute
feature values before the summing [8]
∑ |𝑥 −𝑦 |
Canberra Distance = ∑ |𝑥𝑖+𝑦𝑖|
𝑖 𝑖
9. Cosine (CosD): Also known as angular distance, measures the angle between two
vectors.
∑ 𝑥𝑖 𝑦𝑖
Cosine distance (x, y) =
√𝑥𝑖 2 𝑦𝑖 2
10. BrayCurtis (BRD): It is the distance between the range {0,1} and undefined if the
vector length is zero
∑𝑚
𝑖 |𝑥𝑖 −𝑦𝑖 |
BrayCurtis distance (x, y) = ∑𝑚
𝑖 𝑥𝑖 +𝑦𝑖
11. Correlation Distance (CorD): Correlation distance is a version of the Pearson
distance, where the Pearson distance is scaled in order to obtain a distance measure in
the range between zero and one.
1 ∑𝑚 ′ ′
𝑖=1(𝑥𝑖 −𝑥 )(𝑦𝑖 −𝑦 )
Correlation distance (x, y) = 2 (1 −
√∑𝑚 2 𝑚
𝑖=1(𝑥𝑖 −𝑥′) √∑𝑖=1(𝑦𝑖 −𝑦′)
2
12. Minkowski Distance (MD): The Minkowski distance, which is also known as Lp
norm, is a generalized metric. It is defined as:
Minkowski distance (x, y) = (∑𝑚
𝑖=1(𝑥𝑖 − 𝑦𝑖 ))
𝑝
Figure 2: shows the various distance metrics
3.3 Feature Scaling
Data preprocessing is the technique in data mining to transform raw data into an understandable
format. One of the technique comes under Data preprocessing is Feature scaling. Feature
scaling usually refers to the Normalization or Standardization of data around the certain mean
value or standard deviation value so to increase the rate of convergence of some optimization
algorithms like Gradient Descent, Adam optimizer, etc. required by Linear classifiers and also
to reduce the computational time required for computing distance between the points as in case
of distance measure algorithms like k-NN, k-Means etc. There are many types of feature scaling
methods available but our research is centered around the most commonly used three types of
feature scaling discussed below:
13. Min max Scaler: It scales the data based on the minimum and the maximum of
features within a range. Let us say after scaling our x feature vector becomes x’.
𝑥𝑖’ = xi – min (x) / max(x) – min(x)
Figure3 : shows the feature before and after Min max scaling
14. Standard Scaler: Another rescaling method compared to the Min max scaler is
Standard Scaler. It transforms feature vector x to the standard normally distributed,
with mean zero and standard deviation one.
xi’ = xi- xmean / xstd
Figure 4: shows the feature before and after standard scaling
15. Robust Scaler: This type of feature scaling is commonly used to overcome the
presence of outliers in our data. It rescales features using the interquartile range.
Where X’ is the new feature vector, Q1 is the first quartile and Q3 is the third
quartile. All the above- mentioned feature scaling can easily be done using python
library scikit-learn.
xi’ = xi – Q1 / Q3 – Q1
Figure 5: shows the feature before and after Robust scaling
4. Experimental Framework
4.1 Datasets used for experiments
The experiments were done on three different datasets which represent real-life classification
problems, collected from the Kaggle Machine Learning repository. X-ray images are collected from
Covid-19 related papers from medRxiv, bioRxiv, JAMA, Lancet, etc. Kaggle is an open-source online
community where data scientists and machine learning practitioners, can easily access and publish
their datasets. Each dataset consists of a set of examples, Each example is defined by the number of
attributes and, all the examples in data are represented by the same number of attributes. One of these
attributes is known as the class attribute or the class (label) which has to be predicted for the test
sample.
Figure 6: Framework of our experimental setup
4.2 Experiment Setup
From above figure details of each step is as follows : Each dataset is divided into three data sets, one
for training, the other for validation, and the rest for testing. For, this purpose 60% of the dataset is
used for training, 20% for cross-validation, and the rest 20% for testing. We explicitly set the value
of k = 1 and the 20% of data which were used as a test sample is randomly shuffled and each
experiment is repeated for eight times for each value of distance metric and after each experiment
value of k is incremented and execution goes on till we reach the value of k = 15.
Figure 7. shows X-ray image feeding into a k-NN classifier.
The Experiment also aims to find the best feature scaling used by the k-NN classifier. Hence all the
three feature scaling as discussed above is used and the above-discussed algorithm is repeated three
times each time for each feature scaling.
4.3 Performance Evaluation measures
Different measures were available for evaluating the performance of classifiers. In this paper, two
measures were used accuracy and ROC-AUC curve. Accuracy is calculated to evaluate the overall
performance of the classifier. It is defined as the ratio of test samples correctly classified to the
total number of samples.
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐𝑜𝑟𝑟𝑒𝑐𝑡𝑙𝑦 𝑐𝑙𝑎𝑠𝑠𝑖𝑓𝑖𝑒𝑑 𝑠𝑎𝑚𝑝𝑙𝑒
Accuracy = 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑇𝑜𝑡𝑎𝑙 𝑠𝑎𝑚𝑝𝑙𝑒𝑠
We have plotted the graph of Accuracy v/s Value of K for best feature scaling i.e. Robust Feature
scaling of all three datasets along with each distance metric.
Graph 1: showing the accuracy v/s K for each distance metric with Robust Feature scaling for dataset
1
Graph 1 shows the classification accuracy of k-NN over dataset 1. For the distance function, it is clear
that Canberra distance metric is winner of all with 85.56% accuracy. However, overall speaking,
using the Euclidean distance function is not the best metric for k-NN for dataset 1. The classification
accuracy by Euclidean and Minkowski distance functions are almost the same, which means that
using euclidean or minkowski does not affect the k-NN performance . On the other hand, k-NN by the
Bray Curtis distance function gave a tough competition to best distance function i.e. Canberra distance
function wit classification accuracy of 83.95%.
Graph2:showing the accuracy v/s K for each distance metric with Robust Feature scaling for dataset
2.
Graph 2 shows the classification accuracy of k-NN over dataset 2. For the distance function, it is clear
that Canberra distance metric is winner of all with 77.09% accuracy. However, overall speaking,
using the Euclidean distance function is not the best metric for k-NN for dataset 2. The classification
accuracy by Euclidean and Minkowski distance functions are almost the same, which means that
using euclidean or minkowski does not affect the k-NN performance . On the other hand, k-NN by the
Bray Curtis distance function gave a tough competition to best distance function i.e. Canberra distance
function wit classification accuracy of 77.08%.
Graph3: showing the accuracy v/s K for each distance metric with Robust Feature scaling for dataset
3
For the distance function, it is clear that Canberra distance metric is winner of all with 84.61%
accuracy. However, overall speaking, using the Euclidean distance function is not the best metric
for k-NN for dataset 2. The classification accuracy by Euclidean and Minkowski distance functions
are almost the same, which means that using Euclidean or Minkowski does not affect the k-NN
performance. On the other hand, k-NN by the Bray Curtis, Correlation, Cosine distance function gave
a tough competition to best distance function i.e. Canberra distance function with their average
classification accuracy of 83.4%.
AUC – ROC curve is a performance measurement for the classification problems at various threshold
settings. ROC is a probability curve and AUC represents the degree of measure of separability. It tells
us how much the model is capable of distinguishing between classes. Higher the AUC, better the
model is at predicting 0s as 0s and 1s as 1s. By analogy, higher the AUC better the model is at
distinguishing between patients with disease and no disease.
Figure 8: shows the ROC-AUC evaluation metric
To evaluate performance with every class in a dataset. We compute the ROC-AUC curve for each
dataset for the best distance metric and comparable distance metrics with the best values of k
associated with them. All three curves have best feature scaling i.e. Robust feature scaling
Graph 4 shows the ROC-AUC curve of three comparable distance metric over dataset 3. For the
distance function, it is clear from top right ROC-AUC curve that Canberra distance metric is winner
of all with maximum AUC score of 0.9178. However, overall speaking, using the bray Curtis and
Manhattan distance function gave comparable results with AUC score of 0.9068, 0.8787 for dataset 1
having value of k as 4.
Graph 5 shows the ROC-AUC curve of three comparable distance metric over dataset 2. For the
distance function, it is clear from top right ROC-AUC curve that Canberra distance metric is winner
of all with maximum AUC score of 0.8047. However, overall speaking, using the bray Curtis and
Cosine distance function gave comparable results with AUC score of 0.78, for dataset 2 having value
of k as 13.
Graph 6 shows the ROC-AUC curve of three comparable distance metric over dataset 3. For the
distance function, it is clear from top left ROC-AUC curve that Canberra distance metric is winner of
all with maximum AUC score of 0.907774. However, overall speaking, using the Euclidean and
Manhattan distance function gave comparable results with AUC score of 0.8864, 0.910 for dataset 3
having value of k as 15.
Graph 4: showing the ROC-AUC curve for Dataset 1
Graph 5: showing the ROC-AUC curve for Dataset 2
Graph 6: showing the ROC-AUC curve for Dataset 3
5. Conclusion and Future Prespectives
In this review, the performance (accuracy, ROC-AUC) of the k-NN classifier has been evaluated
using various distance measures and Feature scaling, attempting to search out the most appropriate
distance metric and therefore the feature scaling which will be used with k-NN normally. The
results of those experiments show the following:
• The performance of the k-NN algorithm depends significantly on feature scaling and
distance metric used, the results show the massive gap between the performance of far
metrics. For instance, we have found that the Canberra distance metric is performing best
when applied on most datasets instead of mostly used Euclidean distance metric.
• We get similar classification results when we use distances from the same family having
almost the same equation, some distances are very similar, for example, one is twice the
other, or one is the square of another. In these cases, and since the KNN compares examples
using the same distance, the nearest neighbours will be the same if all distances were
multiplied or divided by the same constant.
• We get the same performance of Robust feature scaling on every dataset instead of a Min-
Max or Standard Scaler. Thus it is considered a decent choice over traditionally used
standard Scaler.
• There was no optimal distance metric, that may be used for all kinds of datasets
because the results show that each of the dataset favors a particular distance metric.
However one can start classification with the Canberra distance metric because it
outperforms the commonly used Euclidean distance.
5.1 Our work has subsequent limitations
• Although we have tested a large number of distance metrics still there are many distance
metrics available within the machine learning area that require to be tested and evaluated
further for optimal performance.
• The 3 datasets we have used belong to medical domain which still might not be enough to
draw significant conclusions in terms of the effectiveness of certain distance measures, and
therefore, there is a need to use a larger number of datasets with varied data types and
should be from different domains.
• We have got reviewed only one style of k-NN during this paper, other variants of k-NN
such as [23,24,25] need to be investigated.
• Distance measures are not used only with the KNN, but also with other machine learning
algorithms, such as different types of clustering, those need to be evaluated under different
distance measures.
5. References
[1] Innovatemedtec, accessed 9 September 2020, https://innovatemedtec.com.
[2] D.Hand, H. Manilla, P. Symth.: Principles of Data Mining. The MIT Press. (2001).
[3] Chomboon, K., Pasapichi, C., Pongsakorn, T., Kerdprasop, K., & Kerdprasop, N. (2015). An
empirical study of distance metrics for k-nearest neighbor algorithm. In The 3rd International
Conference on Industrial Application Engineering 2015 (pp. 280–285).
[4] Hu, L.-Y., Huang, M.-W., Ke, S.-W., & Tsai, C.-F. (2016). The distance function effect on k-
nearest neighbor classification for medical datasets. SpringerPlus, 5 (1), 1304.
[5] Todeschini, R., Ballabio, D., & Consonni, V. (2015). Distances and other dissimilarity measures
in chemometrics. Encyclopedia of Analytical Chemistry.
[6] Todeschini, R., Consonni, V., Grisoni, F. G., & Ballabio, D. (2016). A new concept of higher-
order similarity and the role of distance/similarity measures in local classification methods.
Chemometrics and Intelligent Laboratory Systems, 157, 50–57.
[7] Lopes, N., & Ribeiro, B. (2015). On the Impact of Distance Metrics in Instance-Based Learning
Algorithms. Iberian Conference on Pattern Recognition and Image Analysis (pp. 48–56).
Springer.
[8] Alkasassbeh, M., Altarawneh, G. A., & Hassanat, A. B. (2015). On enhancing the performance of
nearest neighbour classifiers using Hassanat distance metric. Canadian Journal of Pure and
Applied Sciences, 9 (1), 3291–3298.
[9] Hassanat, A. B. (2014). Dimensionality invariant similarity measure. Journal of American
Science, 10 (8), 221–26.
[10] Hassanat, A. B. (2014). Solving the problem of the k parameter in the KNN classifier using an
ensemble learning approach. International Journal of Computer Science and Information Security,
12 (8), 33–39.
[11] Jirina, M., & Jirina, M. (2010). Using singularity exponent in distance based classifier.
Proceedings of the 10th International Conference on Intelligent Systems Design and
Applications (ISDA2010). Cairo.
[12] Hassanat, A. B. (2014). Dimensionality invariant similarity measure. Journal of American
Science, 10 (8), 221–26.
[13] Lindi, G. A. (2016). Development of face recognition system for use on the NAO robot.
Stavanger University, Norway.
[14] Euclid. (1956). The Thirteen Books of Euclid’s Elements. Courier Corporation
[15] Deza, E., and Deza, M. M. (2009). Encyclopedia of distances. Springer.
[16] Grabusts, P. (2011). The choice of metrics for clustering algorithms. Environment.
Technology. Resources, 70-76.
[17] Williams, W. T., & Lance, G. N. (1966). Computer programs for hierarchical
polythetic classification (“ similarity analyses ”). The Computer, 9 (1), 60-64.
[18] Lance, G. N., & Williams, W. T. (1967). Mixed-data classificatory programs 1 –
Agglomerative systems. Australian Computer Journal, 1 (1), 15-20
[19] Akila, A., & Tairi, H. (2016). Combining Jaccard and Mahalanobis Cosine distance to
enhance the face recognition rate. WSEAS Transactions on Signal Processing, 16, 171-178.
[20] Wilcoxon, F. (1945). Individual comparisons by ranking methods. Biometrics Bulletin, 1 (6), 80-
83.
[21] Hassanat, A. B. (2018). Norm-Based Binary Search Trees for Speeding Up KNN Big Data
Classification. Computers, 7(4), 54.
[22] Hassanat, A. B. (2018). Two-point-based binary search trees for accelerating big data
classification using KNN. PloS one, 13(11), e0207772.
[23] Hassanat, A. B. (2018). Furthest-Pair-Based Decision Trees: Experimental Results on Big Data
Classification. Information, 9(11), 284.
[24] Hassanat, A. B. (2018). Furthest-pair-based binary search tree for speeding big data classification
using k nearest neighbors. Big Data, 6(3), 225-235.