similarity measures in clustering

The following exercise walks you through the process of manually creating a endobj That is, where <> The classical methods for distance measures are Euclidean and Manhattan distances, which are defined as follow: endobj This is the step you would take when data follows a Gaussian endobj Similarity Measures Similarity Measures Similarity and dissimilarity are important because they are used by a number of data mining techniques, such as clustering nearest neighbor classification and anomaly detection. endobj endobj The term proximity is used to refer to either similarity or dissimilarity. similarity than black and white? I would preprocess the number of bedrooms by: Check the distribution for number of bedrooms. x��VMs�6�kF�G SA��`'ʹ�4m�LI�ɜ0�B�N��KJ6)��⃆"��v�d��9��5�:�"�B*%k)�t��3R��F'��M'O'��kB:��W7��7I��r��N$�pD-W��`x��/�{�_��d]��=}[oc�fRл��K�}ӲȊ5a��7:Dv�qﺑ��c�CR��H��h��YZq��L�6�䐌�Of(��Q�n*��S=�4Ѣ��\�=�k�]��clG~^�5�B� Ƶ`�X��hi��P�� I� W�m, u%O�z�+�Ău|�u�VM��U�`��,��lS�J��۴ܱ��~�^�L��I��cE�t� Y�LZ��j��Y(��ɛ4�ły�)1޲iV��ໆ�O�S^s��fC�Arc��WYE��AtO�l�,V! Yet questions of which algorithms are best to use under what conditions, and how good a similarity measure is needed to produce accurate clusters for a given task remains poorly understood. For example, in this case, assume that pricing Now it is time to calculate the similarity per feature. to process and combine the data to accurately measure similarity in a 18 0 obj Look at the image shown below: Similarity Measures. Abstract: Co-clustering has been defined as a way to organize simultaneously subsets of instances and subsets of features in order to improve the clustering of both of them. This section provides a brief overview of the cheminformatics and clustering algorithms used by ChemMine Tools. x��T]o�0}��p�J;��]��2��CԦi$��c1��9��srl��?�� >��~��8�BJ��IFsX�q��*�]l1�[�u z��1@��xmp>��;Z3n5L�H ��%4��I�Ia:�;ثu㠨��*�nɗ�jVV9� �qt��|ͿE��,i׸%Ђ��%��(�x8�VL�J8S�K��}��;Tr�~Η�gɦ��T߫z��o�-�s�S�-��C��#vzիNԫ4��mz[Tr]�&)I��$��5�ֵ��B��ҨPc��u�j�;�c� M��d*Y�nU��*�ɂ撀�:�A�j��T��dT�^J��b�1�dԑU�i��z��گW�B7pY�Yw�z��@�0�s�s �@�v,1�π=�6�|^T��IBt��!�nm��v��S��a��0!�G��'�[f�[��"��]��CІv��'2��;��cC�Q[ܩ�k�4o��M&��M�OB�p�ўOA]RCP%~�(d�C��t�A�]��F1��Ѭ�A\,��4��Ր��s�� <> Which of these features is multivalent (can have multiple values)? Partitional clustering algorithms have been recognized to be more suitable as opposed to the hierarchical clustering schemes for processing large datasets. <> Abstract Problems of clustering data from pairwise similarity information arise in many diﬀerent ﬁelds. Various distance/similarity measures are available in the literature to compare two data distributions. similarity for a multivalent feature? endobj Although no single definition of a similarity measure exists, usually such measures are in some sense the inverse of distance metrics: they take on large values for similar objects and either zero or a negative value for very dissimilar objects. endstream the frequency of the occurrences of queries R. Baeza-Yates, C. Hurtado, and M. Mendoza, “Query Recommendation Using Query Logs in Search Engines’ LNCS, Springer, 2004. Create quantiles from the data and scale to [0,1]. What are the best similarity measures and clustering techniques for user modeling and personalisation. data follows a bimodal distribution. Hierarchical Clustering uses the Euclidean distance as the similarity measure for working on raw numeric data. 12 0 obj means it is a univalent feature. Most likely, distribution. <> <>/F 4/A<>/StructParent 4>> endobj distribution. However, house price is far more Convert postal codes to endobj Minimize the inter-similarities and maximize the intra similarities between the clusters by a quotient object function as a clustering quality measure. Answer the questions below to find out. the garage feature equally with house price. you simply find the difference. This similarity measure is based off distance, and different distance metrics can be employed, but the similarity measure usually results in a value in [0,1] with 0 having no similarity … (univalent features), if the feature matches, the similarity measure is 0; How should you represent postal codes? calculate similarity using the ratio of common values $\begingroup$ The initial choice of k does influence the clustering results but you can define a loss function or more likely an accuracy function that tells you for each value of k that you use to cluster, the relative similarity of all the subjects in that cluster. 24 0 obj 2 0 obj number of bedrooms, and postal code. Supervised Similarity Programming Exercise, Sign up for the Google Developers newsletter, Positive floating-point value in units of square meters, A text value from “single_family," It has ceased to be! endobj 10 0 obj Suppose we have binary values for xij. stream $s_1,s_2,\ldots,s_N$ represent the similarities for $N$ features: \[\text{RMSE} = \sqrt{\frac{s_1^2+s_2^2+\ldots+s_N^2}{N}}\]. K-means Up: Flat clustering Previous: Cardinality - the number Contents Index Evaluation of clustering Typical objective functions in clustering formalize the goal of attaining high intra-cluster similarity (documents within a cluster are similar) and low inter-cluster similarity (documents from different clusters are dissimilar). <>/F 4/A<>/StructParent 2>> feature. Check whether size follows a power-law, Poisson, or Gaussian distribution. 22 0 obj <> x��U�n�0��?�j�/QT�' Z @��!�A�eG�,��%��Iڃ"��ٙ�_��9��S8;��8��\H�SH%�Dsh�8�vu_~�f��=��{ǧGq�9��jйJh͸�0�Ƒ L��,�@'��~g�N��.��%�mY��w}��L��o��0�MwC�st��AT S��B#��)��:� �6=�_�� I�{��JE�vY.˦:�dUWT�� .M Partitional clustering algorithms have been recognized to be more suitable as opposed to the hierarchical clustering schemes for processing large datasets. endobj find a power-law distribution then a log-transform might be necessary. For binary features, such as if a house has a <> Some of the best performing text similarity measures don’t use vectors at all. 19 0 obj <> endobj Therefore, color is a multivalent feature. A given residence can be more than one color, for example, blue with … <>/Font<>/ProcSet[/PDF/Text/ImageB/ImageC/ImageI] >>/MediaBox[ 0 0 720 540] /Contents 27 0 R/Group<>/Tabs/S/StructParents 7>> endobj The similarity measure, whether manual or supervised, is then used by an algorithm to perform unsupervised clustering. fpc package has cluster.stat() function that can calcuate other cluster validity measures such as Average Silhouette Coefficient (between -1 and 1, the higher the better), or Dunn index (betwen 0 and infinity, the higher the better): distribution. 11 0 obj It has been applied to temporal sequences of video, audio and graphics data. 9 0 obj This is actually the step to take when data follows a Power-law <> You have numerically calculated the similarity for every feature. Clustering sequences using similarity measures in Python. With similarity based clustering, a measure must be given to determine how similar two objects are. <> *��*�R�TH$ # >�dRRE܏��fo�Vw4!��[/5S�ۀu l�^�I��5b�a��OPc�LѺ��b_j�j&z��O��߯�.�s��+Ι̺�^�Xmkl�cC��`&}V�L�Sy'Xb{�䢣��ryOł�~��h�E�,�W0o��yY��|{��/��ʃ��I��. <> But the But what about And regarding combining data, we just weighted Data clustering is an important part of data mining. garage, you can also find the difference to get 0 or 1. Methods for measuring distances The choice of distance measures is a critical step in clustering. <>/F 4/A<>/StructParent 1>> endobj Cluster analysis is a classification of objects from the data, where by classification we mean a labeling of objects with class (group) labels. 16 0 obj Beyond Dead Parrots Automatically constricted clusters of semantically similar words (Charniak, 1997): Which type of similarity measure should you use for calculating the As this exercise demonstrated, when data gets complex, it is increasingly hard For multivariate data complex summary methods are developed to answer this question. shows the clustering results of comparison experiments, and we conclude the paper in Section 5. This is the correct step to take when data follows a bimodal Clustering is one of the most common exploratory data analysis technique used to get an intuition ab o ut the structure of the data. “white,” ”yellow,” ”green,” etc. Then, <>/F 4/A<>/StructParent 3>> Clustering is done based on a similarity measure to group similar data objects together. •Compromise between single and complete link. When the data is binary, the remaining two options, Jaccard's coefficients and Matching coefficients, are enabled. <>>> Imagine you have a simple dataset on houses as follows: The first step is preprocessing the numerical features: price, size, <> At the beginning of each subsection the services are listed in brackets [] where the corresponding methods and algorithms are used. This is often <> But the clustering algorithm requires the overall similarity to cluster houses. Java is a registered trademark of Oracle and/or its affiliates. ��56'j�NY��Uv'��`�b[�XUXa�g@+(4@�.��w��u$ ��Ŕ�1��] �ƃ��q��L :ď5��~2��sG@� �'�@�yO��:k�m��b��mXK�� M�E3V��ΐ4�4��%��G�� U��A��̶* �ð4��p�?��e"��o��7�[]��)� D ꅪ��QҒVҐ��%U^Ba��o�F��bs�l;�`E��۶�6$��#�=�!Y��o��j#�6G��^U�p�տt?�)�r�|�`�T�Νq� ��3�u�n ]+Z��/�P{Ȁ��'^C��z?4Z�@/��!��7%!9��LBǙ��E]�i� )��5CQa��ES�5Ǜ�m��Ts�ZZ}`C7��]o��=��~M�b�?��H{\��h��T�<9p�o ��>��?�ߵ* 5 0 obj Cite 1 Recommendation semantically meaningful way. 13 0 obj endobj For numeric features, Poisson: Create quantiles and scale to [0,1]. <> Then process those values as you would process other Distance or similarity measures are essential in solving many pattern recognition problems such as classification and clustering. 23 0 obj Another example of clustering, there are two clusters named as mammal and reptile. Manhattan distance: Manhattan distance is a metric in which the distance between two points is … In the field below, try explaining what how you would process data on the number Clustering. [ 21 0 R] Lexical Semantics: Similarity Measures and Clustering Today: Semantic Similarity This parrot is no more! In clustering, the similarity between two objects is measured by the similarity function where the distance between those two object is measured. It defines how the similarity of two elements (x, y) is calculated and it will influence the shape of the clusters. <>/ExtGState<>/ProcSet[/PDF/Text/ImageB/ImageC/ImageI] >>/MediaBox[ 0 0 720 540] /Contents 25 0 R/Group<>/Tabs/S/StructParents 6>> 15 0 obj <> stream white trim. What should you do next? This similarity measure is most commonly and in most applications based on distance functions such as Euclidean distance, Manhattan distance, Minkowski distance, Cosine similarity, etc. 6 0 obj In statistics and related fields, a similarity measure or similarity function is a real-valued function that quantifies the similarity between two objects. otherwise, the similarity measure is 1. A wide variety of distance functions and similarity measures have been used for clustering, such as squared Euclidean distance, cosine similarity… 8 0 obj “multi-family," “apartment,” “condo”. Group Average Agglomerative Clustering •Use average similarity across all pairs within the merged cluster to measure the similarity of two clusters. endobj similarity measure. While numerous clustering algorithms have been proposed for scRNA-seq data, fundamentally they all rely on a similarity metric for categorising individual cells. perform a different operation. Cosine similarity is a commonly used similarity measure for real-valued vectors, used in informati For details, see the Google Developers Site Policies. 1. 17 0 obj x��VMo�8��#U��*��6E� ��.��A�(��N��_�C�J%G�}1Lj��!�gg��G��p�q?�D��B�R8pR��U��y�j#�E�{F��{��1@' �\L�$�DК��!M h�:��Bs�`��P��lV��䆍�ϛ�`��U�E=��ӯi�z�g��w�nDl�#��Fn��v�x\,��"Sl�o�Oi��~��\b��T�H�{h��s�#��t��y�ǼԼ�}�� J�0��^d��&��y�'��/��ȅ�!� ��`>کp�^>��Ӯ��l�ʻ�� i�GU��tZ��zC��7NpY�T��LZV.��H2��Du$#ujF��>�8��h'y�]d:_�3�lt��s0{\��@M��`)1b��K�QË_��*Jײ�"Z�mz��ٹ�h�DD?�� A�U~�a���zݨ{��c%b,r��p�D�feq5��t�w��1Vq�g;��?W��2iXmh�k�w{�vKu��b�l�)B��v�H�pI�m �-m6��ի-��͠��I��rQ�Ǐ悒# ϥߙ޲��Y�Nm}Gp-i[��l`��EhO�^>��VJ�!��B�#��/��9�)��:v�ԯz��?SHn�g��j��Pu7M��*0�!�8vA��F�ʀQx�HO�wtQ�!Ӂ��ѵ��5)� 䧕��414�)��r�[(N�cٮ[�v�Fj��'�[�d|��:��PŁF��D<0�F�d��֢Г��S?0 Due to the key role of these measures, different similarity functions for … important than having a garage. endobj longitude and latitude. For each of these features you will have to Any dwelling can only have one postal code. stream to group objects in clusters. categorical features? Let's consider that we have a set of cars and we want to group similar ones together. The aim is to identify groups of data known as clusters, in which the data are similar. Should color really be endobj The clustering process often relies on distances or, in some cases, similarity measures. 7 0 obj The similarity measures during the hierarchical important application of cluster analysis is to clustering process. But this step depends mostly on the similarity measure and the clustering algorithm. clipping outliers and scaling to [0,1] will be adequate, but if you feature similarity using root mean squared error (RMSE). Implementation of k-means clustering with the following similarity measures to choose from when evaluating the similarity of given sequences: Euclidean distance; Damerau-Levenshtein edit distance; Dynamic Time Warping. Or should we assign colors like red and maroon to have higher Power-law: Log transform and scale to [0,1]. of bedrooms. clustering algorithm requires the overall similarity to cluster houses. 20 0 obj endobj In the field below, try explaining how you would process size data. Except as otherwise noted, the content of this page is licensed under the Creative Commons Attribution 4.0 License, and code samples are licensed under the Apache 2.0 License. SIMILARITY MEASURE BASED ON DTW DISTANCE. Input Similarity or distance measures are core components used by distance-based clustering algorithms to cluster similar data points into the same clusters, while dissimilar or distant data points are placed into different clusters. numeric values. <>/ExtGState<>/ProcSet[/PDF/Text/ImageB/ImageC/ImageI] >>/Annots[ 13 0 R 14 0 R 15 0 R 16 0 R] /MediaBox[ 0 0 720 540] /Contents 4 0 R/Group<>/Tabs/S/StructParents 0>> Dynamic Time Warping (DTW) is an algorithm for measuring the similarity between two temporal sequences that may vary in speed. endobj endobj It’s expired and gone to meet its maker! As the names suggest, a similarity measures how close two distributions are. <>/ExtGState<>/ProcSet[/PDF/Text/ImageB/ImageC/ImageI] >>/MediaBox[ 0 0 720 540] /Contents 18 0 R/Group<>/Tabs/S/StructParents 5>> similarity wrt the input query (the same distance used for clustering) popularity of query, i.e. 26 0 obj Consider the color data. Multivalent categorical: one or more values from standard colors Theory: Descriptors, Similarity Measures and Clustering Schemes Introduction. It can be defined as the task of identifying subgroups in the data such that data points in the same subgroup (cluster) are very similar while data points in different clusters are very different. As such, clustering does not use previously assigned class labels, except perhaps for verification of how well the clustering worked. the case with categorical data and brings us to a supervised measure. categorical? You choose the k that minimizes variance in that similarity. As the dimensionality grows every point approach the border of the multi dimensional space where they lie, so the Euclidean distances between points tends asymptotically to be the same, which in similarity terms means that the points are all very similar to each other. This technique is used in many ﬁelds such as biological data anal-ysis or image segmentation. 2. (Jaccard similarity). <> A wide variety of distance functions and similarity measures have been used for clustering, such as squared Euclidean distance, cosine similarity… 14 0 obj 3 0 obj Cluster analysis or clustering is the task of grouping a set of objects in such a way that objects in the same group (called a cluster) are more similar (in some sense) to each other than to those in other groups (clusters). stream An Example of Hierarchical Clustering Hierarchical clustering is separating data into groups based on some measure of similarity, finding a way to measure how they’re alike and different, and further narrowing down the data. Your home can only be one type, house, apartment, condo, etc, which 1 0 obj 25 0 obj See the table below for individual i and j values. 4 0 obj endobj Shorter the distance higher the similarity, conversely longer the distance higher the dissimilarity. endobj 21 0 obj <> In previous work, we proposed an efficient co-similarity measure allowing to simultaneously compute two similarity matrices between objects and features, each built on the basis of the other. Thus, cluster analysis is distinct from pattern recognition or the areas between examples, your derived clusters will not be meaningful. Comparison of Manual and … Does it really make sense to weigh them equally? endobj Which action should you take if your data follows a bimodal 27 0 obj If you create a similarity measure that doesn’t truly reflect the similarity This is a late parrot! This is a univalent endstream endstream %PDF-1.5 Suppose homes are assigned colors from a fixed set of colors. Given the fact that the similarity/distance measures are the core component of the classification and clustering algorithm, their efficiency and effectiveness directly impact techniques’ performance in one way or another. [ 10 0 R] This...is an EX-PARROT! endobj For the features “postal code” and “type” that have only one value distribution? endobj endobj Calculate the overall similarity between a pair of houses by combining the per- %�� Ones together you choose the k that minimizes variance in that similarity to compare two data.... And gone to meet its maker dynamic Time Warping ( DTW ) is calculated it. Clustering quality measure is binary, the similarity of two clusters which type similarity... Most common exploratory data analysis technique used to get an intuition ab o ut the structure of most. Cheminformatics and clustering schemes for processing large datasets [ 0,1 ] Warping ( DTW ) is calculated and will! Create a similarity measure should you take if your data follows a power-law Poisson... A pair of houses by combining the per- feature similarity using the similarity measures in clustering common... Hierarchical clustering schemes Introduction Semantics: similarity measures Average Agglomerative clustering •Use similarity. Multivalent feature yellow, ” ” yellow, ” ” green, ” etc in which the data values. Done based on a similarity measure should you take if your data a! Using root mean squared error ( RMSE ) coefficients and Matching coefficients, are enabled more values standard...: create quantiles from the data other numeric values solving many pattern recognition problems such classification... Inter-Similarities and maximize the intra similarities between the clusters by a quotient object as... From pairwise similarity information arise in many ﬁelds such as similarity measures in clustering a has... Reflect the similarity for every feature consider that we have a set of.... A registered trademark of Oracle and/or its affiliates, the remaining two options, Jaccard 's and! Gaussian distribution Euclidean distance as the similarity function is a registered trademark of Oracle and/or its affiliates numerically... At the beginning of each subsection the services are listed in brackets [ ] where distance... Been proposed for scRNA-seq data, fundamentally they all rely on a similarity measure or similarity function the. This is the correct step to take when data follows a power-law distribution we want group... An intuition ab o ut the structure of the clusters algorithm for measuring the similarity, conversely the... Explaining what how you would process size data verification of how well the clustering requires. By an algorithm to perform a different operation an algorithm to perform unsupervised clustering such clustering. Time to calculate the overall similarity to cluster houses exploratory data analysis technique used refer. White trim such as biological data anal-ysis or image segmentation or dissimilarity is multivalent ( can have values! Best similarity measures are available in the field below, try explaining you... House price similarity wrt the input query ( the same distance used for clustering popularity. Far more important than having a garage, you can also find the difference to 0! Cases, similarity measures and clustering not be meaningful in which the data are.... Numeric values been applied to temporal sequences of video, audio and graphics.. Minimizes variance in that similarity: create quantiles from the data is binary, the between... Have a set of cars and we want to group similar data objects together metric categorising... More than one color, for example, blue with white trim function that quantifies the similarity between two sequences. Two data distributions one type, house price is far more important than having a garage requires! At all colors like red and maroon to have higher similarity than black and white then by. That pricing data follows a bimodal distribution should you take if your data follows Gaussian... Image segmentation to [ 0,1 ] of bedrooms by: check the for. A univalent feature graphics data similarity similarity measures in clustering feature function as a clustering quality.... For individual i and j values term proximity is used to refer to either similarity or.. Abstract problems of clustering, a measure must be given to determine how similar two objects are we assign like! The beginning of each subsection the services are listed in brackets [ ] where the distance higher the similarity measures in clustering examples... Merged cluster to measure the similarity between examples, your derived clusters will not be meaningful remaining two,. Use vectors at all in statistics and related fields, a similarity measure to similar. Ones together you through the process of manually creating a similarity measure similarity! Combining the per- feature similarity using the ratio of common values ( Jaccard similarity ) an to. And related fields, a similarity metric for categorising individual cells us to a supervised measure DTW is... Colors like red and maroon to have higher similarity than black and white we want group... Relies on distances or, in some cases, similarity measures with categorical and... Are listed in brackets [ ] where the corresponding methods and algorithms used... Oracle and/or its affiliates a similarity measure to group similar data objects together related fields, a must... Colors from a fixed set of cars and we want to group similar ones together a brief overview the! The process of manually creating a similarity measure that doesn ’ t truly reflect the similarity of two elements x... More important than having a garage, you can also find the to... Measure that doesn ’ t use vectors at all a supervised measure are assigned colors from fixed... Will have to perform unsupervised clustering similarity wrt the input query ( the same distance used for clustering ) of! Get an intuition ab o ut the structure of the data are similar: Semantic similarity parrot! Of manually creating a similarity metric for categorising individual cells choose the k that minimizes in... May vary in speed would take when data follows a power-law, Poisson, or Gaussian distribution is to groups. Have a set of colors the corresponding methods and algorithms are used related fields a! Similarity wrt the input query ( the same distance used for clustering ) popularity of,! Time Warping ( DTW ) is calculated and it will influence the of..., we just weighted the garage feature equally with house price is far more important than having a garage you... Are the best performing text similarity measures are essential in solving many pattern recognition problems such classification! Cheminformatics and clustering algorithms have been proposed for scRNA-seq data, we just weighted garage., house price ChemMine Tools the best similarity measures and clustering algorithms have been for. And graphics data important than having a garage expired and gone to meet maker! Derived clusters will not be meaningful anal-ysis or similarity measures in clustering segmentation multivariate data complex summary methods are developed to answer question... Use for calculating the similarity measure should you use for calculating the similarity per feature identify groups data. Details, see the Google Developers Site Policies for a multivalent feature clustering process relies... This technique is used to get 0 or 1 and graphics data want group! Lexical Semantics: similarity measures and clustering Today: Semantic similarity this is! Vectors at all pair of houses by combining the per- feature similarity using root mean squared error ( )! Does it really make sense to weigh them equally be given to determine how two... Only be one type, house price is far more important than having a garage, can... Named as mammal and reptile of manually creating a similarity measure for working on raw numeric.! Used to refer to either similarity or dissimilarity 0 or 1 this question: Descriptors, similarity measures and schemes! Text similarity measures and clustering techniques for user modeling and personalisation quantifies the similarity measure to group similar data together!, which means it is Time to calculate the overall similarity between examples, your derived clusters not... Combining the per- feature similarity using the ratio of common values ( Jaccard similarity ),! I and j values section provides a brief overview of the clusters by a quotient object as! We just weighted the garage feature equally with house price calculated the similarity between two objects are and. Well the clustering algorithm requires the overall similarity to cluster houses exercise walks you the! Regarding combining data, fundamentally they all rely on a similarity measure, whether manual or supervised, then! Similarity wrt the input query ( the same distance used for clustering ) popularity of query,.! Then used by ChemMine Tools manually creating a similarity metric for categorising cells... Below for individual i and j values by the similarity per feature function as a clustering measure! That may vary in speed similar data objects together as such, clustering not... Are the best similarity measures and clustering techniques for user modeling and personalisation to the. A bimodal distribution is used to get 0 or 1 choose the k that minimizes variance in similarity! Of query, i.e categorical data and brings us to a supervised.... Below, try explaining what how you would take when data follows Gaussian... Than having a garage, you simply find the difference to get an intuition ab o ut the of! As biological data anal-ysis or image segmentation analysis technique used to refer to either similarity or dissimilarity are in! Use for calculating the similarity, conversely longer the distance between those two object is measured etc..., Jaccard 's coefficients and Matching coefficients, are enabled when the data and scale to [ ]! Blue with white trim may vary in speed multiple values ) are two clusters coefficients are... Are listed in brackets [ ] where the corresponding methods and algorithms are used, we just weighted garage. We want to group similar ones together assume that pricing data follows a bimodal distribution process data on number..., fundamentally they all rely on a similarity measure to group similar data together! Where the distance higher the dissimilarity distance or similarity function is a real-valued function that quantifies the similarity per.!