1. Random Data Workshops 1 & 2 L. L. Koss

2. Random Data 1 ---- L.L. Koss Random Data Analysis

3. Random Data –L.L. Koss

4. Random Data--- L.L. Koss Methods of characterizing random data: 1.Probability 2.Correlation 3.Frequency spectra System Modeling 1.Frequency response functions 2.Auto-regressive models 3.Impulse response functions

5. Random Data—L.L. Koss 1.Data that can be represented by a mathematical function is called “Deterministic”. Vibration of a spring mass system from an initial displacement. 2.Data that can not be represented by an explicit mathematical function is called “Random”. Wave height of ocean wave. 3.A time series is a random function, e.g. X(t), of an independent variable t where t is time.

6. Random data – L.L. Koss Stochastic Process 1.If we examine random data, h(t), over different periods of time there may not be any visual similarities over different time periods of observation, T1 and T2. 2.Many transducers are placed in the field to observe the random process h1(t), h2(t), … hn(t). Let n approach infinity.

7. Random Data--- L.L. Koss 3. Each of the hi(t) is called a sample time function. The collection of time functions is called an ensemble. 4. The collection of all possible time functions that the random process may have produced is called a “stochastic process”. 5. Usually, only a small number of time records are available to describe the process and they last only for a finite time. 6. When can sample time records be used to describe a process?

8. Random Data--- L.L. Koss Classification of random processes: Random Process Stationary Non-stationary Ergodic Non-ergodic Special classifications

9. Random Data--- L.L. Koss

10. If m(t1) and R(t1+  ) vary in amplitude as t1 is changed, the random process is said to be “non-stationary”. If m(t1) and R(t1+  ) do not vary in amplitude as t1 is changed, the random process is said to be weakly “stationary” or stationary in the wide sense. Many processes fit this description. For stationary data m and R (  ) are independent of absolute time.

11. Random Data--- L.L. Koss Ergodic Random Process Let us examine the “kth” time record and compute m and R (  ) over time “t” rather than over an ensemble. m(k)=  hk(t)dt as t becomes large R(k,  )=  hk(t)*hk(t+  ) dt “””””””””””” If the random process is stationary and m(k) and R(k,  ) are independent of “k” (do not differ) and are equal to the ensemble averaged values the random process is said to be “ergodic”.

12. Random Data--- L.L. Koss For ergodic random processes the time averaged mean value and autocorrelation function are equal to the corresponding ensemble averaged values. Thus, analysis of a single sample time function gives results that describe the random process!!

13. Random Data--- L.L. Koss Example of nearly rectangular/ flat distribution

14. Random Data--- L.L. Koss Gaussian or Normal Distribution P(x)=1/(sqrt(2  )  )*exp-((x-m) 2 /2  2 ) Where m is the mean value and  is the standard deviation. For all probability density functions:  p(x) dx = 1--- integration from – to + infinity

15. Random Data--- L.L. Koss E- expectation operator: The expected or mean value of a random variable. The expected value of wave height is given by E[h(t)] and is an average over an ensemble of time functions. For a stationary process E[(h(t)]=  hp(h)dh from –to + infinity Mean square value: E[h(t) 2 ] E[h(t) 2 ]=  h 2 p(h)dh from – to +infinity 2 If the process is ergodic then the average and mean square value can be calculated along a sample time function also 2

16. Random Data--- L.L. Koss The variance of the process,  2, or standard deviation, , is given by  2 =E[(h-E[h]) 2 ]  2 =E[h 2 ]-(E[h]) 2 or Variance = Mean square value – mean squared

17. Random Data--- L.L. Koss Joint Probability Distributions SISO- Single Input Single Output System

18. Random Data--- L.L. Koss

19. First order probability density function can be obtained from second order probability density function by integrating out the variable not required e.g. P(h)=  p(h,y)dy from – to + infinity Conditional probability density function Given a y value what is the probability of h? P(h|y)=p(h,y)/p(y)

20. Random Data--- L.L. Koss Second order Gaussian distribution

21. Random Data--- L.L. Koss MISO – Multi-Input Single- Output MIMO – Multi-Input Multi-Output

22. Random Data--- L.L. Koss Higher order probability density functions P(x1,u,h,v) – 5 Dimensions Do relationships exist between these variables? Are they linear ? What frequencies exist in the time data? Use Correlation to assist in answering above questions Ordinary correlation between two variables Partial correlation between inputs, inputs and outputs

23. Newland—Chap 3. Newland, D. E. (1993) “An Introduction to Random Vibrations, Spectral and Wavelet Analysis”. Chapter 3 –p. 21-23

24. Newland—Chap 3. Random Structure under load

25. Newland —Chap 3. Newland, D. E. (1993) “An Introduction to Random Vibrations, Spectral and Wavelet Analysis”. Chapter 3 –p. 24-32

26. Circular Correlation Function Recommended References Bendat, J. S. and Piersol, A. G. (1971) “Random data; analysis and measurement procedures” Bendat, J. S. and Piersol, A. G. (1980) “Engineering applications of correlation and spectral analysis”.