AUTOCOVARIANCE STRUCTURES FOR RADIAL AVERAGES IN
SMALL ANGLE X-RAY SCATTERING EXPERIMENTS
Andreea Erciulescu, F. Jay Breidt, Mark van der Woerd
Departments of Statistics and Biochemistry & Molecular Biology, Colorado State University, Fort Collins
Abstract
Remove the mean to study autocorrelation
Small-angle X-ray scattering (SAXS) is a relatively simple experimental method to obtain low-resolution molecular information, using high-intensity X-rays. In analyzing the
recorded SAXS data, it is common to find data quality issues, such as detector saturation,
low signal-to-noise ratio, radiation damage to the sample and adverse effects from sample
concentration. In current analysis methods these issues are found after the experiment is
complete, too late for adjustments in experimental protocols. Developing a rigorous statistical methodology for immediate assessment of data quality in raw SAXS images, without
preprocessing, requires estimation of the autocovariance structure of the errors in the images and their radial averages.
Autocorrelation results are stable
Asymptotic Bartlett Bounds: ±1.96 ÷
√
m, where m = 32 in our case. We would expect
exceedances of about 5%.
SAXS experiments and images
The investigator typically exposes different concentrations of the molecule in solution at
different exposure times, collecting digital images of scattering patterns for each combination of concentration and exposure time. The images are subsequently reduced to a one-
Consider the difference in replicate images to subtract the mean
Problems: heteroskedasticity and residual mean structure
Are local sample autocorrelations significantly different from zero?
dimensional curve, which is interpreted and ultimately used for particle reconstruction.
Nearest neighbors show autocorrelation
Generating a SAXS Image
φ
e
tter
a
c
S
-ray
tX
iden
Inc
kI
kS
ay
-r
dX
2θ
kI
Theoretical model
q
De
te
ct
Sample autocorrelations are consistent with ”kernel convolution” model:
RR 1 s
RR 1 s
I(q) =
G( τ )µ(q + s)ds +
G( τ )σ(q + s)Z(q + s)ds
τ2
τ2
or
Sample
where Z(q) is an iid random field with mean 0 and variance 1.
This has a physical interpretation due to detector engineering.
Data processing
Conclusion
Neigboring pixels in the image data are in fact correlated, while farther removed pixels
are uncorrelated.
Future Directions
Locally-estimated autocorrelations appear small, globally homogeneous, and isotropic (not
direction-dependent).
1D Intensity profile
Fourier Transform
Particle shape reconstruction
Consider a range of experimental conditions
• Investigate model properties
• Incorporate globally-estimated autocorrelations into statistical tests.
The importance of autocorrelation
Samples
2. Protein only
(Glucose Isomerase)
• We need to know confidence intervals on intensity data and derived data interpretation,
ultimately for molecular shape reconstructions
1. Buffer (solvent)
• Investigate various experimental conditions with new data sets.
• Ensure best data quality possible for molecular envelope reconstructions.
• Appropriately track uncertainty throughout the process.
Instruments
3. DNA only
• Current interpretation methods assume Gaussian error model and independent pixels
• We apply correlation analysis to our two-dimensional images to test if individual pixels
in the images are uncorrelated
Two different instruments
4. Both protein
and DNA
Acknowledgements
This research was supported in part by Award #R01GM096192 from the Joint NSF/NIGMS Initiative to Support Research in the Area
of Mathematical Biology. The content is solely the responsibility of the authors and does not necessarily represent the official views of
the National Institute Of General Medical Sciences or the National Institutes of Health.
Bibliography
Brockwell, P.J. and Davis, R.A. (1991). Time Series: Theory and Methods, 2nd ed. Springer-Verlag, New York.