1. Probability and Distributions

2. Deterministic vs. Random Processes In deterministic processes, the outcome can be predicted exactly in advance Eg. Force = mass x acceleration. If we are given values for mass and acceleration, we exactly know the value of force In random processes, the outcome is not known exactly, but we can still describe the probability distribution of possible outcomes Eg. 10 coin tosses: we don’t know exactly how many heads we will get, but we can calculate the probability of getting a certain number of heads

3. Events An event is an outcome or a set of outcomes of a random process Example: Tossing a coin three times Event A = getting exactly two heads = {HTH, HHT, THH} Example: Picking real number X between 1 and 20 Event A = chosen number is at most 8.23 = {X ≤ 8.23} Example: Tossing a fair dice Event A = result is an even number = {2, 4, 6} Notation: P(A) = Probability of event A Probability Rule 1: 0 ≤ P(A) ≤ 1 for any event A

4. 4 Sample Space The sample space S of a random process is the set of all possible outcomes Example: one coin toss S = {H,T} Example: three coin tosses S = {HHH, HTH, HHT, TTT, HTT, THT, TTH, THH} Example: roll a six-sided dice S = {1, 2, 3, 4, 5, 6} Example: Pick a real number X between 1 and 20 S = all real numbers between 1 and 20 Probability Rule 2: The probability of the whole sample space is 1 P(S) = 1

5. Equally Likely Outcomes Rule If all possible outcomes from a random process have the same probability, then P(A) = (# of outcomes in A)/(# of outcomes in S) Example: One Dice Tossed P(even number) = |2,4,6| / |1,2,3,4,5,6| = 3/6 = 1/2 Note: equal outcomes rule only works if the number of outcomes is “countable” Eg. of an uncountable process is sampling any fraction between 0 and 1. Impossible to count all possible fractions !

6. Combinations of Events The complement A c of an event A is the event that A does not occur Probability Rule 3: P(A c ) = 1 - P(A) The union of two events A and B is the event that either A or B or both occurs The intersection of two events A and B is the event that both A and B occur Event AComplement of AUnion of A and BIntersection of A and B

7. Disjoint Events Two events are called disjoint if they can not happen at the same time Events A and B are disjoint means that the intersection of A and B is zero Example: coin is tossed twice S = {HH,TH,HT,TT} Events A={HH} and B={TT} are disjoint Events A={HH,HT} and B = {HH} are not disjoint Probability Rule 4: If A and B are disjoint events then P(A or B) = P(A) + P(B)

8. Independent events Events A and B are independent if knowing that A occurs does not affect the probability that B occurs Example: tossing two coins Event A = first coin is a head Event B = second coin is a head Disjoint events cannot be independent! If A and B can not occur together (disjoint), then knowing that A occurs does change probability that B occurs Probability Rule 5: If A and B are independent P(A and B) = P(A) x P(B) P( 2 H in two Tosses) = 0.5 * 0.5 = 0.25 Independent multiplication rule for independent events

9. Distributions The magnitude of an event will vary over a range of values with time. This variation can be described by some type of distribution function. –Frequency –Cumulative

10. Frequency Distribution A frequency distribution is an arrangement of the values that one or more variables take in a sample. Each entry in the table contains the frequency or count of the occurrences of values within a particular group or interval.

11. Cumulative Distribution Function (CDF) CDF is the probability of Variable X, taking on a number that is less than or equal to number X. This may also be known as the "area in so far" function. Median Flow is at 0.5 value on the CDF

12. Normal Distribution

13. Probability Distribution A probability is a numerical value that measures the uncertainty that a particular event will occur. The probability of an event ordinarily represents the proportion of times under identical circumstances that the outcome can be expected to occur. A probability distribution of a random variable X provides a probability for each possible value. Those probabilities must sum to 1, and they are denoted by: P[X = x] where x represents any one of the possible values that the random variable may assume.

14. Types of Distributions Discrete (binary, nominal, ordinal): –Bernoulli –Binomial –Poisson –Geometric Continuous distributions (interval, ratio): –Uniform –Normal (Gaussian) –Gamma –Chi Square –Student t

15. Statistics of a Distribution Central Value –Mean –Medium –Mode Variability –Min, Max and Range –Variance –Standard Deviation –Coefficient of Variation (CV) - a measure of dispersion of a probability distribution (Standard Deviation / Mean) Shape -Skewness - a measure of symmetry -Kurtosis - a measure of whether the data are peaked or flat relative to a normal distribution.

16. Basic Statistics Mean - Variance - Standard Deviation - Coefficient of Variation - Skew Coefficient - n = number of observations x i = observation i Excel function: AVERAGE Excel function: VAR Excel Function: STDEV Excel Function: Skew

17. Other Metrics Central Tendency –Mean –Median Point in the distribution where half of the values in the distribution lie below the point, and half lie above the point –Mode Value of x at which the distribution is at its maximum

18. Continuous Uniform Distribution All events within a range has a equal chance of occurrence. Frequency Cumulative Probability density function Used in stochastic modeling

19. Normal Distribution Symmetrical – equal number of events on either side of the mean value. Mean, medium and mode values are equal. f(x) =

20. Gamma Distribution A skewed distribution, not symmetric. Mean, medium and mode are not equal. f(x, k, Θ) =

21. Inference Most spatial analysis is based on comparing sample events to theoretical distributions. With a normal distribution –+/- 1 standard deviations = 0.68 of the events –+/- 2 standard deviations = 0.955 of the events –+/- 3 standard deviations = 0.997 of the events P(x > +3SD) = 0.0015 Z statistic – normal deviate transformation –Z = (X – Expect Mean of X)/ Expected SD of X –Z = (10 – 5) / 1.5 = +3.33

22. Nearest Neighbor Analysis Nearest neighbor analysis examines the distances between each point and the closest point to it, and then compares these to expected values for a random sample of points from a CSR (complete spatial randomness) pattern. CSR is generated by means of two assumptions: 1) that all places are equally likely to be the recipient of a case (event) and 2) all cases are located independently of one another. The mean nearest neighbor distance = where N is the number of points. d i is the nearest neighbor distance for point i.

23. The expected value of the nearest neighbor distance in a random pattern = where A is the area and B is the length of the perimeter of the study area. The variance =

24. Nearest Neighbor Distance R < 1 R > 1 R = 1

25. And the Z statistic = This approach assumes: Equations for the expected mean and variance cannot be used for irregularly shaped study areas. The study area is a regular rectangle or square. Area (A) is calculated by (Xmax – Xmin) * (Ymax – Ymin), where these represent the study area boundaries. R statistic = Observed Mean d / Expect d R = 1 random, R  0 cluster, R  2+ uniform

26. 2 x 0.5 A = 1, B = 5 E (di) = 0.05277 Var (d) = 8.85 x 10 -6 1 x 1 A = 1, B = 4 E(di) = 0.05222 Var(d) = 8.48 x 10 -6 2 x 2: E(di) = 0.10444

27. Real world study areas are complex and violate the assumptions of most equations for expected values. Wilderness Campsites

28. Solution * Simulate randomization using Monte Carlo Methods. Compare simulated distribution to observed. * If possible use the “true” area and perimeter to compute the expected value. * Software that does not ask for area/perimeter or a shapefile of the study area will assume a rectangle based on the minimum and maximum coordinates.