Counting Hawks

Sylvia R. MoriPSW2005

A small, jay-size hawk (avg. 10-14" long), with dark gray back, a rusty-barred breast,

a slender square-tipped tail.

A medium crow-size hawk (avg.14-20" long) with a dark gray back, a rusty-barred breast, dark cap,and a long, rounded tail.

(Accipiter striatus) (Accipiter cooperii)

(Accipiter striatus) (Accipiter cooperii)

Immature Sharp-shinned Hawk Immature Cooper's

Immature Northern goshawk (Accipiter gentilis )

Northern goshawk

Cooper’s hawk

Relative AbundanceSharp-shinned hawk

Number of sightings

The Golden Gate Raptor Observatory (GGRO) is three staff members and more than 250 community volunteers, all dedicated to studying the autumn hawk migration

The GGRO is a program of the Golden Gate NationalParks Conservancy in cooperation with the National

Park Service

http://www.ggro.org/

Every autumn, thousands of migrating birds of prey appear over the Golden Gate near

San Francisco, California

How is the Hawk population doing? Are they increasing, decreasing or the same?

Statistical technique: Trend analysis

What are the proportions of misidentification

for Copper’s hawks and Sharp-shinned hawks

by age class and sex?

Statistical technique:Estimation of misclassification probabilities

Questions

Measured variables for trend analysis:

Number of birds counted by a group of trained observers (2-20),Year of the observation (Y), Age class (adult, juvenal or unknown) (A) ,Time of the year in Julian days (J), Time of the observation in hours on day of the observation (T),Duration of the observation in minutes (D),Number of observers’ class (1 if <7; 2 if ≥ 7) (O),Visibility class (1if <10 miles; 2 if ≥ 10 miles) (V).

Number of birds counted is the response of interest

Counting Hawks 18 years of count data (1986-2003)

Statistical model

The parameters in the functions are estimated via Maximum Likelihood Estimation (MLE)

les)ory variabg(Explanatf(year))count(E e

Example:

ˆˆˆˆˆˆˆ2cb2ˆˆE(count)

2 VOηAφDψTγTδJJYβYαe

^=estimate

Trend analysis

Observed as COHA

At noon for 1 hour and peak julian day (A:298, j:276, U:280)

with < 7 observers and visibility > 10

Year

20052000199519901985

Mea

n co

unt

20

15

10

5

0

95% CL

95% CL

Unid. age

95% CL

95% CL

Juvenal

95% CL

95% CL

Adult

Example

Julian Day of maximum abundance calculation

VηOφAψDγTδTJJβYαYeo

y22cb2

E(count)

2cblog *J*Ja)o

(yo

z

Fix all the explanatory variables except for the Julian day J, therefore the above equality is equivalent to

Using the derivative of zo with respect to J, the maximum is given by:

298)9.13(*2

10000*83.

c2

b-Max(J) Estimated

The Julian day 298 corresponds to October 25

The delta method expands a (non-linear) function of a random variable about its mean with a one-step Taylor approximation, and then takes the variance.

f(x) ≈ f(mu) + (x-mu)f'(mu) so that

Var(f(x)) = Var(x)*[f'mu)]2

where f() is differentiable and f'() = df/dx.

Expanded to vector-valued functions of random vectors,

Var(f(X)) = f'(mu) Var(X) [f'(mu)]T

and that in fact is the basis for deriving the asymptotic variance of maximum likelihood estimators. X is a 1 x p column vector; Var(X) is its p x p variance–covariance matrix; f() is a vector function returning a 1 x n column vector; and f'() is its n x p matrix of first (partial) derivatives. T is the transpose operator. Var(f(X)) is the resulting n x n variance–covariance matrix of f(X).

Julian Day of Maximum abundance and variance

The delta method

)c(c

)c,b(f*

)b(b

)c,b(f*)c,b(Covariance*2

2

)c(c

)c,b(f*)c(Variance

2

)b(b

)c,b(f*)b(Variance

c2

b-Variance

2c

1*

2

b-

c

)c,b(f

2c

1-

b

)c,b(f

2c

b-)c,b(ff(X)

Vector: X=(a,b)

Using the Delta Method, the estimated variance of the Julian dayof maximum abundance is approximated by:

A statistical computing procedure estimates the parameters of the statistical model via MLE . This procedure can also estimate the variance of the estimated Julian day of maximum abundance and its 95% confidence interval:

])Max(J tedvar[Estima2Max(J) Estimated

Adult

Juvenal

Unknown age

Sept. 30

Julian Day of maximum abundanceand confidence intervals

Species 1Species 2