MONTE CARLO SIMULATION
In a dynamic system simulation one can study the effect of random noise on parameter/state estimates to evaluate the performance of the estimator. We first get one set of estimated parameters; then we change the seed number for random number generator and add these random numbers to measurements as noise. We get estimates of the parameters with the new data. We can formulate a number of such data sets with different seeds and obtain parameters to establish the variability of the estimates across different realizations of the data. We next obtain the mean value and the variance of the parameter estimates using all the individual estimates from all these realizations. The mean of the estimates should converge to the true values. The approach can be used for any type of system. Depending on the problem’s complexity 500 simulation runs or as small as 20 runs could be used to generate average results.
B19 PROBABILITY AND RELATIVE FREQUENCY
Relative frequency is the value calculated by dividing the number of times an event occurs by the total number of times an experiment has been carried out. The probability of an event is then thought of as its limiting value when the experiment is carried out several times; the relative frequency of any particular event will settle down to this value. Probability is expressed on a scale from 0 to 1; a rare event has a probability close to 0 and a very common event has a probability of 1.
If X is a continuous random variable, there is a function p(x) (PDF) such that for every pair of numbers a < = b, P(a < = X < = b) = (area under p between a and b). The probability density function of a random variable with a standard normal/ Gaussian distribution is the normal curve. Only continuous random variables have probability density functions.
The probability distribution of a random variable specifies the chance that the variable takes a value in any subset of the real numbers, and the probability distribution of a random variable is completely characterized by the cumulative probability distribution function. The probability distribution of a discrete random variable can be characterized by the chance that the random variable takes each of its possible values. The probability distribution of a continuous random variable is characterized by its PDF. The Gaussian PDF is given by p(x) = exP (— ), with m as the mean and s2 as the variance of the
distribution. For the measured random variables, given the state x (or parameters), the PDF is given by p(z|x) = (2p)„/2|R|1/2 exp(- 2 (z – Hx)TR-1(z – Hx)), with R as the covariance matrix of measurement noise.