Distance Measure For Time Series Research Articles

Parameterizing the cost function of dynamic time warping with application to time series classification

Dynamic time warping (DTW) is a popular time series distance measure that aligns the points in two series with one another. These alignments support warping of the time dimension to allow for processes that unfold at differing rates. The distance is the minimum sum of costs of the resulting alignments over any allowable warping of the time dimension. The cost of an alignment of two points is a function of the difference in the values of those points. The original cost function was the absolute value of this difference. Other cost functions have been proposed. A popular alternative is the square of the difference. However, to our knowledge, this is the first investigation of both the relative impacts of using different cost functions and the potential to tune cost functions to different time series classification tasks. We do so in this paper by using a tunable cost function lambda _{gamma } with parameter gamma . We show that higher values of gamma place greater weight on larger pairwise differences, while lower values place greater weight on smaller pairwise differences. We demonstrate that training gamma significantly improves the accuracy of both the { DTW } nearest neighbor and Proximity Forest classifiers.

Elastic similarity and distance measures for multivariate time series

This paper contributes multivariate versions of seven commonly used elastic similarity and distance measures for time series data analytics. Elastic similarity and distance measures can compensate for misalignments in the time axis of time series data. We adapt two existing strategies used in a multivariate version of the well-known Dynamic Time Warping (DTW), namely, Independent and Dependent DTW, to these seven measures. While these measures can be applied to various time series analysis tasks, we demonstrate their utility on multivariate time series classification using the nearest neighbor classifier. On 23 well-known datasets, we demonstrate that each of the measures but one achieves the highest accuracy relative to others on at least one dataset, supporting the value of developing a suite of multivariate similarity and distance measures. We also demonstrate that there are datasets for which either the dependent versions of all measures are more accurate than their independent counterparts or vice versa. In addition, we also construct a nearest neighbor-based ensemble of the measures and show that it is competitive to other state-of-the-art single-strategy multivariate time series classifiers.

Amercing: An intuitive and effective constraint for dynamic time warping

Dynamic Time Warping (DTW) is a time series distance measure that allows non-linear alignments between series. Constraints on the alignments in the form of windows and weights have been introduced because unconstrained DTW is too permissive in its alignments. However, windowing introduces a crude step function, allowing unconstrained flexibility within the window, and none beyond it. While not entailing a step function, a multiplicative weight is relative to the distances between aligned points along a warped path, rather than being a direct function of the amount of warping that is introduced. In this paper, we introduce Amerced Dynamic Time Warping (ADTW), a new, intuitive, DTW variant that penalizes the act of warping by a fixed additive cost. Like windowing and weighting, ADTW constrains the amount of warping. However, it avoids both abrupt discontinuities in the amount of warping allowed and the limitations of a multiplicative penalty. We formally introduce ADTW, prove some of its properties, and discuss its parameterization. We show on a simple example how it can be parameterized to achieve an intuitive outcome, and demonstrate its usefulness on a standard time series classification benchmark. We provide a demonstration application in C++ Herrmann(2021)[1].

Ultra-fast meta-parameter optimization for time series similarity measures with application to nearest neighbour classification

Nearest neighbour similarity measures are widely used in many time series data analysis applications. They compute a measure of similarity between two time series. Most applications require tuning of these measures’ meta-parameters in order to achieve good performance. However, most measures have at least O(L^2) complexity, making them computationally expensive and the process of learning their meta-parameters burdensome, requiring days even for datasets containing only a few thousand series. In this paper, we propose UltraFastMPSearch, a family of algorithms to learn the meta-parameters for different types of time series distance measures. These algorithms are significantly faster than the prior state of the art. Our algorithms build upon the state of the art, exploiting the properties of a new efficient exact algorithm which supports early abandoning and pruning for most time series distance measures. We show on 128 datasets from the UCR archive that our new family of algorithms are up to an order of magnitude faster than the previous state of the art.

A Multivariate-Time-Series-Prediction-Based Adaptive Data Transmission Period Control Algorithm for IoT Networks

In order to reduce unnecessary data transmissions from Internet of Things (IoT) sensors, this article proposes a multivariate-time-series-prediction-based adaptive data transmission period control (PBATPC) algorithm for IoT networks. Based on the spatio-temporal correlation between multivariate time-series data, we developed a novel multivariate time-series data encoding scheme utilizing the proposed time-series distance measure <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\textit {ADMWD}$ </tex-math></inline-formula> . Composed of two significant factors for a multivariate time-series prediction, i.e., the absolute deviation from the mean (ADM) and the weighted differential (WD) distance, the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\textit {ADMWD}$ </tex-math></inline-formula> considers both the time distance from a prediction point and a negative correlation between the time-series data concurrently. Utilizing the convolutional neural network (CNN) model, a subset of IoT sensor readings can be predicted from encoded multivariate time-series measurements, and we compared the predicted sensor values with actual readings to obtain the adaptive data transmission period. Extensive performance evaluations show a substantial performance gain of the proposed algorithm in terms of the average power reduction ratio (approximately 12%) and average data reconstruction error (approximately 8.32% MAPE). Finally, this article also provides a practical implementation of the proposed PBATPC algorithm via the HTTP protocol under the IEEE 802.11-based WLAN network.

Kernel Distance Measures for Time Series, Random Fields and Other Structured Data

This paper introduces kdiff, a novel kernel-based measure for estimating distances between instances of time series, random fields and other forms of structured data. This measure is based on the idea of matching distributions that only overlap over a portion of their region of support. Our proposed measure is inspired by MPdist which has been previously proposed for such datasets and is constructed using Euclidean metrics, whereas kdiff is constructed using non-linear kernel distances. Also, kdiff accounts for both self and cross similarities across the instances and is defined using a lower quantile of the distance distribution. Comparing the cross similarity to self similarity allows for measures of similarity that are more robust to noise and partial occlusions of the relevant signals. Our proposed measure kdiff is a more general form of the well known kernel-based Maximum Mean Discrepancy distance estimated over the embeddings. Some theoretical results are provided for separability conditions using kdiff as a distance measure for clustering and classification problems where the embedding distributions can be modeled as two component mixtures. Applications are demonstrated for clustering of synthetic and real-life time series and image data, and the performance of kdiff is compared to competing distance measures for clustering.

Causality Distance Measures for Multivariate Time Series with Applications

In this work, we focus on the development of new distance measure algorithms, namely, the Causality Within Groups (CAWG), the Generalized Causality Within Groups (GCAWG) and the Causality Between Groups (CABG), all of which are based on the well-known Granger causality. The proposed distances together with the associated algorithms are suitable for multivariate statistical data analysis including unsupervised classification (clustering) purposes for the analysis of multivariate time series data with emphasis on financial and economic data where causal relationships are frequently present. For exploring the appropriateness of the proposed methodology, we implement, for illustrative purposes, the proposed algorithms to hierarchical clustering for the classification of 19 EU countries based on seven variables related to health resources in healthcare systems.

An ultra-fast time series distance measure to allow data mining in more complex real-world deployments

At their core, many time series data mining algorithms reduce to reasoning about the shapes of time series subsequences. This requires an effective distance measure, and for last two decades most algorithms use Euclidean distance or DTW as their core subroutine. We argue that these distance measures are not as robust as the community seems to believe. The undue faith in these measures perhaps derives from an overreliance on the benchmark datasets and self-selection bias. The community is simply reluctant to address more difficult domains, for which current distance measures are ill-suited. In this work, we introduce a novel distance measure MPdist. We show that our proposed distance measure is much more robust than current distance measures. For example, it can handle data with missing values or spurious regions. Furthermore, it allows us to successfully mine datasets that would defeat any Euclidean or DTW distance-based algorithm. Additionally, we show that our distance measure can be computed so efficiently as to allow analytics on very fast arriving streams.

Time-Series Clustering Based on the Characterization of Segment Typologies.

Time-series clustering is the process of grouping time series with respect to their similarity or characteristics. Previous approaches usually combine a specific distance measure for time series and a standard clustering method. However, these approaches do not take the similarity of the different subsequences of each time series into account, which can be used to better compare the time-series objects of the dataset. In this article, we propose a novel technique of time-series clustering consisting of two clustering stages. In a first step, a least-squares polynomial segmentation procedure is applied to each time series, which is based on a growing window technique that returns different-length segments. Then, all of the segments are projected into the same dimensional space, based on the coefficients of the model that approximates the segment and a set of statistical features. After mapping, a first hierarchical clustering phase is applied to all mapped segments, returning groups of segments for each time series. These clusters are used to represent all time series in the same dimensional space, after defining another specific mapping process. In a second and final clustering stage, all the time-series objects are grouped. We consider internal clustering quality to automatically adjust the main parameter of the algorithm, which is an error threshold for the segmentation. The results obtained on 84 datasets from the UCR Time Series Classification Archive have been compared against three state-of-the-art methods, showing that the performance of this methodology is very promising, especially on larger datasets.

SynSys: A Synthetic Data Generation System for Healthcare Applications.

Creation of realistic synthetic behavior-based sensor data is an important aspect of testing machine learning techniques for healthcare applications. Many of the existing approaches for generating synthetic data are often limited in terms of complexity and realism. We introduce SynSys, a machine learning-based synthetic data generation method, to improve upon these limitations. We use this method to generate synthetic time series data that is composed of nested sequences using hidden Markov models and regression models which are initially trained on real datasets. We test our synthetic data generation technique on a real annotated smart home dataset. We use time series distance measures as a baseline to determine how realistic the generated data is compared to real data and demonstrate that SynSys produces more realistic data in terms of distance compared to random data generation, data from another home, and data from another time period. Finally, we apply our synthetic data generation technique to the problem of generating data when only a small amount of ground truth data is available. Using semi-supervised learning we demonstrate that SynSys is able to improve activity recognition accuracy compared to using the small amount of real data alone.

A novel distance measure for time series: Maximum shifting correlation distance

Time series distance or similarity measurement is one of the most important problems in time series data mining, including representation, clustering, classification, and outlier detection. The existing distance measures may not efficiently deal with the drifts in the series in both the phase and amplitude. In this study, a novel measurement, maximum shifting correlation distance (MSCD), is proposed to improve the accuracy and efficiency of the time series distance measure. By integrating the curve registration and correlation, the misalignments or drifts in the phase and amplitude can be eliminated, respectively. In addition, the second distance of the MSCD (MSCD-2nd), which has the “shrinkage effect”, enhances the similarity of the samples within a cluster. The second distance has the same effect on the dynamic time warping (DTW) distance. The experimental results demonstrate that MSCD-2nd is the preferred measure in terms of the accuracy and efficiency for time series clustering and classification.

Time-series clustering based on linear fuzzy information granules

In this paper, time-series clustering is discussed. At first ℓ1 trend filtering method is used to produce an optimal segmentation of time series. Next optimized fuzzy information granulation is completed for each segment to form a linear fuzzy information granule, which includes both average and trend information. Once the optimal segmentation and granulation have been completed, the original time series is transformed into a granular time series. To finalize time-series clustering, a distance measure for granular time series is established, and a linear fuzzy information granule-based dynamic time warping (LFIG_DTW) algorithm is developed for calculating the distance of two equal-length or unequal-length granular time series. Furthermore, the distance realized by the LFIG_DTW algorithm can detect not only the increasing or decreasing trends, but also the changing periods and rates of changes. After calculating all the distances between any two granular time series, a LFIG_DTW distance-based hierarchical clustering method is designed for time-series clustering. Experiment results involving several real datasets show the effectiveness of the proposed method.

Two Faces of the Framework for Analysis and Prediction, Part 2 - Research

Continuing Two Faces of the Framework for Analysis and Prediction, Part 1 - Education, this article overviews the successful applications of the Framework for Analysis and Prediction (FAP) – a Java-based tool dedicated to time-series analysis – in research, highlighting the influence applications had on subsequent development and enhancements of the FAP software. Research applications include applications in data mining (development of a new time-series representation and various interactions between time-series distance measures and classification) and multidisciplinary research (in psychology, medicine, emotion recognition and agent-based distributed computing). DOI: http://dx.doi.org/10.5755/j01.itc.47.3.18747

Asphalt pavement classification using smartphone accelerometer and Complexity Invariant Distance

Most modern smartphones have a variety of built-in sensors, as an accelerometer, gyroscope, GPS, proximity and a magnetic sensor. The large variety of sensors makes these devices powerful measurement tools, allowing the emergence of new systems and applications. In this paper, is presented a real data stream application related to asphalt pavement evaluation using acceleration data gathered by the accelerometer sensor of smartphones. The quality of the pavement has a significant influence on the final price of goods and services, the safety of drivers, pedestrians and passengers, and driver’s comfort. Thus, it is essential the use of tools, as proposed in this work, that allows the constant monitoring of pavement conditions by the Government authorities or private entities for more precise interventions in the maintenance planning with fewer expenses. This task is mainly important in developing countries where there is a lack of technology and a reduced budget for maintenance. Due to the popularity of smartphones, this tool can make possible that different users help to monitor the pavement quality ubiquitously during driving periods without effort. The application of asphalt evaluation is modeled in this work as a multi-dimensional time series classification problem, where the time series are the data from the three-axis accelerometer sensor. Given the characteristics of the data, we discuss and propose the combination of some classical distance measure for time series as Dynamic Time Warping or Longest Common Subsequence Similarity with the Complexity Invariant Distance. A comprehensive experimental evaluation was performed on three datasets that represent different scenarios of asphalt pavement classification. The proposed approach reaches a classification accuracy of 80% to 98% in the three evaluated problems and we experimentally show that the complexity invariance substantially improves the results achieved by classical distance measures given our classification tasks.

A Time Series Distance Measure for Efficient Clustering of Input/Output Signals by Their Underlying Dynamics

Starting from a dataset with input/output time series generated by multiple deterministic linear dynamical systems, this paper tackles the problem of automatically clustering these time series. We propose an extension to the so-called Martin cepstral distance, that allows to efficiently cluster these time series, and apply it to simulated electrical circuits data. Traditionally, two ways of handling the problem are used. The first class of methods employs a distance measure on time series (e.g. Euclidean, Dynamic Time Warping) and a clustering technique (e.g. k-means, k-medoids, hierarchical clustering) to find natural groups in the dataset. It is, however, often not clear whether these distance measures effectively take into account the specific temporal correlations in these time series. The second class of methods uses the input/output data to identify a dynamic system using an identification scheme, and then applies a model norm-based distance (e.g. H2, H-infinity) to find out which systems are similar. This, however, can be very time consuming for large amounts of long time series data. We show that the new distance measure presented in this paper performs as good as when every input/output pair is modelled explicitly, but remains computationally much less complex. The complexity of calculating this distance between two time series of length N is O(N logN).

Distance Measures for Time Series in R: The TSdist Package

Distance Measures for Time Series in R: The TSdist Package

Using dynamic time warping distances as features for improved time series classification

Dynamic time warping (DTW) has proven itself to be an exceptionally strong distance measure for time series. DTW in combination with one-nearest neighbor, one of the simplest machine learning metho...

Fuzzy clustering of time series data: A particle swarm optimization approach

With rapid development in information gathering technologies and access to large amounts of data, we always require methods for data analyzing and extracting useful information from large raw dataset and data mining is an important method for solving this problem. Clustering analysis as the most commonly used function of data mining, has attracted many researchers in computer science. Because of different applications, the problem of clustering the time series data has become highly popular and many algorithms have been proposed in this field. Recently Swarm Intelligence (SI) as a family of nature inspired algorithms has gained huge popularity in the field of pattern recognition and clustering. In this paper, a technique for clustering time series data using a particle swarm optimization (PSO) approach has been proposed, and Pearson Correlation Coefficient as one of the most commonly-used distance measures for time series is considered. The proposed technique is able to find (near) optimal cluster centers during the clustering process. To reduce the dimensionality of the search space and improve the performance of the proposed method, a singular value decomposition (SVD) representation of cluster centers is considered. Experimental results over three popular data sets indicate the superiority of the proposed technique in comparing with fuzzy C-means and fuzzy K-medoids clustering techniques.

Query-sensitive distance measure selection for time series nearest neighbor classification

Many distance or similarity measures have been proposed for time series similarity search. However, none of these measures is guaranteed to be optimal when used for 1-Nearest Neighbor (NN) classification. In this paper we study the problem of selecting the most appropriate distance measure, given a pool of time series distance measures and a query, so as to perform NN classification of the query. We propose a framework for solving this problem, by identifying, given the query, the distance measure most likely to produce the correct classification result for that query. From this proposed framework, we derive three specific methods, that differ from each other inthe way they estimatethe probability that a distance measure correctly classifiesa query object. In our experiments, our pool of measures consists of Dynamic Time Warping (DTW), Move-Split-Merge (MSM), and Edit distance with Real Penalty (ERP). Based on experimental evaluation with 45 datasets, the best-performing of the three proposed methods provides the best results in terms of classification error rate, compared to the competitors, which include using the Cross Validation method for selecting the distance measure in each dataset, as well as using a single specific distance measure (DTW, MSM, or ERP) across all datasets.

Time-series analysis in the medical domain: A study of Tacrolimus administration and influence on kidney graft function

There exists a major concern regarding toxic effects of immunosuppressive medication on the kidney graft during post-transplant care, with observed variation in individual susceptibility to adverse drug effects amongst patients. To date, there has been no possibility to identify susceptible patients prospectively. This study analyzes medical data which includes time series of measures of renal function and trough levels of immunosuppressive drug Tacrolimus, with the main aim of identifying patients susceptible to drug toxicity. We evaluate a plethora of time-series distance measures, determining their appropriateness to the domain based on two criteria: (1) preserving the expected correlations between distances, and (2) ability to detect the expected patterns of interaction between immunosuppressive drug levels and renal function. Besides identifying the most suitable time-series distance measures, we observed that the majority of patients do not exhibit an association between impaired graft function and higher Tacrolimus dosing. On the other hand, the minority of patients determined most sensitive to varying Tacrolimus levels showed a strong tendency to prefer low Tacrolimus dosing.