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# SAMPL ANIM POPNS ST 506

NCS

GPA 3.79

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This 55 page Class Notes was uploaded by Jordane Kemmer on Thursday October 15, 2015. The Class Notes belongs to ST 506 at North Carolina State University taught by K. Pollock in Fall. Since its upload, it has received 32 views. For similar materials see /class/223941/st-506-north-carolina-state-university in Statistics at North Carolina State University.

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Date Created: 10/15/15

Lecture 5 DISTANCE SAMPLING Ch 13 LILquot Key Concepts in Distance Analysis An observer travels along a line and counts animals plus measure their perpendicular distance Detection probability is related to perpendicular distance There is a detection function g X which starts at l on the line and decreases monotonically with distance We estimate g X and use it to compute the probability of detecting an animal in the strip P J gemw 2 area under curve area if perfect detection Probability of Detection in Strip area under curve 33 w B c x dx 3 Area under Curve P J1 E w a 2 1 9Xdxit w E D 9 3 5 o 1 I 39 i V area 1f perfect detectlon I 7 1 4 quotx 391 4 4 7 W o m4 f v Perpendicular distance x Fig 31 The unconditional probability that an animal within distance in of the line is detected is the area under the detection function it divided by the area of the rectangle 10711 Key Concepts in Distance Analysis We estimate g X and use it to compute the probability of detecting an animal in the strip We can then estimate density using Perfect Detection D 2 na 2 n2wL Imperfect Detection D naf a n2wLI3a Key Concepts in Distance Analysis To t gx we need to t a series of models based on a key function plus a series expansion This is to give us more and more exible functions We distinguish between competing models using the AIC we minimize AIC 2 log L 2 pars This will be covered later today For the chosen model with min AIC ML Estimation is used to estimate the parameters and hence the density parameter with its SE and Con dence Interval based on approx normality Key Concepts in Distance Analysis We can also look at the chisquare goodness of t test as a way of seeing if the best model using AIC actually explains the observed data very wellThis will be covered later today 12 2 20139 Ei2 Ei Oi observed value for celli E i expected value for cell i Key Functions ASSUMPTIONS AND MUDI IIJJKG PHILOSOPHY Uniform Half Normal Uniform eciian lunctian 90 Negative Exponential Del m Not usually used I will explain this 50 Distance y The Hazard Key Function Has a Range of Shapes Fig26 Key functions useful in modelling distance data a u i ml m quot1ndb 39 quotforl39ourd Iliform illurlrnl MORE ROBUST MODELS Program DISTANCE gx keyx 1 seriesx Usual Kev functions Uniform W no parameters 2 2 Halfnormal eXp39 X 2039 one parameter Hazardrate 1 exp Xa39b two parameters Series expansion Simple polynomials Hermite polynomials Fourier series These approaches all involve speci cation of some exible parametric form for gxThey also involve complex model choices ROBUST MODELS Program DISTANCE gx keyx 1 seriesx Example Key function Halfnormal exp X2 2039 2 Example Series expansion Simple polynomialsjust a way to get a more complex function m 2 39 serles 2 ijla xa J j1 611xc72 j2 611xc72 azxc74 ASSUMPTIONS OF LINE TRANSECT SAMPLING 1 g0 l ie certain detection on line 2 Objects are detected at their initial location prior to any movement in response to the observer 3 Distances measured accurately ungrouped data or objects counted in proper distance category grouped data 4 Objects are detected independently 5 We are also assuming transects are placed randomly or systematic random perhaps Within strata DUCK NEST DATA Example of a Line Transect To show intuition behind the method Anderson and Pospahala 1970 JWM 34141146 Survey of duck nests at the Monde Vista National Wildlife Refuge 1600 miles of transect with 8 feet on either side of transect 534 nests seen in 1967 and 1968 7473796678585254 in each 1 foot grouping Simple Quadratic Model worked well and led to the theoretical advances that came later Anderson And Pospahala Grouped Data g3 17 72 S 50 Frequency Anderson and Pospahala Curve Fitting This Led to all the Later Theory 100 Frequency 7705 0409 x 2 PO888 quot Correction 1126 x Frequency U1 0 Disiance ft DUCK NEST DATA The rough quadratic model Frequency 7705 04039 X2 Allows calculation of Pa as 13 unshaded area a area of rectangle The unshaded area could be computed using calculus or other methods A Pa 0888 DUCK NEST DATA f n 2LW a 534 2x1 600x85280x0888 12403 A D 124 nests per square mile This example is only intuitive Better analysis methods eXist for estimating the parameter Pa Hwk 2 Duck Nest Data Analysis 1 In the Table below I present results of a set of 4 models starting with the uniform key function and adding simple polynomial adjustment terms for the duck nest data truncated at 7 ft Model pars AIC D SEGA Zzpvalue Uniform 1 186807 113 14 516 016 Uniform 1 186353 128 14 808 067 2 Uniform 2 3 186484 12353 1024 066 Uniform 3 4 186684 12363 1226 049 Explain clearly and in detail all terms and concepts used and the then please indicate what model should be used and what the density estimate and its 95 con dence interval would be Also Anderson and Posphahala intuitive estimate was 12403 It is similar to the estimates here except for the uniform Without any adjustment 11314 Note how the SE s increase as the models get more complex by adding parameters Stake Data Example 277278 True D 375 stakeshectacre Discuss Table 132 Page 278 in detail to illustrate the methodology and use of program DISTANCE First we need to review MLEs and Goodness of Fit Tests We also need to have a discussion of Akaike s Information Criteria AIC now This is covered in 44 5557 TABLE 132 Example of Line Transact Estimation Using Laake s Wooden Stake Data Number of Goodness of fit 95 CI Key adjustment A A A function Adjustment terms AlC AAIC x2 df P D CL CU cv Uniform Cosine 2 38214 0 887 9 045 40577 29213 56361 01658 Uniform Cosine 1 38411 197 1342 10 020 33079 25379 43116 01334 Uniform Cosine 3 38414 200 465 7 070 40416 27363 59695 01973 Half normal Hermite polynomial 2 38416 202 888 8 035 40793 28574 58236 01798 Half normal 0 38578 364 1621 10 009 34561 25942 46044 01445 Half normal llermite polynomial 1 38773 559 1613 9 006 34589 23668 50549 01919 Uniform 0 40924 271 3937 11 lt00 18817 14839 23862 01195 After Burnham ct 11 1980 The analysis is based on a sample of 68 wooden stakes of known density D 375 stakes m utilizing the complete data set no right censoring SAMPLING ANIMAL POPULATIONS Statistical Concepts Chapter 4 Methods of Estimation Method of Maximum Likelihood See Also Handout Derive the probability distribution of the observed data as a function of the parameters px9 View this as a function of the parameters this is the likelihood function L6 Find the values of the parameters which maximise this functionThese are the maximum likelihood estimators MLEs 6 Method of Maximum Likelihood Methods of Maximising a Function Mathematical Set partial derivatives equal to O Solve the resulting equations You may have seen this in a calculus class Computer Package Use a computer package which may use a variety of algorithmns Often necessary as the likelihoods are so complex with many parameters Program DISTANCE does this for us Goodness of Fit Tests 433 ChiSquare Goodness of Fit Tests are the simplest approach is to use 12 220iEi2Ei Oi observed value for cell i Ei expected value for celli based on the model if k l q k no cells q no of parameters tted Example 22 20 E2 E 01 74 E1 723 model prediction 12 74 7232 723 other terms df k 1 q k no cells 8 in duck nest data q no of parameters tted 0 uniform 1 uniform plus 1 term etc df 8 11 6 for uniform with one term added MODEL SELECTION In many problems in statistics we don t just t one model We may t many models and estimate parameters with maximum likelihood and then have to decide which is the best one Remember the duck nest data you have to look at for homework There are 4 models The first is uniform but then additional terms are added to get a more realistically shaped curve Model pars AIC f SEGA zzpvalue Uniform 1 186807 11314 516 016 Uniform 1 2 186353 12814 808 067 Uniform 2 3 186484 12353 1024 066 Uniform 3 4 186684 12363 1226 049 Note For the stake data there are even more models compared 7 In what other area in statistics have you seen where there are many models tted MODEL SELECTION Simple Model Reality Precision Low bias Smaller SEs Complex Model Key Points Principle of parsimony Trading off Reality vs Precision MODEL SELECTION Akaike Information Criteria Reduce parameters to achieve parsimony and higher precision of estimates Need enough parameters to be realistic model We need a way of trading off reality more parameters vs precision less parameters MODEL SELECTION Akaike Information Criteria Use Model selection criteria which is to Minimize AIC 2 log L 2 pars First term 2 log Lnote L is Likelihood function evaluated at the MLE values Second Term Penalty for over parameterization so we add a term of twice the no of parameters TABLE 132 Example of Line Transect Estimation Using Laake s Wooden Stake Data Number of Goodness of fit 95 Cl Key adjustment A function Adjustment terms AIC AAIC x2 df P D 1 CU cv Uniform Cosine 2 38214 0 887 9 045 40577 29213 56361 01658 Uniform Cosine 1 38411 197 1342 10 020 33079 25379 43116 01334 Uniform Cosine 38414 200 465 7 070 40416 27363 59695 01973 38416 202 888 8 035 40793 28574 58236 01798 38578 364 1621 10 009 34561 25942 46044 01445 38773 559 1613 9 006 34589 23668 50549 01919 40924 271 3937 11 lt0039l 18817 14839 23862 01195 Half normal Hermite polynomial Half normal llalf normal llermite polynomial O CDNLQ Uniform quot After Burnham at al 1980 The analysis is based on a sample of 68 wooden stakes of known density D 375 stakesin utilizing the complete data set no right censoring Background on the Laake Stake Data Field Test of the Line Transects Method Wooden stakes at a density of 375 stakes per hectacre were placed in a sagebrush meadow in Utah Jeff Laake who was a graduate student of David Anderson ran the experiment A whole class of students went along transect lines and detected stakes Nice example to see how well the method works in a eld setting with objects that don t move Stake Data of Table 132 This was data for just one of the many observers and was used to illustrate the methodology Remember that he detected 68 stakes and each one would have a perpendicular distance This data is not shown Also the true density is 375 stakesha A whole suite of model results 7 models from DISTANCE are shown and compared based on Delta AIC AAIC AICmod AICmm Stake Data of Table 132 Two different key functions Uniform and Half Normal and then different numbers of adjustment terms Total of 7 models shown 4 based on uniform and 3 based on the Half Normal For each model the density estimate and its con dence limits are shown For the best one the estimate is 40577 with Cl 29213 56361 The estimate is slightly above truth but not unusually so if you look at the confidence interval Note the perfect detectability model is the uniform with no adjustment terms It has delta AIC of 271 by far the largest and a density estimate of 18817 which is about half of the truth of 375 Stake Data of Table 132 We also need to notice the results of the chi squared goodness of t tests as well Higher p values the model ts the data adequately Whereas small p values the model does not t the data adequately Note in particular how badly the uniform with no adjustment terms ts the data p lt00l Recall that is the model With assumed perfect detectability TABLE 132 Example of Line Transect Estimation Using Laake s Wooden Stake Data Number of Goodness of fit 95 Cl Key adjustment A A A function Adjustment terms AlC AAIC x2 df P D CL CU cv Uniform Cosine 2 38214 0 887 9 045 40577 29213 56361 01658 Uniform Cosine 1 38411 197 1342 10 020 33079 25379 43116 01334 Uniform Cosine 3 38414 200 465 7 070 40416 27363 59695 01973 Half normal Hermite polynomial 2 38416 202 888 8 035 40793 28574 58236 01798 Half normal 0 38578 364 1621 10 009 34561 25942 46044 01445 llalf normal llermite polynomial 1 38773 559 1613 9 006 34589 23668 50549 01919 Uniform 0 40924 271 3937 11 lt001 18817 14839 23862 01195 After Burnham of a1 1980 The analysis is based on a sample of 68 wooden stakes of known density D 375 stakesm utilizing the complete data set no right censoring More On Stake Data from Distance Book Multiple Observers on the Stake Data 328 Table 83 Summary of stake data taken in 1978 in a sagebrushegrass eld near ILLUSTRATIVE EXAMPLES Logan Utah Laake 1978 Density in each of the 11 surveys was 375 stakesha Cosine adjustments were added as required in modelling f m For each survey L 1000111 and w 2 20m Survey Key Sample Density Log based 95 no function size estimate cu con dence interval 1 Half normal 72 3711 193 2551 5399 Uniform 3000 145 2263 3977 2 Half normal 48 3518 199 2390 5178 Uniform 3601 201 2436 5323 3 Half normal 74 2876 158 2116 3910 Uniform 2926 148 2192 3907 4 Halfnormal 59 3831 191 2642 5552 Uniform 3330 175 2368 4681 5 Half normal 59 3441 199 2337 5066 Uniform 2958 190 2047 4276 6 Half normal 72 2638 162 1924 3617 Uniform 2708 156 1998 3669 7 Half normal 55 3448 109 2344 5072 Uniform 3469 211 2306 5218 8 Halfrnormal 61 3331 202 2251 4930 Uniform 3448 213 2282 5209 9 Half normal 46 2832 219 1851 4331 Uniform 2352 212 1560 3548 10 Halfnormal 43 3416 201 2315 5042 Uniform 3269 211 2171 4922 11 Half normal 53 2980 174 2125 4180 Uniform 3145 178 2223 4450 Mean Half normal 642 3275 36 3054 3511 Uniform 3110 36 2809 3336 Pooled Half normal 642 3437 72 2986 3955 Uniform 3460 75 2986 4008 Note Observers vary a lot in ability On average they underestimate True Density Objects near line missed plus some measurement error on distances MORE EXAMPLES Immobile Objects Aerial Surveys Bird Surveys IMMOBILE OBJECTS Duck Nest Data Example C0uld be used to estimate deaths of animals after a re Pr0minent nest structures of large birds like eagles quot a r R 3 Line Transects discussed rst Flushing species sighted Terrestrial forest birds sight and sound Concerns gt movement before detection gt high canopy g0 7quot 1 gt measurement errors on distance gt terrain too rugged to walkampcount point counts better then AERIAL LINE TRANSECTS Porpoise and Dolphins in the OceansMany surveys Southern Blue n Tuna Schools Australia Polar Bears Alaska Terrestrial MammalsMany Surveys Concerns Logistics Fatigue of Observers g0 may not be 1 The issue of availability arises underwater also even if available there still could be animals on line missed Measurement Error in Distance Clusters of Animals Clustered Animals Section 1327 in Text In many situations animals may occur in groups or clusters For example dolphins in an aerial line transect survey Protocol then involves rst detecting the clusters measuring the distance to the center of the cluster and then counting accurately how many animals are in the cluster One problem is if clusters are very large and diffuse This is not usually the case for say dolphins Clustered Animals Estimation Issues If clusters all equally detectable then use a simple approach D D E 2 density of clusters X mean cluster size S Potential Problem Sometimes clusters have different detection probabilities depending on cluster sizeThere are analyses which take account of this but I will not discuss in this class Point Transects 0r Point Counts for Estimating Bird Numbers EEPoint Counts or Transects Brief Summary Notes 1 The Method V Sampling oints are set up in study area They could be Placed sys ematlcally along transects or scattered randomly hroughout area V Thedistance between stations should be such that the same blrd IS yery unllkely to be counted at more than one stat10n Thls Wlll vary dependmg on the habltat V Count all the birds seen or heard with their distances in a xed t1me Interval say 3 mmutes Coun r all birds seen or heard during a fixed Time in rerval No es rima re of de rec rion probabili ry Analyses generally assume de rec rion probabili ry is invarian r Coun rs considered as an index Coun rs very misleading Scarle r Tanagers vs Golden Crowned Kingle rs discussed in lec rure 3 on Indicies Goldencrowned Kingle r d U C N 10 392 3 Q 08 7 247 PaIred Pomts N 30 detectionsspecies 2 D T 05 Primary Forest El Secondary Forest 3 n c 04 7 N d E 02 00 7 DJ BB BT WW SV VE OB RV ST GK RN CN BL BC PA HW BR RT Density Kinglef jumps To Third when defection probability accounted for using distance samplingll HE A n C m n a V gt r m c o D BEE WEE IE EPoint Counts or Transects Brief Summary Notes 1 The Method V Each bird seen or heard during a xed period around a station IS cgunted and the horizontal dlstance to its location IS measure V then the bird IS heard rather than seen and thus the distance has to be estimated rather 1mprec1sely as some birds may never be seen Training programs to try to minimize the errors V DISTANCE used for analyses V Multiple species are counted at the same time Note Fisheries exam les Individuals 0fa variet of species could be counted rom underwater pomts in c ear water eg coral reefs Recommendations Measure distances to birds rather than xed radius points I Allows more options for analyses i Full distance analysis ii Fixed radius for individual species ie different radii I Allows detectability to be modeled and hence should be less biased I If concerned about disruption have a waiting period before counting begins Simons Avian Study in GSMNP Point counts Multiple species 10 minute interval Single observer Distance sampling Birds Heard amp Rarely Seen Point locations Low use hiking trails Stratified by vegetation type Distance Sampling Use of sheets to help with recording and distance estimation Distance used to estimate detection probabilities DisTance Sampling The Sigh ring func rion gr39 1 Analogous To in gr Line Tr39ansec rs O Declines wi rh distance distance r We assume gO 10 Es rima rion of Densi ry Similar Equa rions To Line Tr39ansec rs bu r geome rr39y of circles a Ii r rle differen r f HA APa WithA k7Tw2 and I 2 r rdrw2 0 3 Variable Circular Plot Assumptions same as for line transects i Birds that are very close to the station will always be detected ii There is no movement of birds in response to the observer attraction or repulsion and none are counted twice iii All distances are measured Without error iv Sightings of different birds are independent events v Points placed randomly or systematic random House Wrens Point Count Study Description of Study 155 points in 10 16 ha plots Analysis from DISTANCETable 133 Data fitted a little better when truncated but still not great This often happens with point count data Use of AIC to choose models like for the line transect example The best model had an estimate of 814 birds per square km CI was 6441030 The next two models had similar estimates and were close in terms of AIC TABLE 133 Example of Point Data from Surveys of House Wrens Troglodytes aedon Surveyed along the South Platte River Colorado Number of Goodness of fit 95 CI adjustment A Key function Adjustment terms AIC AAIC x2 df P D It CU Data untruncated mz 5m Half normal Hermite polynomial 3 66248 0 108 4 003 828 698 982 Half normal Cosine 3 66299 51 107 4 003 847 724 991 Uniform Cosine 4 66334 86 188 3 lt0001 672 595 758 Hazard Simple polynomial 1 66655 408 393 4 lt0001 605 528 693 Data truncated w 425 m Hazard Simple polynomial 1 55238 0 71 3 007 814 644 1030 Half normal Cosine 1 55248 1 76 4 011 901 743 1092 Uniform Cosine 3 55260 22 70 3 007 905 748 1095 Half normal Hermite polynomial 1 55280 42 121 4 002 784 677 907 From Buckland ST Anderson DR Burnham KB and Laake L 1993 quotDistance Sampling Estimation of Biological Populations Chapman and Hall New York with kind permission from Kluwer Academic Publishers Comparison of Distance Methods For Birds Line Transects Variable Circular Plots c0ver larger areas more quickly More practical in rough terrain have to assign line segments to Variable habitats it is easier to habitats assign point to a habitat type speed may vary xed time at each station more time to see or hear birds in high canopy

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