Statistical design and modeling of
experiments with high-tech applications
C. F. Jeff Wu
School of Industrial and Systems Engineering
Georgia Institute of Technology
•A statistical trilogy: data collection, analysis,
decision making
•Examples in high-tech applications:
 nano technology
 cell biology
 complex system simulations
1
A Statistical Trilogy
I. Data collection:
II. Data modeling (incl. inference):
III. Optimization and decision making:
2
A Statistical Trilogy
I. Data collection: experimental design,
sample surveys.
II. Data modeling (incl. inference): regression,
analysis of variance, time series analysis, survival
data analysis.
III. Optimization and decision making:
decision analysis, Bayesian method.
3
What’s Next?
The High-Tech Revolution
• Availability of massive data: cannot do design of
experiments, but can do data mining and data
experimentation.
• "The sexy job in the next 10 years will be
statisticians,” Google chief economist (NY Times,
2009/8/5)
• Physical experiments replaced by computer experiments
(savings in cost and time, more feasible): a definite
opportunity.
• Other opportunities abound (nanotechnology, molecular
medicine, biotech devices, alternative fuel): unknown
territory, tremendous promises.
4
Statistical Work in Nano Technology
The nano part is based on two papers:
– A Statistical Approach to Quantifying the Elastic Deformation of
Nanomaterials (X. Deng, V. R. Joseph, W. Mai*, Z. L. Wang* ,
C. F. J. Wu). Proc. Nat. Acad. Sciences, 106, 11845-50, 2009.
– Robust optimization of the output voltage of nanogenerators by
statistical design of experiments (J.Song*, H.Xie, W.Wu*,
V.R.Joseph, C.F.J.Wu, Z.L.Wang*). Nano Research, 3(9) , 613-9,
2010.
*School of Materials Science and Engineering, Georgia Tech
5
A Statistical Approach to Quantifying the
Elastic Deformation of Nanomaterials
• Existing method and drawbacks
• A new method: Sequential Profile Adjustment
by Regression (SPAR)
• Demonstration on nanobelt data
6
Introduction
• One-dimensional (1D) nanomaterials: fundamental building
blocks for constructing nanodevices and nanosystems.
• Important to quantify mechanical property such as elastic
modulus of 1D nanomaterials: dictate their applications in
nanotechnology.
• A common strategy is to deform a 1D nanostructure using an
AFM (Atomic Force Microscopy) tip.
Schematic diagram of AFM
7
Method of Experimentation and Modeling
• Mai and Wang (2006, Appl. Phys. Lett.) proposed a new approach to
measure the elastic modulus of ZnO nanobelt (NB).
AFM images of a suspended ZnO nanobelt
• The AFM tip scans along the length of the NB under a constant
applied force.
• A series of bending profiles of the same NB are obtained by
sequentially changing the magnitude of the contact force.
8
Free-Free Beam Model
• Mai and Wang (2006) suggested a free-free beam model (FFBM) to quantify
the elastic deflection (with free boundary condition):
F
h
x
L
A
B
F
h
x
L
• The deflection v of NB at x is determined by
where E is the elastic modulus, L is the width of trench, and I is the moment of
inertia.
• FFBM gives better fit than clamped-clamped beam model.
9
FFBM Profiles Example
• The profiles are calculated based on FFBM. The
force F changes from low 78 nN to high 261 nN.
10
Profiles of the Nanobelt Experiment
• AFM image profiles of NB under load forces from low 78 nN to high
261 nN.
• Initial bias of the nanobelt:
– The NB is not perfectly straight: initial bending during sample
manipulation.
– The profile curves in Figure are not smooth: caused by a small surface
roughness (around 1 nm) of the NB.
11
MW Method
• Eliminate the initial bias: Normalize profiles by subtracting the first profile
(acquired at 78 nN) from the profiles in (a).
• The elastic modulus is estimated by fitting the normalized AFM image
profiles using the FFBM. (MW method)
12
Problem with MW Method
• Subtracting the first profile to normalize the data can result in poor
estimation if the first profile behaves poorly.
• Systematic biases can occur during the measurement,
• Inconsistent (order reversal) pattern: profiles at applied force 235, 248
and 261 nN lie above on those obtained at lower force F = 209 and 222
nN. This pattern persists in the normalized profiles.
13
Problem with MW Method
• Subtracting the first profile to normalize the data can result in poor
estimation if the first profile behaves poorly.
• Systematic biases can occur during the measurement.
235 nN
248 nN
261 nN
209 nN
222 nN
• Inconsistent (order reversal) pattern: profiles at applied force 235, 248
and 261 nN lie above on those obtained at lower force F = 209 and 222
nN. This pattern persists in the normalized profiles.
14
Problem with MW Method
• Subtracting the first profile to normalize the data can result in poor
estimation if the first profile behaves poorly.
• Systematic biases can occur during the measurement.
157 nN
170 nN
235 nN
183 nN
248 nN
261 nN
209 nN
131 nN
222 nN
144 nN
• Inconsistent (order reversal) pattern: profiles at applied force 235, 248
and 261 nN lie above on those obtained at lower force F = 209 and 222
nN. This pattern persists in the normalized profiles.
15
Counter Measures
• Experimenters: drop the data (i.e., five belts)
that exhibit inconsistency.
– loss of data and waste of information.
• Statisticians: keep the data, use statistical
modeling to remove the inconsistency.
– remaining information in data be utilized.
16
SPAR: A New Method
• The FFBM itself cannot explain the inconsistency.
– Requires a more general model to include other factors
besides the initial bias.
• Propose a general model to incorporate the initial bias
and other potential systematic biases.
• Use model selection to choose an appropriate model.
• The method is called sequential profile adjustment by
regression (SPAR).
17
18
Causes of Systematic Biases
• The changes of boundary conditions:
– Can be nonlinear and irreversible during the measurement.
– Can cause the occasional stick-slip events.
• The wear and tear of AFM tip and the nanobelt
surface.
• The lateral shifting and sliding, and other artifacts.
• Because of the nano scale, such causes are more acute
in nano experiment and can occur at any stage of the
experiment.
19
Model Selected from Deflection Data
20
F13 = 235 nN
F14 = 248 nN
F15 = 261 nN
F11 = 209 nN
F12 = 222 nN
21
F13 = 235 nN
F14 = 248 nN
F15 = 261 nN
F11 = 209 nN
F12 = 222 nN
• Matching the FFBM better, but inconsistent pattern persists 
22
F11 = 209 nN
F12 = 222 nN
F13 = 235 nN
F14 = 248 nN
F15 = 261 nN
• Inconsistent pattern removed 
23
• The δ12 term over-corrects and moves the curves down; this is rectified
by adding δ10; curves are moved up, middle part smoothed  better
match with FFBM.
24
std reduced by 50% .
25
Mechanistic vs. Statistical Modeling
• The error and noise of the experiment are stochastic in
nature.
• It is difficult to develop a catch-all mechanistic model.
– The mechanistic model is deterministic and predictive.
• A purely statistical model lacks prediction power.
• The proposed mechanistic-empirical modeling strategy
can be a useful approach.
– Make the statistical corrections physically meaningful.
– Improve the estimation of physical parameters.
26
Understanding Cell Adhesion State
Using Hidden Markov Model
C. F. Jeff Wu+
(joint with Y. Hung*, V. Zarnitsyna§,
Yijie Wang+, & C. Zhu§)
+
Georgia Tech, Industrial & Systems Engineering
*Rutgers, the State University of New Jersey
§ Georgia Tech, Biomedical Engineering
Based on NIH-GMS Grant
27
Cell adhesion
• Motivated by the statistical analysis of biomechanical
experiments at Georgia Tech.
• Cell adhesion: binding of a cell to another cell or
surface.
 Mediated by interaction between cell adhesion
proteins (receptors) and the molecules that they bind
to (ligands).
• Biologists describe the receptor-ligand binding as a
key-to-lock type relation.
• What makes cells sticky? When, how, and to what cells
adhere?
• Why important? It plays an important role in many
physiological and pathological processes and in tumor
metastasis in cancer study.
28
Thermal fluctuation experiment
• It uses reduced thermal fluctuations to indicate
the presence of receptor-ligand bonds.
• Objective: Identify association and
dissociation points for receptor-ligand bonds.
• Accurate estimation of these points is essential
because
it is required for precise measurement of bond
lifetimes and waiting times,
it forms the basis for subsequent estimation of the
kinetic parameters.
29
Experimental setting
• A micropipette red blood cell with a bead (probe) glued to its apex (left)
was aligned against another bead (target) aspirated by another pipette (right).
(Developed at Georgia Tech.)
• Driven by a piezoelectric translator, a computer-programmed test cycle
consisted of an approach-push-retract-hold-return cycle.
• During the holding period, the left pipette was held stationary to allow the
probe and the target to contact via thermal fluctuations, thereby providing an
opportunity for the receptors and ligands to interact.
• Position of probe was tracked by image analysis software to produce data.
30
Data
• Interested in the thermal fluctuation during the holding period.
• Bond formation is equivalent to adding a molecular spring in
parallel to the force transducer spring to stiffen the system
the fluctuation decreases when a receptor-ligand bond forms and
resumes when the bond dissociates.
Bond
Bond
forms
dissociates
31
Challenges
• Challenges in identifying the bond association/dissociation
points:
 Points are not directly observable.
 Observations are not independent.
 In practice, data contains an unknown number of bond types and
each bond associated with different fluctuation decreases due to
their string strength difference.
32
Challenges
• Challenges in identifying the bond association/dissociation
points:
 Points are not directly observable. Can only be detected by
variance changes.
 Observations are not independent.
 In practice, data contains an unknown number of bond types and
each bond associated with different fluctuation decreases due to
their string strength difference.
33
Challenges
• Challenges in identifying the bond association/dissociation
points:
 Points are not directly observable. Can only be detected by
variance changes.
 Observations are not independent. Need to take into
account cell memory effect. Binding probability increases if
there is a binding in the immediate past.
 In practice, data contains an unknown number of bond types and
each bond associated with different fluctuation decreases due to
their string strength difference.
34
Challenges
• Challenges in identifying the bond association/dissociation
points:
 Points are not directly observable. Can only be detected by
variance changes.
 Observations are not independent. Need to take into
account cell memory effect. Binding probability increases if
there is a binding in the immediate past.
 In practice, data contains an unknown number of bond types and
each bond associated with different fluctuation decreases due to
their string strength difference.
35
Hidden Markov Models (HMM) Framework
• Assume the probe fluctuates with different
variances that correspond to different underlying
binding states.
• These states, including no bond and a number of
distinct types of bonds, are not observable but the
process of these binding states change can be
captured by a Markov chain model.
• Such Markov chain process can also be used to
capture the cell memory effect.
36
Hidden Markov Model with two states
37
Hidden Markov Model with two states
38
Hidden Markov Model with two states
39
Hidden Markov Model with two states
40
Hidden Markov Model with two states
41
Transition Probability in HMM
• aij , i, j  0,1 denotes the prob. of going from
state i to state j
• A large a11 indicates a memory effect
• Called “Hidden” because the Markov chain
transition works underneath the normal
distribution N(μi,σi²) for state i
42
Analysis Results for Two States
43
HMM with three states
• No bond, P-selectin bond, L-selectin bond:
P/L-selectin are different proteins on cell
surface. They play an important role in
transiently rolling process of cell.
• It is known that L-selectin has a more stiff
string than P-selectin
σL² < σp² . This
physical knowledge allows us to focus the
HMM on the variance change as an indication
of chang of bond type.
44
Thermal fluctuation data:
Three states
Three States Experiment Data
50
40
30
Position x (nm)
20
10
0
-10
-20
-30
-40
-50
0
100
200
300
400
500
observation
600
700
800
45
Estimation for HMM
•

 a00

 a10
 a
 20


a01  a02


a01 a02   0.9499 0.0498
0 


  
a11 a12    0.0018 0.8953 0.1029 


a21 a22   0.0449 0.0636 0.8915 
: No bond (state 0) more likely transits to
P-bond (state 1) than to L-bond (state 2)
•


a12  a10
: P-bond more likely transits to L-bond
than to no bond
• a20

a21
: not much difference
• Estimates attached with statistical significance
46
Analysis for three states
47
Why computer experiments?
48
Some examples
49
Statistical Meta-Modeling of
Computer Experiments
Uncertainty
Quantification
50
GP with quanti/quali factors:
Data Center Thermal Distribution
51
Configuration Variables for
Data Center Example
• Five quantitative factors:
rack temperature rise, rack power, diffuser
angle, diffuser flow rate, ceiling height
• Three qualitative factors:
diffusor location, hot-air return-vent
location, power allocation
52
Gaussian Process Models with
Quantitative and Qualitative Factors
53
Summary
• Statistics not used in some high-tech applications, e.g.,
Nobel-winning experimental effects (or Science,
Nature) should be “obvious”.
• It has made impact in industrial work when
“incremental” improvement needs statistical tools;
increasingly popular for high-tech work when “subtle”
effects need to be ascertained.
• Massive online data is the biggest opportunity for stat,
e.g., webpage design and optimization using stat doe.
• Major role in complex stochastic system study.
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