1. MEGN 537 – Probabilistic Biomechanics Ch.3 – Quantifying Uncertainty
Anthony J Petrella, PhD

2. Big Picture Why study traditional probability (Ch. 2)?
Unions and intersections allow us to conceptualize system level variability with multiple sources
Chapter 3 deals with uncertainty in variables
Chapter 2 presupposes failure probabilities, but often times these are not absolute or fixed
This chapter allows us to characterize variability in parameters

3. Random Variables Continuous (variable) Data
Data measured on an infinitely divisible scale or continuum
No gaps between possible values
Examples
Height, weight
Blood glycogen level
Joint contact force
Kinematic measures
Response time (milliseconds)
Discrete (attribute) Data
Discrete data measures attributes, qualitative conditions, counts
Gaps between possible values
Examples
Surgery or placebo
Fusion or dynamic
Number of surgeries per week or per year
Pain/satisfaction surveys
Number of patients

4. Population vs. Sample POPULATION Sample
Population - A total set of all process results
Sample - A subset of a population
POPULATION
Sample
Population Measures N m s
Sample Measures n x s
number of data points
mean
standard deviation _

5. Frequency Histograms
Histogram visually represent data centering, variability, and shape
Histograms are a graphical tool used to depict the frequency of numerical data by categories (classes or bins)
Properties
All data will fall into a class or bin
No data will overlap

6. Descriptors of Uncertainty
Used to characterize measured data and distributions
Common descriptors
Mean
Standard deviation
Coefficient of variation
Skewness

7. Measures of Location (Central Tendency )
Mean – also known as average; sum of all values divided by number of values
Grand Average: overall average or average of the averages
Median - midpoint of the data
Arrange the data from lowest to highest, the median is the middle data point number
50% of the data points will fall below the median and the other 50% will fall above the median
Mode - the most frequent data point, or value occurring the most often

8. Example - Measures of Location
Given the following data on salaries: $50k, $30k, $170k, $45k, $30k, $55k, $40k
Mean = ( )/7 = 420/7 = $60k
Median (midpoint): 30,30,40, 45, 50,55,170
Mode (most frequent) 1 2
# data pts
170 55 50 45 40 30
Value
Mean
Median
Mode

9. Measures of Dispersion (Variation or Spread)
Range - Total width of a distribution.
Range = Maximum Value - Minimum Value
Variance (V) – Measure of the spread in data about the mean
Second central moment
Standard Deviation (S) - The most common measure of dispersion
Range

10. Standard Deviation
Standard deviation is a measure of variation telling us about consistency around the mean
High spread
High standard deviation
Poor performance
Low spread
Low standard deviation
Consistent performance X

11. Other Descriptors
Coefficient of variation (COV) - Relative indicator of uncertainty in a variable
Ratio of standard deviation and the mean
Skewness – Measure of the spread of data about the mean emphasizing the shape of the distribution
Third central moment

12. Skewness Skewness Coefficient (qx) – Non dimensional measure
0 = symmetric
+ skewness  Most values below the mean
- skewness  Most values above the mean
qx = +
qx = -
f(x) x

13. Skewness Skewness Coefficient (qx) – Non dimensional measure
0 = symmetric
+ skewness  Most values below the mean
- skewness  Most values above the mean
mode
qx = +
qx = -
f(x)
median
mean x

14. Probability Density Function
PDF is the typical histogram or bell curve
fX(x) = Probability x is in a specific bin

15. Cumulative Distribution Function
CDF ranges from 0 to 1
Integral of the pdf f(x)
CDF
PDF
m = 0
s = 1

16. Cumulative Distribution Function
The CDF gives the probability of a continuous random variable having a value less than or equal to a specific value
Relationship between pdf and cdf:
The CDF has the following values
F(x  -∞) = 0
F(x = median) = 0.5
F(x  +∞) =1

17. Creating Histograms or PDFs
Arrange data in increasing order
Create evenly spaced bins and count how many data points occur in each bin
Note: the # of bins can affect the appearance of the histogram
Rule of thumb: k = log10 n where k = # of bins and n = number of data points
Plot the number of observations versus the variable
Note: for PDFs, plot frequency = (# of observations) / n

18. Creating CDFs Arrange data in increasing order For each data point
Create an index i = 1, 2, 3,…, n
Compute F(x) = i / (n+1)
Plot F(x) as a function of the variable x
Demo in Excel (come back to this)

19. Multiple Random Variables
Joint distributions or joint PDF’s can be defined for multiple variables
Joint PDF in 3D represents how the variables are dependent on each other

20. Covariance and Correlation
Covariance indicates the degree of linear relationship between two random variables, denoted as:
Cov(X,Y) = E(XY) – E(X)*E(Y) where E( ) is the expected value
Covariance is the second moment about the respective means
Covariance = 0 for statistically independent events
Correlation coefficient (non-dimensional) represents the degree of linear dependence between two random variables
ρx,y = Cov(X,Y)/(σx * σy)
Correlation coefficient can range from -1 to +1 (Haldar p. 53)
ρ = 0 no correlation
ρ = +1 perfectly correlated / proportionate
ρ = -1 perfectly correlated / inversely proportionate

21. Project Demo
Assume the following parameters are normally distributed with a coefficient of variation of 0.05: |rquad.knee|, rtubercle.x, rtubercle.y, rham.knee.y.

22. Demos… NESSUS Excel for random trials, PDF, CDF
Matlab demo with trials from #2
Reverse engineer Excel and repeat for rtx