Abstract: A heart-rate detection system can receive heartbeat data generated by a wearable heart-rate sensor worn by a wearer. The system can then execute a noise-reduction process for reducing noise in the heartbeat data. The noise-reduction process can involve applying a lowpass filter to the heartbeat data, generating wavelet coefficients by applying a wavelet transform to the filtered heartbeat data, and generating a reduced set of wavelet coefficients by thresholding the wavelet coefficients. An inverse wavelet signal can then be generated by applying an inverse wavelet transform to the reduced set of wavelet coefficients. R-peaks can be identified by performing peak detection on the instantaneous amplitudes of the data points in the inverse wavelet signal. A heart rate curve can then be generated based on the R-peaks and modified by applying a Hampel filter. Heartbeat data can then be generated based on the modified heart rate curve for output.

Abstract: Anomalies in a target object can be detected and diagnosed using improved Mahalanobis-Taguchi system (MTS) techniques. For example, an anomaly detection and diagnosis (ADD) system can receive a set of measurements associated with attributes of a target object. A Mahalanobis distance (MD) can be determined using a generalized inverse matrix. An abnormal condition can be detected when the MD is greater than a predetermined threshold value. The ADD system can determine an importance score for each measurement of a corresponding attribute. The attribute whose measurement has the highest importance score can be determined to be responsible for the abnormal condition.

Abstract: Anomalies in a target object can be detected and diagnosed using improved Mahalanobis-Taguchi system (MTS) techniques. For example, an anomaly detection and diagnosis (ADD) system can receive a set of measurements associated with attributes of a target object. A Mahalanobis distance (MD) can be determined using a generalized inverse matrix. An abnormal condition can be detected when the MD is greater than a predetermined threshold value. The ADD system can determine an importance score for each measurement of a corresponding attribute. The attribute whose measurement has the highest importance score can be determined to be responsible for the abnormal condition.

Abstract: A singular value decomposition (SVD) is computed of a first matrix to define a left matrix, a diagonal matrix, and a right matrix. The left matrix, the diagonal matrix, and the right matrix are updated using an arrowhead matrix structure defined from the diagonal matrix and by adding a next observation vector to a last row of the first matrix. The updated left matrix, the updated diagonal matrix, and the updated right matrix are updated using a diagonal-plus-rank-one (DPR1) matrix structure defined from the updated diagonal matrix and by removing an observation vector from a first row of the first matrix. Eigenpairs of the DPR1 matrix are computed based on whether a value computed from the updated left matrix is positive or negative. The left matrix updated in (C), the diagonal matrix updated in (C), and the right matrix updated in (C) are output.

Abstract: A computer transforms high-dimensional data into low-dimensional data. (A) A distance matrix is computed from observation vectors. (B) A kernel matrix is computed from the distance matrix using a bandwidth value. (C) The kernel matrix is decomposed using an eigen decomposition to define eigenvalues. (D) A predefined number of largest eigenvalues are selected from the eigenvalues. (E) The selected largest eigenvalues are summed. (F) A next bandwidth value is computed based on the summed eigenvalues. (A) through (F) are repeated with the next bandwidth value until a stop criterion is satisfied. Each observation vector of the observation vectors is transformed into a second space using a kernel principal component analysis with the next bandwidth value and the kernel matrix. The second space has a dimension defined by the predefined number of first eigenvalues. Each transformed observation vector is output.

Abstract: A computing system determines if an event has occurred. A first window is defined that includes a subset of a plurality of observation vectors modeled as an output of an autoregressive causal system. A magnitude adjustment vector is computed from a mean computed for a matrix of magnitude values that includes a column for each window of a plurality of windows. The first window is stored in a next column of the matrix of magnitude values. Each cell of the matrix of magnitude values includes an estimated power spectrum value for a respective window and a respective frequency. A second matrix of magnitude values is updated using the magnitude adjustment vector. Each cell of the second matrix of magnitude values includes an adjusted power spectrum value for the respective window and the respective frequency. A peak is detected from the next column of the second matrix of magnitude values.

Abstract: A computer transforms high-dimensional data into low-dimensional data. A distance is computed between a selected observation vector and each observation vector of a plurality of observation vectors, a nearest neighbors are selected using the computed distances, and a first sigmoid function is applied to compute a distance similarity value between the selected observation vector and each of the selected nearest neighbors where each of the computed distance similarity values is added to a first matrix. The process is repeated with each observation vector of the plurality of observation vectors as the selected observation vector. An optimization method is executed with an initial matrix, the first matrix, and a gradient of a second sigmoid function that computes a second distance similarity value between the selected observation vector and each of the nearest neighbors to transform each observation vector of the plurality of observation vectors into the low-dimensional space.

Abstract: A computing device determines hyperparameter values for outlier detection. An LOF score is computed for observation vectors using a neighborhood size value. Outlier observation vectors are selected from the observation vectors. Outlier mean and outlier variance values are computed of the LOF scores of the outlier observation vectors. Inlier observation vectors are selected from the observation vectors that have highest computed LOF scores of the observation vectors that are not included in the outlier observation vectors. Inlier mean and inlier variance values are computed of the LOF scores of the inlier observation vectors. A difference value is computed using the outlier mean and variance values and the inlier mean and variance values. The process is repeated with each neighborhood size value of a plurality of neighborhood size values. A tuned neighborhood size value is selected as the neighborhood size value associated with an extremum value of the difference value.

Abstract: A computing device determines a bandwidth parameter value for outlier detection or data classification. A mean pairwise distance value is computed between observation vectors. A tolerance value is computed based on a number of observation vectors. A scaling factor value is computed based on a number of observation vectors and the tolerance value. A Gaussian bandwidth parameter value is computed using the mean pairwise distance value and the scaling factor value. An optimal value of an objective function is computed that includes a Gaussian kernel function that uses the computed Gaussian bandwidth parameter value. The objective function defines a support vector data description model using the observation vectors to define a set of support vectors. The Gaussian bandwidth parameter value and the set of support vectors are output for determining if a new observation vector is an outlier or for classifying the new observation vector.

Abstract: A computing device updates an estimate of one or more principal components for a next observation vector. An initial observation matrix is defined with first observation vectors. A number of the first observation vectors is a predefined window length. Each observation vector of the first observation vectors includes a plurality of values. A principal components decomposition is computed using the initial observation matrix. The principal components decomposition includes a sparse noise vector s, a first singular value decomposition vector U, and a second singular value decomposition vector v for each observation vector of the first observation vectors. A rank r is determined based on the principal components decomposition. A next principal components decomposition is computed for a next observation vector using the determined rank r. The next principal components decomposition is output for the next observation vector and monitored to determine a status of a physical object.

Abstract: A Gaussian similarity matrix is computed between observation vectors. An inverse Gaussian similarity matrix is computed from the Gaussian similarity matrix. A row sum vector is computed that includes a row sum value computed from each row of the inverse Gaussian similarity matrix. (a) A new observation vector is selected. (b) An acceptance value is computed for the new observation vector using the set of boundary support vectors, the row sum vector, and the new observation vector. (c) (a) and (b) are repeated when the computed acceptance value is less than or equal to zero. (d) An incremental vector is computed from the inverse Gaussian similarity matrix and the new observation vector when the computed acceptance value is greater than zero. (e) the selected new observation vector is output as an outlier observation vector when a maximum value of the incremental vector is less than a first predefined tolerance value.

Abstract: A computing device determines a bandwidth parameter value for outlier detection or data classification. A mean pairwise distance value is computed between observation vectors. A tolerance value is computed based on a number of observation vectors. A scaling factor value is computed based on a number of observation vectors and the tolerance value. A Gaussian bandwidth parameter value is computed using the mean pairwise distance value and the scaling factor value. An optimal value of an objective function is computed that includes a Gaussian kernel function that uses the computed Gaussian bandwidth parameter value. The objective function defines a support vector data description model using the observation vectors to define a set of support vectors. The Gaussian bandwidth parameter value and the set of support vectors are output for determining if a new observation vector is an outlier or for classifying the new observation vector.

Abstract: A computing device employs machine learning and determines a bandwidth parameter value for a support vector data description (SVDD). A mean pairwise distance value is computed between observation vectors. A scaling factor value is computed based on a number of the plurality of observation vectors and a predefined tolerance value. A Gaussian bandwidth parameter value is computed using the computed mean pairwise distance value and the computed scaling factor value. An optimal value of an objective function is computed that includes a Gaussian kernel function that uses the computed Gaussian bandwidth parameter value. The objective function defines a SVDD model using the plurality of observation vectors to define a set of support vectors. The computed Gaussian bandwidth parameter value and the defined a set of support vectors are output for determining if a new observation vector is an outlier.

Abstract: A computing device detects an abnormal observation vector using a principal components decomposition. The principal components decomposition includes a sparse noise vector st computed for the observation vector that includes a plurality of values, wherein each value is associated with a variable to define a plurality of variables. The sparse noise vector st has a dimension equal to m a number of the plurality of variables. A zero counter time series value ?t is computed using ?t=?i=1mst[i]. A probability value for ?t is computed using p=?i=?t+1m+1Hc[i]/?i=0m+1Hc[i], where Hc[i] includes a count of a number of times each value of ?t occurred for previous observation vectors. The probability value is compared with a predefined abnormal observation probability value. An abnormal observation indicator is set when the probability value indicates the observation vector is abnormal. The observation vector is output when the probability value indicates the observation vector is abnormal.

Abstract: A computing device updates an estimate of one or more principal components for a next observation vector. An initial observation matrix is defined with first observation vectors. A number of the first observation vectors is a predefined window length. Each observation vector of the first observation vectors includes a plurality of values. A principal components decomposition is computed using the initial observation matrix. The principal components decomposition includes a sparse noise vector s, a first singular value decomposition vector U, and a second singular value decomposition vector ? for each observation vector of the first observation vectors. A rank r is determined based on the principal components decomposition. A next principal components decomposition is computed for a next observation vector using the determined rank r. The next principal components decomposition is output for the next observation vector and monitored to determine a status of a physical object.

Abstract: A computing device detects an abnormal observation vector using a principal components decomposition. The principal components decomposition includes a sparse noise vector st computed for the observation vector that includes a plurality of values, wherein each value is associated with a variable to define a plurality of variables. The sparse noise vector st has a dimension equal to m a number of the plurality of variables. A zero counter time series value ?t is computed using ?t=?i=1mst[i]. A probability value for ?t is computed using p=?i=?t+1m+1Hc[i]/?i=0m+1Hc[i], where Hc[i] includes a count of a number of times each value of ?t occurred for previous observation vectors. The probability value is compared with a predefined abnormal observation probability value. An abnormal observation indicator is set when the probability value indicates the observation vector is abnormal. The observation vector is output when the probability value indicates the observation vector is abnormal.

Abstract: A computing device determines a kernel parameter value for a support vector data description for outlier identification. A first candidate optimal kernel parameter value is computed by computing a first optimal value of a first objective function that includes a kernel function for each of a plurality of kernel parameter values from a starting kernel parameter value to an ending kernel parameter value using an incremental kernel parameter value. The first objective function is defined for a SVDD model using observation vectors to define support vectors. A number of the observation vectors is a predefined sample size. The predefined sample size is incremented by adding a sample size increment. A next candidate optimal kernel parameter value is computed with an incremented number of vectors until a computed difference value is less than or equal to a predefined convergence value.

Abstract: A computing device determines an SVDD to identify an outlier in a dataset. First and second sets of observation vectors of a predefined sample size are randomly selected from a training dataset. First and second optimal values are computed using the first and second observation vectors to define a first set of support vectors and a second set of support vectors. A third optimal value is computed using the first set of support vectors updated to include the second set of support vectors to define a third set of support vectors. Whether or not a stop condition is satisfied is determined by comparing a computed value to a stop criterion. When the stop condition is not satisfied, the first set of support vectors is defined as the third set of support vectors, and operations are repeated until the stop condition is satisfied. The third set of support vectors is output.

Abstract: A computing device determines an SVDD to identify an outlier in a dataset. First and second sets of observation vectors of a predefined sample size are randomly selected from a training dataset. First and second optimal values are computed using the first and second observation vectors to define a first set of support vectors and a second set of support vectors. A third optimal value is computed using the first set of support vectors updated to include the second set of support vectors to define a third set of support vectors. Whether or not a stop condition is satisfied is determined by comparing a computed value to a stop criterion. When the stop condition is not satisfied, the first set of support vectors is defined as the third set of support vectors, and operations are repeated until the stop condition is satisfied. The third set of support vectors is output.

Abstract: A computing device determines a kernel parameter value for a support vector data description for outlier identification. A first candidate optimal kernel parameter value is computed by computing a first optimal value of a first objective function that includes a kernel function for each of a plurality of kernel parameter values from a starting kernel parameter value to an ending kernel parameter value using an incremental kernel parameter value. The first objective function is defined for a SVDD model using observation vectors to define support vectors. A number of the observation vectors is a predefined sample size. The predefined sample size is incremented by adding a sample size increment. A next candidate optimal kernel parameter value is computed with an incremented number of vectors until a computed difference value is less than or equal to a predefined convergence value.