Spectral Analysis and Filtering

Shumway, Robert H.; Stoffer, David S.

doi:10.1007/978-1-4419-7865-3_4

Cited by 6 publications

(7 citation statements)

References 19 publications

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“…The Fourier transform (FT) of a discrete stationary series of length M equispaced at intervals Δ s ( x s , s = 0,1,…, M −1) (where x is the variable to be transformed and s is the spatial or temporal location on the series) is defined as (Shumway, ):

X ((), k) = M^{prefix- 1} true {truefalse\sum}_{s = 0}^{M - 1} ((), x_{s} - \overset{x}{true}) normale normalx normalp ((), prefix- 2 π i ν_{k} s),

…”

Section: Discussionmentioning

confidence: 99%

“…The periodogram may be written as the squared modulus of the FT:

P_{x} ((), ν_{k}) = {|X (k)|}^{2} = ([], X_{C}^{2} ((), k) + X_{S}^{2} ((), k)) = X ((), k) \overset{X ((), k)}{true},

where the overbar denotes a complex conjugate, which is an approximately unbiased estimator for the spectrum (Shumway, ). It is common practice to average adjacent values of the periodogram to obtain estimates with more degrees of freedom, and so create a smoothed power spectrum.…”

Section: Discussionmentioning

confidence: 99%

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Calibration of an electromagnetic induction sensor with time‐domain reflectometry data to monitor rootzone electrical conductivity under saline water irrigation

Coppola

Smettem

Ajeel

et al. 2016

European J Soil Science

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Management of saline water irrigation at the field scale requires electrical conductivity to be monitored regularly in the generally shallow soil layer explored by plant roots. Non-invasive electromagnetic induction (EMI) sensors might be valuable for evaluating large-scale soil salinity. However, obtaining information on soil surface salinity from depth-integrated EMI measurements requires either an inversion of the electromagnetic signal or an empirical calibration. We opted for an empirical calibration of an EM38 sensor, which requires a reference dataset of local bulk electrical conductivity, sigma(b), for comparison with EMI readings for estimating regression coefficients. We used time-domain reflectometry (TDR) to replace direct sampling for local sigma(b) measurements. With empirical approaches, the different soil volumes involved with the EMI and TDR sensors become problematic. We resolved this issue by analysing large EMI and TDR datasets recorded at several times along three transects irrigated with water at 1, 3 and 6dSm(-1) degrees of salinity. A Fourier filtering technique was applied to remove the high frequency part (at small spatial scales) of the variation in the original data, which was the main source of dissimilarity between the two datasets. Therefore, calibration focused on the lower frequency information only; that is, information at a spatial scale larger than the observation volume of the sensors. We show that information from the TDR observations derives from a combination of local and larger scale heterogeneities, and how it should be managed for calibration of the EMI sensor. Analysis enabled us to identify characteristics of the calibration data that should be included to improve predictio

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X ((), k) = M^{prefix- 1} true {truefalse\sum}_{s = 0}^{M - 1} ((), x_{s} - \overset{x}{true}) normale normalx normalp ((), prefix- 2 π i ν_{k} s),

…”

Section: Discussionmentioning

confidence: 99%

“…The periodogram may be written as the squared modulus of the FT:

P_{x} ((), ν_{k}) = {|X (k)|}^{2} = ([], X_{C}^{2} ((), k) + X_{S}^{2} ((), k)) = X ((), k) \overset{X ((), k)}{true},

Section: Discussionmentioning

confidence: 99%

Calibration of an electromagnetic induction sensor with time‐domain reflectometry data to monitor rootzone electrical conductivity under saline water irrigation

Coppola

Smettem

Ajeel

et al. 2016

European J Soil Science

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“…The coherence between Y i and Y j at frequency λ is a measure of correlation in frequency and is defined as coh i j (λ ) = | f i j (λ )| 2 /( f i (λ ) * f j (λ )) where f i j (λ ) is the cross-spectral density between Y i and Y j and f i (λ ), f j (λ ) are the corresponding spectral densities for each process (see e.g. Shumway and Stoffer, 2011). The mutual information within frequency band [λ 1 , λ 2 ] is then…”

Section: Mutual Information Networkmentioning

confidence: 99%

High-Dimensional Classification for Brain Decoding

Croteau

Nathoo

Cao

et al. 2017

Contributions to Statistics

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Brain decoding involves the determination of a subject's cognitive state or an associated stimulus from functional neuroimaging data measuring brain activity. In this setting the cognitive state is typically characterized by an element of a finite set, and the neuroimaging data comprise voluminous amounts of spatiotemporal data measuring some aspect of the neural signal. The associated statistical problem is one of classification from high-dimensional data. We explore the use of functional principal component analysis, mutual information networks, and persistent homology for examining the data through exploratory analysis and for constructing features characterizing the neural signal for brain decoding. We review each approach from this perspective, and we incorporate the features into a classifier based on symmetric multinomial logistic regression with elastic net regularization. The approaches are illustrated in an application where the task is to infer, from brain activity measured with magnetoencephalography (MEG), the type of video stimulus shown to a subject.

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“…The ordinary spectrum is a critical feature of Gaussian time series, and it is often used to classify or cluster time series. For example, Orhan et al (2011) Motivated by the function approximation capability of the autoregressive (AR) spectrum (Shumway and Stoffer, 2011), we propose a new method that uses the parametric form of the AR spectrum to approximate the quantile spectrum at each quantile level. The novelty of our approach is threefold.…”

Section: Introductionmentioning

confidence: 99%

A Semi-Parametric Estimation Method for the Quantile Spectrum with an Application to Earthquake Classification Using Convolutional Neural Network

Chen

Sun

Li³

2019

Preprint

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In this paper, a new estimation method is introduced for the quantile spectrum, which uses a parametric form of the autoregressive (AR) spectrum coupled with nonparametric smoothing. The method begins with quantile periodograms which are constructed by trigonometric quantile regression at different quantile levels, to represent the serial dependence of time series at various quantiles. At each quantile level, we approximate the quantile spectrum by a function in the form of an ordinary AR spectrum. In this model, we first compute what we call the quantile autocovariance function (QACF) by the inverse Fourier transformation of the quantile periodogram at each quantile level. Then, we solve the Yule-Walker equations formed by the QACF to obtain the quantile partial autocorrelation function (QPACF) and the scale parameter. Finally, we smooth QPACF and the scale parameter across the quantile levels using a nonparametric smoother, convert the smoothed QPACF to AR coefficients, and obtain the AR spectral density function.Numerical results show that the proposed method outperforms other conventional smoothing techniques. We take advantage of the two-dimensional property of the estimators and train a convolutional neural network (CNN) to classify smoothed quantile periodogram of earthquake data and achieve a higher accuracy than a similar classifier using ordinary periodograms.

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Spectral Analysis and Filtering

Cited by 6 publications

References 19 publications

Calibration of an electromagnetic induction sensor with time‐domain reflectometry data to monitor rootzone electrical conductivity under saline water irrigation

Calibration of an electromagnetic induction sensor with time‐domain reflectometry data to monitor rootzone electrical conductivity under saline water irrigation

High-Dimensional Classification for Brain Decoding

A Semi-Parametric Estimation Method for the Quantile Spectrum with an Application to Earthquake Classification Using Convolutional Neural Network

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