1. Signal processing

2. Example data – ChIP-Seq

3. Gaussian peak with normal noise
Frequency
Frequency
Frequency

4. Removing High Frequences
Frequency

5. Smoothing w=ones(2*width+1,'d') convolve(w/w.sum(),y,'valid‘)
Intensity
w=ones(2*width+1,'d')
convolve(w/w.sum(),y,'valid‘)
Frequency
Frequency
Frequency

6. Peak Finding The derivative of a function is zero at its
minima and maxima.
The second derivative is negative at maxima and
positive at minima.

7. Detection of steps Motivation: To demonstrate a general strategy for
Intensity
Motivation: To demonstrate a general strategy for
separating signal from noise:
Characterize the signal and the noise
Make a model of the data
Select detection method
Select parameters using simulations

8. Detection of steps: Characterization of noise
Remove signal by
subtracting a
moving average

9. Detection of steps: Model of data
S/N=0.75
S/N=1
S/N=2
points=1000
x = linspace(-1,1,points)
y=noise*random.normal(size=len(x))
y[points/2:]+=signal

10. Detection of steps: Detection method
S/N=0.75
S/N=1
S/N=2
Steps can be converted into peaks by calculating
the difference between the moving average in
two windows

11. Detection of steps: Detection method
S/N=0.75
S/N=1
S/N=2
Bin size = 10
Average Intensity
Average Intensity
Average Intensity
Bin size = 30
Average Intensity
Average Intensity
Average Intensity
Bin size = 100
Average Intensity
Average Intensity
Average Intensity

12. Detection of steps: Simulations - peak location
S/N=0.05
S/N=0.25
S/N=1
Bin size = 10
Bin size = 30
Bin size = 100

13. Detection of steps: Simulations – correct peak
S/N=0.05
S/N=0.25
S/N=1
Bin size = 10
Frequency
Frequency
Frequency
Score
Score
Score
Bin size = 30
Frequency
Frequency
Frequency
Score
Score
Score
Bin size = 100
Frequency
Frequency
Frequency
Score
Score
Score

14. Detection of steps: Simulations - FDR and FNR
S/N=0.05
S/N=0.25
S/N=1
Bin size = 10
False Rate
False Rate
False Rate
Threshold
Threshold
Threshold
False
Negative
Rate
Bin size = 30
False Rate
False Rate
False Rate
False
Discovery
Rate
Threshold
Threshold
Threshold
Bin size = 100
False Rate
False Rate
False Rate
Threshold
Threshold
Threshold

15. Peak Finding Characterize the signal and the noise
Intensity
Characterize the signal and the noise
Make a model of the data
Select detection method
Select parameters using simulations

16. Peak Finding: Characterizing the noise
Intensity
Let’s first try without removing the peaks

17. Peak Finding: Characterizing the noise
Removing the peaks
by looking for outliers
in the root mean
square deviation
(RMSD)
Intensity
RMSD

18. Peak Finding: Characterizing the peaks
Intensity

19. Peak Finding: Model of data
S/N=1
S/N=2
S/N=4
points=1000
x = linspace(-1,1,points)
y=noise*random.normal(size=len(x))
y+=signal*gaussian(x,0,0.01)

20. Peak Finding: Detection method
S/N=1
S/N=2
S/N=4
Peaks can be detected by finding maxima in the moving average with a window size similar to the peak width

21. Peak Finding: Detection method – moving average
Signal
Bin size = 5
Bin size = 20
Bin size = 80
S/N=1
S/N=2
S/N=4

22. Peak Finding: Detection method – RMSD
Signal
Bin size = 5
Bin size = 20
Bin size = 80
S/N=1
S/N=2
S/N=4

23. Peak Finding: Information about the Peak
maximum
mean
variance
skewness
kurtosis
full width
at half
maximum
(FWHM)
Intensity
height
centroid
(mean)
area

24. Information about a Peak
A peak is defined by
Centroid or mean
To calculate any of these measures we need
to know where the peak starts and ends.

25. Where does a peak start and end?

26. Estimating quantity Peak height Peak height Curve fitting
Intensity
Peak area
m/z

27. What is the best way to estimate quantity?
Peak height - resistant to interference
- poor statistics
Peak area - better statistics
- more sensitive to interference
Curve fitting - better statistics
- needs to know the peak shape
- slow

28. Sampling
Intensity
Retention Time

29. Sampling

30. Summary Fourier transform - transformation to frequency space and back
Signal – how do we detect and characterize signals?
Noise – how do we characterize noise?
Modeling signal and noise
Simulation to select thresholds and select parameters
Filters – fitering by low-pass (i.e. smoothing) and high-pass filters (e.g. adaptive background correction)
Detection methods based on moving average and RMSD
Convolution - describes the response of a linear and
time-invariant system to an input signal
Cross-correlation is a measure of similarity of two signals
Autocorrelation can be used for finding periodic signals obscured by noise
The dot product can be used to determine how similar two signals are
Coincidence measurements enhance the signal and supresses noise
The quantity associated with a peak – height and area
Sampling – how often do we need to sample a peak to get a good estimate of its area?

31. Peak Finding Examples

32. Peak Finding Example 1
RMSD
Window Size 40 80
160

33. Peak Finding Example 1
RMSD
Window
Size 40 80
160

34. Peak Finding Example 1
RMSD
Window
Size 40 80
160

35. Peak Finding Example 1
RMSD
Window Size 40 80
160

36. Peak Finding Example 1
Smoothing
Window Size 40 80
160

37. Peak Finding Example 1
Smoothing
Window Size 40 80
160

38. Peak Finding Example 1
Smoothing
Window Size 40 80
160

39. Peak Finding Example 1
Smoothing
Window Size 40 80
160

40. Peak Finding Example 1
Smoothing
Window Size 40 80
160

41. Peak Finding Example 2
RMSD
Window Size 40 80
160

42. Peak Finding Example 2
RMSD
Window
Size 40 80
160

43. Peak Finding Example 2
RMSD
Window
Size 40 80
160

44. Peak Finding Example 2
RMSD
Window Size 40 80
160

45. Peak Finding Example 2
Smoothing
Window Size 40 80
160

46. Peak Finding Example 2
Smoothing
Window Size 40 80
160

47. Peak Finding Example 2
Smoothing
Window Size 40 80
160

48. Peak Finding Example 2
Smoothing
Window Size 40 80
160

49. Peak Finding Example 2
Smoothing
Window Size 40 80
160

50. Peak Finding Example 3

51. Peak Finding Example 4

52. Peak Finding Example 5

53. Homework: Background Subtraction
Using Smoothing

54. Extra Homework