Signal processing methods for drift compensation

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2nd NOSE II Workshop
Linköping, 18 - 21 May 2003
Signal processing methods
for drift compensation
Ricardo Gutierrez-Osuna
Department of Computer Science
Texas A&M University
Outline
‰
‰
‰
‰
‰
Sources of "drift"
Compensation approaches
Univariate methods
Multivariate methods
Discussion
This material is based upon work supported
by the National Science Foundation under
CAREER Grant No. 9984426 / 0229598
2nd Workshop
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Signal processing methods for drift compensation - 2 -
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What is drift?
"A gradual change in any
quantitative characteristic
that is supposed to remain
constant"
[Holmberg and Artursson, 2003],
and references therein
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Signal processing methods for drift compensation - 3 -
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Sources of "drift"
‰
True drift
• Aging
• Poisoning
‰
(reorganization of sensing layer)
(irreversible binding)
Experimental noise
•
•
•
•
•
•
Short-term drift
(warm-up, thermal trends)
Memory effects (hysteresis, sampling sequence)
Environmental
(pressure, temp., seasonal)
Odor delivery (flow rate, outgass, condensation)
Matrix effects
(background, humidity)
Sample degradation
(oxidation, decarbonation)
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Approaches for drift compensation
‰
‰
‰
‰
‰
‰
‰
Drift-free sensors
• Duh!
Reference sensors
• Differential or ratiometric measurements [Choi et al., 1985]
Excitation
• Temperature modulation [Roth et al., 1996]
Frequent re-calibration
• Unavoidable
Careful experimental design
• Avoid systematic errors
Feature extraction
• Transient response analysis [Wilson and DeWeerth, 1995]
Signal processing
• The focus of this tutorial
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Signal processing for drift compensation
‰
Univariate
• Compensation applied to each sensor independently
•
•
•
•
‰
Frequency analysis
Baseline manipulation
Differential measurements (w/ calibrant)
Multiplicative correction (w/ calibrant)
Multivariate
• Compensation applied to the response across sensors
•
•
•
•
Adaptive clustering
System identification
Calibration transfer (w/ calibrant)
Orthogonal signal correction and deflation (w/ calibrant)
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2nd NOSE II Workshop
Linköping, 18 - 21 May 2003
Univariate Techniques
Frequency analysis
‰
Drift, noise and odor information occur at
different time scales [Artursson et al., 2000]
• Noise = high frequencies
• Drift = low frequencies
• Perform separation in the frequency domain
• Filter banks [Davide et al., 1996]
• Discrete Wavelet Transform
• Drawbacks
• Requires long-term time series to be collected
• Time series has "gaps“, variable sampling rates
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Signal processing methods for drift compensation - 8 -
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Baseline manipulation (1)
‰
Basics
• The simplest form of drift compensation
• “Remove" the sensor response in the recovery
cycle prior to sample delivery
• Also used for pre-processing [Gardner and Bartlett, 1999]
• A local technique, processes one "sniff" at a time
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Baseline manipulation (2)
‰
Differential
• Corrects additive δA or baseline drift
xˆ s (t) = y s (t) - y s (0) = (x s (t) + δ A ) − (x s (0) + δ A ) = x s (t) − x s (0)
{
{
measured
response
ideal
response
(w/o drift)
y(t)
x(t)
t=0
t=0
t=0
time
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Signal processing methods for drift compensation - 10 -
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Baseline manipulation (3)
‰
Relative
• Corrects multiplicative (1+δM) or sensitivity drift
xˆ s (t) =
y s (t) x s (t) (1+ δM ) x s (t)
=
=
y s (0) x s (0) (1+ δM ) x s (0)
y(t)
x(t)
t=0
t=0
t=0
time
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Baseline manipulation (4)
‰
Fractional
• Percentual change in the sensor response
x s (t) - x s (0)
y s (t) =
x s (0)
• Properties
• Yields a dimensionless measurement
• Normalizes sensor responses, but can amplify noisy
channels
• Fractional conductance shown to be the most
suitable method for MOS sensors [Gardner, 1991]
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Reference gas [Fryder et al., 1995]
‰
Equivalent to diff. BM, except a calibrant is used
• Calibrant must be chemically stable over time AND
highly correlated with samples [Haugen et al., 2000]
y(t)
Sample
Calibrant
x(t)
New
baseline
time
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Calibration schedule
Calibrant
Samples
Traditional
(time intensive)
Efficient
(systematic memory errors)
Efficient
(random memory errors)
Time, €
adapted from [Salit and Turk, 1998]
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Multiplicative correction [Haugen et al., 2000]
‰
Basic idea
• Model temporal variations in a calibration gas with
a multiplicative correction factor
• Apply the same correction to the samples
• Perform this process first on short-term trends,
then on long-term fluctuations
‰
Properties
• Heuristic, global technique
• Practical for industrial applications
• Compensates for short- and long-term drift
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Short-term correction (1)
U
STC
Sequence 1
Sequence 2
Sequence 3
index
Sequence 1
Sequence 2
Sequence 3
index
USCT
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Short-term correction (2)
‰
For each sequence
• Compute a correction factor for
each calibration sample
qn,seq =
U1,calseq
Un,calseq
• Build regression model for
series {q1,seq, q6,seq, q11,seq,…}
qˆ n,seq = ax n + b
with x n = mod(n,Nsam + 1)
• Correct all samples
ˆ
USTC
n,seq = Un,seq qn,seq
Name
U1cal
U2
q6,seq U3
U4
U5
U6cal
q̂9,seq
U7
U8
U9
U10
U11cal
U12
U13
U14
U15
…
Class
Index xn
Calibration
1
0
Sample
2
0
Sample
3
0
Sample
4
0
Sample
5
0
Calibration
6
1
Sample
7
1
Sample
8
1
Sample
9
1
Sample
10
1
Calibration
11
2
Sample
12
2
Sample
13
2
Sample
14
2
Sample
15
2
…
…
…
(seq subindex omitted for clarity)
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Long-term correction (1)
USCT
LTC
Sequence 1
Sequence 2
Sequence 3
index
Sequence 1
Sequence 2
Sequence 3
index
ULTC
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Long-term correction (2)
‰
For each sequence
• Compute a correction factor for
the first calibration sample
fseq
cal
1,1
cal
1,seq
U
=
U
• Correct all samples
ˆ
ULTC
n,seq = Un,seq qn,seq fseq
14243
USTC
n,seq
Name
U1,1cal
U2,1
U3,1
…
cal
f2 U6,1
U7,1
U8,1
…
U1,2cal
U2,2
f3
U3,2
…
U6,2cal
U7,2
U8,2
…
U1,3cal
U2,3
U3,3
…
U6,3cal
U7,3
U8,3
…
Class
Index
Calibration
1
Sample
2
Sample
3
xn
0
0
0
Calibration
Sample
Sample
6
7
8
1
1
1
Calibration
Sample
Sample
1
2
3
0
0
0
Calibration
Sample
Sample
…
Calibration
Sample
Sample
6
7
8
…
1
2
3
1
1
1
…
0
0
0
Calibration
Sample
Sample
…
6
7
8
…
1
1
1
…
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Signal processing methods for drift compensation - 19 -
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Seq
1
1
1
1
1
1
1
2
2
2
2
2
2
2
3
3
3
3
3
3
3
Performance of multiplicative correction
PCA plot of uncorrected milk samples measured over
2 days: pasteurized milk ({) , oxidized pasteurized
milk (‹), calibration samples (U).
PCA plot of drift corrected milk samples measured
over 2 days: reference milk ({), oxidized milk (‹),
calibration samples (U).
from [Haugen et al., 2000]
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2nd NOSE II Workshop
Linköping, 18 - 21 May 2003
Multivariate Techniques
Adaptive clustering
‰
Basic idea
• Model the distribution of examples with a
codebook
• Assign an incoming (unknown) sample to the
"closest" class
• Adapt class parameters to incorporate information
from the newly classified example
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Adaptive clustering (1)
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Signal processing methods for drift compensation - 23 -
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Adaptive clustering (2)
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Signal processing methods for drift compensation - 24 -
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Adaptive clustering (3)
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Signal processing methods for drift compensation - 25 -
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Adaptive clustering (4)
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Signal processing methods for drift compensation - 26 -
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Adaptive clustering (5)
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Signal processing methods for drift compensation - 27 -
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Adaptive clustering (6)
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Signal processing methods for drift compensation - 28 -
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Adaptive clustering (7)
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Signal processing methods for drift compensation - 29 -
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Adaptive clustering (8)
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Signal processing methods for drift compensation - 30 -
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Adaptive clustering (9)
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Signal processing methods for drift compensation - 31 -
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Adaptive clustering (10)
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Signal processing methods for drift compensation - 32 -
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Adaptive clustering (11)
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Signal processing methods for drift compensation - 33 -
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Adaptive clustering (12)
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Adaptive clustering methods
‰
Mean updating
• One cluster center per class [Holmberg et al., 1996]
‰
Kohonen self-organizing maps
• One SOM common to all classes [Davide et al., 1994;
Marco et al., 1997]
• A separate SOM for each class [Distante et al., 2002]
‰
Adaptive Resonance Theory
• ART is slightly different; new clusters can be created
[Gardner et al., 1996]
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Adaptive clustering: discussion
‰
Algorithm relies on correct classification
• Miss-classifications will eventually cause the
model to lose track of the class patterns
‰
‰
All odors need to be sampled frequently to
prevent their patterns to drift too far
This problem has also been addressed in the
machine learning literature
• see [Freund and Mansour, 1997] and refs. therein
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System identification [Holmberg et al., 1996]
‰
Basic idea
• Chemical sensor responses co-vary over time
• This "common-mode" behavior can be modeled
with a dynamic model (e.g., ARMAX)
A
S
C
B
k − n) = ∑∑ b y (k − n) + ∑ c e(k − n)
∑ a y14(2
1
424
3
1
424
3
4
3
n =0
n
s
sensor s
i=1 n = 0
i≠ s
in
i
all other
sensors
n =0
n
white noise
• where ys(k) is the response of sensor 's' at time k, and
yi(k) is the response from all other sensors
• Model parameters {A,B,C} may be adapted over time
with a recursive least-squares procedure [Holmberg et
al., 1997]
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Signal processing methods for drift compensation - 37 -
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System identification: classification
‰
Each odor/sensor pair has a unique dynamic
behavior that can be used as a fingerprint
• Build a dynamic model for each sensor/odor
• When an unknown odor is presented, predict its
behavior of each sensor with each of the models
• The method requires that multiple (consecutive)
samples of the unknown odor be collected
• The model with lowest prediction error
corresponds to the true odor
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System identification: illustration
MODEL DYNAMIC BEHAVIOR
FOR EACH ODOR/SENSOR
APPLY EACH O/S MODEL
TO UNKNOWN SAMPLE
CHOOSE ODOR MODEL
WITH LOWEST ERROR
Unknown
ERROR 1
Unknown
ERROR 2
Unknown
Odor A
A1Y=B1Y’+C1E
ERROR 3
Odor B
A2Y=B2Y’+C2E
Odor C
A3Y=B3Y’+C3E
time
time
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Calibration transfer
‰
Learn a regression mapping (e.g., MLPs, PLS)
from drifting calibrant samples onto a baseline t0
• MLPs for PyMS [Goodacre and Kell, 1996]
• PLS for e-noses [Tomic et al., 2002]
• MLPs for e-noses [Balaban et al., 2000]
Training phase:
Drifting calibrant samples
at times t1, t2, …tN
Training phase:
Baseline calibrant sample
at time t0
Recall phase:
Drifting odor samples
at times t1, t2, …tN
Recall phase:
Corrected odor samples
at times t1, t2, …tN
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Orthogonal signal correction [Wold et al., 1998]
‰
Basic idea
• Assume dataset matrices
• Y: sensor-array data (independent variables)
• C: concentration vector or class label (dependent)
• Subtract from Y factors that account for as much
of the variance in Y as possible AND are
orthogonal to C
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Orthogonal signal correction: intuition
Y3
Direction orthogonal
to concentration vector
Y1
Subspace correlated with
concentration vector C
Y2
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Orthogonal signal correction: an example
6
4
2
PCA2
0
-2
-4
-6
-8
2
3
4 4 3 2222
odors
3
4
2
4
4
4
2
3
2
42
2
4423242224
4
3
4
3
3
4
2
3
4
1 11
3
3
1
22
111
4333233 222
3
4 422433233442 22
11 11 1
433
5
55 5 1 1 111
32 34 23 3 324
55 5 5 51 1 1
424 2
34
14
2343
5
5
5
5
1
5
1
55
1
51
4 3 4 4
5 5 55 11111
3
3
234
4342 4
55 5
4
1
1
3
1
2
1
1
4
5
5
2
5
3
3
3
5
322
1
55 5 1
24 2
5
3
5
4
1
5 5
55
4
5
1
5
5
3 2
5
3
11
5 1
drift
1
-10
-10
-5
0
1
5
10
15
PCA1
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Component correction [Artursson et al., 2000]
‰
Basic idea
• Drift is associated with the principal components
of variance in a calibration gas
• These directions are removed from the
multivariate sensor response by means of a
bilinear transformation
‰
Algorithm
T
x corrected = x − (x ⋅ v cal )v cal
• where vcal is the first eigenvector of the calibration data xcal
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Component correction results
from [Art, 2000]
PCA scatter plot of uncorrected samples for eight gas
mixtures. Arrows indicate the direction of drift. The
center cluster (+) is the calibration gas.
PCA scatter plot after component correction. The
calibration gas (+), no longer shown in the figure, has
been used to estimate and remove the principal
direction of drift (vcal).
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Component correction results (2)
3
2
4
3
3
4
4
4
3
3
2 4
523 4
555 8
2
6
5
1516
1
1
616
61
1
PCA2
0
-1
2
2
2
1
777 7 7
-4
-4
Rioja Joven 2000
Rioja Crianza 1999
Rioja Reserva 1998
Ribera Joven 2000
Ribera Crianza 1999
Ribera Reserva 1998
Water
12% v/v EtOH/Water
3
444
4
3 34 4
3 3
3
3
0
-3
-6
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
3
8
8
-2
7
777
4
-2
0
PCA1
2
4
6
2 2
666
61 5 55 5 2
22
1161
1
66 55 5
-1
8
8
-5
-4
-3
-2
-1
0
1
PCA1
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2
2
2
3
4
Component deflation [Gutierrez, 2000]
‰
Basic idea
• Identify variables 'x' whose variance can be attributed to
drift or interferents
• E.g., response to a wash/reference gas, time stamps,
temperature, pressure, humidity, etc.
• Measure 'y', the sensor-array response to an odor
• Remove variance in 'y' that can be explained by 'x' (by
means of regression/deflation)
‰
Related to target rotation [Esbensen et al., 1987]
• "…removal of undesired information provided that there
are variables uniquely connected to that information"
[Christie, 1996]
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Component deflation algorithm
‰
Find linear projections x’=Ax and y’=By that are maximally correlated
{A,B} = argmax[ρ(Ax,By )]
• How? Canonical Correlation Analysis (CCA) or PLS
• Interpretation: x’ and y’ are low-dimensional projections that summarize
the linear dependencies between x and y
‰
Find regression model ypred=Wy’
W = argmin y − Wy'
2
• Interpretation: ypred contains the variance in the odor vector y that
can be explained by y’ and, as a result of the CCA stage, by x
‰
Deflate y and use the residual z as a corrected sensor response
z = y − y pred = y − WBy
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Component deflation: interpretation
x3
Ax
x2
x1
B
W
z=y
y
WBy
By
class1
class3
z2
y2
class2
z1
y1
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Component deflation: motivation
Exploit transient information in wash/reference cycle
• To capture temporal trends, augment vector x with the time
stamp of each sample (also in [Artursson et al., 2000])
Sensor response (V)
Wash
Reference
Odour
3
2.5
2
1.5
50
WTS Kernels
‰
100
xW
150
xR
200
250
Time (s)
y
x
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Component deflation performance
‰
Database
• Three months of data collection
• Four cooking spices
• Twenty four days
• Four samples per day and spice
• Ten metal-oxide sensors
‰
Class 1
Class 2
Class 3
Class 4
Date
Date
PLS
CCA
Plot shows one particular
transient feature
• Examples are sorted
RAW
• By odor class
• Within a class, by date
Date
Date
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Influence of aging and training set size
Training set
Day: 1 2
i
Test set
j
k
W
Width=1
Predictive accuracy
1
M
D
Width=5
Width=10
0.9
CCA
0.8
PLS
0.7
RAW
0.6
0.5
2
4
6
Distance
8
10
2
4
6
Distance
8
10
2
4
6
Distance
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8
10
Discussion of univariate methods
‰
‰
‰
‰
Frequency decomposition
• Mostly useful for diagnostics and analysis
Baseline manipulation
• Attractive for its simplicity, but not an effective driftcorrection approach for chemical sensors
Differential measurements wrt calibrant
• First-order approximation, but does not exploit crosscorrelations
Multiplicative correction wrt calibrant
• Empirically shown to work, heuristic, does not exploit
cross-correlations
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Discussion of multivariate methods (1)
‰
‰
Adaptive clustering
• Requires equiprobable and frequent sampling
• Correct identification is essential to ensure that the
adapting distributions track the odors
• Does not use information from a calibrant
System identification
• Builds a separate drift model for each odor, but
requires multiple consecutive samples
• Can be used as a template-matching classifier
• Does not use information from a calibrant
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Discussion of multivariate methods (2)
‰
‰
Calibration transfer
• Shown to work in MS and e-nose data
• Black-box approach
Orthogonal Signal Correction and Deflation
• In my experience, the best approach
• Why? Time tested in chemometrics, uses multivariate
and calibrant information, and IT IS SIMPLE
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References (1)
‰
‰
‰
‰
‰
‰
‰
‰
‰
‰
M. Holmberg and T. Artursson, "Drift Compensation, Standards and Calibration Methods," in T. C.
Pearce, S. S. Schiffman, H. T. Nagle and J. W. Gardner (Eds.), Handbook of Machine Olfaction:
Electronic Nose Technology, Weinheim, Germany: Wiley-VCH, 2002, pp.325-346.
S.-Y. Choi, K. Takahashi, M. Esani and T. Matsuo, "Stability and sensitivity of MISFET hydrogen
sensors," in Proc. Int. Conf. Solid State Sensors and Actuators, Philadelphia, PA, 1985.
M. Roth, R. Hartinger, R. Faul and H.-E. Endres, "Drift reduction of organic coated gas-sensors by
temperature modulation," Sensors and Actuators B 36(1-3), pp. 358-362, 1996.
D.M. Wilson and S.P. DeWeerth, "Odor discrimination using steady-state and transient
characteristics of tin-oxide sensors," Sensors and Actuators B 28(2), pp. 123-128. 1995.
J.W. Gardner and P.N. Bartlett, Electronic Noses. Principles and Applications, New York: Oxford
University Press, 1999.
J.W. Gadner, "Detection of Vapours and Odours from a Multisensor Array using Pattern
recognition. Part 1. Principal Components and Cluster Analysis," in Sensors and Actuators B 4, pp.
109-115, 1991.
M. Fryder, M. Holmberg, F. Winquist and I. Lündstrom, "A calibration technique for an electronic
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M.L. Salit and G.C Turk, "A Drift Correction Procedure," Analytical Chemistry 70(15), pp. 31843190, 1998.
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2nd Workshop
Ricardo Gutierrez-Osuna
Signal processing methods for drift compensation - 56 -
Texas A&M University
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Ricardo Gutierrez-Osuna
Signal processing methods for drift compensation - 57 -
Texas A&M University
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2nd Workshop
Ricardo Gutierrez-Osuna
Signal processing methods for drift compensation - 58 -
Texas A&M University
2nd NOSE II Workshop
Linköping, 18 - 21 May 2003
Questions
2nd NOSE II Workshop
Linköping, 18 - 21 May 2003
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