The invention concerns four methodologies regarding the unsupervised clustering of a set of observations in multidimensional space, considering a defined number of clusters. The invention comprises a special procedure for calculating the minimum distance of a given point to a set of points in a multidimensional space, the main component of the first methodology.
Please help me write a proper abstract based on the patent claims. CLAIM: 1. A method for optimizing the clustering of a given set of 171 observations sj, j≦1, . . . , m, each represented by n components, into a specified number of groups q, represented by a set of centroids xi, i=1, . . . , q, following a criterion of minimizing the sum of the values of a monotonic increasing function f applied to each argument zj, defined as the shortest distance from each generic observation sj to the nearest centroid, according to a general optimization formulation: minimize   F  ( x ) = ∑ j = 1 m  f  ( z j ) subject   to  :   z j = min x i ∈ X   s j - x i  , j = 1 , …  , m , where x=(x1, x2, . . . , x4)∈ Rnq and f(zj) is an arbitrary monotonic increasing function f applied to the distance zi measured following one of four different metrics, namely Euclidian norm, Minkowski norm, Manhattan norm and Chebychev norm, comprising the steps of: performing an initialization step; and performing an iterative procedure until an arbitrary stopping rule is attained, with a counter of iterations k, comprises the steps of splitting the set of observations into two non-overlapping parts: the first set JB, corresponds to the observations that are relatively close to two or more centroids and the second set JG corresponds to the observations that are significantly close to a unique centroid in comparison to the other ones, then; F  ( x ) = F B  ( x ) + F G  ( x ) = ∑ j ∈ J B  f  ( z j ) + ∑ j ∈ J G  f  ( z j ) ; solving an optimization problem; determining the q hard clustering; determining membership levels of each observation to each particular cluster within a fuzzy clustering referential; and updating the smoothing parameters. 2. The method of claim 1 wherein the step of solving an optimization problem consists of solving a completely differentiable unconstrained optimization problem: minimize   F  ( x ) = ∑ j = 1 m  f  ( z j ) , starting at the initial starting point xl−1 and obtaining an optimal solution x* used as the next starting point xk, where the calculation of the objective function and its gradient is performed in two independent procedures: performing the calculation of the first part associated to the boundary band zone; and performing the calculation of the second part associated to the gravitational regions. 3. The method of claim 2 wherein the procedure for calculation of the first part of the objective function and its gradient, associated to the boundary band zone, consists of calculations F B = ∑ j ∈ J B  f  ( z j  ( x ) ) where each zj(x) results from the calculation of the zero of a general smooth equation hj(zj, x)=0, j ∈ JB which assumes a particular smoothing procedure for each considered metric: Euclidian, Minkowski, Manhattan and Chebychev, and where the gradient of this part of the objective function can always be exactly calculated by the expression ∇ F B  ( x ) = ∑ j ∈ J B   f  ( z j  ( x ) )  z j  ∇ z j  ( x ) where ∇ z j  ( x ) = - ∇ h j  ( z j , x ) / ∂ h j  ( z j , x ) ∂ z j . 4. The method of claim 3 wherein the smoothing procedure for the case of the Euclidian metric considers the general smooth equation under the particular form: h j  ( z j , x ) = ∑ i = 1 q  φ  ( z j - θ 2  ( s j , x i , γ ) , τ ) - ɛ = 0 , j ∈ J B ,  where   φ  ( y , τ ) = ( y + y 2 + τ 2 ) / 2   and θ 2  ( s j , x i , γ ) = ∑ i = 1 n  ( ? - ? ) 2 + γ 2 . ?  indicates text missing or illegible when filed 5. The method of claim 3 wherein the smoothing procedure for the Minkowski metric considers the general smooth equation under the particular form: h j  ( z j , x ) = ∑ i = 1 q  φ  ( z j - θ p  ( s j , x i , γ ) , τ ) - ɛ = 0 , j ∈ J B ,  where   φ  ( y , τ ) = ( y + y 2 + τ 2 ) / 2   and θ p  ( s j , x i , γ ) = ( ∑ i = 1 n  ( ? - ? ) p + γ p ) 1 / p .  ?  indicates text missing or illegible when filed 6. The method of claim 3 wherein the smoothing procedure for the Manhattan metric considers the general smooth equation under the particular form: h j  ( z j , x ) = ∑ i = 1 q  φ  ( z j - θ 1  ( s j , x i , γ ) , τ ) - ɛ = 0 , j ∈ J B ,  where   φ  ( y , τ ) = ( y + y 2 + τ 2 ) / 2   and θ 1  ( s j , x i , γ ) = ∑ i = 1 n  ( ? - ? ) 2 + γ 2 .  ?  indicates text missing or illegible when filed 7. The method of claim 3 wherein the smoothing procedure fob the Chebychev metric considers the general smooth equation under the particular form: h j  ( z j , x ) = ∑ i = 1 q  ∑ i = 1 n  φ  ( θ ∞  ( ? , ? , γ ) - z j , τ ) - ɛ = 0 , j = 1 , …  , m ,  where   φ  ( y , τ ) = ( y + y 2 + τ 2 ) / 2   and θ ∞  ( ? , ? , γ ) = ( ? - ? ) 2 + γ 2 .  ?  indicates text missing or illegible when filed 8. The method of claim 2 wherein the procedure for calculation of the second part of the objective function and its gradient, associated to the gravitational regions, is indeed performed by the following straightforward form F G = ∑ j ∈ J G  f  ( z j ) ,  ∇ F G = ∑ j ∈ J G  ∇ f  ( z j ) , where zj=∥sj−xi(i)∥, j=1, . . . , m, and i(j) represents the index of the gravitational region to which observation j belongs. 9. The method of claim 1 wherein the step of determining the q hard clustering consists of calculating the set of q hard clusters, Ci, i=1, . . . , q, where each cluster Ci at each iteration k is formed by the set of observations for which the closest centroid is xi*; the step of determining membership levels pij of each observation j to each particular cluster i within a fuzzy clustering referential consists of calculating at each iteration k values pij by the expression p ij = g  ( v ij / ɛ ) / ∑ q = 1 q  g  ( v ij / ɛ ) , j = 1 , …  , m , i = 1 , …  , q , where g: R→R, g(0)=0, g(1)=1 is an arbitrary monotonic crescent function and vij is the contribution of centroid i to the general smooth equation associated to observation j that assumes the common form: vij=φ(zj−θ(sj, xi+, γ), τ) for the Euclidian, Minkowski and Manhattan cases; and the step of updating the smoothing parameters consists of any manipulation procedure of parameters ε, τ, γ, including the most traditional decreasing of the three parameters to zero in order to gradually approach the original problem formulation. 10. A method of data mining using the clustering method of claim 1. 11. A method for optimizing the clustering of a given set of m observations sj, j=1, . . . , m, each represented by n components, into a specified number of groups q, represented by a set of centroids xi, i=1, . . . , q, following the criterion of minimizing the sum of squares of Euclidian distances zj, defined as the shortest distance from each generic observation sj to the nearest centroid, according to the optimization formulation: minimize    F  ( x ) = ∑ j = 1 m   ( z j ) 2 subject   to  :   z j = min x i ∈ X   s j - x i  2 ,  j = 1 , …  , m , where x=(x1, x2, . . . , xq)∈ Rnq, comprising the steps of: performing an initialization step; and performing an iterative procedure until an arbitrary stopping rule is attained, with a counter of iterations k, comprises the steps of: splitting the set of observations into two non-overlapping parts: the first set JB corresponds to the observations that are relatively close to two or more centroids and the second set JG to the observations that are significantly close to a unique centroid in comparison to the other ones, then: F(x)=FB(x)+FG(x); solving an optimization problem; determining the q hard clustering; determining membership levels of each observation to each particular cluster within a fuzzy clustering referential; and updating the smoothing parameters. 12. The method of claim 11 wherein the step of solving an optimization problem consists of solving a completely differentiable unconstrained optimization problem: minimize    F  ( x ) = ∑ j = 1 m   ( z j ) 2 starting at the initial starting point xk−i and obtaining an optimal solution x* used as the next starting point xk, where the calculation of the objective function and its gradient is performed in two independent procedures; performing the calculation of the first part associated to the boundary band zone; and performing the calculation of the second part associated to the gravitational regions. 13. The method of claim 12 wherein the procedure for calculation of the first part of the objective function and its gradient, associated to the boundary band zone, consists of calculations F B = ∑ j ∈ J s   ( z j  ( x ) ) 2 , where each zj(x) results from the calculation of the zero of a general smooth equation hi(zj, x)=0, j ∈ JB, which assumes a particular smoothing procedure for the considered Euclidian metric, where the gradient of this part of the objective function can always be exactly calculated by the expression ∇ F B  ( x ) = ∑ j ∈ J s   2  z j  ( x )  ∇ z j  ( x ) where ∇ z j  ( x ) = - ∇ h j  ( z j , x ) / ∂ h j  ( z j , x ) ∂ z j . 14. The method of claim 13 wherein the smoothing procedure for the Euclidian metric considers the general smooth equation under the particular form: h j  ( z j , x ) = ∑ i = 1 q   φ  ( z j - θ 2  ( s j , x i , γ ) , τ ) - ɛ = 0 ,  j ∈ J B ,  where   φ  ( y , τ ) = ( y + y 2 + τ 2 ) / 2   and θ 2  ( s j , x i , γ ) = ∑ i = 1 n   ( s j i - x i i ) 2 + γ 2 . 15. The method of claim 12 wherein the procedure for calculation of the second part of the objective function and its gradient, associated to the gravitational regions, is indeed performed by the following straightforward form F  G = ∑ i  =  1  q   ∑ j  ∈  J  i     s  j  -  v  i  2 + ∑ i  =  1  q    J  i     x  i  -  v  i  2 ,  ∇ F  G  ( x ) = ∑ i  =  1  q   2   J  i  ( x  i - v  i )  where vi is the center of the observations of each non-empty gravitational subset and |Ji| is the number of gravitational observations of each subset. 16. The method of claim 11 wherein the step of determining the q hard clustering consists of calculating the set of q hard clusters, Ci, i=1, . . . , q, where each cluster Ci at each iteration k is formed by the set of observations for which the closest centroid is xi*; the step of determining membership levels pij of each observation j to each particular cluster i within a fuzzy clustering referential consists of calculating at each iteration k values pii by the expression p ij = g  ( v ij / ɛ ) / ∑ i = 1 q   g  ( v ij / ɛ ) ,  j = 1 , …  , m ,  i = 1 , …  , q , where g:R→R g(0)=0, g(1)=1 is an arbitrary monotonic crescent function and vij, the contribution of centroid i to the general smooth equation associated to observation j, is given by the expression: vij=φ(zi−θ2(sj, xio, γ), τ); and the step of updating the smoothing parameters consists of any manipulation procedure of parameters ε, τ, γ, including the most traditional decreasing of the three parameters to zero in order to gradually approach the original problem formulation. 17. A method for optimizing the clustering of a given set of m observations sj, j=1, . . . , m, each represented by n components, into a specified number of groups q, represented by a set of centroids xi, i=1, . . . , q, following the criterion of minimizing the sum of Manhattan distances zi, defined as the shortest distance from each generic observation sj to the nearest centroid, according to the optimization formulation: minimize    F  ( x ) = ∑ j = 1 m   z j subject   to  :   z j = min x i ∈ X   s j - x i  2 ,  j = 1 , …  , m , where x=(x1, x2, . . . , xq)∈ Rnq, comprising the steps of: performing an initialization step; and performing an iterative procedure until an arbitrary topping rule is attained, with a counter of iterations k, comprises the steps of: splitting the set of observations into two non-overlapping parts: the first set JB corresponds to the observations that are relatively close to two or more centroids and the second set JG corresponds to the observations that are significantly close to a unique centroid in comparison to the other ones, then: F(x)=FB(x)+FG(x); solving an optimization problem; determining the q hard clustering; determining membership levels of each observation to each particular cluster within a fuzzy clustering referential; and updating the smoothing parameters. 18. The method of claim 17 wherein the step of solving an optimization problem consists of solving a completely differentiable unconstrained optimization problem: minimize    F  ( x ) = ∑ j = 1 m   z j starting at the initial starting point xk−1 and obtaining an optimal solution xo used as the next starting point xk, where the calculation of the objective function and its gradient is performed in two independent procedures: performing the calculation of the first part associated to the boundary band zone; and performing the calculation of the second part associated to the gravitational regions. 19. The method of claim 18 wherein the procedure for calculation of the first part of the objective function and its gradient, associated to the boundary band zone, consists of calculations F B = ∑ j ∈ J s   z j  ( x ) where each zj(x) results from the calculation of the zero of a general smooth equation hj(zj, x)=0, j ∈ JB, which assumes a particular smoothing procedure for the considered Manhattan metric, where the gradient of the objective function can be exactly calculated by the expression ∇ F B  ( x ) = ∑ j ∈ J s   ∇ z j  ( x ) where ∇ z j  ( x ) = - ∇ h j  ( z j , x ) / ∂ h j  ( z j , x ) ∂ z j . 20. The method of claim 19 wherein the smoothing procedure for the Manhattan metric considers the general smooth equation under the particular form: h j  ( z j , x ) = ∑ i = 1 q   φ  ( z j - θ 1  ( s j , x i , γ ) , τ ) - ɛ = 0 ,  j ∈ J B ,  where   φ  ( y , τ ) = ( y + y 2 + τ 2 ) / 2   and θ 1  ( s j , x i , γ ) = ∑ i = 1 n   ( s j i - x i i ) 2 + γ 2 . 21. The method of claim 19 wherein the procedure for calculation of the second part of the objective function and its gradient, associated to the gravitational regions, is indeed performed by the following straightforward form F G = ∑ i = 1 q   ∑ l = 1 n   [ ∑ j ∈ J B +    s j i - v i l  -  J B +   Δ   x i l + ∑ j ∈ J B -    s j i - v i l  +  J B -   Δ   x i l + ∑ j ∈ J B σ   σ  ( s j i , v i l  γ ) ] ,   ∇ F G = ∑ i = 1 q   ∑ l = 1 n   [ -  J B +  +  J B -  + ∑ j ∈ J B σ   ( s j i - v i l ) / σ  ( s j i , v i l  γ ) ] where vi the referential centroid position used for the partition of the set of observations into two non-overlapping parts and |Jil+|, |Jil−| and |Jilθ| are the cardinality of the pertinent subsets. 22. The method of claim 17 wherein the step of determining the q hard clustering consists of calculating the set of q hard clusters, Ci, i=1, . . . , q, where each cluster Ci at each iteration k is formed by the set of observations for which the closest centroid is xi*; the step of determining membership levels pil of each observation j to each particular cluster i within a fuzzy clustering referential consists of calculating at each iteration k values pij by the expression p ij = g  ( v ij / ɛ ) / ∑ i = 1 q   g  ( v ij / ɛ ) ,  j = 1 , …  , m ,  i = 1 , …  , q , where g:R→R, g(0)=0, g(1)=1 is an arbitrary monotonic crescent function and vij, the contribution of centroid i to the general smooth equation associated to observation j, is given by the expression vij=φ(zj−θ1(sj, xi*, γ), τ); and the step of updating the smoothing parameters consists of any manipulation procedure of parameters ε, τ, γ, including the most traditional decreasing of the three parameters to zero in order to gradually approach the original problem formulation.