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Case Studies
Published in Abhijit Pandit, Mathematical Modeling using Fuzzy Logic, 2021
The smoothing method (stable series) is when the time series does not show any significant influence of trend, periodic or seasonal components. In this case, the goal is to smooth out the irregular components of the time series using an averaging process. The moving-average method is the most widely used smoothing technique. In this method, the prediction is the stereotype of the last “x” observation, where “x” is an appropriate number. Suppose that the predictor wants to generate a three period moving average. In the three-period example, the moving-average method uses the average of the three most recent data observations in the time series as predictions for the next time period. These predicted values for the next period, along with the last two observations from the past time series, create a stereotype that can be used as a prediction for the second period in the future. Three-cycle moving stereotype numbering is described in the tracking table. According to the third moving average, you can predict that in 2008, 2.5 million new homes will be most likely to be sold in the United States (Table 4.1).
Intelligent Robotic Vision Systems
Published in Spyros G. Tzafestas, Intelligent Robotic Systems, 2020
L. Van Gool, P. Wambacq, A. Oosterlinck
In the description of contours, another noteworthy development has retained a great deal of attention over the last years. Scale-space approaches describe contours at different resolutions by subsequent smoothing operations. The smoothing is carried out by convolution with a Gaussian filter. When obtained with Gaussian filtering, it can be proven that zero crossings of the signal (e.g., curvature or its derivative with arc length) find their origin in the original signal. The technique thus yields a list of gross features (e.g., inflection points or curvature extrema), which really stem from the original high-resolution shape and are not merely spurious artifacts. An interesting theoretical result is the possibility of reconstructing the original contour from the zero crossings (e.g., curvature extrema) by making the smoothing process continuous, that is, by widening the Gaussian filter continuously. Smoothing one-dimensional contour descriptions has been criticized because such an approach is blind to the actual two-dimensional shape in the image, and therefore may unfortunately yield unnatural results. Closed contours are not necessarily rendered as closed, and the topology of the original contours is often not taken into account. Nevertheless, the idea of coarse-to-fine analysis is important. Not only have some remedies for these problems been presented, this idea of descriptions at at multiple resolutions is applied to several other applications in vision. We may remind the reader of the pyramid architecture presented in Section 5.
Edge Analytics
Published in Chandrasekar Vuppalapati, Building Enterprise IoT Applications, 2019
Smoothing is a technique applied to time series in order to remove the fine-grained variation between time steps. The goal of smoothing is to remove noise and better expose the signal of the underlying causal processes. Moving averages are a simple and common type of smoothing used in time series analysis and time series forecasting. Calculating a moving average involves creating a new series where the values are comprised of the average of raw observations in the original time series. A moving average requires that you specify a window size called the window width. This defines the number of raw observations used to calculate the moving average value. The “moving” part in the moving average refers to the fact that the window defined by the window width is slid along the time series in order to calculate the average values in the new series.
Recognition and analysis of fabric texture by double-sided fusion of transmission and reflection images under compound light source
Published in The Journal of The Textile Institute, 2022
Mingzhu Fan, Na Deng, Binjie Xin, Runhu Zhu
The sharp noise in transmission image and reflection image will affect the results of yarn location and tissue point classification respectively, so it is necessary to denoise the image. Common image smoothing methods include median filter, mean filter and Gaussian filter. This paper adopts mean filtering, which selects a template (composed of several adjacent elements) for a pixel in the currently processed image and replaces the value of the original pixel with the mean of all pixels in the template (Wang et al., 2010). The formula can be expressed as: where f (x, y) is the pixel value of the image, s is the filter template, n is the number of pixels contained in the current template and G (x, y) is the filtered image. The effect of mean filtering is directly related to the size and shape of the selected filtering template. The commonly used shapes are square and cross. The size of square template is generally 3 × 3 and different effects can be achieved by changing the shape of the template and the step size of the filter.
Denoising Algorithm for Subtle Anomaly Detection
Published in Nuclear Technology, 2022
Arvind Sundaram, Yeni Li, Hany Abdel-Khalik
The moving average filter replaces a measurement by computing an unweighted average of its neighboring values, thus achieving a denoising effect. The smoothing is enhanced by increasing the size of the window in this technique; however, information about relevant peak behavior may be lost to excessive smoothing. Weighted moving average and exponential moving average (EMA) filters are closely related concepts that assign weights to the values in the window. For example, in exponential smoothing, recent values are assigned exponentially larger weights than past values.26 Fourier smoothing involves a domain transformation via the Fourier transform. Here, the time series is decomposed into low-frequency and high-frequency components, the latter of which typically contain most of the noise. The high-frequency components are then attenuated with a low-pass filter using a desired cutoff frequency.27 The Fourier transform is further generalized by the wavelet transform, in which the dominant coefficients are unaltered while the small coefficients typically describing noise are zeroed to achieve a denoising effect.28 Generally speaking, the above methods are advantageous when there is no adequate physical model to describe the process and can be used to preprocess the data to reduce noise for feature extraction. They are also simple to implement algorithmically and may run significantly faster than their model-based counterparts.29
Trajectory reconstruction using locally weighted regression: a new methodology to identify the optimum window size and polynomial order
Published in Transportmetrica A: Transport Science, 2018
Suvin P. Venthuruthiyil, Mallikarjuna Chunchu
Some of the other smoothing techniques such as the moving average technique, rlowess, and Savitzky–Golay filter were also considered for the comparison. Figures 10–12 show the observed and the reconstructed position, speed and acceleration values corresponding to the proposed method as well as the other methods. A comparison of jerk was also performed and shown in Figure 13. Table 1 shows that most of the methods result in comparatively lesser error than the proposed method for the estimated position, speed, and acceleration. However, from Figures 9 to 13, it is clear that the error statistics are not appropriate for comparing the trajectory smoothing techniques. Hence, the minimization of the error is not the best way for the reconstruction of the vehicle trajectories. Internal consistency evaluation for all these methods are made and the results are as shown in Table 2. It is evident from the table that the proposed methodology gives better smoothing of the observed trajectory even though it is resulting in comparatively higher MAE and RMSE values shown in Table 1.