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  User-defined function

  A user-defined function (UDF) is a function provided by the user of a program or environment, in a context where the usual assumption is that functions are built into the program or environment. UDFs are usually written for the requirement of its creator. == BASIC language == In some old implementations of the BASIC programming language, user-defined functions are defined using the "DEF FN" syntax. More modern dialects of BASIC are influenced by the structured programming paradigm, where most or all of the code is written as user-defined functions or procedures, and the concept becomes practically redundant. == COBOL language == In the COBOL programming language, a user-defined function is an entity that is defined by the user by specifying a FUNCTION-ID paragraph. A user-defined function must return a value by specifying the RETURNING phrase of the procedure division header and they are invoked using the function-identifier syntax. See the ISO/IEC 1989:2014 Programming Language COBOL standard for details. As of May 2022, the IBM Enterprise COBOL for z/OS 6.4 (IBM COBOL) compiler contains support for user-defined functions. == Databases == In relational database management systems, a user-defined function provides a mechanism for extending the functionality of the database server by adding a function, that can be evaluated in standard query language (usually SQL) statements. The SQL standard distinguishes between scalar and table functions. A scalar function returns only a single value (or NULL), whereas a table function returns a (relational) table comprising zero or more rows, each row with one or more columns. User-defined functions in SQL are declared using the CREATE FUNCTION statement. For example, a user-defined function that converts Celsius to Fahrenheit (a temperature scale used in USA) might be declared like this: Once created, a user-defined function may be used in expressions in SQL statements. For example, it can be invoked where most other intrinsic functions are allowed. This also includes SELECT statements, where the function can be used against data stored in tables in the database. Conceptually, the function is evaluated once per row in such usage. For example, assume a table named Elements, with a row for each known chemical element. The table has a column named BoilingPoint for the boiling point of that element, in Celsius. The query would retrieve the name and the boiling point from each row. It invokes the CtoF user-defined function as declared above in order to convert the value in the column to a value in Fahrenheit. Each user-defined function carries certain properties or characteristics. The SQL standard defines the following properties: Language - defines the programming language in which the user-defined function is implemented; examples include SQL, C, C# and Java. Parameter style - defines the conventions that are used to pass the function parameters and results between the implementation of the function and the database system (only applicable if language is not SQL). Specific name - a name for the function that is unique within the database. Note that the function name does not have to be unique, considering overloaded functions. Some SQL implementations require that function names are unique within a database, and overloaded functions are not allowed. Determinism - specifies whether the function is deterministic or not. The determinism characteristic has an influence on the query optimizer when compiling a SQL statement. SQL-data access - tells the database management system whether the function contains no SQL statements (NO SQL), contains SQL statements but does not access any tables or views (CONTAINS SQL), reads data from tables or views (READS SQL DATA), or actually modifies data in the database (MODIFIES SQL DATA). User-defined functions should not be confused with stored procedures. Stored procedures allow the user to group a set of SQL commands. A procedure can accept parameters and execute its SQL statements depending on those parameters. A procedure is not an expression and, thus, cannot be used like user-defined functions. Some database management systems allow the creation of user defined functions in languages other than SQL. Microsoft SQL Server, for example, allows the user to use .NET languages including C# for this purpose. DB2 and Oracle support user-defined functions written in C or Java programming languages. === SQL Server 2000 === There are three types of UDF in Microsoft SQL Server 2000: scalar functions, inline table-valued functions, and multistatement table-valued functions. Scalar functions return a single data value (not a table) with RETURNS clause. Scalar functions can use all scalar data types, with exception of timestamp and user-defined data types. Inline table-valued functions return the result set of a single SELECT statement. Multistatement table-valued functions return a table, which was built with many TRANSACT-SQL statements. User-defined functions can be invoked from a query like built‑in functions such as OBJECT_ID, LEN, DATEDIFF, or can be executed through an EXECUTE statement like stored procedures. Performance Notes: User-defined functions are subroutines made of one or more Transact-SQL statements that can be used to encapsulate code for reuse. It takes zero or more arguments and evaluates a return value. Has both control-flow and DML statements in its body similar to stored procedures. Does not allow changes to any Global Session State, like modifications to database or external resource, such as a file or network. Does not support output parameter. DEFAULT keyword must be specified to pass the default value of parameter. Errors in UDF cause UDF to abort which, in turn, aborts the statement that invoked the UDF. === Apache Hive === Apache Hive defines, in addition to the regular user-defined functions (UDF), also user-defined aggregate functions (UDAF) and table-generating functions (UDTF). Hive enables developers to create their own custom functions with Java. === Apache Doris === Apache Doris, an open-source real-time analytical database, allows external users to contribute their own UDFs written in C++ to it.

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  Witness set

  In combinatorics and computational learning theory, a witness set is a set of elements that distinguishes a given Boolean function from a given class of other Boolean functions. Let C {\displaystyle C} be a concept class over a domain X {\displaystyle X} (that is, a family of Boolean functions over X {\displaystyle X} ) and c {\displaystyle c} be a concept in X {\displaystyle X} (a single Boolean function). A subset S {\displaystyle S} of X {\displaystyle X} is a witness set for c {\displaystyle c} in X {\displaystyle X} if S {\displaystyle S} distinguishes c {\displaystyle c} from all the other functions in C {\displaystyle C} , in the sense that no other function in C {\displaystyle C} has the same values on S {\displaystyle S} . For a concept class with | C | {\displaystyle |C|} concepts, there exists a concept that has a witness of size at most log 2 ⁡ | C | {\displaystyle \log _{2}|C|} ; this bound is tight when C {\displaystyle C} consists of all Boolean functions over X {\displaystyle X} . By a result of Bondy (1972) there exists a single witness set of size at most | C | − 1 {\displaystyle |C|-1} that is valid for all concepts in C {\displaystyle C} ; this bound is tight when C {\displaystyle C} consists of the indicator functions of the empty set and some singleton sets. One way to construct this set is to interpret the concepts as bitstrings, and the domain elements as positions in these bitstrings. Then the set of positions at which a trie of the bitstrings branches forms the desired witness set. This construction is central to the operation of the fusion tree data structure. The minimum size of a witness set for c {\displaystyle c} is called the witness size or specification number and is denoted by w C ( c ) {\displaystyle w_{C}(c)} . The value max { w C ( c ) : c ∈ C } {\displaystyle \max\{w_{C}(c):c\in C\}} is called the teaching dimension of C {\displaystyle C} . It represents the number of examples of a concept that need to be presented by a teacher to a learner, in the worst case, to enable the learner to determine which concept is being presented. Witness sets have also been called teaching sets, keys, specifying sets, or discriminants. The "witness set" terminology is from Kushilevitz et al. (1996), who trace the concept of witness sets to work by Cover (1965).

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  Sliced inverse regression

  Sliced inverse regression (SIR) is a tool for dimensionality reduction in the field of multivariate statistics. In statistics, regression analysis is a method of studying the relationship between a response variable y and its input variable x _ {\displaystyle {\underline {x}}} , which is a p-dimensional vector. There are several approaches in the category of regression. For example, parametric methods include multiple linear regression, and non-parametric methods include local smoothing. As the number of observations needed to use local smoothing methods scales exponentially with high-dimensional data (as p grows), reducing the number of dimensions can make the operation computable. Dimensionality reduction aims to achieve this by showing only the most important dimension of the data. SIR uses the inverse regression curve, E ( x _ | y ) {\displaystyle E({\underline {x}}\,|\,y)} , to perform a weighted principal component analysis. == Model == Given a response variable Y {\displaystyle \,Y} and a (random) vector X ∈ R p {\displaystyle X\in \mathbb {R} ^{p}} of explanatory variables, SIR is based on the model Y = f ( β 1 ⊤ X , … , β k ⊤ X , ε ) ( 1 ) {\displaystyle Y=f(\beta _{1}^{\top }X,\ldots ,\beta _{k}^{\top }X,\varepsilon )\quad \quad \quad \quad \quad (1)} where β 1 , … , β k {\displaystyle \beta _{1},\ldots ,\beta _{k}} are unknown projection vectors, k {\displaystyle \,k} is an unknown number smaller than p {\displaystyle \,p} , f {\displaystyle \;f} is an unknown function on R k + 1 {\displaystyle \mathbb {R} ^{k+1}} as it only depends on k {\displaystyle \,k} arguments, and ε {\displaystyle \varepsilon } is a random variable representing error with E [ ε | X ] = 0 {\displaystyle E[\varepsilon |X]=0} and a finite variance of σ 2 {\displaystyle \sigma ^{2}} . The model describes an ideal solution, where Y {\displaystyle \,Y} depends on X ∈ R p {\displaystyle X\in \mathbb {R} ^{p}} only through a k {\displaystyle \,k} dimensional subspace; i.e., one can reduce the dimension of the explanatory variables from p {\displaystyle \,p} to a smaller number k {\displaystyle \,k} without losing any information. An equivalent version of ( 1 ) {\displaystyle \,(1)} is: the conditional distribution of Y {\displaystyle \,Y} given X {\displaystyle \,X} depends on X {\displaystyle \,X} only through the k {\displaystyle \,k} dimensional random vector ( β 1 ⊤ X , … , β k ⊤ X ) {\displaystyle (\beta _{1}^{\top }X,\ldots ,\beta _{k}^{\top }X)} . It is assumed that this reduced vector is as informative as the original X {\displaystyle \,X} in explaining Y {\displaystyle \,Y} . The unknown β i ′ s {\displaystyle \,\beta _{i}'s} are called the effective dimension reducing directions (EDR-directions). The space that is spanned by these vectors is denoted by the effective dimension reducing space (EDR-space). == Relevant linear algebra background == Given a _ 1 , … , a _ r ∈ R n {\displaystyle {\underline {a}}_{1},\ldots ,{\underline {a}}_{r}\in \mathbb {R} ^{n}} , then V := L ( a _ 1 , … , a _ r ) {\displaystyle V:=L({\underline {a}}_{1},\ldots ,{\underline {a}}_{r})} , the set of all linear combinations of these vectors is called a linear subspace and is therefore a vector space. The equation says that vectors a _ 1 , … , a _ r {\displaystyle {\underline {a}}_{1},\ldots ,{\underline {a}}_{r}} span V {\displaystyle \,V} , but the vectors that span space V {\displaystyle \,V} are not unique. The dimension of V ( ∈ R n ) {\displaystyle \,V(\in \mathbb {R} ^{n})} is equal to the maximum number of linearly independent vectors in V {\displaystyle \,V} . A set of n {\displaystyle \,n} linear independent vectors of R n {\displaystyle \mathbb {R} ^{n}} makes up a basis of R n {\displaystyle \mathbb {R} ^{n}} . The dimension of a vector space is unique, but the basis itself is not. Several bases can span the same space. Dependent vectors can still span a space, but the linear combinations of the latter are only suitable to a set of vectors lying on a straight line. == Inverse regression == Computing the inverse regression curve (IR) means instead of looking for E [ Y | X = x ] {\displaystyle \,E[Y|X=x]} , which is a curve in R p {\displaystyle \mathbb {R} ^{p}} it is actually E [ X | Y = y ] {\displaystyle \,E[X|Y=y]} , which is also a curve in R p {\displaystyle \mathbb {R} ^{p}} , but consisting of p {\displaystyle \,p} one-dimensional regressions. The center of the inverse regression curve is located at E [ E [ X | Y ] ] = E [ X ] {\displaystyle \,E[E[X|Y]]=E[X]} . Therefore, the centered inverse regression curve is E [ X | Y = y ] − E [ X ] {\displaystyle \,E[X|Y=y]-E[X]} which is a p {\displaystyle \,p} dimensional curve in R p {\displaystyle \mathbb {R} ^{p}} . == Inverse regression versus dimension reduction == The centered inverse regression curve lies on a k {\displaystyle \,k} -dimensional subspace spanned by Σ x x β i ′ s {\displaystyle \,\Sigma _{xx}\beta _{i}\,'s} . This is a connection between the model and inverse regression. Given this condition and ( 1 ) {\displaystyle \,(1)} , the centered inverse regression curve E [ X | Y = y ] − E [ X ] {\displaystyle \,E[X|Y=y]-E[X]} is contained in the linear subspace spanned by Σ x x β k ( k = 1 , … , K ) {\displaystyle \,\Sigma _{xx}\beta _{k}(k=1,\ldots ,K)} , where Σ x x = C o v ( X ) {\displaystyle \,\Sigma _{xx}=Cov(X)} . == Estimation of the EDR-directions == After having had a look at all the theoretical properties, the aim now is to estimate the EDR-directions. For that purpose, weighted principal component analyses are needed. If the sample means m ^ h ′ s {\displaystyle \,{\hat {m}}_{h}\,'s} , X {\displaystyle \,X} would have been standardized to Z = Σ x x − 1 / 2 { X − E ( X ) } {\displaystyle \,Z=\Sigma _{xx}^{-1/2}\{X-E(X)\}} . Corresponding to the theorem above, the IR-curve m 1 ( y ) = E [ Z | Y = y ] {\displaystyle \,m_{1}(y)=E[Z|Y=y]} lies in the space spanned by ( η 1 , … , η k ) {\displaystyle \,(\eta _{1},\ldots ,\eta _{k})} , where η i = Σ x x 1 / 2 β i {\displaystyle \,\eta _{i}=\Sigma _{xx}^{1/2}\beta _{i}} . As a consequence, the covariance matrix c o v [ E [ Z | Y ] ] {\displaystyle \,cov[E[Z|Y]]} is degenerate in any direction orthogonal to the η i ′ s {\displaystyle \,\eta _{i}\,'s} . Therefore, the eigenvectors η k ( k = 1 , … , K ) {\displaystyle \,\eta _{k}(k=1,\ldots ,K)} associated with the largest K {\displaystyle \,K} eigenvalues are the standardized EDR-directions. == Algorithm == === SIR algorithm === The algorithm from Li, K-C. (1991) to estimate the EDR-directions via SIR is as follows. 1. Let Σ x x {\displaystyle \,\Sigma _{xx}} be the covariance matrix of X {\displaystyle \,X} . Standardize X {\displaystyle \,X} to Z = Σ x x − 1 / 2 { X − E ( X ) } {\displaystyle \,Z=\Sigma _{xx}^{-1/2}\{X-E(X)\}} ( 1 ) {\displaystyle \,(1)} can also be rewritten as Y = f ( η 1 ⊤ Z , … , η k ⊤ Z , ε ) {\displaystyle Y=f(\eta _{1}^{\top }Z,\ldots ,\eta _{k}^{\top }Z,\varepsilon )} where η k = β k Σ x x 1 / 2 ∀ k {\displaystyle \,\eta _{k}=\beta _{k}\Sigma _{xx}^{1/2}\quad \forall \;k} .) 2. Divide the range of y i {\displaystyle \,y_{i}} into S {\displaystyle \,S} non-overlapping slices H s ( s = 1 , … , S ) . n s {\displaystyle \,H_{s}(s=1,\ldots ,S).\;n_{s}} is the number of observations within each slice and I H s {\displaystyle \,I_{H_{s}}} is the indicator function for the slice: n s = ∑ i = 1 n I H s ( y i ) {\displaystyle n_{s}=\sum _{i=1}^{n}I_{H_{s}}(y_{i})} 3. Compute the mean of z i {\displaystyle \,z_{i}} over all slices, which is a crude estimate m ^ 1 {\displaystyle \,{\hat {m}}_{1}} of the inverse regression curve m 1 {\displaystyle \,m_{1}} : z ¯ s = n s − 1 ∑ i = 1 n z i I H s ( y i ) {\displaystyle \,{\bar {z}}_{s}=n_{s}^{-1}\sum _{i=1}^{n}z_{i}I_{H_{s}}(y_{i})} 4. Calculate the estimate for C o v { m 1 ( y ) } {\displaystyle \,Cov\{m_{1}(y)\}} : V ^ = n − 1 ∑ i = 1 S n s z ¯ s z ¯ s ⊤ {\displaystyle \,{\hat {V}}=n^{-1}\sum _{i=1}^{S}n_{s}{\bar {z}}_{s}{\bar {z}}_{s}^{\top }} 5. Identify the eigenvalues λ ^ i {\displaystyle \,{\hat {\lambda }}_{i}} and the eigenvectors η ^ i {\displaystyle \,{\hat {\eta }}_{i}} of V ^ {\displaystyle \,{\hat {V}}} , which are the standardized EDR-directions. 6. Transform the standardized EDR-directions back to the original scale. The estimates for the EDR-directions are given by: β ^ i = Σ ^ x x − 1 / 2 η ^ i {\displaystyle \,{\hat {\beta }}_{i}={\hat {\Sigma }}_{xx}^{-1/2}{\hat {\eta }}_{i}} (which are not necessarily orthogonal.)

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  Silhouette (clustering)

  Silhouette is a method of interpretation and validation of consistency within clusters of data. The technique provides a succinct graphical representation of how well each object has been classified. It was proposed by Belgian statistician Peter Rousseeuw in 1987. The silhouette value is a measure of how similar an object is to its own cluster (cohesion) compared to other clusters (separation). The silhouette value ranges from −1 to +1, where a high value indicates that the object is well matched to its own cluster and poorly matched to neighboring clusters. If most objects have a high value, then the clustering configuration is appropriate. If many points have a low or negative value, then the clustering configuration may have too many or too few clusters. A clustering with an average silhouette width of over 0.7 is considered to be "strong", a value over 0.5 "reasonable", and over 0.25 "weak". However, with an increasing dimensionality of the data, it becomes difficult to achieve such high values because of the curse of dimensionality, as the distances become more similar. The silhouette score is specialized for measuring cluster quality when the clusters are convex-shaped, and may not perform well if the data clusters have irregular shapes or are of varying sizes. The silhouette value can be calculated with any distance metric, such as Euclidean distance or Manhattan distance. == Definition == Assume the data have been clustered via any technique, such as k-medoids or k-means, into k {\displaystyle k} clusters. For data point i ∈ C i {\displaystyle i\in C_{i}} (data point i {\displaystyle i} in the cluster C i {\displaystyle C_{i}} ), calculate a ( i ) {\displaystyle a(i)} , the average distance that i {\displaystyle i} is from all other points in that cluster: a ( i ) = 1 | C i | − 1 ∑ j ∈ C i , i ≠ j d ( i , j ) {\displaystyle a(i)={\frac {1}{|C_{i}|-1}}\sum _{j\in C_{i},i\neq j}d(i,j)} where | C i | {\displaystyle |C_{i}|} is the number of points belonging to cluster C i {\displaystyle C_{i}} , and d ( i , j ) {\displaystyle d(i,j)} is the distance between data points i {\displaystyle i} and j {\displaystyle j} in the cluster C i {\displaystyle C_{i}} (we divide by | C i | − 1 {\displaystyle |C_{i}|-1} because the distance d ( i , i ) {\displaystyle d(i,i)} is not included in the sum). a ( i ) {\displaystyle a(i)} can be interpreted as a measure of how well i {\displaystyle i} is assigned to its cluster (the smaller the value, the better the assignment). We then define the mean dissimilarity of point i {\displaystyle i} to some cluster C j {\displaystyle C_{j}} as the mean of the distance from i {\displaystyle i} to all points in C j {\displaystyle C_{j}} (where C j ≠ C i {\displaystyle C_{j}\neq C_{i}} ). For each data point i ∈ C i {\displaystyle i\in C_{i}} , we now define b ( i ) {\displaystyle b(i)} as the average distance between i {\displaystyle i} and the points in the closest cluster (hence: "min") that i {\displaystyle i} does not belong to: b ( i ) = min j ≠ i 1 | C j | ∑ l ∈ C j d ( i , l ) {\displaystyle b(i)=\min _{j\neq i}{\frac {1}{|C_{j}|}}\sum _{l\in C_{j}}d(i,l)} The cluster with the smallest mean dissimilarity is said to be the "neighboring cluster" of i {\displaystyle i} because it is the next best fit cluster for point i {\displaystyle i} . We now define a silhouette (value) of one data point i {\displaystyle i} s ( i ) = b ( i ) − a ( i ) max { a ( i ) , b ( i ) } {\displaystyle s(i)={\frac {b(i)-a(i)}{\max\{a(i),b(i)\}}}} , if | C i | > 1 {\displaystyle |C_{i}|>1} and s ( i ) = 0 {\displaystyle s(i)=0} , if | C i | = 1 {\displaystyle |C_{i}|=1} , which can also be written as s ( i ) = { 1 − a ( i ) b ( i ) , if a ( i ) < b ( i ) 0 , if a ( i ) = b ( i ) b ( i ) a ( i ) − 1 , if a ( i ) > b ( i ) {\displaystyle s(i)={\begin{cases}1-{\frac {a(i)}{b(i)}},&{\mbox{ if }}a(i)b(i)\\\end{cases}}} From the above definition, s ( i ) {\displaystyle s(i)} is bounded to the interval [ − 1 , 1 ] {\displaystyle [-1,1]} , i.e. − 1 ≤ s ( i ) ≤ 1. {\displaystyle -1\leq s(i)\leq 1.} Note that a ( i ) {\displaystyle a(i)} is not clearly defined for clusters with size = 1, in which case we set s ( i ) = 0 {\displaystyle s(i)=0} . This choice is arbitrary, but neutral in the sense that it is at the midpoint of the bounds, -1 and 1. For s ( i ) {\displaystyle s(i)} to be close to 1 we require a ( i ) ≪ b ( i ) {\displaystyle a(i)\ll b(i)} . As a ( i ) {\displaystyle a(i)} is a measure of how dissimilar i {\displaystyle i} is to its own cluster, a small value means it is well matched. Furthermore, a large b ( i ) {\displaystyle b(i)} implies that i {\displaystyle i} is badly matched to its neighbouring cluster. Thus an s ( i ) {\displaystyle s(i)} close to 1 means that the data is appropriately clustered. If s ( i ) {\displaystyle s(i)} is close to -1, then by the same logic we see that i {\displaystyle i} would be more appropriate if it was clustered in its neighbouring cluster. An s ( i ) {\displaystyle s(i)} near zero means that the datum is on the border of two natural clusters. The mean s ( i ) {\displaystyle s(i)} over all points of a cluster is a measure of how tightly grouped all the points in the cluster are. Thus the mean s ( i ) {\displaystyle s(i)} over all data of the entire dataset is a measure of how appropriately the data have been clustered. If there are too many or too few clusters, as may occur when a poor choice of k {\displaystyle k} is used in the clustering algorithm (e.g., k-means), some of the clusters will typically display much narrower silhouettes than the rest. Thus silhouette plots and means may be used to determine the natural number of clusters within a dataset. One can also increase the likelihood of the silhouette being maximized at the correct number of clusters by re-scaling the data using feature weights that are cluster specific. Kaufman et al. introduced the term silhouette coefficient for the maximum value of the mean s ( i ) {\displaystyle s(i)} over all data of the entire dataset, i.e., S C = max k s ~ ( k ) , {\displaystyle SC=\max _{k}{\tilde {s}}\left(k\right),} where s ~ ( k ) {\displaystyle {\tilde {s}}\left(k\right)} represents the mean s ( i ) {\displaystyle s(i)} over all data of the entire dataset for a specific number of clusters k {\displaystyle k} . The silhouette coefficient describes the best possible clustering possible for a given number of clusters, as measured by the highest average silhouette score for all points in the dataset. == Simplified and medoid silhouette == Computing the silhouette coefficient needs all O ( N 2 ) {\displaystyle {\mathcal {O}}(N^{2})} pairwise distances, making this evaluation much more costly than clustering with k-means. For a clustering with centers μ C I {\displaystyle \mu _{C_{I}}} for each cluster C I {\displaystyle C_{I}} , we can use the following simplified Silhouette for each point i ∈ C I {\displaystyle i\in C_{I}} instead, which can be computed using only O ( N k ) {\displaystyle {\mathcal {O}}(Nk)} distances: a ′ ( i ) = d ( i , μ C I ) {\displaystyle a'(i)=d(i,\mu _{C_{I}})} and b ′ ( i ) = min C J ≠ C I d ( i , μ C J ) {\displaystyle b'(i)=\min _{C_{J}\neq C_{I}}d(i,\mu _{C_{J}})} , which has the additional benefit that a ′ ( i ) {\displaystyle a'(i)} is always defined, then define accordingly the simplified silhouette and simplified silhouette coefficient s ′ ( i ) = b ′ ( i ) − a ′ ( i ) max { a ′ ( i ) , b ′ ( i ) } {\displaystyle s'(i)={\frac {b'(i)-a'(i)}{\max\{a'(i),b'(i)\}}}} S C ′ = max k 1 N ∑ i s ′ ( i ) {\displaystyle SC'=\max _{k}{\frac {1}{N}}\sum _{i}s'\left(i\right)} . If the cluster centers are medoids (as in k-medoids clustering) instead of arithmetic means (as in k-means clustering), this is also called the medoid-based silhouette or medoid silhouette. If every object is assigned to the nearest medoid (as in k-medoids clustering), we know that a ′ ( i ) ≤ b ′ ( i ) {\displaystyle a'(i)\leq b'(i)} , and hence s ′ ( i ) = b ′ ( i ) − a ′ ( i ) b ′ ( i ) = 1 − a ′ ( i ) b ′ ( i ) {\displaystyle s'(i)={\frac {b'(i)-a'(i)}{b'(i)}}=1-{\frac {a'(i)}{b'(i)}}} . == Silhouette clustering == Instead of using the average silhouette to evaluate a clustering obtained from, e.g., k-medoids or k-means, we can try to directly find a solution that maximizes the Silhouette. We do not have a closed form solution to maximize this, but it will usually be best to assign points to the nearest cluster as done by these methods. Van der Laan et al. proposed to adapt the standard algorithm for k-medoids, PAM, for this purpose and call this algorithm PAMSIL: Choose initial medoids by using PAM Compute the average silhouette of this initial solution For each pair of a medoid m and a non-medoid x swap m and x compute the average silhouette of the resulting solution remember the best swap un-swap m and x for the next iteration Perform the best swap and return to

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  Deluxe Paint

  Deluxe Paint, often referred to as DPaint, is a bitmap graphics editor created by Dan Silva for Electronic Arts and published for the then-new Amiga 1000 in November 1985. A series of updated versions followed, some of which were ported to other platforms. An MS-DOS release with support for the 256 color VGA standard became popular for creating pixel graphics in video games in the 1990s. Author Dan Silva previously worked on the Cut & Paste word processor (1984), also from Electronic Arts. == History == Deluxe Paint began as an in-house art development tool called Prism. As author Dan Silva added features to Prism, it was developed as a showcase product to coincide with the Amiga's debut in 1985. Upon release, it was quickly embraced by the Amiga community and became the de facto graphics (and later animation) editor for the platform. Amiga manufacturer Commodore International later commissioned EA to create version 4.5 AGA to bundle with the new Advanced Graphics Architecture chipset (A1200, A4000) capable Amigas. Version 5 was the last release after Commodore's bankruptcy in 1994. Early versions of Deluxe Paint were available in protected and non copy-protected versions, the latter retailing for a slightly higher price. The copy protection scheme was later dropped. Deluxe Paint was first in a series of products from the Electronic Arts Tools group—then later moved to the ICE (for Interactivity, Creativity, and Education) group—which included such Amiga programs as Deluxe Music Construction Set (preceded by Music Construction Set for the Apple II), Deluxe Video, and the Studio series of paint programs for the Mac. With the development of Deluxe Paint, EA introduced the ILBM and ANIM file format standards for graphics. While widely used on the Amiga, these formats never gained widespread end user acceptance on other platforms, but were heavily used by game development companies. Deluxe Paint was used by LucasArts to make graphics for their adventure games such as The Secret of Monkey Island, and the name of a particular filename used to store the main protagonist Guybrush Threepwood was probably at the origin of his peculiar name. One of the main artist developer of the game, Mark Ferrari, in an interview for The Making of Monkey Island 30th Anniversary Documentary remembers that "there was a pulldown menu in DPaint called brushes, so character sprites were referred to as brushes", and the male protagonist was simply "the guy.brush" until the artist Steve Purcell suggested to take the very name "Guybrush". The author Ron Gilbert remembers that the PC DOS version of the file was named "guybrush.bbm". == Versions == === Amiga === Deluxe Paint I was released in 1985. A major feature was animation by using color cycling. The Amiga natively supports indexed color, where a pixel's color value does not carry any RGB hue information but instead is an index to a color palette (a collection of unique color values). By adjusting the color value in the palette, all pixels with that palette value change simultaneously in the image or animation, creating cyclic movement in the image. In the Christmas demo files on the Deluxe Paint I disk, this kind of animation (which is toggled by pressing the tab key) is used to depict falling snowflakes, a blinking Christmas tree, and a roaring fire in the fireplace. In 1986, Deluxe Paint II was introduced, which added many convenient features such as pattern and gradient fill, which could be selected by right-clicking on a fill tool. An effects menu with e.g. perspective transformation was also added. The screen format could now be changed from a dedicated selection page. Deluxe Paint III appeared in 1989 and added support for Extra Halfbrite. New editing modes allowed one to stencil certain colors to protect them, so it is possible to e.g. paint a landscape from front to back, with the foreground protected by a stencil. A major new feature of Deluxe Paint III was the ability to create cel-like animation, and animbrushes (1MB of RAM is needed for animation). These let the user pick up a section of an animation as an "animbrush", which can then be placed onto the canvas while it animates. Deluxe Paint III was one of the first paint programs to support animbrushes. This is similar to copy and paste, except one can pick up more than one image. Deluxe Paint IV (introduced in 1991), which did not include Silva as the lead programmer, offered significant new features like non-bitplane-indexed Hold-and-Modify support for creating images with up to 4,096 colors. Animation support was improved by adding a light table, i.e. onion skinning, and AnimBrush morphing. The color mixer was now a HAM region at the bottom of the screen (instead of a floating window as before) and allowed mixing adjacent colors similar to a real palette. Deluxe Paint 4.5 AGA appeared the following year, addressing the stability issues and providing support for the new A1200 and A4000 AGA machines and a revamped screen mode interface. It appeared in both standalone and Commodore-bundled versions. The final release, Deluxe Paint V, in 1995, supported true 24-bit RGB images. However, using only the AGA native chipset, the 24-bit RGB color was only held in computer memory, the on-screen image was displayed in HAM8 (18-bit color). === Apple IIGS === DeluxePaint II for the Apple IIGS was developed by Brent Iverson and released in 1987. === MS-DOS === Deluxe Paint II for MS-DOS was released in 1988, It required MS-DOS 2.0 and 640 kB of RAM. It supports CGA, EGA, MCGA, VGA, Hercules and Tandy IBM PC-compatible graphic cards. Deluxe Paint II Enhanced was released in 1989, requiring MS-DOS 2.11 and 640 kB of RAM. It supports resolutions up to 800x600 pixels with 256 colors. Deluxe Paint II Enhanced 2.0, released in 1994, was the most successful MS-DOS version, and was compatible with PC Paintbrush PCX image files. The MS-DOS conversion was done by Brent Iverson with the enhanced features by Steve Shaw. It supports CGA, EGA, MCGA, VGA, Hercules, Tandy, and Amstrad video cards, as well as early Super VGA video cards enabling it to support up to 800 × 600 with 256 (from 262,144) colors and 1024 × 768 with 16 colors. The sister product Deluxe Paint Animation (only for 320×200 pixels and 256 colors) was widely used, especially in video game development. === Atari ST === Deluxe Paint ST was developed by ArtisTech Development, published by Electronic Arts, and was released in 1990. It supports the Atari STE 4096 color palette and animated graphics. Features advertised for the Atari ST version include 3D perspective, design your own fonts, mirror symmetry, multi-color airbrushing & animations, printing up to poster size, split-screen magnification with variable zoom, and working on animations (including multiple animations). == Workflow == "[" and "]" hotkeys step through the indexed palette, turning indexed-pixel-painting into a fast two-handed mouse+keys process, and the right mouse button paints with the background color. For example, transparency is obtained as simply as selecting a background color index (a single right click on the palette GUI to change). colors could be locked from editing by use of a stencil (a list of color indices whose pixels should not be altered in the image data) and simple color-cycling animations could be created using contiguous entries in the palette. This was easy to change the hue and tone of a section of the image by altering the corresponding colors in the palette. (The specific section needed to use a dedicated part of the palette for this technique to work.) Brushes can be cut from the background by using the box, freehand, or polygon selection tools. They can then be used in the same manner as any other brush or pen. This functionality is simpler to use than the "stamp" tool of Photoshop or Alpha Channels as provided in later programs. Brushes can be rotated and scaled, even in 3D. After a brush is selected, it appears attached to the mouse cursor, providing an exact preview of what will be drawn. This allows precise pixel positioning of brushes. Animations stored in IFF ANIM format are delta compressed making animations both smaller and faster to playback. == Reception == Compute! criticized the documentation of the first release of DeluxePaint as inadequate, but stated that "DeluxePaint is a visual arts program of immense scope and flexibility". In later versions the documentation was much improved; for instance DeluxePaint IV came with a 300-page manual. Deluxe Paint was a hit for EA. The main line of the series, particularly installments one to three, has won a total of at least nine awards from independent publications and organizations, including three Amiga-specific awards. Deluxe Paint III also won Commodore International's Enterprise and Vision award in 1990, becoming the first software to win the award, for what the company's judges believed to be best utilizing the Amiga's graphical capabilities. Deluxe Pai

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  Modern Hopfield network

  Modern Hopfield networks (also known as Dense Associative Memories) are generalizations of the classical Hopfield networks that break the linear scaling relationship between the number of input features and the number of stored memories. This is achieved by introducing stronger non-linearities (either in the energy function or neurons’ activation functions) leading to super-linear (even an exponential) memory storage capacity as a function of the number of feature neurons. The network still requires a sufficient number of hidden neurons. The key theoretical idea behind the modern Hopfield networks is to use an energy function and an update rule that is more sharply peaked around the stored memories in the space of neuron’s configurations compared to the classical Hopfield network. == Classical Hopfield networks == Hopfield networks are recurrent neural networks with dynamical trajectories converging to fixed point attractor states and described by an energy function. The state of each model neuron i {\textstyle i} is defined by a time-dependent variable V i {\displaystyle V_{i}} , which can be chosen to be either discrete or continuous. A complete model describes the mathematics of how the future state of activity of each neuron depends on the known present or previous activity of all the neurons. In the original Hopfield model of associative memory, the variables were binary, and the dynamics were described by a one-at-a-time update of the state of the neurons. An energy function quadratic in the V i {\displaystyle V_{i}} was defined, and the dynamics consisted of changing the activity of each single neuron i {\displaystyle i} only if doing so would lower the total energy of the system. This same idea was extended to the case of V i {\displaystyle V_{i}} being a continuous variable representing the output of neuron i {\displaystyle i} , and V i {\displaystyle V_{i}} being a monotonic function of an input current. The dynamics became expressed as a set of first-order differential equations for which the "energy" of the system always decreased. The energy in the continuous case has one term which is quadratic in the V i {\displaystyle V_{i}} (as in the binary model), and a second term which depends on the gain function (neuron's activation function). While having many desirable properties of associative memory, both of these classical systems suffer from a small memory storage capacity, which scales linearly with the number of input features. == Discrete variables == A simple example of the Modern Hopfield network can be written in terms of binary variables V i {\displaystyle V_{i}} that represent the active V i = + 1 {\displaystyle V_{i}=+1} and inactive V i = − 1 {\displaystyle V_{i}=-1} state of the model neuron i {\displaystyle i} . E = − ∑ μ = 1 N mem F ( ∑ i = 1 N f ξ μ i V i ) {\displaystyle E=-\sum \limits _{\mu =1}^{N_{\text{mem}}}F{\Big (}\sum \limits _{i=1}^{N_{f}}\xi _{\mu i}V_{i}{\Big )}} In this formula the weights ξ μ i {\textstyle \xi _{\mu i}} represent the matrix of memory vectors (index μ = 1... N mem {\displaystyle \mu =1...N_{\text{mem}}} enumerates different memories, and index i = 1... N f {\displaystyle i=1...N_{f}} enumerates the content of each memory corresponding to the i {\displaystyle i} -th feature neuron), and the function F ( x ) {\displaystyle F(x)} is a rapidly growing non-linear function. The update rule for individual neurons (in the asynchronous case) can be written in the following form V i ( t + 1 ) = sign ⁡ [ ∑ μ = 1 N mem ( F ( ξ μ i + ∑ j ≠ i ξ μ j V j ( t ) ) − F ( − ξ μ i + ∑ j ≠ i ξ μ j V j ( t ) ) ) ] {\displaystyle V_{i}^{(t+1)}=\operatorname {sign} {\bigg [}\sum \limits _{\mu =1}^{N_{\text{mem}}}{\bigg (}F{\Big (}\xi _{\mu i}+\sum \limits _{j\neq i}\xi _{\mu j}V_{j}^{(t)}{\Big )}-F{\Big (}-\xi _{\mu i}+\sum \limits _{j\neq i}\xi _{\mu j}V_{j}^{(t)}{\Big )}{\bigg )}{\bigg ]}} which states that in order to calculate the updated state of the i {\textstyle i} -th neuron the network compares two energies: the energy of the network with the i {\displaystyle i} -th neuron in the ON state and the energy of the network with the i {\displaystyle i} -th neuron in the OFF state, given the states of the remaining neuron. The updated state of the i {\displaystyle i} -th neuron selects the state that has the lowest of the two energies. In the limiting case when the non-linear energy function is quadratic F ( x ) = x 2 {\displaystyle F(x)=x^{2}} these equations reduce to the familiar energy function and the update rule for the classical binary Hopfield network. The memory storage capacity of these networks can be calculated for random binary patterns. For the power energy function F ( x ) = x n {\displaystyle F(x)=x^{n}} the maximal number of memories that can be stored and retrieved from this network without errors is given by N mem max ≈ 1 2 ( 2 n − 3 ) ! ! N f n − 1 ln ⁡ ( N f ) {\displaystyle N_{\text{mem}}^{\max }\approx {\frac {1}{2(2n-3)!!}}{\frac {N_{f}^{n-1}}{\ln(N_{f})}}} For an exponential energy function F ( x ) = e x {\textstyle F(x)=e^{x}} the memory storage capacity is exponential in the number of feature neurons N mem max ≈ 2 N f / 2 {\displaystyle N_{\text{mem}}^{\max }\approx 2^{N_{f}/2}} == Continuous variables == Modern Hopfield networks or Dense Associative Memories can be best understood in continuous variables and continuous time. Consider the network architecture, shown in Fig.1, and the equations for the neurons' state evolutionwhere the currents of the feature neurons are denoted by x i {\textstyle x_{i}} , and the currents of the memory neurons are denoted by h μ {\displaystyle h_{\mu }} ( h {\displaystyle h} stands for hidden neurons). There are no synaptic connections among the feature neurons or the memory neurons. A matrix ξ μ i {\displaystyle \xi _{\mu i}} denotes the strength of synapses from a feature neuron i {\displaystyle i} to the memory neuron μ {\displaystyle \mu } . The synapses are assumed to be symmetric, so that the same value characterizes a different physical synapse from the memory neuron μ {\displaystyle \mu } to the feature neuron i {\displaystyle i} . The outputs of the memory neurons and the feature neurons are denoted by f μ {\displaystyle f_{\mu }} and g i {\displaystyle g_{i}} , which are non-linear functions of the corresponding currents. In general these outputs can depend on the currents of all the neurons in that layer so that f μ = f ( { h μ } ) {\displaystyle f_{\mu }=f(\{h_{\mu }\})} and g i = g ( { x i } ) {\textstyle g_{i}=g(\{x_{i}\})} . It is convenient to define these activation function as derivatives of the Lagrangian functions for the two groups of neuronsThis way the specific form of the equations for neuron's states is completely defined once the Lagrangian functions are specified. Finally, the time constants for the two groups of neurons are denoted by τ f {\displaystyle \tau _{f}} and τ h {\displaystyle \tau _{h}} , I i {\displaystyle I_{i}} is the input current to the network that can be driven by the presented data. General systems of non-linear differential equations can have many complicated behaviors that can depend on the choice of the non-linearities and the initial conditions. For Hopfield networks, however, this is not the case - the dynamical trajectories always converge to a fixed point attractor state. This property is achieved because these equations are specifically engineered so that they have an underlying energy function The terms grouped into square brackets represent a Legendre transform of the Lagrangian function with respect to the states of the neurons. If the Hessian matrices of the Lagrangian functions are positive semi-definite, the energy function is guaranteed to decrease on the dynamical trajectory This property makes it possible to prove that the system of dynamical equations describing temporal evolution of neurons' activities will eventually reach a fixed point attractor state. In certain situations one can assume that the dynamics of hidden neurons equilibrates at a much faster time scale compared to the feature neurons, τ h ≪ τ f {\textstyle \tau _{h}\ll \tau _{f}} . In this case the steady state solution of the second equation in the system (1) can be used to express the currents of the hidden units through the outputs of the feature neurons. This makes it possible to reduce the general theory (1) to an effective theory for feature neurons only. The resulting effective update rules and the energies for various common choices of the Lagrangian functions are shown in Fig.2. In the case of log-sum-exponential Lagrangian function the update rule (if applied once) for the states of the feature neurons is the attention mechanism commonly used in many modern AI systems (see Ref. for the derivation of this result from the continuous time formulation). == Relationship to classical Hopfield network with continuous variables == Classical formulation of continuous Hopfield networks can be understood as a

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  Ordination (statistics)

  Ordination or gradient analysis, in multivariate analysis, is a method complementary to data clustering, and used mainly in exploratory data analysis (rather than in hypothesis testing). In contrast to cluster analysis, ordination orders quantities in a (usually lower-dimensional) latent space. In the ordination space, quantities that are near each other share attributes (i.e., are similar to some degree), and dissimilar objects are farther from each other. Such relationships between the objects, on each of several axes or latent variables, are then characterized numerically and/or graphically in a biplot. The first ordination method, principal components analysis, was suggested by Karl Pearson in 1901. == Methods == Ordination methods can broadly be categorized in eigenvector-, algorithm-, or model-based methods. Many classical ordination techniques, including principal components analysis, correspondence analysis (CA) and its derivatives (detrended correspondence analysis, canonical correspondence analysis, and redundancy analysis, belong to the first group). The second group includes some distance-based methods such as non-metric multidimensional scaling, and machine learning methods such as T-distributed stochastic neighbor embedding and nonlinear dimensionality reduction. The third group includes model-based ordination methods, which can be considered as multivariate extensions of Generalized Linear Models. Model-based ordination methods are more flexible in their application than classical ordination methods, so that it is for example possible to include random-effects. Unlike in the aforementioned two groups, there is no (implicit or explicit) distance measure in the ordination. Instead, a distribution needs to be specified for the responses as is typical for statistical models. These and other assumptions, such as the assumed mean-variance relationship, can be validated with the use of residual diagnostics, unlike in other ordination methods. == Applications == Ordination can be used on the analysis of any set of multivariate objects. It is frequently used in several environmental or ecological sciences, particularly plant community ecology. It is also used in genetics and systems biology for microarray data analysis and in psychometrics.

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  U-matrix

  The U-matrix (unified distance matrix) is a representation of a self-organizing map (SOM) where the Euclidean distance between the codebook vectors of neighboring neurons is depicted in a grayscale image. This image is used to visualize the data in a high-dimensional space using a 2D image. == Construction procedure == Once the SOM is trained using the input data, the final map is not expected to have any twists. If the map is twist-free, the distance between the codebook vectors of neighboring neurons gives an approximation of the distance between different parts of the underlying data. When such distances are depicted in a grayscale image, light colors depict closely spaced node codebook vectors and darker colors indicate more widely separated node codebook vectors. Thus, groups of light colors can be considered as clusters, and the dark parts as the boundaries between the clusters. This representation can help to visualize the clusters in the high-dimensional spaces, or to automatically recognize them using relatively simple image processing techniques.

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  Scenery generator

  A scenery generator (or terrain generator) is a software used to create landscape images, 3D models, and animations. These programs often use procedural generation to generate the landscapes, or sometimes created and rendered by a 3D artist. These programs are often used in video games or movies. Basic elements of landscapes created by scenery generators include terrain, water, foliage, and clouds. The process for basic random generation uses a diamond square algorithm. == Common features == Most scenery generators can create basic heightmaps to simulate the variation of elevation in basic terrain. Common techniques include Simplex noise, fractals, or the diamond-square algorithm, which can generate 2-dimensional heightmaps. A version of scenery generator can be very simplistic. Using a diamond-square algorithm with some extra steps involving fractals, an algorithm for random generation of terrain can be made with only 120 lines of code. The program in example takes a grid and then divides the grid repeatedly. Each smaller grid is then split into squares and diamonds and the algorithm then makes the randomized terrain for each square and diamond. Most programs for creating landscapes also allow for adjustment and editing of the landscape. For example, World Creator allows for terrain sculpting, which uses a similar brush system as Photoshop, and allows for additional terrain enhancement with its procedural techniques such as erosion, sediments, and more. Other tools in the World Creator program include terrain stamping, which allows you to import elevation maps and use them as a base. The programs tend to also allow for additional placement of rocks, trees, etc. These can be done procedurally or by hand depending on the program. Typically the models used for the placement objects are the same as to lessen the amount of work that would be done if the user was to create a multitude of different trees. The terrain generated the computer does a generation of multifractals then integrates them until finally rendering them onto the screen. These techniques are typically done “on-the-fly” which typically for a 128 × 128 resolution terrain would mean 1.5 seconds on a CPU from the early 1990s. == Applications == Scenery generators are commonly used in movies, animations, 3D rendering, and video games. For example, Industrial Light & Magic used E-on Vue to create the fictional environments for Pirates of the Caribbean: Dead Man's Chest. In such live-action cases, a 3D model of the generated environment is rendered and blended with live-action footage. Scenery generated by the software may also be used to create completely computer-generated scenes. In the case of animated movies such as Kung Fu Panda, the raw generation is assisted by hand-painting to accentuate subtle details. Environmental elements not commonly associated with landscapes, such as ocean waves, have also been handled by the software. Scenery generation is used in most 3D based video-games. These typically use either custom or purchased engines that contain their own scenery generators. For some games they tend to use a procedurally generated terrain. These typically use a form of height mapping and use of Perlin noise. This will create a grid that with one point in a 2D coordinate will create the same heightmap as it is pseudorandom, meaning it will result in the same output with the same input. This can then easily be translated into the product 3D image. These can then be changed from the editor tools in most engines if the terrain will be custom built. With recent developments neural networks can be built to create or texture the terrain based on previously suggested artwork or heightmap data. These would be generated using algorithms that have been able to identify images and similarities between them. With the info the machine can take other heightmaps and render a very similar looking image to the style image. This can be used to create similar images in example a Studio Ghibli or Van Gogh art-style. == Software == Most game engines, whether custom or proprietary, will have terrain generation built in. Some terrain generator programs include, Terragen, which can create terrain, water, atmosphere and lighting; L3DT, which provides similar functions to Terragen, and has a 2048 × 2048 resolution limit; and World Creator, which can create terrain, and is fully GPU powered. === List of 3D terrain generation software ===

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  VITAL (machine learning software)

  VITAL (Validating Investment Tool for Advancing Life Sciences) was a Board Management Software machine learning proprietary software developed by Aging Analytics, a company registered in Bristol (England) and dissolved in 2017. Andrew Garazha (the firm's Senior Analyst) declared that the project aimed "through iterative releases and updates to create a piece of software capable of making autonomous investment decisions." According to Nick Dyer-Witheford, VITAL 1.0 was a "basic algorithm". On 13 May 2014, Deep Knowledge Ventures, a Hong Kong venture capital firm, claimed to have appointed VITAL to its board of directors in order to prove that artificial intelligence could be an instrument for investment decision-making. The announcement received great press coverage despite the fact commentators consider this a publicity stunt. Fortune reported in 2019 that VITAL is no longer used. == Criticism == Academics and journalists viewed VITAL's board appointment with skepticism. University of Sheffield computer science professor Noel Sharkey called it "a publicity hype". Michael Osborne, a University of Oxford associate professor in machine learning, found it is "a gimmick to call that an actual board member". Simon Sharwood of The Register, wrote there is "a strong whiff of stunt and/or promotion about this". In a 2019 speech, the Chief Scientist of Australia, Alan Finkel, commented, "At the time, most of us probably dismissed Vital as a PR exercise. I admit, I used her story three years ago to get a laugh in one of my speeches." Florian Möslein, a law professor at the University of Marburg, wrote in 2018 that "Vital has widely been acknowledged as the 'world's first artificial intelligence company director'". Vice journalist Jason Koebler suggested that the software did not have any article intelligence capabilities and concluded "VITAL can’t talk, and it can’t hear, and it can’t be a real, functional executive of a company." Sharwood of The Register noted that because VITAL was not a natural person, it could not be a board member under Hong Kong's corporate governance laws. However, in a 2017 interview to The Nikkei, Dmitry Kaminskiy, managing partner of Deep Knowledge Ventures, stated that VITAL had observer status on the board and no voting rights. University of Sheffield computer science professor Noel Sharkey said of VITAL, "On first sight, it looks like a futuristic idea but on reflection it is really a little bit of publicity hype." Vice journalist Jason Koebler said "this is a gimmick" and said "There is literally nothing to suggest that VITAL has any sort of capabilities beyond any other proprietary analysis software". Michael Osborne, a University of Oxford associate professor in machine learning, found VITAL's appointment to be noncredible, saying it is "a bit of a gimmick to call that an actual board member". Osborne said that a core duty of board members to converse with each other, which the algorithm is incapable of doing, so its more likely functionality is to serve as a springboard for conversation among other board members. In a 2019 speech, the Chief Scientist of Australia, Alan Finkel, commented, "At the time, most of us probably dismissed Vital as a PR exercise. I admit, I used her story three years ago to get a laugh in one of my speeches." == Machine intelligence as board member == VITAL was created by a group of programmers employed by Aging Analytics According to Andrew Garazh, Aging Analytics Senior Analyst, VITAL was not a machine learning algorithm as the necessary datasets on investment rounds, intellectual property and clinical trial outcomes are generally not disclosed. Rather, VITAL used fuzzy logic based on 50 parameters to assess risk factors. Aging Analytics licensed the software to Deep Knowledge Ventures. It was used to help the human board members of Deep Knowledge Venture make investment decisions in biotechnology companies. For instance, it supported investments in Insilico Medicine, which creates ways for computers to help find drugs in research into aging. VITAL also supported investing in Pathway Pharmaceuticals, which uses the OncoFinder algorithm to choose and appraise cancer treatments. According to Dmitry Kaminskiy, managing partner of Deep Knowledge Ventures, the motivation for using VITAL was the large number of failed investments in the biotechnology sector and the desire to avoid investing in companies likely to fail. == Ethical and legal implications == Scholars addressed questions around the safety, privacy, accountability transparency and bias in algorithms. Writing in the philosophical journal Multitudes, the academic Ariel Kyrou raised questions about the consequences of a mistake made by an algorithm recommending a dangerous investment. He raised the hypothetical where VITAL was able to persuade the board to invest in a startup that had the facade of doing research into treatment for age-associated ills, but in actuality was run by terrorists who were raising funds. Kyrou raised a series of questions about who society would fault for VITAL's mistake. As the owner of VITAL, should Deep Knowledge Ventures be held accountable, or rather should the companies that supplied data to VITAL or the people who created VITAL be held liable? Simon Sharwood of The Register wrote that because the appointment of a software program to the board directors is not legally feasible in Hong Kong, there is "a strong whiff of stunt and/or promotion about this". Quoting a Thomson Reuters website describing Hong Kong legislation related to corporate governance, Sharwood pointed out that in Hong Kong "the board comprises all of the directors of the company" and "a director must normally be a natural person, except that a private company may have a body corporate as its director if the company is not a member of a listed group." He concluded that since VITAL cannot be considered a "natural person", it is merely a "cosmetic" appointment to the board and that "this software is no more a Board member than Caligula's horse was a senator". Sharwood further argued that corporations frequently purchase directors and officers liability insurance but that it would be practically impossible to get such insurance for VITAL. Sharwood also wrote that were VITAL to be hacked, any misinformation it outputs could be considered "false and misleading communications". In the book Research Handbook on the Law of Artificial Intelligence, Florian Mölein wrote that VITAL could not become a director as defined in Hong Kong's corporate laws, so the other directors just were approaching it as "a member of [the] board with observer status". Lin Shaowei raised concerns in a Journal of East China University of Political Science and Law article about how the software's appearance inspired a complex question about the relationship between corporate law and artificial intelligence. VITAL could be considered either a board director who has voting rights or an observer who does not. Lin said either choice raised questions about whether VITAL is subject to corporate law and who would be held accountable if VITAL recommends a choice that turns out to be damaging to the company. David Theo Goldberg in the Critical Times, a peer reviewed journal in Critical Global Theory, argues that VITAL processed a dataset to predict the most remunerative investment opportunities. Drawing his analysis on an article from Business Insider, Goldberg describes VITAL's decision-making predictiveness based "on surface pattern recognition and the identification of regularities and/or irregularities". In other words, Goldberg asserts that "the normativity of the surface" explains algorithmic knowledge of a "product" like VITAL. In Homo Deus, Yuval Noah Harari mentions VITAL as an example of the future risks that humankind faces. Harari argues that the human mind is being replaced by a world in which algorithms and data make the decisions. Specifically, it is argued that "as algorithms push humans out of the job market," executive boards driven by artificial intelligence are more likely to give priority to algorithms over the humans.

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  Wake-sleep algorithm

  The wake-sleep algorithm is an unsupervised learning algorithm for deep generative models, especially Helmholtz Machines. The algorithm is similar to the expectation-maximization algorithm, and optimizes the model likelihood for observed data. The name of the algorithm derives from its use of two learning phases, the “wake” phase and the “sleep” phase, which are performed alternately. It can be conceived as a model for learning in the brain, but is also being applied for machine learning. == Description == The goal of the wake-sleep algorithm is to find a hierarchical representation of observed data. In a graphical representation of the algorithm, data is applied to the algorithm at the bottom, while higher layers form gradually more abstract representations. Between each pair of layers are two sets of weights: Recognition weights, which define how representations are inferred from data, and generative weights, which define how these representations relate to data. == Training == Training consists of two phases – the “wake” phase and the “sleep” phase. It has been proven that this learning algorithm is convergent. === The "wake" phase === Neurons are fired by recognition connections (from what would be input to what would be output). Generative connections (leading from outputs to inputs) are then modified to increase probability that they would recreate the correct activity in the layer below – closer to actual data from sensory input. === The "sleep" phase === The process is reversed in the “sleep” phase – neurons are fired by generative connections while recognition connections are being modified to increase probability that they would recreate the correct activity in the layer above – further to actual data from sensory input. == Extensions == Since the recognition network is limited in its flexibility, it might not be able to approximate the posterior distribution of latent variables well. To better approximate the posterior distribution, it is possible to employ importance sampling, with the recognition network as the proposal distribution. This improved approximation of the posterior distribution also improves the overall performance of the model.

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  Density-based clustering validation

  Density-Based Clustering Validation (DBCV) is a metric designed to assess the quality of clustering solutions, particularly for density-based clustering algorithms like DBSCAN, Mean shift, and OPTICS. This metric is particularly suited for identifying concave and nested clusters, where traditional metrics such as the Silhouette coefficient, Davies–Bouldin index, or Calinski–Harabasz index often struggle to provide meaningful evaluations. Unlike traditional validation measures, which often rely on compact and well-separated clusters, DBCV index evaluates how well clusters are defined in terms of local density variations and structural coherence. This metric was introduced in 2014 by David Moulavi and colleagues in their work. It utilizes density connectivity principles to quantify clustering structures, making it especially effective at detecting arbitrarily shaped clusters in concave datasets, where traditional metrics may be less reliable. The DBCV index has been employed for clustering analysis in bioinformatics, ecology, techno-economy, and health informatics , as well as in numerous other fields. == Definition == DBCV index evaluates clustering structures by analyzing the relationships between data points within and across clusters. Given a dataset X = x 1 , x 2 , . . . , x n {\displaystyle X={x_{1},x_{2},...,x_{n}}} , a density-based algorithm partitions it into K clusters C 1 , C 2 , . . . , C K {\displaystyle {C_{1},C_{2},...,C_{K}}} . Each point x i {\displaystyle x_{i}} belongs to a specific cluster, denoted as C c l u s t e r ( x i ) {\displaystyle C_{cluster(x_{i})}} A key concept in DBCV index is the notion of density-connected paths. Two points within the same cluster are considered density-connected if there exists a sequence of intermediate points linking them, where each consecutive pair meets a predefined density criterion. The density-based distance between two points is determined by identifying the optimal path that minimizes the maximum local reachability distance along its trajectory. DBCV index extends the Silhouette coefficient by redefining cluster cohesion and separation using density-based distances: Within-cluster density distance measures how closely a point is related to other members of its cluster: a i = 1 | C c l u s t e r ( x i ) | − 1 ∑ x j ∈ C c l u s t e r ( x i ) , y ≠ x d d e n s i t y ( x j , x i ) {\displaystyle a_{i}={\frac {1}{|C_{cluster(x_{i})}|-1}}\sum _{x_{j}\in C_{cluster(x_{i})},y\neq x}d_{density}(x_{j},x_{i})} Nearest-cluster density distance quantifies how far a point is from the closest external cluster: b i = min C ≠ C cluster ( x i ) C ∈ { C 1 , … , C K } ( 1 | C | ∑ x j ∈ C d density ( x i , x j ) ) . {\displaystyle b_{i}=\min _{C\neq C_{{\text{cluster}}(x_{i})} \atop C\in \{C_{1},\dots ,C_{K}\}}\left({\frac {1}{|C|}}\sum _{x_{j}\in C}d_{\text{density}}(x_{i},x_{j})\right).} Using these measures, the DBCV index is computed as: D B C V = 1 n ∑ i = 1 n b i − a i max ( a i , b i ) {\displaystyle DBCV={\frac {1}{n}}\sum _{i=1}^{n}{\frac {b_{i}-a_{i}}{\max(a_{i},b_{i})}}} == Explanation == DBCV index values range between −1 and +1: +1: Strongly cohesive and well-separated clusters. 0: Ambiguous clustering structure. −1: Poorly formed clusters or incorrect assignments. By leveraging density-based distances instead of traditional Euclidean measures, DBCV index provides a more robust evaluation of clustering performance in datasets with irregular or non-spherical distributions.

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  Diia

  Diia (Ukrainian: Дія [ˈd⁽ʲ⁾ijɐ] , lit. 'Action'; also an acronym for Держава і Я, Derzhava i Ya, IPA: [derˈʒɑwɐ i ˈjɑ], lit. 'State and Me') is a mobile app, a web portal and a brand of e-governance in Ukraine. Launched in 2020, the Diia app allows Ukrainian citizens to use digital documents on their smartphones instead of physical ones for identification and sharing purposes. The Diia portal allows access to over 130 government services. Eventually, the government plans to make all kinds of state-person interactions available through Diia. Diia was built in partnership with the United States and is poised to be shared with other countries. On the sidelines of the 2023 World Economic Forum in Davos, USAID Administrator Samantha Power said the US hopes to replicate the success of Diia in other countries. == History == Diia was first presented on September 27, 2019, by the Ministry of Digital Transformation of Ukraine as a brand of the State in a Smartphone project. Vice Prime Minister and Minister of Digital Transformation Mykhailo Fedorov announced the creation of a mobile app and a web portal that would unite in a single place all the services provided by the state to citizens and businesses. On February 6, 2020, the mobile app Diia was officially launched. During the presentation, Ukrainian President Volodymyr Zelensky said that 9 million Ukrainians now have access to their driver's license and car registration documents on their phones, while Prime Minister Oleksiy Honcharuk called the implementation of the State in a Smartphone project a priority for the government. In April 2020, the Ukrainian government approved a resolution for experimental usage of digital ID-cards and passports which would be issued to all Ukrainians via the Diia. On October 5, 2020, during the Diia Summit, the government presented a first major update of the app and web portal branded "Diia 2.0". More types of documents were added to the app as well as the ability to share documents with others via a single tap on a push-message. The web portal in turn expanded the number of available services to 27, including the ability to register a private limited company in half an hour. President Zelensky who opened the summit, announced that in 2021 Ukraine will enter the "paper less" mode by prohibiting civil servants from demanding paper documents. By the end of 2020, the app had more than six million users, while the portal had 50 available services. In March 2021, the Ukrainian parliament adopted a bill equating digital identity documents with their physical analogues. Starting on August 23, Ukrainian citizens can use digital ID-cards and passports for all purposes while in Ukraine. According to Minister of Digital Transformation Mykhailo Fedorov, Ukraine will become the first country in the world where digital identity documents are considered legally equivalent to ordinary ones. In September 2024, Diia launched an online marriage registration service, which can be beneficial especially for military personnel who spend much time on the frontline separated from their partners. In October 2024, Diia's online marriage service appeared in Time's Inventions of the 2024 list. In the first month of its operations over 1.1 million Ukrainians tried to make proposals using the technology, and 435 couples got married. == Benefits and challenges == The first and most obvious benefit is the convenience of such a platform. Citizens can have many documents on their smartphones at once, without concern about losing or damaging them. Whenever needed, they can just open an app on their smartphones and show/check the document they need. The idea is that Diia will help cut the bureaucracy associated with public services, which in turn will help fight corruption and increase government savings. Fewer people are needed to be employed in the public sector and fewer human to human interactions are supposed to happen. With the start of the program, already 10% of government employees were reduced, which contributes to hundreds of millions of dollars in savings, but besides this, the initiative also improves the speed, efficiency, and transparency of government services. In addition, the digitalization of the government sector helps to develop the whole IT industry in the country, people become more digitally aware and educated, this affects other sectors as well, increasing the spread of digital infrastructure and expediting the speed of overall digitalization. The UN E-government Development Index, which assesses the capabilities of governments to integrate its functions electronically, such as the use of internet and mobile devices, ranked Ukraine 69th in 193 countries surveyed in 2020. Despite its low ranking in the e-government development index, Ukraine made a big jump on the e-participation index, which they ranked 43rd out of 193 countries from 0.66 in 2018 to 0.81 in 2020 (un.org, 2020), suggesting that the government and its citizens are adapting the IT-based government functions. The main goal of e-government according to Perez-Morote et.al. (2020) is to have accountability and transparency among the countries involved. But to do so, there are several challenges that a country should assess first prior to implementing e-government. In the research written by Heeks (2001), the author identified 2 main challenges that countries face in the development of e-government, first is the strategic challenge which involves the preparedness (e-readiness) of the entire government system for electronic transformation, and second challenge is the tactical challenge where the government must design (e-governance design) a system where it can be understood by every user, it's important that the information that needs to be communicated to the consumers is received clearly. For the first challenge (e-readiness), Ukraine had an internet penetration rate of 76% in 2020 and is expected to grow to 82%, it is important that consumers have the internet access for it to enable the consumers to utilize the service. Another factor is the readiness of its institutional infrastructure, which means that the government has its own organization which is solely focused on implementing the e-government project. In the case of Ukraine, the e-governance team is led by Oleksandr Ryzhenko, and the country's e-governance initiative is even further strengthened by ensuring that the data and legal infrastructure are already prepared. Ukraine has done this by modernizing their legislation that is more appropriate in the digital service, and the data exchange solution used by Ukraine is called Trembita. The human infrastructure is also being updated, as competent individuals must be the one doing the task, hence, EGOV4UKRAINE was launched, this aims to get IT developers for developing a system for administrative services. These efforts by the Ukrainian government did not go unnoticed, and they received an award from the e-Governance Academy as "partner of the year 2017". For the second challenge, which deals with the system design, the success of Ukraine can be seen on the latest data of UNDP, where it shows a high increase in the E-participation index. In 2018, Ukraine ranked 75th it ranked 46th in 2020 (un.org, 2020). Despite visible success, the implementation of the e-government was accompanied by problems. Data leakage became the main one. In May 2020, the data of 26 million driver's licenses appeared in the public domain on the Internet. The Ukrainian government said the Diia app was not linked to a data breach, but it is impossible to say for certain. Any storage of official documents in electronic format is associated with the risk of their leakage. In addition, the Diia application still has data protection issues, as the required protection system has not been implemented. This is also compounded by the country's weak data protection legal regime. In addition, since 2023, Ukrainians are able to register their cars with this app. Issued license plates are not using regional codes, but they are using special codes starting with DI or PD. == Diia City == In May 2020, the government presented Diia City headed by Oleksandr Borniakov, a large-scale project which would establish a virtual model of a free economic zone for representatives of the creative economy. It would provide for special digital residency with a particular taxation regime, intellectual property protection and simplified regulations. Diia City concurrently imposes certain constraints on contracts involving individual entrepreneurs (FOPs). It also offers the benefit of tax rebates. Diia City garners endorsement from the Ukrainian government, believing it will support the country's position in the IT market. As of July 30, 2023, the program had more than 600 residents, including companies like iGama, Avenga, SBRobotiks, and Intellectsoft.

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  Ground truth

  Ground truth is information that is known to be real or true, provided by direct observation and measurement (i.e. empirical evidence) as opposed to information provided by inference. The term ground truth appeared in remote sensing literature as early as 1972, when NASA described it as essential "data about ... materials on the earth's surface" used to calibrate measurements. It was later adopted by the statistical modeling and machine learning communities. == Etymology == The Oxford English Dictionary (s.v. ground truth) records the use of the word Groundtruth in the sense of 'fundamental truth' from Henry Ellison's poem "The Siberian Exile's Tale", published in 1833. == Usage == The term "ground truth" can be used as a noun, adjective, and verb. Noun: "ground truth" (no hyphen). Example: "The ground truth is essential for training accurate models." Adjective: "ground-truth" (hyphenated compound adjective). Example: "We need to use ground-truth data to validate the model." Verb: "to ground-truth" or "to groundtruth" (compound verb,). Example: "We need to ground-truth the results to ensure their accuracy." == Statistics and machine learning == In statistics and machine learning, ground truth is the ideal expected result, used in statistical models to prove or disprove research hypotheses. "Ground truthing" is the process of gathering the good data for this test. Ground truth is typically included in labeled data. In machine learning, "ground truth" is not necessarily objectively correct or true. For example, in training AI models or relevance rankers, it may be a set of judgments made by people or inferred from user behavior, which may depend on context. For example, in Bayesian spam filtering, a supervised learning system is typically trained by examples labeled as spam and non-spam. Although these labels may be subjective or inaccurate, they are considered ground truth. True ground truth in machine learning is objective data. For example, suppose we are testing a stereo vision system to see how well it can estimate 3D positions. A calibrated laser rangefinder may provide accurate distances as ground truth. == Remote sensing == In remote sensing, "ground truth" refers to information collected at the imaged location. Ground truth allows image data to be related to real features and materials on the ground. The collection of ground truth data enables calibration of remote-sensing data, and aids in the interpretation and analysis of what is being sensed. Examples include cartography, meteorology, analysis of aerial photographs, satellite imagery and other techniques in which data are gathered at a distance. More specifically, ground truth may refer to a process in which "pixels" on a satellite image are compared to what is imaged (at the time of capture) in order to verify the contents of the "pixels" in the image (noting that the concept of "pixel" is imaging-system-dependent). In the case of a classified image, supervised classification can help to determine the accuracy of the classification by the remote sensing system which can minimize error in the classification. Ground truth is usually done on site, correlating what is known with surface observations and measurements of various properties of the features of the ground resolution cells under study in the remotely sensed digital image. The process also involves taking geographic coordinates of the ground resolution cell with GPS technology and comparing those with the coordinates of the "pixel" being studied provided by the remote sensing software to understand and analyze the location errors and how it may affect a particular study. Ground truth is important in the initial supervised classification of an image. When the identity and location of land cover types are known through a combination of field work, maps, and personal experience these areas are known as training sites. The spectral characteristics of these areas are used to train the remote sensing software using decision rules for classifying the rest of the image. These decision rules such as Maximum Likelihood Classification, Parallelopiped Classification, and Minimum Distance Classification offer different techniques to classify an image. Additional ground truth sites allow the remote sensor to establish an error matrix that validates the accuracy of the classification method used. Different classification methods may have different percentages of error for a given classification project. It is important that the remote sensor chooses a classification method that works best with the number of classifications used while providing the least amount of error. Ground truth also helps with atmospheric correction. Since images from satellites have to pass through the atmosphere, they can get distorted because of absorption in the atmosphere. So ground truth can help fully identify objects in satellite photos. === Errors of commission === An example of an error of commission is when a pixel reports the presence of a feature (such a tree) that, in reality, is absent (no tree is actually present). Ground truthing ensures that the error matrices have a higher accuracy percentage than would be the case if no pixels were ground-truthed. This value is the complement of the user's accuracy, i.e. Commission Error = 1 - user's accuracy. === Errors of omission === An example of an error of omission is when pixels of a certain type, for example, maple trees, are not classified as maple trees. The process of ground-truthing helps to ensure that the pixel is classified correctly and the error matrices are more accurate. This value is the complement of the producer's accuracy, i.e. Omission Error = 1 - producer's accuracy == Geographical information systems == In GIS the spatial data is modeled as field (like in remote sensing raster images) or as object (like in vectorial map representation). They are modeled from the real world (also named geographical reality), typically by a cartographic process (illustrated). Geographic information systems such as GIS, GPS, and GNSS, have become so widespread that the term "ground truth" has taken on special meaning in that context. If the location coordinates returned by a location method such as GPS are an estimate of a location, then the "ground truth" is the actual location on Earth. A smart phone might return a set of estimated location coordinates such as 43.87870, −103.45901. The ground truth being estimated by those coordinates is the tip of George Washington's nose on Mount Rushmore. The accuracy of the estimate is the maximum distance between the location coordinates and the ground truth. We could say in this case that the estimate accuracy is 10 meters, meaning that the point on Earth represented by the location coordinates is thought to be within 10 meters of George's nose—the ground truth. In slang, the coordinates indicate where we think George Washington's nose is located, and the ground truth is where it really is. In practice a smart phone or hand-held GPS unit is routinely able to estimate the ground truth within 6–10 meters. Specialized instruments can reduce GPS measurement error to under a centimeter. == Military usage == US military slang uses "ground truth" to refer to the facts comprising a tactical situation—as opposed to intelligence reports, mission plans, and other descriptions reflecting the conative or policy-based projections of the industrial·military complex. The term appears in the title of the Iraq War documentary film The Ground Truth (2006), and also in military publications, for example Stars and Stripes saying: "Stripes decided to figure out what the ground truth was in Iraq."

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  Local tangent space alignment

  Local tangent space alignment (LTSA) is a method for manifold learning, which can efficiently learn a nonlinear embedding into low-dimensional coordinates from high-dimensional data, and can also reconstruct high-dimensional coordinates from embedding coordinates. It is based on the intuition that when a manifold is correctly unfolded, all of the tangent hyperplanes to the manifold will become aligned. It begins by computing the k-nearest neighbors of every point. It computes the tangent space at every point by computing the d-first principal components in each local neighborhood. It then optimizes to find an embedding that aligns the tangent spaces, but it ignores the label information conveyed by data samples, and thus can not be used for classification directly.

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