1. BMI2 SS07 – Class 8 “Image Processing 2” Slide 1 Biomedical Imaging 2 Class 8 – Time Series Analysis (Pt. 2); Image Post-processing (Pt. 2) 03/20/07

2. BMI2 SS07 – Class 8 “Image Processing 2” Slide 2 Flowchart for Imaging Data Analysis Measurement → Raw Data Pre- processing, or pre- conditioning Image Reconstruction Post- processing “Post-post- processing” “Post-post- post- processing” Filter, normalize, SNR threshold Integrate in space and/or time, define metrics Develop metrics into diagnostic indicators Time-series analysis (TSA) (FT, corr., SSS, GLM) TSA

3. BMI2 SS07 – Class 8 “Image Processing 2” Slide 3 Time Series Analysis… Definitions The branch of quantitative forecasting in which data for one variable are examined for patterns of trend, seasonality, and cycle. nces.ed.gov/programs/projections/appendix_D.asp nces.ed.gov/programs/projections/appendix_D.asp Analysis of any variable classified by time, in which the values of the variable are functions of the time periods. www.indiainfoline.com/bisc/matt.htmlwww.indiainfoline.com/bisc/matt.html An analysis conducted on people observed over multiple time periods. www.rwjf.org/reports/npreports/hcrig.html www.rwjf.org/reports/npreports/hcrig.html A type of forecast in which data relating to past demand are used to predict future demand. highered.mcgraw- hill.com/sites/0072506369/student_view0/chapter12/glossary.htmlhighered.mcgraw- hill.com/sites/0072506369/student_view0/chapter12/glossary.html In statistics and signal processing, a time series is a sequence of data points, measured typically at successive times, spaced apart at uniform time intervals. Time series analysis comprises methods that attempt to understand such time series, often either to understand the underlying theory of the data points (where did they come from? what generated them?), or to make forecasts (predictions). en.wikipedia.org/wiki/Time_series_analysisen.wikipedia.org/wiki/Time_series_analysis

4. BMI2 SS07 – Class 8 “Image Processing 2” Slide 4 Time Series Analysis… Varieties Frequency (spectral) analysis –Fourier transform: amplitude and phase –Power spectrum; power spectral density Auto-spectral density –Cross-spectral density –Coherence Correlation Analysis –Cross-correlation function Cross-covariance Correlation coefficient function –Autocorrelation function –Cross-spectral density Auto-spectral density

5. BMI2 SS07 – Class 8 “Image Processing 2” Slide 5 Time Series Analysis… Varieties Time-frequency analysis –Short-time Fourier transform –Wavelet analysis Descriptive Statistics –Mean / median; standard deviation / variance / range –Short-time mean, standard deviation, etc. Forecasting / Prediction –Autoregressive (AR) –Moving Average (MA) –Autoregressive moving average (ARMA) –Autoregressive integrated moving average (ARIMA) Random walk, random trend Exponential weighted moving average

6. BMI2 SS07 – Class 8 “Image Processing 2” Slide 6 Time Series Analysis… Varieties Signal separation –Data-driven [blind source separation (BSS), signal source separation (SSS)] Principal component analysis (PCA) Independent component analysis (ICA) Extended spatial decomposition, extended temporal decomposition Canonical correlation analysis (CCA) Singular-value decomposition (SVD) an essential ingredient of all –Model-based General linear model (GLM) Analysis of variance (ANOVA, ANCOVA, MANOVA, MANCOVA) –e.g., Statistical Parametric Mapping, BrainVoyager, AFNI

7. BMI2 SS07 – Class 8 “Image Processing 2” Slide 7 A “Family Secret” of Time Series Analysis… Scary-looking formulas, such as –Are useful and important to learn at some stage, but not really essential for understanding how all these methods work All the math you really need to know, for understanding, is –How to add: 3 + 5 = 8, 2 - 7 = 2 + (-7) = -5 –How to multiply: 3 × 5 = 15, 2 × (-7) = -14 Multiplication distributes over addition u × (v 1 + v 2 + v 3 + …) = u×v 1 + u×v 2 + u×v 3 + … –Pythagorean theorem: a 2 + b 2 = c 2 a b c

8. BMI2 SS07 – Class 8 “Image Processing 2” Slide 8 A “Family Secret” of Time Series Analysis… A most fundamental mathematical operation for time series analysis: The x i time series is measurement or image data. The y i time series depends on what type of analysis we’re doing: Fourier analysis: y i is a sinusoidal function Correlation analysis: y i is a second data or image time series Wavelet or short-time FT: non-zero y i values are concentrated in a small range of i, while most of the y i s are 0. GLM: y i is an ideal, or model, time series that we expect some of the x i time series to resemble

9. BMI2 SS07 – Class 8 “Image Processing 2” Slide 9 Correlation Analysis

10. BMI2 SS07 – Class 8 “Image Processing 2” Slide 10 Hb-oxy Hb-deoxy

11. BMI2 SS07 – Class 8 “Image Processing 2” Slide 11 Hb-oxy Hb-deoxy mean value standard deviation

12. BMI2 SS07 – Class 8 “Image Processing 2” Slide 12 Hb-oxy Hb-deoxy mean value standard deviation +k+k

13. BMI2 SS07 – Class 8 “Image Processing 2” Slide 13 Hb-oxy Hb-deoxy

14. BMI2 SS07 – Class 8 “Image Processing 2” Slide 14

15. BMI2 SS07 – Class 8 “Image Processing 2” Slide 15

16. BMI2 SS07 – Class 8 “Image Processing 2” Slide 16

17. BMI2 SS07 – Class 8 “Image Processing 2” Slide 17

18. BMI2 SS07 – Class 8 “Image Processing 2” Slide 18

19. BMI2 SS07 – Class 8 “Image Processing 2” Slide 19

20. BMI2 SS07 – Class 8 “Image Processing 2” Slide 20 Time-Frequency Analysis

21. BMI2 SS07 – Class 8 “Image Processing 2” Slide 21 (a) (b) (c) Figure 9. Illustration of Morlet wavelet analysis concept. The complex wavelet (solid and dashed sinusoidal curves denote real and imaginary part, respectively) shown in 9(a) is superimposed on the time-varying measurement depicted in 9(b). A new function, equivalent to the covariance between the wavelet and measured signal, as a function of the time point about which the wavelet is centered, is generated. (See Figure 10 for an example of such a computation.) Varying the width of the wavelet, as shown in 9(c), changes the frequency whose time-varying amplitude is computed.

22. BMI2 SS07 – Class 8 “Image Processing 2” Slide 22 Figure 10. Result of wavelet analysis (see Fig. 9) applied to (a) an unmodulated 0.1-Hz sine wave and (b) a frequency-modulated 0.1-Hz sine wave. In 10(a) it is seen that the amplitude and frequency both are constant over time, while in 10(b) it is seen that the amplitude is fixed but the frequency varies.

23. BMI2 SS07 – Class 8 “Image Processing 2” Slide 23 Data “Post-Post-Processing” and “Post- Post-Post-processing”

24. BMI2 SS07 – Class 8 “Image Processing 2” Slide 24 Starting point: Time Series of Reconstructed Images Physiological parameters: 1) Hb oxy, 2) Hb deoxy, 3) Blood volume 4) HbO 2 Sat Time Position 1.Temporal Averaging  Spatial Averaging 2.Spatial Averaging  Temporal Averaging 3.Wavelet Analysis

25. BMI2 SS07 – Class 8 “Image Processing 2” Slide 25 Method 1: Temporal  Spatial Averaging Time Position (IV) Spatial map of temporal standard deviation (SD) (III) Baseline temporal mean is 0, by definition temporal integration drop position information sorted parameter value 100 0 Hb deoxy Hb oxy (II) spatial integration meanSD scalar quantities (I)

26. BMI2 SS07 – Class 8 “Image Processing 2” Slide 26 Method 2: Spatial  Temporal Averaging Time Position (IV) spatial integration (II) (I) Time series of spatial mean → O 2 demand / metabolic responsiveness Time series of spatial SD → Spatial heterogeneity temporal integration Temporal mean of spatial mean time series: 0, by definition Temporal SD of spatial mean time series Temporal mean of spatial SD time series Temporal SD of spatial SD time series scalar quantities

27. BMI2 SS07 – Class 8 “Image Processing 2” Slide 27 1.Starting point is reconstructed image time series (IV) 2.Use (complex Morlet) wavelet transform as a time-domain bandpass filter operation A.Output is an image time series (IV) of amplitude vs. time vs. spatial position, for the frequency band of interest B.Filtered time series can be obtained for more than one frequency band 3.Recompute previously considered Class-II and Class-I results, using Methods 1 and 2, but starting with the wavelet amplitude time series Method 3: Time-frequency (wavelet) analysis time f1f1 f2f2

28. BMI2 SS07 – Class 8 “Image Processing 2” Slide 28 Baseline GTC: Healthy Volunteer Class IV results: normalized wavelet amplitude, right breast Temporal coherence index = 25.7% (26.3% for left breast (not shown))

29. BMI2 SS07 – Class 8 “Image Processing 2” Slide 29 Baseline GTC: Ductal Carcinoma in Right Breast Class IV results: normalized wavelet amplitude, left (-CA) breast Temporal coherence index = 18.4% Sharp, deep troughs are indicative of strong spatial coordination

30. BMI2 SS07 – Class 8 “Image Processing 2” Slide 30 Example 2: Ductal Carcinoma in Right Breast Class IV results: normalized wavelet amplitude, right (+CA) breast Temporal coherence index = 13.5% Troughs (and peaks) appreciably reduced, or absent

31. BMI2 SS07 – Class 8 “Image Processing 2” Slide 31 Specificity and Sensitivity Presence of Disease Test Result True Positive Disease (+)Disease (–) Test (+) Test (–) False Positive False Negative True Negative Given disease, what is the probability of a positive test result? Given no disease, what is the probability of a negative test?

32. BMI2 SS07 – Class 8 “Image Processing 2” Slide 32 Given negative test result, what is the probability of not having disease? Predictive Values Presence of Disease Test Result True Positive Disease (+)Disease (–) Test (+) Test (–) False Positive False Negative True Negative Given positive test result, what is the probability of disease?

33. BMI2 SS07 – Class 8 “Image Processing 2” Slide 33 Diagnostic Threshold ROC (Receiver Operating Characteristic) Analysis

34. BMI2 SS07 – Class 8 “Image Processing 2” Slide 34 ROC Curves for Metrics – 1 Area0.854 (0.708) Area0.780 (0.560)

35. BMI2 SS07 – Class 8 “Image Processing 2” Slide 35 ROC Curves – 2 Area0.786 (0.572) Area0.665 (0.330)

36. BMI2 SS07 – Class 8 “Image Processing 2” Slide 36 ROC Curves – 3 Area0.826 (0.652) Area0.809 (0.618)

37. BMI2 SS07 – Class 8 “Image Processing 2” Slide 37 ROC Curves - 4 Area0.818 (0.636) Area0.800 (0.600)

38. BMI2 SS07 – Class 8 “Image Processing 2” Slide 38 ROC Curves - 5 Area0.227 (0.546) Area0.205 (0.590)

39. BMI2 SS07 – Class 8 “Image Processing 2” Slide 39 Summary of Calculated Metrics Data reduction yielded 16 “metrics” Paired t-tests and ROC curves were used to select metrics that can distinguish between cancer and non-cancer subjects Selected metrics used in Logistic Regression Baseline MeasurementsValsalva TMSSDTSDSMTSDSSDSMTSDAreaHeightWavelet HbOXYXX X XXX HbRED XX XXX X: 0.01 ≤ p < 0.05, for difference between Cancer and Non-Cancer Subjects XX: p < 0.01

40. BMI2 SS07 – Class 8 “Image Processing 2” Slide 40 Logistic Regression Binary Distributions (Cancer vs. Non- Cancer) are non-linear Logistic regression expresses probability of event as a linear combination of “metrics” X i and coefficients  i

41. BMI2 SS07 – Class 8 “Image Processing 2” Slide 41 Logistic Regression Applied Metrics Probability Metrics calculated and selected based on t-tests & ROC curves Metrics used as inputs into logistic regression model Logistic regression model calculates  i for each metric (X i ) Using  i, a predicted probability distribution can be created New patient’s X i used to generate probability of cancer in patient X 1 =.43; X 2 = -.05 New Patient’s Values Linear Model: P(cancer) = 0.75 Logistic Regression: P(cancer) = 0.90

42. BMI2 SS07 – Class 8 “Image Processing 2” Slide 42 Limitations of Logistic Regression Metrics X i must be independent of each other  orthogonalization may be needed Consequently, biologically relevant phenomenology may be ignored by model Model may be mathematically unstable if the number of cases is low

43. BMI2 SS07 – Class 8 “Image Processing 2” Slide 43 Orthogonalization The logistic regression model excluded several metrics due to inherent co-linearity (not all are linearly independent) Transforming excluded metrics to be orthogonal to each other caused a loss of magnitude and of significance Result using orthogonalized metrics was very similar to original result

44. BMI2 SS07 – Class 8 “Image Processing 2” Slide 44 The final predicted probabilities were established by averaging the predicted probabilities for the N=21 and N=37 results Predicted probabilities for patients within the N = 37 group and not in the N = 21 group were unchanged Final Result Sensitivity0.93 Specificity0.96 PPV0.93 NPV0.96 Combined Metrics (N=21 & N=37)