DOCSLIB.ORG

On the Equivalence of Time and Frequency Domain Maximum Likelihood Estimation ?

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
On the Equivalence of Time and Frequency Domain Maximum Likelihood Estimation ? On the Equivalence of Time and Frequency Domain Maximum Likelihood Estimation ? Juan C. Ag¨uero a , Juan I. Yuz b , Graham C. Goodwin a , and Ram´onA. Delgado b aARC Centre for Complex Dynamics Systems and Control (CDSC) School of Electrical Engineering and Computer Science, The University of Newcastle, Australia bDepartment of Electronic Engineering Universidad T´ecnica Federico Santa Mar´ıa,Valpara´ıso,Chile Abstract Maximum likelihood estimation has a rich history. It has been successfully applied to many problems including dynamical system identification. Different approaches have been proposed in the time and in the frequency domains. In this paper we discuss the relationship between these approaches and we establish conditions under which the different formulations are equivalent for finite length data. A key point in this context is how initial (and final) conditions are considered and how they are introduced in the likelihood function. Key words: Maximum likelihood estimation, frequency domain identification, system identification. 1 Introduction In [34], Box-Jenkins identification has been analyzed. Extensions to identification in closed loop have also been Maximum Likelihood (ML) estimation methods have be- presented [34]. Also the case of reduced bandwidth esti- come a popular approach to dynamic system identifica- mation has been considered. A surprising result, in this tion [10,40,18]. Different approaches have been proposed context, is that, for processes operating in open loop, in the time and frequency domains [19,17,21,33,34]. the commonly used frequency domain ML method re- A commonly occurring question is how time- and quires exact knowledge of the noise model in order to frequency-domain versions of ML estimation are related. obtain consistent estimates for the plant parameters. Some insights into the relationship between the methods On the other hand, it is well known that the commonly have been given in past literature. However, to the best used ML in the time domain (for systems driven by a of the authors knowledge, there has not previously been quasi-stationary input and Gaussian white noise that a comprehensive account of the equivalence between the are mutually uncorrelated) provides consistent estimates two approaches, in particular, for finite length data. For for the transfer function from input to output irrespec- example, [17,21,23] have shown that, when working in tive of possible under-modelling of the transfer func- the frequency domain, an extra term arises in the like- tion from noise to output [18]. This fact suggests that lihood function that depends on the noise model. This there could be key differences between the time- and term vanishes asymptotically for long data sets when frequency-domain approaches in the usual formats. In considering uniformly spaced frequency points over the the current paper we will see that the apparent differ- full bandwidth [−π; π] (see, for example, [34]). ences are a result of inconsistent formulations rather than fundamental issues between the use of time or fre- ? quency domain data. In particular, we establish in this This paper was not presented at any IFAC meeting. Corre- paper that the domain chosen to describe the available sponding author: Tel. +61-2-49216351, Fax +61-2-49601712. Work supported by CDSC and FONDECYT-Chile through data (i.e., time or frequency) does not change the result grant 11070158. of the estimation problem (see also [39,19]). Instead, it is Email addresses: [email protected] (Juan the choice of the likelihood function, i.e., which parame- C. Ag¨uero), [email protected] (Juan I. Yuz), ters are to be estimated and what data is assumed avail- [email protected] (Graham C. able, that leads to perceived differences in the estimation Goodwin), [email protected] (Ram´onA. problems. This issue has previously been highlighted for Delgado). Preprint submitted to Automatica 26 October 2009 time domain methods in the statistics literature where, stable, with no poles on the unit circle; (ii) H−1(q) for example, the way in which initial conditions are con- is stable (i.e., H(q) is minimum phase, with no zeros sidered defines different likelihood functions and, thus, on the unit circle); and (iii) limq!1 H(q) = 1. The different estimation problems (see e.g. [36, chapter 22]). transfer function description of the system in (1) can More specifically, for dynamic system identification, the equivalently be represented in state space form as: time domain likelihood function is different depending on the assumptions made regarding the initial state (x0), e.g. xt+1 = A xt + B ut + K wt (2) yt = C xt + D ut + wt (3) (T1) x0 is assumed to be zero, (T2) x0 is assumed as a deterministic parameter to be estimated, or The system parameter vector θ contains the coeffi- (T3) x is assumed to be a random vector. cients of the transfer functions G(q) and H(q) in (1). 0 This parameter vector also uniquely defines the matri- On the other hand, if we convert the data to the fre- ces A; B; C; D; K in the state space representation in quency domain by applying the discrete Fourier trans- (2){(3) for controllable or observable canonical forms form (DFT) then a term arises which depends on the dif- [16,22,5]. The two alternative models are related by ference between the initial and final states, α = x0 −xN . Different assumptions can be made about this term in G(q) = C(qI − A)−1B + D (4) the frequency domain, e.g. H(q) = C(qI − A)−1K + 1 (5) (F1) α is assumed to be zero (equivalent to assuming periodicity in the state) The initial conditions in (2) summarize the past of the (F2) α is estimated as a deterministic parameter (as in, system prior to time t = 0. To include the effect of ini- e.g., [1,34]), or tial conditions on the system response, we note that the (F3) α is considered as a hidden random variable. solution of the state-space model (2){(3) can be written as In this paper we show that the case when the term α is considered as a random variable is the most general. In fact, we show that each of the six cases described yt = F (q)st + G(q)ut + H(q)wt (6) above, i.e., (T1){(T3) and (F1){(F3), can be obtained "t−1 # X by making particular assumptions regarding the statis- = CAt x + CAt−1−`B u + Du tical properties of the random variable α and x . In par- 0 ` t 0 `=0 ticular, our analysis shows that the same solution is ob- "t−1 # X tained (in the time and in the frequency domain, and + CAt−1−`K w + w (7) using finite data) only if the properties of α as a ran- ` t dom variable are chosen such that they are consistent `=0 with the system dynamics and the way the initial state is considered (Theorem 21 in Section 4.2). Throughout the where the additional term st captures the effect of an paper we assume that the system is operating in open initial state x0 on the system response. If we interpret loop. Closed loop data can be treated in a similar fash- st as a Kronecker delta function, i.e., st = x0δK [t], then ion by the inclusion of additional terms (see for example the transfer functions in (6) are given by (4), (5), and Remark 2 below). F (q) = C(qI − A)−1q (8) 2 Time Domain Maximum Likelihood 2.1 Time-domain model and data For the sake of simplicity, we represent the system re- sponse using block matrices. Equation (7) can then be We consider the following Single-Input Single-Output rewritten as (SISO) linear system model: ~y = Γx0 + Λ~u + Ω~w (9) yt = G(q)ut + H(q)wt (1) where where futg and fytg are the (sampled time-domain) T input and output signals, respectively, and fwtg is zero ~y = [y0; : : : ; yN−1] (10) mean Gaussian noise with variance σ2 . G(q; θ) and T w ~u = [u0; : : : ; uN−1] (11) H(q; θ) are rational functions in the forward shift op- T erator q. We also assume that: (i) G(q) and H(q) are ~w = [w0; : : : ; wN−1] (12) 2 and matrix for the output data given the parameters, i.e. 2 3 2 3 C D 0 ::: 0 µ~y = Λ~u + Γµx0 (17) 6 7 6 7 Σ = ΓΣ ΓT + σ2 ΩΩT (18) 6 CA 7 6 CBD::: 0 7 ~y 0 w 6 7 6 7 Γ = 6 . 7 ; Λ = 6 . 7 6 . 7 6 . .. 7 4 5 4 5 CAN−1 CAN−2BCAN−3B:::D PROOF. The likelihood of the data given the param- (13) eter vector θ and the random vector x0 can be obtained from (9): 2 3 I 0 ::: 0 " # h i x0 6 7 ~y = Λ~u + ΓΩ (19) 6 CKI::: 07 6 7 ~w Ω = 6 . .7 (14) 6 . .. .7 4 5 Using the PDF of a transformation of random variables CAN−2KCAN−3K:::I (see, for example, [15, page 34]), we have that the PDF for ~y is readily obtained from the affine transformation in (19), i.e. 2.2 Time domain maximum likelihood n 1 T −1 o exp − 2 (~y − µ~y) Σ~y (~y − µ~y) p~y(~y) = (20) p N The time-domain likelihood function is defined as the (2π) det Σ~y conditional probability density function (PDF) of the data given the parameters, i.e. where the mean and covariance are given by equations (17) and (18). The negative log-likelihood function is readily obtained from (20).
Load more

Recommended publications
  • Moving Average Filters
    CHAPTER 15 Moving Average Filters The moving average is the most common filter in DSP, mainly because it is the easiest digital filter to understand and use. In spite of its simplicity, the moving average filter is optimal for a common task: reducing random noise while retaining a sharp step response. This makes it the premier filter for time domain encoded signals. However, the moving average is the worst filter for frequency domain encoded signals, with little ability to separate one band of frequencies from another. Relatives of the moving average filter include the Gaussian, Blackman, and multiple- pass moving average. These have slightly better performance in the frequency domain, at the expense of increased computation time. Implementation by Convolution As the name implies, the moving average filter operates by averaging a number of points from the input signal to produce each point in the output signal. In equation form, this is written: EQUATION 15-1 Equation of the moving average filter. In M &1 this equation, x[ ] is the input signal, y[ ] is ' 1 % y[i] j x [i j ] the output signal, and M is the number of M j'0 points used in the moving average. This equation only uses points on one side of the output sample being calculated. Where x[ ] is the input signal, y[ ] is the output signal, and M is the number of points in the average. For example, in a 5 point moving average filter, point 80 in the output signal is given by: x [80] % x [81] % x [82] % x [83] % x [84] y [80] ' 5 277 278 The Scientist and Engineer's Guide to Digital Signal Processing As an alternative, the group of points from the input signal can be chosen symmetrically around the output point: x[78] % x[79] % x[80] % x[81] % x[82] y[80] ' 5 This corresponds to changing the summation in Eq.
    [Show full text]
  • Power Grid Topology Classification Based on Time Domain Analysis
    Power Grid Topology Classification Based on Time Domain Analysis Jia He, Maggie X. Cheng and Mariesa L. Crow, Fellow, IEEE Abstract—Power system monitoring is significantly im- all depend on the correct network topology information. proved with the use of Phasor Measurement Units (PMUs). Topology change, no matter what caused it, must be The availability of PMU data and recent advances in ma- immediately updated. An unattended topology error may chine learning together enable advanced data analysis that lead to cascading power outages and cause large scale is critical for real-time fault detection and diagnosis. This blackout ( [2], [3]). In this paper, we demonstrate that paper focuses on the problem of power line outage. We use a machine learning framework and consider the problem using a machine learning approach and real-time mea- of line outage identification as a classification problem in surement data we can timely and accurately identify the machine learning. The method is data-driven and does outage location. We leverage PMU data and consider not rely on the results of other analysis such as power the task of identifying the location of line outage as flow analysis and solving power system equations. Since it a classification problem. The methods can be applied does not involve using power system parameters to solve in real-time. Although training a classifier can be time- equations, it is applicable even when such parameters are consuming, the prediction of power line status can be unavailable. The proposed method uses only voltage phasor done in real-time. angles obtained from PMUs.
    [Show full text]
  • Logistic-Weighted Regression Improves Decoding of Finger Flexion from Electrocorticographic Signals
    Logistic-weighted Regression Improves Decoding of Finger Flexion from Electrocorticographic Signals Weixuan Chen*-IEEE Student Member, Xilin Liu-IEEE Student Member, and Brian Litt-IEEE Senior Member Abstract— One of the most interesting applications of brain linear Wiener filter [7–10]. To take into account the computer interfaces (BCIs) is movement prediction. With the physiological, physical, and mechanical constraints that development of invasive recording techniques and decoding affect the flexion of limbs, some studies applied switching algorithms in the past ten years, many single neuron-based and models [11] or Bayesian models [12,13] to the results of electrocorticography (ECoG)-based studies have been able to linear regressions above. Other studies have explored the decode trajectories of limb movements. As the output variables utility of non-linear methods, including neural networks [14– are continuous in these studies, a regression model is commonly 17], multilinear perceptrons [18], and support vector used. However, the decoding of limb movements is not a pure machines [18], but they tend to have difficulty with high regression problem, because the trajectories can be apparently dimensional features and limited training data [13]. classified into a motion state and a resting state, which result in a binary property overlooked by previous studies. In this Nevertheless, the studies of limb movement translation paper, we propose an algorithm called logistic-weighted are in fact not pure regression problems, because the limbs regression to make use of the property, and apply the algorithm are not always under the motion state. Whether it is during an to a BCI system decoding flexion of human fingers from ECoG experiment or in the daily life, the resting state of the limbs is signals.
    [Show full text]
  • STATISTICAL FOURIER ANALYSIS: CLARIFICATIONS and INTERPRETATIONS by DSG Pollock
    STATISTICAL FOURIER ANALYSIS: CLARIFICATIONS AND INTERPRETATIONS by D.S.G. Pollock (University of Leicester) Email: stephen [email protected] This paper expounds some of the results of Fourier theory that are es- sential to the statistical analysis of time series. It employs the algebra of circulant matrices to expose the structure of the discrete Fourier transform and to elucidate the filtering operations that may be applied to finite data sequences. An ideal filter with a gain of unity throughout the pass band and a gain of zero throughout the stop band is commonly regarded as incapable of being realised in finite samples. It is shown here that, to the contrary, such a filter can be realised both in the time domain and in the frequency domain. The algebra of circulant matrices is also helpful in revealing the nature of statistical processes that are band limited in the frequency domain. In order to apply the conventional techniques of autoregressive moving-average modelling, the data generated by such processes must be subjected to anti- aliasing filtering and sub sampling. These techniques are also described. It is argued that band-limited processes are more prevalent in statis- tical and econometric time series than is commonly recognised. 1 D.S.G. POLLOCK: Statistical Fourier Analysis 1. Introduction Statistical Fourier analysis is an important part of modern time-series analysis, yet it frequently poses an impediment that prevents a full understanding of temporal stochastic processes and of the manipulations to which their data are amenable. This paper provides a survey of the theory that is not overburdened by inessential complications, and it addresses some enduring misapprehensions.
    [Show full text]
  • Validity of Spatial Covariance Matrices Over Time and Frequency
    Validity of Spatial Covariance Matrices over Time and Frequency Ingo Viering'1,Helmu t Hofstette?,W olfgang Utschick 3, ')Forschungszenmm ')Institute for Circuit Theory and Signal Processing ')Siemens AG Telekommunikation Wien Munich University of Technology Lise-Meitner-StraRe 13 Donau-City-StraRe I Arcisstraae 21 89081 Ulm, Germany 1220 Vienna, Austria 80290 Munich, Germany [email protected] [email protected] [email protected] Abstract-A MIMOchaonrl measurement campaign with s moving mo- measurement data in section IV. Finally, section V draws some bile has heen moducted in Vienna. The measund data rill be used to conclusions. investigate covariance matrices with rapst to their dependence on lime and frequency. This document fmra 00 the description ofthe WaIUatiO1I 11. MEASUREMENTSETUP techniqoa which rill be applied to the measurement data in fhe future. The F-eigen-raHo is defined expressing the degradation due to outdated The measurements were done with the MlMO capable wide- covariance maMcer. IUuitntiag the de*& methods, first nsulti based band vector channel sounder RUSK-ATM, manufactured by 00 the mrasored daU are rho- for a simple line4-sight scemlio. MEDAV [I]. The sounder was specifically adapted to operate at a center frequency of 2GHz with an output power of 2 Wan. The transmitted signal is generated in frequency domain to en- 1. INTRODUCTION sure a pre-defined spectrum over 120MHz bandwidth, and ap- proximately a constant envelope over time. In the receiver the All mobile communication systems incorporating multiple input signal is correlated with the transmitted pulse-shape in antennas on one or on both sides of the transmission link the frequency domain resulting in the specific transfer func- strongly depend on the spatial structure of the mobile chan- tions.
    [Show full text]
  • Ultrasonic Velocity Measurement Using Phase-Slope and Cross- Correlation Methods
    H84 - NASA Technical Memorandum 83794 Ultrasonic Velocity Measurement Using Phase-Slope and Cross- Correlation Methods David R. Hull, Harold E. Kautz, and Alex Vary Lewis Research Center Cleveland, Ohio Prepared for the 1984 Spring Conference of the American Society for Nondestructive Testing Denver, Colorado, May 21-24, 1984 NASA ULTRASONIC VELOCITY MEASUREMENT USING PHASE SLOPE AND CROSS CORRELATION HE I HODS David R. Hull, Harold E. Kautz, and Alex Vary National Aeronautics and Space Administration Lewis Research Center Cleveland, Ohio 44135 SUMMARY Computer Implemented phase slope and cross-correlation methods are In- troduced for measuring time delays between pairs of broadband ultrasonic pulse-echo signals for determining velocity 1n engineering materials. The phase slope and cross-correlation methods are compared with the overlap method §5 which 1s currently 1n wide use. Comparison of digital versions of the three £J methods shows similar results for most materials having low ultrasonic attenu- uj atlon. However, the cross-correlation method 1s preferred for highly attenua- ting materials. An analytical basis for the cross-correlation method 1s presented. Examples are given for the three methods investigated to measure velocity in representative materials in the megahertz range. INTRODUCTION Ultrasonic velocity measurements are widely used to determine properties and states of materials, tn the case of engineering solids measurements of ultrasonic wave propagation velocities are routinely used to determine elastic constants (refs;. 1 to 5). There has been an Increasing use of ultrasonic velocity measurements for nondestructive characterization of material micro- structures and mechanical properties (refs. 6 to 10). Therefore, 1t 1s Impor- tant to have appropriate practical methods for making velocity measurements on a variety of material samples.
    [Show full text]
  • Advanced Antenna Measurement Techniques Using Time Domain Gating
    Advanced Antenna Measurement Techniques using Time Domain Gating Zhong Chen Director, RF Engineering ETS-Lindgren ©2019 ETS-LINDGREN ©2019 ETS-LINDGREN Where do we use TD gating in EMC? TD Gating is a function provided in VNAs: • Antenna measurements • Chamber Qualifications – the new C63.25 TD sVSWR • Cable/Signal integrity measurement • General RF/Microwave loss and reflection measurements • It is a common tool in labs, but rarely fully understood and can be misused. ©2019 ETS-LINDGREN 2 1 Goals of this Presentation • Understand how VNA performs time domain transform and gating. • Understand the nuances of the different parameters, and their effects on time domain gating. • Discuss gating band edge errors, mitigation techniques and limitations of the post-gate renormalization used in a VNA. • Application Example: C63.25 Time Domain site VSWR. ©2019 ETS-LINDGREN 3 Background on Frequency/Time • Time domain data is obtained mathematically from frequency domain. Vector antenna responses in frequency domain can be transformed to time domain. This is a function in commercial Vector Network Analyzers (VNA). • Time Domain and frequency domain are in reciprocal space (via Fourier Transform), transformed from one to the other without any loss of information. They are two ways of viewing the same information. • Bandlimited frequency signals (no DC) is transformed to impulse response in TD. TD step response requires DC, and integration of the impulse response. ©2019 ETS-LINDGREN 2 Two views of the same function https://tex.stackexchange.com/questions /127375/replicate-the-fourier-transform -time-frequency-domains-correspondence Source: wikipedia -illustrati?noredirect=1&lq=1 ©2019 ETS-LINDGREN 5 Do more with time domain gating • Time gating can be thought of as a bandpass filter in time.
    [Show full text]
  • The Wavelet Tutorial Second Edition Part I by Robi Polikar
    THE WAVELET TUTORIAL SECOND EDITION PART I BY ROBI POLIKAR FUNDAMENTAL CONCEPTS & AN OVERVIEW OF THE WAVELET THEORY Welcome to this introductory tutorial on wavelet transforms. The wavelet transform is a relatively new concept (about 10 years old), but yet there are quite a few articles and books written on them. However, most of these books and articles are written by math people, for the other math people; still most of the math people don't know what the other math people are 1 talking about (a math professor of mine made this confession). In other words, majority of the literature available on wavelet transforms are of little help, if any, to those who are new to this subject (this is my personal opinion). When I first started working on wavelet transforms I have struggled for many hours and days to figure out what was going on in this mysterious world of wavelet transforms, due to the lack of introductory level text(s) in this subject. Therefore, I have decided to write this tutorial for the ones who are new to the topic. I consider myself quite new to the subject too, and I have to confess that I have not figured out all the theoretical details yet. However, as far as the engineering applications are concerned, I think all the theoretical details are not necessarily necessary (!). In this tutorial I will try to give basic principles underlying the wavelet theory. The proofs of the theorems and related equations will not be given in this tutorial due to the simple assumption that the intended readers of this tutorial do not need them at this time.
    [Show full text]
  • Covariance Matrix Estimation in Time Series Wei Biao Wu and Han Xiao June 15, 2011
    Covariance Matrix Estimation in Time Series Wei Biao Wu and Han Xiao June 15, 2011 Abstract Covariances play a fundamental role in the theory of time series and they are critical quantities that are needed in both spectral and time domain analysis. Es- timation of covariance matrices is needed in the construction of confidence regions for unknown parameters, hypothesis testing, principal component analysis, predic- tion, discriminant analysis among others. In this paper we consider both low- and high-dimensional covariance matrix estimation problems and present a review for asymptotic properties of sample covariances and covariance matrix estimates. In particular, we shall provide an asymptotic theory for estimates of high dimensional covariance matrices in time series, and a consistency result for covariance matrix estimates for estimated parameters. 1 Introduction Covariances and covariance matrices play a fundamental role in the theory and practice of time series. They are critical quantities that are needed in both spectral and time domain analysis. One encounters the issue of covariance matrix estimation in many problems, for example, the construction of confidence regions for unknown parameters, hypothesis testing, principal component analysis, prediction, discriminant analysis among others. It is particularly relevant in time series analysis in which the observations are dependent and the covariance matrix characterizes the second order dependence of the process. If the underlying process is Gaussian, then the covariances completely capture its dependence structure. In this paper we shall provide an asymptotic distributional theory for sample covariances and convergence rates for covariance matrix estimates of time series. In Section 2 we shall present a review for asymptotic theory for sample covariances of stationary processes.
    [Show full text]
  • Time-Domain Analysis of Continuous-Time Systems*
    Time-Domain Analysis of Continuous-Time Systems* *Systems are LTI from now on unless otherwise stated Recall course objectives Main Course Objective: Fundamentals of systems/signals interaction (we’d like to understand how systems transform or affect signals) Specific Course Topics: -Basic test signals and their properties -Systems and their properties -Signals and systems interaction Time Domain: convolution Frequency Domain: frequency response -Signals & systems applications: audio effects, filtering, AM/FM radio -Signal sampling and signal reconstruction Signals & Systems interaction in the TD Goals I. Impulse Response (IR) and Convolution Formula -Definition of IR and its use for system identification -Convolution formula and its graphical interpretation II. Properties of systems from IR and convolution -Impulse response as a measure of system memory/stability -Alternative measures of memory/stability: step response III. Applications of convolution -Audio effects: reverberation -Noise removal (i.e. signal filtering or smoothing) The Impulse Response Problem: Find the response of a system to an impulsive input. "(t) S h(t) h ( t) is called the Impulse Response. ! ! Use what you know about LTI systems to compute h ( t ). ! Easy to “compute” in state space. ! The Impulse Response Problem: Find the response of an LTI system in state space to an impulsive input. Solution: If the LTI system is causal it can be represented in state space. We are looking for a solution to Since the system is causal, and assuming zero initial conditions, we have that for all . In particular . Integrating from to The Impulse Response Solution (continued): Fact: does not contain impulses! Therefore From this point on the system has zero input and its response is the homogenous solution: Putting it all together: where is the impulse response.
    [Show full text]
  • What Is the Frequency Domain? We Live in the Time Domain. the Sounds
    What is the Frequency Domain? We live in the time domain. The sounds that our ears perceive exist in time, and we have no other way to perceive them. Sound cannot be frozen like a moving image. Stop a movie, and you see a fully discernible image. Stop a sound and it ceases to exist. However, we can use computers to analyze frozen signals. We do this by transforming sound from the time domain into the frequency domain using the Fourier Transform. With the Fourier Transform, we can pause a signal to look at it’s momentary frequency content. We can also alter and filter thin slices of the frequency spectrum. In the time domain, we think about sound as a series of pulse waves that hit our ear over a period of time. We also often picture a sound like a sine wave or more complex signal. In the frequency domain, we are able to look at a small slice of a signal and see what frequencies are present. We care able to look at the entire spectrum, from below the range of human hearing up to ultraviolet waves and beyond. FFT: We use an FFT or Fast Fourier Transform to convert a signal from the time domain (our world) to the frequency domain (another dimension). An FFT takes a very small section of sound from the time domain - 512, 1024, 2048 or any power of 2 number of samples - and converts this into the frequency domain. The math of the Fourier Transform is beyond the scope of this class, but you can still use it as a powerful tool for sound transformation.
    [Show full text]
  • Condition Monitoring Using Accelerometer Readings
    Condition Monitoring Using Accelerometer Readings Cheryl Danner, Kelly Gov, Simon Xu Stanford University Abstract—In the field of condition monitoring, machine learn- around the frequencies that are indicative of faults [3]. This is ing techniques are actively being developed to detect and/or good for certain fault patterns, such as the raised harmonics predict faults in equipment. In this paper, the authors prototype present in mechanical looseness, but not so good for other a process to monitor the condition of rotating machinery using accelerometer readings. Time domain and frequency domain common problems, such as rolling bearing wear, which are features are used to categorize data samples, with performance present in subharmonic frequencies. While there are simpler compared for three supervised learning algorithms: logistic techniques, a trained machine learning algorithm can be the regression, Naive Bayes, and SVM. Two unsupervised learning most effective at classifying machinery problems from spectral algorithms, k-means and mixture of Gaussians, are investigated patterns. for use in filtering data initially and identifying machine mal- functions. Logistic regression gives the lowest error rate in tests run on the available dataset, classifying all test samples correctly III. DATASET AND FEATURES using time or frequency features. A combination of k-means and mixture of Gaussians algorithms shows potential for filtering data To train and test our models, we have obtained several and identifying machine malfunctions. datasets from different rotating machines, labeled M1, M3, M4, and SN41, which refer to a feed pump, a condensate I. INTRODUCTION pump, the motor driving the condensate pump, and a shaker In the industrial and manufacturing world, a vibration motor, respectively.
    [Show full text]