1

Time-Frequency Complexity Based Remaining

Useful Life (RUL) Estimation for Bearing FaultsRodney K. Singleton II, Elias G. Strangas and Selin Aviyente

Abstract—Reliable operation of electrical machines dependson the timely detection and diagnosis of faults as well as onprognosis, i.e. estimating the remaining useful life (RUL) ofthe components. Bearings are the most common components inrotary machines and usually constitute a large portion of thefailure cases in these machines. Although there has been a lotof work in the study of bearing life failure mechanisms andmodeling, the problem is still far from being solved. In thispaper, we introduce a time-frequency feature extraction basedmethod for estimating remaining useful life of bearings fromvibration signals. The proposed approach extracts measures thatquantify the complexity of time-frequency surfaces correspondingto vibration signals. The extracted features are then trackedthrough the life time of a bearing using curve fitting and ExtendedKalman Filtering algorithms. The proposed methodology is testedon a publicly available bearing data set with known RULs.

Index Terms—Bearing Faults, Prognosis, Remaining UsefulLife, Time-Frequency Analysis, Entropy, Extended Kalman Filter

I. INTRODUCTION

The ability to accurately predict the remaining useful life

of electromechanical systems is critical for affordable system

operation and can also be used to enhance system safety.

The theme of condition-based maintenance (CBM) is that

maintenance is performed based on an assessment or pre-

diction of the component health instead of its service time,

which achieves objectives of cost reduction and safety en-

hancement. If one can predict the degradation of a component

before it actually fails, then it will provide ample time for

maintenance engineers to schedule a repair, and to acquire

replacement components before the components actually fail.

Bearings are of paramount importance to almost all forms of

rotating machinery, and are among the most common machine

elements. The failures of bearing without warning will result

in catastrophic consequences in many situations, such as in

helicopters, transportation vehicles, etc. Most of the current

maintenance procedure includes periodic visual inspections

and replacement of the components at fixed time intervals.

According to early surveys bearing faults represent the most

common cause for mechanical failure. Consequently, majority

of the proposed fault detection methods are focused on detect-

ing bearing faults. Despite such a variety of approaches, most

of them focus on extracting a set of well-established features

that indicate bearing surface faults.

R. K. Singleton, E. G. Strangas and S. Aviyente are with the Department ofElectrical and Computer Engineering, Michigan State University, East Lans-ing, MI 48824, USA (e-mail: strangas@egr.msu.edu; aviyente@egr.msu.edu).

This material is based in part upon work supported by the National ScienceFoundation under Grant No. EECS-1102316 and by the National ScienceFoundation Graduate Research Fellowship under Grant No. DGE-0802267.

Much work has been done on the diagnosis of bearing faults

using data-driven and model-based methods, such as Hidden

Markov Modeling and Particle Filtering. It has been shown

that by using current signals you can detect the presence of

a fault in the bearing as well as diagnose the severity [16],

[20]. Further work has been done on the prognosis of bearing

faults in order to obtain the Remaining Useful Life (RUL)

of bearings, or motors in general. Particle Filters have also

been used for the prognosis of fault severity [2], [5], [7],

[13]. However, one drawback of using particle filtering is

that you must have a reliable physical model for the fault

degradation. For most real- life signals and systems, including

bearings, a reliable physical model for the degradation process

is not available. Kalman Filtering has also been used for

fault diagnosis and prognosis, using an n-step ahead Kalman

Predictor for data extrapolation [12], [14].

Particularly, with bearing fault analysis, vibration data is

used over current signal since vibration data is more robust to

operating conditions. Previous studies have shown that bear-

ing diagnostics can also be performed using current signals,

but only at certain frequency rates [8]. In this paper, we

propose a data-driven method for estimating the remaining

useful life of bearings from accelerometer recordings. The

proposed method relies on extraction of features from time-

frequency distribution of the vibration signals as the signals

are nonstationary in nature. The extracted features, entropy and

concentration, quantify the spread of the energy across time

and frequency and relies on the observation that the vibration

signals become more chaotic or impulsive as the severity of

the fault progresses. The extracted features are then used by

Extended Kalman Filtering to build a degradation model and

determine a threshold value for the features from training

data. This threshold is then applied to testing data to estimate

the RUL as well as a confidence interval associated with the

estimated RUL.

II. BACKGROUND

A. Time-Frequency Analysis

In literature, different time-frequency transform methods

have been proposed for the analysis of nonstationary sig-

nals [4]. The most common approaches include the Short-

Time Fourier Transform (STFT), Wigner Distribution, wavelet

transform and Cohen’s class of time-frequency distributions.

Cohen’s class of distributions are bilinear time-frequency

distributions (TFDs) that are expressed as 1 [4]:

1All integrals are from −∞ to ∞ unless otherwise stated.

978-1-4799-0025-1/13/$31.00 ©2013 IEEE600600

2

C(t, ω) =∫ ∫ ∫

φ(θ, τ)s(u + τ2 ) (1)

s∗(u− τ2 )e

j(θu−θt−τω)du dθ dτ,

where the function φ(θ, τ) is the kernel function and s is the

signal. TFDs represent the energy distribution of a signal over

time and frequency, simultaneously. The kernel completely

determines the properties of the corresponding TFD.

The major differences between Cohen’s class of TFDs

compared to other time-frequency representations such as the

wavelet transform are the nonlinearity of the distribution,

energy preservation and the uniform resolution over time and

frequency. The wavelet transform provides a representation

over time and scale where the frequency resolution is high at

low frequencies and low at high frequencies. Although this

property makes wavelet transform attractive in detecting high

frequency transients in a given signal, it inherently imposes a

non-uniform time-frequency tiling on the analyzed signal and

thus results in biased energy representations. Cohen’s class of

bilinear TFDs on the other hand assumes uniform resolution

over the entire time-frequency plane. One popular member of

Cohen’s class of distributions is Choi-Williams distribution,

which offers both high time-frequency resolution and reduced

interference for multi-component signals. The Choi-Williams

distribution of a signal s(t) is defined as:

C(t, ω) =∫ ∫ ∫

φ(θ, τ)s(u + τ2 )s

∗(u − τ2 ) (2)

ej(θu−θt−τω)du dθ dτ,

where φ(θ, τ) = exp(− (θτ)2

σ) is the kernel function that

acts as a filter on the signal’s autocorrelation function. This

distribution can be thought of as a filtered/smoothed version

of the Wigner distribution and the amount of smoothing is

controlled by σ. This smoothing removes the cross-terms seen

in the Wigner distribution at the expense of reduced resolution.

B. Extended Kalman Filter

Kalman Filters have been used to estimate the state of a

system given a finite-length data stream. However, Kalman

Filters are only useful for linear degradation models with

additive white noise. This proves to be very disadvantageous

since most signals in engineering are non-linear. The Extended

Kalman Filter (EKF) proposes a solution to this problem

by approximating the state using a local linearization of a

nonlinear function [6]. In the EKF, the state transition and

observation models must be differentiable but not necessarily

linear functions.

xk = f(xk−1, uk−1) + wk−1 (3)

zk = h(xk) + vk (4)

where xk is the state, zk is the observaion, uk is the input at

time sample k, f and h are the nonlinear functions, with wk

and vk being zero-mean, Gaussian noise with some covariance

matrices Qk and Rk, respectively. Prediction of the state is

computed via the following:

xk|k−1 = f(xk−1|k−1, uk−1) (5)

Pk|k−1 = Fk−1Pk−1|k−1FTk−1 +Qk−1 (6)

The estimate is then updated

yk = zk − h(xk|k−1) (7)

Sk = HkPk|k−1HTk +Rk (8)

Kk = Pk|k−1HTk S

−1k (9)

xk|k = xk|k−1 +Kkyk (10)

Pk|k = Pk|k−1 −KkHkPk|k−1 (11)

where Fk and Hk are the local linearizations of the state

transition and observation model, respectively, given by:

Fk−1 =∂f

∂x

∣

∣

∣

∣

xk−1|k−1,uk−1

(12)

Hk =∂h

∂x

∣

∣

∣

∣

xk|k−1

(13)

It is also important to note that the p(xk|z1:k) is approximated

by a Gaussian distribution.

C. Information Theoretic Measures on the Time-Frequency

Plane

In recent years, there has been an interest in adapting

information-theoretic measures to the time-frequency plane in

order to quantify signal complexity [1], [3], [10], [19]. The

application of information-theoretic measures such as entropy

and divergence have made it easier to quantify the complexity

of non-stationary signals on the time-frequency plane as well

as differentiate between different signals. These measures have

been shown to be effective in quantifying the number of signal

components. Some of the most desired properties of TFDs are

the energy preservation and the marginals. They are given as

follows and are satisfied when φ(θ, 0) = φ(0, τ) = 1 ∀τ, θ.

∫ ∫

C(t, ω) dt dω =

∫

|s(t)|2 dt =

∫

|S(ω)|2 dω∫

C(t, ω) dω = |s(t)|2 ,

∫

C(t, ω) dt = |S(ω)|2.

(14)

The formulas given above evoke an analogy between a TFD

and the probability density function (pdf) of a two-dimensional

random variable. This analogy has inspired the adaptation of

information-theoretic measures such as entropy to the time-

frequency plane. The main difference between TFDs and

pdfs is that TFDs are not always positive and thus, not all

information measures are well-defined on the time-frequency

plane. Another important point is that distributions have to be

normalized by their energy before applying any information

theoretic measures on them.

The well-known Shannon entropy as applied to TFDs can

be defined as:

H(C) = −

∫ ∫

C(t, ω) logC(t, ω)dt dω. (15)

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This measure is not well-defined when the time-frequency dis-

tribution takes on a negative value and can only be applied to

positive distributions such as the spectrogram. For this reason,

generalized entropy measures such as Renyi entropy have been

considered for quantifying the information content of signals

on the time-frequency plane [15]. Renyi’s generalized entropy

of order α for TFDs is:

Hα(C) =1

1− αlog2

∫ ∫

C(t, ω)∫ ∫

C(u, v)du dv

α

dt dω

(16)

where α > 0 is the order of Renyi entropy, and entropy is well-

defined as long as∫ ∫

Cα(t, ω)dt dω > 0. This condition does

not require the TFDs to be positive for all time and frequency

points, and is well-defined for a large class of distributions

and signals [3]. As α → 1, Renyi entropy approaches the

well-known Shannon entropy. Since in actual implementations

we are interested in the discrete-time implementation of this

measure, we will define Renyi entropy in discrete time-

frequency domain as:

Hα(C) =1

1− αlog2

∑

n

∑

k

(

C[n, k]∑

n′

∑

k′ C[n′, k′]

)α

+ (17)

log2 δtδω

where δt and δω are the sampling step size in time and

frequency, respectively.

Another measure that is commonly used to quantify the

spread of the signals in the time-frequency plane is the

concentration measure. Contrary to the entropy, concentration

measure is a statistic on how concentrated a signal is in the

time-frequency plane and is given below:

M [Cnorm[n, k]] =

(

∑

n

∑

k

|Cnorm[n, k]|1

p

)p

(18)

where p > 1, and Cnorm[n, k] must be a probability distribu-

tion function characterized by

Cnorm [n, k] =C [n, k]

∑

n

∑

k C [n, k](19)

where C[n,k] is the original TFD. Furthermore, p < 4 is

chosen since higher values of p will emphasize the small

energy regions in the TFD [17]. This paper uses p = 2.

III. METHOD

A. Feature Extraction

The vibration signals (see Figs. 1-3) were first transformed

into the time-frequency plane using the Choi-Williams dis-

tribution. Each sample, which is of variable length, was

transformed into the time-frequency (TF) domain with a 256point FFT. By looking at the transformed signals, we observed

that the horizontal accelerometer was more informative of the

progression of the fault compared to the vertical one (See

Figs. 4 - 6). For the horizontal accelerometer data, a clear

trend in the time-frequency distribution across all 6 data sets

was observed. At the beginning of each training run, there

0 0.02 0.04 0.06 0.08 0.1−2

0

2

4

Horizontal Raw Vibration Data

Time (s)

Acc

eler

atio

n

0 0.02 0.04 0.06 0.08 0.1−2

−1

0

1

2

Vertical Raw Vibration Data

Time (s)

Acc

eler

atio

n

Fig. 1. Raw Data of Initial Vibration Signal

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1−4

−2

0

2

4

Horizontal Raw Vibration Data

Time (s)

Acc

eler

atio

n

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1−3

−2

−1

0

1

2

Vertical Raw Vibration Data

Time (s)

Acc

eler

atio

n

Fig. 2. Raw Data of Intermediate Vibration Signal

was significant energy in the frequency range of 160-200 Hz.

As the severity of the fault increased, this band moved up

in frequency, around frequency band 236-256 Hz. Finally, as

the motor neared failure state, the concentration of energy

distributed itself across the entire plot, in a less uniform

manner. The progression of the faults, from the first sample,

to an intermediate, then finally the last, or failure sample, can

be seen in figures 1 - 3, for the raw signals, and figures 4 - 6

in the time-frequency domain, respectively.

Based on these observations, we quantify the spread of

energy in the time-frequency plane using the entropy and

concentration measures. Since over time, the time frequency

energy distribution plot goes from a uniform distribution

0 0.02 0.04 0.06 0.08 0.1−40

−20

0

20

40

Horizontal Raw Vibration Data

Time (s)

Acc

eler

atio

n

0 0.02 0.04 0.06 0.08 0.1−40

−20

0

20

40

Vertical Raw Vibration Data

Time (s)

Acc

eler

atio

n

Fig. 3. Raw Data of Final Vibration Signal

602602

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Horizontal Vibrations

Time (samples)

Freq

uenc

y (H

z)

500 1000 1500 2000 2500

40

80

120

160

200

240

−0.2

−0.1

0

0.1

0.2

Vertical Vibrations

Time (samples)

Freq

uenc

y (H

z)

500 1000 1500 2000 2500

40

80

120

160

200

240

−0.1

−0.05

0

0.05

0.1

Fig. 4. Choi-WIlliams Transformation of Initial Vibration Signal

Horizontal Vibrations

Time (samples)

Freq

uenc

y (H

z)

500 1000 1500 2000 2500

40

80

120

160

200

240

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

Vertical Vibrations

Time (samples)

Freq

uenc

y (H

z)

500 1000 1500 2000 2500

40

80

120

160

200

240

−0.1

−0.05

0

0.05

0.1

Fig. 5. Choi-WIlliams Transformation of Intermediate Vibration Signal

to a more impulsive distribution near failure state, entropy

and concentration measures are suitable. The measures were

extracted over selected frequency bands across the samples.

Four different frequency bands, 160-200 Hz, 236-256 Hz, 0-

40 Hz, and the entire surface 0-256 Hz were considered.

B. Fault Prognosis and RUL Prediction

The extracted features are tracked across time using the

Extended Kalman Filter (EKF). First, the trend of the feature

across training samples is approximated by an analytical

signal, such as a linear, quadratic, or exponential model.

Once such an approximation is found, EKF can be used to

estimate the parameters of the model by altering equation (3)

Horizontal Vibrations

Time (samples)

Freq

uenc

y (H

z)

500 1000 1500 2000 2500

40

80

120

160

200

240

−0.2

−0.1

0

0.1

0.2

Vertical Vibrations

Time (samples)

Freq

uenc

y (H

z)

500 1000 1500 2000 2500

40

80

120

160

200

240

−0.1

−0.05

0

0.05

0.1

Fig. 6. Choi-WIlliams Transformation of Final Vibration Signal

to track the parameters as well as the overall state, by allowing

x = [x, θ] [9].

[xk, θk] = f([xk−1, θk−1], uk−1) + wk−1 (20)

where θ is the set of all parameters in the equation. It is also

assumed that

θk = θk−1 + nk (21)

where nk is some zero-mean, Gaussian noise with a covariance

Mk. Once the parameters of the model are obtained, the value

of the parametric prediction model, obtained through EKF,

at the failure sample is extracted as the threshold, γ . The

threshold is extracted from each training set individually and

then the final threshold for testing is computed as the average

across training sets

γ =1

K

K∑

i=1

γi (22)

where K is the number of training samples, equal to 6 in this

case.

In order to predict the RUL of the bearing, the time it will

take to reach the failure threshold, γ, is computed. At each

sample, k, the degradation equation found by EKF, gk(τk) is

computed by plugging in the estimated parameters, θk. Using

gk(τk) = γ (23)

we can solve for τk. The RUL, then, simply becomes

RUL(k) = τk − k (24)

C. Confidence Intervals of RULs

The main focus of this paper is to not only obtain RUL infor-

mation, but to add confidence to the prediction based on how

likely the prediction is. By the end of the algorithm, we have

obtained RUL estimates {RUL(1), RUL(2), ..., RUL(N)},

where N is the number of samples in the test data. From

these estimates, we can build a histogram for each test case

and feature separately. Confidence intervals are then given in

the 95% level rounded to the nearest hundred. This gives the

user time estimates in which the motor will fail as well as

the likelihood of the bearing failing around a range of time

centered at the RUL estimate [18]. This feature is useful to

the user because there may be some outlying RUL predictions

which may not be realistic. These ”outliers” will have a low

probability of occurring and the user will be notified about

it. For example, in the initial stages of testing the estimated

RULs based on just a few sample points may not be very

reliable. Similarly, towards the end of the lifetime of the

bearing, the accelerometer samples may be noisy making the

extracted features unreliable. The RUL predictions with the

highest probabilities will be provided to the user.

IV. DATA

In this paper, the data provided by FEMTO-ST Institute

(Besancon - France, http: //www.femto-st.fr/) was used. Ex-

periments were carried out on a laboratory experimental plat-

form (PRONOSTIA) that enables accelerated degradation of

603603

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Fig. 7. Overview of PRONOSTIA set up

bearings under constant and/or variable operating conditions,

while gathering online health monitoring data (rotating speed,

load force, temperature, vibration). PRONOSTIA is an exper-

imentation platform dedicated to test and validate bearings

fault detection, diagnostic and prognostic approaches. The

main objective of PRONOSTIA is to provide real experimental

data that characterize the degradation of ball bearings along

their whole operational life (until their total failure). Data

representing 3 different loads were considered (rotating speed

and load force): First operating conditions: 1800 rpm and 4000

N; second operating conditions: 1650 rpm and 4200 N; third

operating conditions: 1500 rpm and 5000 N. 6 run-to-failure

datasets were used to build the prognostics models, and the 11

remaining bearings were used for testing. The characterization

of the bearing’s degradation is based on two data types of

sensors: vibration and temperature. In this paper, we are

only using the data from the vibration sensor for prognosis.

The vibration sensors consist of two miniature accelerometers

positioned at 900 to each other; the first is placed on the

vertical axis and the second is placed on the horizontal axis.

The two accelerometers are placed radially on the external

race of the bearing. The acceleration measures are sampled at

25.6 kHz [11]. An overview of the entire PRONOSTIA set up

can be found in figure 7.

V. RESULTS

For the selected frequency bands of 160-200 Hz, 236-

256 Hz, 0-40 Hz, and 0-256 Hz, the two features discussed

above, entropy and concentration measure, were extracted for

each recorded sample. Median filtering was performed for

smoothing the features using a window size of 3, determined

empirically. Out of all the features that were tested, the feature

that had the clearest trend across samples was entropy (See

Figs. 8 and 9), and in particular, the entropy of the 160-200

Hz band. Across all 6 training sets, this feature looks to have a

linear degradation trend which the EKF parameter estimation

method can exploit. Median filtering was also performed for

smoothing the features using a window size of 3, determined

empirically. However, the entropy over the 160-200 Hz band

is not exactly linear as seen in figure 10. For feature data

extrapolation, two different analytical models were viewed:

linear and exponential. The results of the RUL estimations

over the 11 testing sets can be found in Table 1.

0 500 1000 1500 2000 2500 30000

2

4

6

8

10

12

Time (samples)

Entropy of "Healthy" Band − Training Set 1

En

tro

py

(a) Training Set 1

0 100 200 300 400 500 600 700 800 9009

9.2

9.4

9.6

9.8

10

10.2

10.4

10.6

10.8

11

Time (samples)

Entropy of "Healthy" Band − Training Set 2

En

tro

py

(b) Training Set 2

0 200 400 600 800 10007.5

8

8.5

9

9.5

10

10.5

11

11.5

Time (samples)

Entropy of "Healthy" Band − Training Set 3

En

tro

py

(c) Training Set 3

0 100 200 300 400 500 600 700 8006.5

7

7.5

8

8.5

9

9.5

10

10.5

11

Time (samples)

Entropy of "Healthy" Band − Training Set 4

En

tro

py

(d) Training Set 4

0 100 200 300 400 500 6008.5

9

9.5

10

10.5

11

11.5

Time (samples)

Entropy of "Healthy" Band − Training Set 5

En

tro

py

(e) Training Set 5

0 200 400 600 800 1000 1200 1400 1600 18009

9.5

10

10.5

11

11.5

Time (samples)

Entropy of "Healthy" Band − Training Set 6

En

tro

py

(f) Training Set 6

Fig. 8. Median filtered entropy over the ”healthy” band across all 6 trainingsets

0 500 1000 1500 2000 2500 30000

0.5

1

1.5

2

2.5

3x 10

6

Time (samples)

Concentration Measure of "Healthy" Band − Training Set 1

Co

nce

ntr

atio

n M

ea

su

re

(a) Training Set 1

0 100 200 300 400 500 600 700 800 9001

1.2

1.4

1.6

1.8

2

2.2

2.4x 10

5

Time (samples)

Concentration Measure of "Healthy" Band − Training Set 2

Co

nce

ntr

atio

n M

ea

su

re

(b) Training Set 2

0 200 400 600 800 10001

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3x 10

5

Time (samples)

Concentration Measure of "Healthy" Band − Training Set 3

Co

nce

ntr

atio

n M

ea

su

re

(c) Training Set 3

0 100 200 300 400 500 600 700 8001

1.5

2

2.5

3

3.5

4

4.5

5x 10

5

Time (samples)

Concentration Measure of "Healthy" Band − Training Set 4

Co

nce

ntr

atio

n M

ea

su

re

(d) Training Set 4

0 100 200 300 400 500 6001

1.5

2

2.5

3

3.5x 10

5

Time (samples)

Concentration Measure of "Healthy" Band − Training Set 5

Co

nce

ntr

atio

n M

ea

su

re

(e) Training Set 5

0 200 400 600 800 1000 1200 1400 1600 18001

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8x 10

5

Time (samples)

Concentration Measure of "Healthy" Band − Training Set 6

Co

nce

ntr

atio

n M

ea

su

re

(f) Training Set 6

Fig. 9. Median filtered concentration measure over the ”healthy” band acrossall 6 training sets

604604

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TABLE IRUL ESTIMATIONS AND PDF

Test Set True RUL (s) Est. RUL (s) Confidence Int. Error

1 5730 N/A N/A N/A2 339 1422.5 [200, 1800] 319.62%

3 1610 1117.9 [400, 2300] 30.57%

4 1460 3721.4 [1300, 6800] 154.8%

5 7570 11734.4 [7300, 13000] 55.01%

6 7530 6608.6 [6000, 8400] 12.24%

7 1390 2691.3 [1200, 3100] 93.62%

8 3090 3678.9 [2800, 4200] 19.06%

9 1290 4367.8 [1200, 6900] 238.5%

10 N/A 580 N/A N/A11 N/A 820 N/A N/A

0 200 400 600 800 1000 12008

8.5

9

9.5

10

10.5

11

Sample Number

Ent

ropy

Estimated

Actual

Fig. 10. Linear EKF Parameter Estimation for Bearing 1 5

Out of 11 testing datasets, 3 of them were deemed to already

be in failure state because at the time of RUL estimation,

the feature was already below the threshold value. These

cases are marked as N/A, or not attainable. For the remaining

testing sets, the RUL estimation and corresponding confidence

intervals were calculated. In particular, we can see the results

from bearing 1 5 in the figures 10 and 11. The error can be

calculated by

error =estimatedRUL− trueRUL

trueRUL∗ 100 (25)

and is also given in Table 1 along with the true estimates.

For the 8 testing cases out of 11 that were assigned an RUL,

6 of them had errors less than 100%. One reason for the

relatively high errors in RUL estimates is adopting a linear

−500 0 500 1000 1500 2000 2500 3000 35000

10

20

30

40

50

60

70

PDF of RUL Estimation

RUL Estimations

No.

Of O

ccur

renc

es

Fig. 11. RUL PDF for Bearing 1 5 in seconds

degradation curve for a nonlinear phenomenon. Furthermore,

the RUL estimates at each time point should not be weighted

the same since there is less data and more uncertainty at

the initial samples. This made it difficult to perform RUL

predictions. Even though the RUL estimation error is high for

certain test cases, in all of the cases the true RUL lies within

the 95% confidence interval computed from the distribution

of the estimated RULs. This indicates that not all RUL

estimates should be equally weighted and the actual estimate

should rather be an interval or a weighted mean based on the

reliabilities of each predicted RUL value.

VI. CONCLUSIONS

In this paper, we presented a new method for RUL estima-

tion from bearing data. The proposed method uses complexity

based features in the time-frequency plane to predict RUL for

bearings as well as to provide how likely a particular outcome

is. These RUL notifications, however, are dependent on the

extracted features. The methodology requires the extraction of

a feature that resembles or can be modeled by an analytical

signal across time. This proves to be problematic as the

trajectory of the extracted fault is not always a monotonic

function and may show fluctuations which makes it harder

to fit an analytic signal model. For future work, model-free

techniques will be examined to compensate for this drawback.

Furthermore, performing prognosis using multiple features

instead of one may prove to be more beneficial and improve

the reliability of the RUL estimates. Moreover, alternative

ways of selecting the threshold, other than the mean, can

improve the robustness of the method. The proposed method

can also be applied to other types of sensor data such as the

current from the bearings.

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VII. BIOGRAPHIES

Rodney K. Singleton II received his B.S. in electrical en-

gineering with honors from Michigan State University (MSU)

in East Lansing, Michigan in 2010. During his undergradu-

ate career, he completed summer internships at Ford Motor

Company and The Johns Hopkins University Applied Physics

Laboratory. Currently, he is pursuing his M.S. and Ph.D. in

Electrical Engineering, also at MSU, as a GEM and NSF

fellow. His graduate research includes the prognosis of bearing

faults in electrical motors, as well as remaining useful life

estimation of a motor.

Selin Aviyente received her B.S. degree with high hon-

ors in electrical and electronics engineering from Bogazici

University, Istanbul in 1997. She received her M.S. and

Ph.D. degrees, both in Electrical Engineering: Systems, from

the University of Michigan, Ann Arbor, in 1999 and 2002,

respectively. In August 2002, she joined the Department

of Electrical and Computer Engineering at Michigan State

University where she is currently an associate professor. Her

research focuses on the theory and applications of statistical

signal processing, in particular non-stationary signal analysis.

She is interested in developing methods for efficient signal

representation, detection and classification. She has published

over 80 refereed journal articles and conference proceedings

on time-frequency analysis, signal detection and classification.

She is the recipient of 2005 Withrow Teaching Excellence

Award and 2008 NSF CAREER Award.

Elias Strangas received the Dipl. Eng. degree in electrical

engineering from the National Technical University of Greece,

Athens, Greece, in 1975 and the Ph.D. degree from the

University of Pittsburgh, Pittsburgh, PA, in 1980. He was

with Schneider Electric (ELVIM), Athens, from 1981 to 1983

and the University of Missouri, Rolla, from 1983 to 1986.

Since 1986, he has been with the Department of Electrical

and Computer Engineering, Michigan State University, East

Lansing, MI, where he heads the Electrical Machines and

Drives Laboratory. His research interests include the design

and control of electrical machines and drives, finite-element

methods for electromagnetics, and fault prognosis and mitiga-

tion of electrical drive systems.

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