Time Series Data Cleaning: A Survey <s> A. ABNORMAL POINT DETECTION <s> Preface 1 INTRODUCTION 1.1 Examples of Time Series 1.2 Objectives of Time Series Analysis 1.3 Some Simple Time Series Models 1.3.3 A General Approach to Time Series Modelling 1.4 Stationary Models and the Autocorrelation Function 1.4.1 The Sample Autocorrelation Function 1.4.2 A Model for the Lake Huron Data 1.5 Estimation and Elimination of Trend and Seasonal Components 1.5.1 Estimation and Elimination of Trend in the Absence of Seasonality 1.5.2 Estimation and Elimination of Both Trend and Seasonality 1.6 Testing the Estimated Noise Sequence 1.7 Problems 2 STATIONARY PROCESSES 2.1 Basic Properties 2.2 Linear Processes 2.3 Introduction to ARMA Processes 2.4 Properties of the Sample Mean and Autocorrelation Function 2.4.2 Estimation of $\gamma(\cdot)$ and $\rho(\cdot)$ 2.5 Forecasting Stationary Time Series 2.5.3 Prediction of a Stationary Process in Terms of Infinitely Many Past Values 2.6 The Wold Decomposition 1.7 Problems 3 ARMA MODELS 3.1 ARMA($p,q$) Processes 3.2 The ACF and PACF of an ARMA$(p,q)$ Process 3.2.1 Calculation of the ACVF 3.2.2 The Autocorrelation Function 3.2.3 The Partial Autocorrelation Function 3.3 Forecasting ARMA Processes 1.7 Problems 4 SPECTRAL ANALYSIS 4.1 Spectral Densities 4.2 The Periodogram 4.3 Time-Invariant Linear Filters 4.4 The Spectral Density of an ARMA Process 1.7 Problems 5 MODELLING AND PREDICTION WITH ARMA PROCESSES 5.1 Preliminary Estimation 5.1.1 Yule-Walker Estimation 5.1.3 The Innovations Algorithm 5.1.4 The Hannan-Rissanen Algorithm 5.2 Maximum Likelihood Estimation 5.3 Diagnostic Checking 5.3.1 The Graph of $\t=1,\ldots,n\ 5.3.2 The Sample ACF of the Residuals <s> BIB001 </s> Time Series Data Cleaning: A Survey <s> A. ABNORMAL POINT DETECTION <s> Abstract Autoregressive integrated moving average (ARIMA) is one of the popular linear models in time series forecasting during the past three decades. Recent research activities in forecasting with artificial neural networks (ANNs) suggest that ANNs can be a promising alternative to the traditional linear methods. ARIMA models and ANNs are often compared with mixed conclusions in terms of the superiority in forecasting performance. In this paper, a hybrid methodology that combines both ARIMA and ANN models is proposed to take advantage of the unique strength of ARIMA and ANN models in linear and nonlinear modeling. Experimental results with real data sets indicate that the combined model can be an effective way to improve forecasting accuracy achieved by either of the models used separately. <s> BIB002 </s> Time Series Data Cleaning: A Survey <s> A. ABNORMAL POINT DETECTION <s> In this article we consider the problem of detecting unusual values or outliers from time series data where the process by which the data are created is difficult to model. The main consideration is the fact that data closer in time are more correlated to each other than those farther apart. We propose two variations of a method that uses the median from a neighborhood of a data point and a threshold value to compare the difference between the median and the observed data value. Both variations of the method are fast and can be used for data streams that occur in quick succession such as sensor data on an airplane. <s> BIB003 </s> Time Series Data Cleaning: A Survey <s> A. ABNORMAL POINT DETECTION <s> The deployment of environmental sensors has generated an interest in real-time applications of the data they collect. This research develops a real-time anomaly detection method for environmental data streams that can be used to identify data that deviate from historical patterns. The method is based on an autoregressive data-driven model of the data stream and its corresponding prediction interval. It performs fast, incremental evaluation of data as it becomes available, scales to large quantities of data, and requires no pre-classification of anomalies. Furthermore, this method can be easily deployed on a large heterogeneous sensor network. Sixteen instantiations of this method are compared based on their ability to identify measurement errors in a windspeed data stream from Corpus Christi, Texas. The results indicate that a multilayer perceptron model of the data stream, coupled with replacement of anomalous data points, performs well at identifying erroneous data in this data stream. <s> BIB004 </s> Time Series Data Cleaning: A Survey <s> A. ABNORMAL POINT DETECTION <s> Anomaly detection is considered an important data mining task, aiming at the discovery of elements (also known as outliers) that show significant diversion from the expected case. More specifically, given a set of objects the problem is to return the suspicious objects that deviate significantly from the typical behavior. As in the case of clustering, the application of different criteria lead to different definitions for an outlier. In this work, we focus on distance-based outliers: an object x is an outlier if there are less than k objects lying at distance at most R from x. The problem offers significant challenges when a stream-based environment is considered, where data arrive continuously and outliers must be detected on-the-fly. There are a few research works studying the problem of continuous outlier detection. However, none of these proposals meets the requirements of modern stream-based applications for the following reasons: (i) they demand a significant storage overhead, (ii) their efficiency is limited and (iii) they lack flexibility. In this work, we propose new algorithms for continuous outlier monitoring in data streams, based on sliding windows. Our techniques are able to reduce the required storage overhead, run faster than previously proposed techniques and offer significant flexibility. Experiments performed on real-life as well as synthetic data sets verify our theoretical study. <s> BIB005
For single-point anomalies, the most common idea is to use predictive models for detection. That is, the predicted value of the established model and the observed value for each data point is compared, and if the difference between the two values is greater than a certain threshold, the observed value is considered to be an abnormal value. Specifically, Basu and Meckesheimer BIB003 select all data points with timestamps t − k to t + k with the timestamp t as the center point, and the median of these data points is considered to be the predicted value of data points with timestamp t value. Hill and Minsker BIB004 first cluster the data points and take the average of the clusters as the predicted value of the point. The AR model and the ARX model are widely used for anomaly detection in various fields, such as economics, social surveys  , BIB001 , and so on. The ARX model takes advantage of manually labeled information, so it is more accurate than the AR model when cleaning data. The ARIMA model BIB002 represents a type of time series model consisting of AR and MA mentioned above, which can be used for data cleaning of non-stationary time series. Kontaki et al. BIB005 propose continuous monitoring of distance-based outliers over data streams. One of the most widely used definitions is the one based on distance as shown in Figure 5 : an object p is marked as an outlier, if there are less than k objects in given distance. Here k = 4, q is the normal point and p is the abnormal point. 