1. Statistical Distributions

2. Why do we care about probability?
Foundation of theory of statistics.
Description of uncertainty associated with random variables.
Measurement error
Process error
Needed to understand:
Estimating model parameters.
Model validation
Prediction

3. Types of random variables
Discrete random variables take only integer values (e.g. counts, memberships in categories). They are represented by probability mass functions.
Continuous random variables are represented by probability density functions.

4. Terminology This notation is equivalent.
You will see it all in this course.
These are general stand-ins for distributions.

5. Probability Mass Functions
For a discrete random variable, X, the probability that x takes on
a value x is a discrete density function, f(x) also known as
probability mass or distribution function.
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Event (z)
Probability
S=support is the domain
of function [z]

6. Probability Density Functions: Continuous variables
A probability density function [z] gives the probability that a
random variable Z takes on values within a range.
[z]≥0
Pr(a ≤ z ≤ b)
𝑎 𝑏 𝑧 𝑑𝑧
−∞ ∞ 𝑧 𝑑𝑧 =1
Non-negative…

7. Probability Mass Functions: First Moment
The expectation of a random variable z is the weighted value of the possible values that z can take, each value weighted by the probability that z assumes it. For discrete variables:
Analogous to “center of gravity”. First moment.
p(-1)= p(0)= p(1)= p(2)=0.35

8. Probability Mass Functions: Second Central Moment
The variance of a random variable reflects the spread of Z values around the expected value. For discrete variables:
Second central moment of a distribution.

9. Continuous distributions

10. Cumulative distribution and quantile function
Z takes on a value less than or equal to u

11. Probability Distributions & Stochasticity
To build stochastic models for ecological data, we need a toolbox of [z]s. This toolbox contains probability functions and probability density functions that describe the way in which different types of data arise.
The [z]’s link our deterministic model with the data in a way that reveals uncertainty.

12. A toolbox of [z]’s Discrete Bernoulli: binary outcome of one trial
Binomial: Outcome of multiple trials.
Poisson: Counts.
Negative binomial: Overdispersed counts.
Multinomial: Multiple categorical outcomes.
Continuous
Normal.
Lognormal.
Exponential
Gamma
Beta
Others (Uniform, Dirilecht, Wishart).

13. Binomial distribution: Number of successes in n trials (Discrete events can take one of two values)
p = 0.5
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Event (xz
Probability
E[z] = np
Variance =np(1-p)
n = number of trials
p = prob. of success
Example: Probability of survival derived from population data

14. Binomial distribution: Number of successes in n trials (Discrete events can take one of two values)
What are the data?
What parameters do we want to estimate?
What terms control the uncertainty?
E[x] = np
Variance =np(1-p)
n = number of trials
p = prob. of success
0 ≤ p ≤ 1
Data are count (z==y). Prob (p),

15. Binomial distribution

16. Bernoulli=Binomial with n=1 trial

17. Multinomial distribution: Number of successes in n trials with > 2 possible outcomes for “success”

18. Poisson Distribution: Counts (or getting hit in the head by a horse)
500
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POISSON
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Count
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Proportion per Bar
Number of Seedlings/quadrat
y= number of units
per sampling effort
λ=average number of units
**Alt param= λ=rt

19. Poisson distribution

20. Exercise What are the data? What parameters do we want to estimate?
What terms control the uncertainty?
Data=z’s, we want to estimate the lambda/////

21. Clustering in space or time (overdispersion)
Poisson process
E[z]=Variance[z]
Negative binomial?
Poisson process
E[z]<Variance[z]
Overdispersed Clumped or patchy

22. Negative binomial: Table 4.2 & 4.3 in H&M Bycatch Data
E[Z]=0.279
Variance[Z]=1.56
Suggests temporal or spatial aggregation
in the data

23. Negative Binomial: Counts
100
0.2 90 80 70 60
Count 50
0.1
Proportion per Bar 40
N=total number of trials.
R=total number of successes
P=Probability of success 30 20 10
0.0 10 20 30 40 50
Number of Seeds
NEGBIN

24. Negative Binomial: Counts
100
0.2 90 80 70 60
Count 50
0.1
Proportion per Bar 40 30 20 10
0.0 10 20 30 40 50
Number of Seeds
NEGBIN

25. Negative binomial

26. Normal Distribution E[x] = μ Variance = σ2 Normal PDF with mean = 0 y
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Prob(x)
Var = 0.25
Var = 0.5
Var = 1
Var = 2
Var = 5
Var = 10
Reuls

27. Normal Distribution How do the data arise?
Often, the process we model represents the sum of continuous variables (ex., growth increment in trees increases on soil C content).
The fact that sums of things tend to be normally distributed is what stands behind the central limit theorem, which shows that the mean of random variables will be normally distributed even when the variables themselves follow other distributions.
With large sample sizes, many other distributions (Poisson, Gamma etc) can be approximated by a normal

28. Normal Distribution What are the data?
What parameters do we want to estimate?
What terms control the uncertainty?

29. A JAGS aside
Note that in JAGS, the normal distribution is parameterized in terms of precision, τ (tau).
Precision is the inverse of variance, so a when we use a small precision as a prior, it’s equivalent to a large variance (i.e., the uncertainty for that parameter is large).

30. Lognormal: One tail and no negative values
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y is always positive x
Product of processes.

31. Lognormal: Radial growth data1 2 3 4
HEMLOCK 50
100
150
Count
0.0
0.1
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Proportion per Bar
REDCEDAR 10 20 30 40
Growth (cm/yr)
Red cedar
Hemlock
(Data from Date Creek, British Columbia)

32. Exponential 1 2 3 4 5 6 10 20 30 40 50 60 70 80
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Proportion per Bar
Count
Data for weather events. Rainfall, probability of extreme events. Exponential is a special case of the gamma..
Variable

33. Exponential: Growth data (negatives assumed 0)1 2 3 4 5 6 7 8
Growth (mm/yr)
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600
800
1000
1200
Count
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Proportion per Bar
Beilschemedia pendula
(Data from BCI, Panama)

34. Gamma: One tail and flexibility
mean.gamma<-mean(sapling$Growth)
var.gamma<-var(sapling$Growth)
# now match moments to distribution parameters.
#get prior shape parameters for prior gamma distribution of lambda:
a=mean.gamma^2/var.gamma # shape parameter
b=mean.gamma/var.gamma #rate parameter

35. Gamma: “raw” growth data1 2 3 4 5 6 7 8 9
Growth (mm/yr)
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400
600
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1000
Count 10 20 30 50
100
150
200
Alseis blackiana
Cordia bicolor
Growth (mm/yr)
(Data from BCI, Panama)

36. Beta distribution: proportions
Survival, proportions.

37. Beta: Light interception by crown trees
(Data from Luquillo, PR)

38. Others
Dirilecht: multivariate version of beta
Multivariate normal

39. Bolker 2007

40. Bolker 2007

41. The Method of Moments
You can calculate the sample values of the moments of the distributions and match them up with the theoretical moments.
Recall that:
The MOM is a good way to get a first (but biased) estimate of the parameters of a distribution.

42. Why is MOM important?
We often need to account for uncertainty in our models but we only have data on sample mean and variance.
How can we make predictions when the mean and variance are not the parameters of the distribution (i.e., the normal distribution)?
The MOM can give us initial estimates of distribution parameters by matching moments (shape parameters) with sample mean and variance.

43. Haul & capture data (ED)
What distribution?
What are the parameters?
What are the moments of this distribution? Catch per haul

44. MOM: Negative binomial
We can get these from the data…

45. MOM Gamma Model biomass production (μ) as a function of rainfall (x).
Production cannot be negative, so you need a distribution for data that are continuous and strictly positive. Moreover, a plot of the data shows that the spread of the residuals increases with increasing production, casting doubt on the assumption that variance is constant
How do you represent uncertainty in production?
Why is normal not appropriate?

46. MOM: Gamma
Match moments (mean and variance) to parameters of gamma distribution, scale (α) and rate (β).

47. Mixture models
What do you do when your data don’t fit any known distribution?
Add covariates
Mixture models
Discrete
Continuous

48. Zero-inflated models
Zero-inflated models are a common type of finite mixture models.
Combine a standard discrete probability distribution (e.g. binomial, Poisson, or negative binomial), which typically include some probability of sampling zero counts even when some individuals are present with some additional process that can also lead to a zero count (e.g. complete absence of the species or trap failure).
Extremely useful in ecology.

49. An example: Seed predation
y =no seeds taken out of N available t1
t2 ( )
Assume each seed has equal probability (p) of being taken. Then:
Normalization constant

50. Zero-inflated binomial

51. Discrete mixtures

52. Discrete mixture: Zero-inflated binomial

53. Continuous mixture Seed retention times Turdus albicollis
Regurgitation
Defecation
Combination of a lognormal and a truncated normal distribution
Retention time (min)
Uriarte et al. 2011

54. Many other distributions…..
rvm(100,30,2)
rose.diag
Zimmerman et al. 2007

55. Some intuition for likelihood and stochasticity
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P(z|θ)
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Observations of z
What is the probability of obtaining the observations (z) conditional
on the value of θ?

56. Some intuition for likelihood and stochasticity
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“Unlikely” values
for θ
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P(z|θ)
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Observations of z

57. Some intuition for likelihood and stochasticity
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The most “likely”
Values for θ
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P(z|θ)
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Observations of z