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Concept# Cross-correlation

Summary

In signal processing, cross-correlation is a measure of similarity of two series as a function of the displacement of one relative to the other. This is also known as a sliding dot product or sliding inner-product. It is commonly used for searching a long signal for a shorter, known feature. It has applications in pattern recognition, single particle analysis, electron tomography, averaging, cryptanalysis, and neurophysiology. The cross-correlation is similar in nature to the convolution of two functions. In an autocorrelation, which is the cross-correlation of a signal with itself, there will always be a peak at a lag of zero, and its size will be the signal energy.
In probability and statistics, the term cross-correlations refers to the correlations between the entries of two random vectors and , while the correlations of a random vector are the correlations between the entries of itself, those forming the correlation matrix of . If each of and is a scalar random variable which is realized repeatedly in a time series, then the correlations of the various temporal instances of are known as autocorrelations of , and the cross-correlations of with across time are temporal cross-correlations. In probability and statistics, the definition of correlation always includes a standardising factor in such a way that correlations have values between −1 and +1.
If and are two independent random variables with probability density functions and , respectively, then the probability density of the difference is formally given by the cross-correlation (in the signal-processing sense) ; however, this terminology is not used in probability and statistics. In contrast, the convolution (equivalent to the cross-correlation of and ) gives the probability density function of the sum .
For continuous functions and , the cross-correlation is defined as:which is equivalent towhere denotes the complex conjugate of , and is called displacement or lag. For highly-correlated and which have a maximum cross-correlation at a particular , a feature in at also occurs later in at , hence could be described to lag by .

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In signal processing, cross-correlation is a measure of similarity of two series as a function of the displacement of one relative to the other. This is also known as a sliding dot product or sliding inner-product. It is commonly used for searching a long signal for a shorter, known feature. It has applications in pattern recognition, single particle analysis, electron tomography, averaging, cryptanalysis, and neurophysiology. The cross-correlation is similar in nature to the convolution of two functions.

In statistics, scaled correlation is a form of a coefficient of correlation applicable to data that have a temporal component such as time series. It is the average short-term correlation. If the signals have multiple components (slow and fast), scaled coefficient of correlation can be computed only for the fast components of the signals, ignoring the contributions of the slow components. This filtering-like operation has the advantages of not having to make assumptions about the sinusoidal nature of the signals.

In mathematics, the discrete-time Fourier transform (DTFT), also called the finite Fourier transform, is a form of Fourier analysis that is applicable to a sequence of values. The DTFT is often used to analyze samples of a continuous function. The term discrete-time refers to the fact that the transform operates on discrete data, often samples whose interval has units of time. From uniformly spaced samples it produces a function of frequency that is a periodic summation of the continuous Fourier transform of the original continuous function.

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This dissertation introduces traffic forecasting methods for different network configurations and data availability.Chapter 2 focuses on single freeway cases.Although its topology is simple, the non-linearity of traffic features makes this prediction still a challenging task.We propose the dynamic linear model (DLM) to approximate the non-linear traffic features. Unlike a static linear regression model, the DLM assumes that its parameters change over time.We design the DLM with time-dependent model parameters to describe the spatiotemporal characteristics of time-series traffic data. Based on our DLM and its model parameters analytically trained using historical data, we suggest the optimal linear predictor in the minimum mean square error (MMSE) sense.We compare our prediction accuracy by estimating expected travel time based on the traffic prediction for freeways in California (I210-E and I5-S) under highly congested traffic conditions with other baselines. We show significant improvements in accuracy, especially for short-term prediction.Chapter 3 aims to generalize the DLM to extensive freeway networks with more complex topologies.Most resources would be consumed to estimate unnecessary spatiotemporal correlations if the DLM was directly used for a large-scale network.Defining features on graphs relaxes such issues by cutting unnecessary connections in advance based on predefined topology information.Exploiting the graph signal processing, we represent traffic dynamics over freeway networks using multiple graph heat diffusion kernels and integrate the kernels into DLM with Bayes' rule. We optimize the model parameters using Bayesian inference to minimize the prediction errors.The proposed model demonstrates prediction accuracy comparable to state-of-the-art deep neural networks with lower computational effort. It notably achieves excellent performance for long-term prediction through the inheritance of periodicity modeling in DLM.Chapter 4 proposes a deep neural network model to predict traffic features on large-scale freeway networks.These days, deep learning methods have heavily tackled traffic forecasting problems of freeway networks because they are outstanding at learning highly complex correlations between variables both in time and space, which the linear models might be limited to.Adopting a graph convolutional network (GCN) becomes a standard to extract spatial correlations; therefore, most works have achieved great prediction accuracy by implanting it into their architecture.However, the conventional GCN has the drawback that receptive field size should be small, i.e., barely refers to traffic features of remote sensors, resulting in inaccurate long-term prediction.We suggest a forecasting model called two-level resolution deep neural network (TwoResNet) that overcomes the limitation.It consists of two resolution blocks:The low-resolution block predicts traffic on a macroscopic scale, such as regional traffic changes.On the other hand, the high-resolution block predicts traffic on a microscopic scale by using GCN to extract spatial correlations, referring to the regional changes produced by the low-resolution block.This process allows the GCN to refer to the traffic features from remote sensors.As a result, TwoResNet achieves competitive prediction accuracy compared to state-of-the-art methods, especially showing excellent performance for long-term predictions.