Probability and Distributions

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Probability and Distributions
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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 dont know exactly how
  many heads we will get, but we can calculate the
  probability of getting a certain number of heads

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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

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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

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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 !

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Combinations of Events

• The complement Ac of an event A is the event that
  A does not occur
• Probability Rule 3
• P(Ac) 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 A
Complement of A
Union of A and B
Intersection of A and B
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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 AHH and BTT are disjoint
• Events AHH,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)

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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
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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

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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.

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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
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Normal Distribution
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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 PX x where x represents any one
  of the possible values that the random variable
  may assume.

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Types of Distributions

• Discrete (binary, nominal, ordinal)
• Bernoulli
• Binomial
• Poisson
• Geometric
• Continuous distributions (interval, ratio)
• Uniform
• Normal (Gaussian)
• Gamma
• Chi Square
• Student t

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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.
  

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Basic Statistics
n number of observations xi observation i
Excel function AVERAGE

• Mean -
• Variance -
• Standard Deviation -
• Coefficient of Variation -
• Skew Coefficient -

Excel function VAR
Excel Function STDEV
Excel Function Skew
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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

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Continuous Uniform Distribution

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

• Symmetrical equal number of events on either
  side of the mean value.
• Mean, medium and mode values are equal.
• f(x)

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Gamma Distribution

• A skewed distribution, not symmetric.
• Mean, medium and mode are not equal.
• f(x, k, T)

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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 gt 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

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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. di is the
nearest neighbor distance for point i.
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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
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Nearest Neighbor Distance
R lt 1
R gt 1
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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
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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
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Wilderness Campsites
Real world study areas are complex and violate
the assumptions of most equations for expected
values.
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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.