class: left, title-slide .title[ # .white[Counting Animals Part II: Sample Counts] ] .subtitle[ ## .white[EFB 390: Wildlife Ecology and Management] ] .author[ ### .white[Dr. Elie Gurarie] ] .date[ ### .white[September 12, 2023] ] --- <!-- https://bookdown.org/yihui/rmarkdown/xaringan-format.html --> ## Drawbacks of total counts / censusing .pull-left-60.large[ Expensive & labor-time intensive Impractical for MOST species / systems - need to ALL be **visible** - the **ENTIRE** study area needs to be survey-able Hard to assess precision ] .pull-right-40[ ![](images/hippos.png) .center[**Hippos**] .small.grey[(Marc Mol/Mercury Press/Caters)] ] --- ### Is the great Elephant Census a Census? <iframe src="https://www.youtube.com/embed/imvehfydUpc?controls=0" width="900px" height="500px"> </iframe> --- ## Sample counts ### Simple idea: - count *some* of the individuals - extrapolate! ### In practice: - Involves some tricky statistics and modeling! - Necessarily - less *precise* due to *sampling error*. - BUT ... if properly done ... more *accurate* and **much less effort**. --- ## A random population .pull-left-60[ ![](images/Pop1.png) ] -- .pull-right-40[ ### Population density .large[$$N = A \times D$$] - `\(N\)` - total count - `\(A\)` - total area - `\(D\)` - overall density ] --- ## Sampling from the population .pull-left-60[ ![](images/Pop2.png) **Squares**, aka, **quadrats** ] -- .pull-right-40[ ### *Sample* density: `$$n_{sample} = \sum_{i=1}^k n_i$$` `$$a_{sample} = \sum_{i=1} a_i$$` `$$d_{sample} = {n_{sample} \over a_{sample}}$$` ] --- ### Sample vs. Population | Population | Sample --|:--:|:-- size | `\(N\)` | `\(n_s\)` area | `\(A\)` | `\(a_s\)` density | `\(D\)` | `\(d_s\)` Note: sample density is an *estimate* of total density. So `\(\widehat{D} = d_s\)`. -- True population: .green.large[$$N = A \times D$$] Population **estimate** (best guess for `\(N\)`): just replace true (unknown) density `\(D\)` with *sampling estimate* of density `\(d_s\)`: .red.large[$$\widehat{N} = A \times \widehat{D} = A \times d_s = A \times {n_s \over a_s}$$] --- ## Example .pull-left[ ![](images/Pop2.png) ] ### Data .blue[10 quadrats; 10x10 km each] .blue[ `n = {0,0,5,0,3,1,2,3,6,1}`] .red[**note:** *variability / randomness!*] -- ### Analysis .green[ `\(n_s = \sum n_i = 21\)` `\(d_s = \widehat{D} = {21 \over 10 \times 10 \times 10} = 0.021\)` `\(A = 100 \times 100\)` ] -- #### final estimate: .large.green[ `$$\widehat{N} = \widehat{D} \times A = 100\times100\times0.021 = \textbf{210}$$`] --- ## What happens when we do this many times? .pull-left[ ![](images/popSims.png) ] .pull-right[ Every time you do this, you get a different value for `\(\widehat{N}\)`. ![](images/SimHist.png) ] --- ### Statistics .pull-left[ **Mean of estimates:** `$$\widehat{N} = 301.5$$` **S.D. of estimate:** `$$s_{\widehat{N}} = 54.6$$` .red[**important**: the *standard deviation* of an *estimate* = **standard error**, SE] **95% Confidence Interval:** `$$\widehat{N} \pm 1.96 \times SE = \{195-408\}$$` .green[**note:** the 1.96 is the number of standard deviatinos that captures 95% of a Normal distribution.] ] .pull-right[ ![](images/SimHist2.png) ] -- Conclusion: this estimate is **accurate** (unbiased), but not very **precise** (big confidence interval). --- ### General principle: The bigger the sample, the smaller the error. `1.` If `\(a_s \ll A\)` (i.e. low sampling intensity) `$$SE(\widehat{N}) = {A \over a} {\sqrt{\sum n_i} \over k}$$` **remember:** - `\(n_s = \sum n_i\)` is the total **sample count** - `\(k\)` is the total number of samples: `\(i =\{1,2,...,k\}\)` ) .center.red[ in our example: `\(SE = 100²/10^2 \times \sqrt{21}/10 = 54.8\)` ] -- `2.` If you are NOT resampling previously sampled locations: `$$SE(\widehat{N}) = {A \over ak} \sqrt{\sum n_i \left(1 - {a_s \over A}\right)}$$` This is the .blue[Finite Area Correction]. If `\(a = A\)` - you sampled everything - SE goes to 0 as expected. .center.red[ in our example: `\(SE = 54.5\)` ... Almost no difference (because `\(a \ll A\)`). ] --- ## Some more complex formulae from Fryxell book Chapter 12: ![](images/fryxellformulae.png) These are used when **sampling areas** are unequal, and account for differences when sampling **with replacement** or **without replacement**. --- ### Poisson process Models *counts*. If you have a perfectly random process with mean *density* (aka *intensity*) 1, you might have some 0 counts, you might have some higher counts. The *average* will be 1: ![](images/Poisson1.png) --- ### Poisson process Here, the intensity is 4 ... ![](images/Poisson4.png) --- ### Poisson process ... and 10. Note, the bigger the intensity, the more "bell-shaped" the curve. ![](images/Poisson10.png) Here's the formula of the Poisson Distribution: `\(\!f(k; \lambda)= \Pr(X{=}k)= \frac{\lambda^k e^{-\lambda}}{k!}\)` --- ### Poisson distribution holds if process is truly random ... not **clustered** or **inhibited** ![](images/processes.png) If you **sample** from these kinds of spatial distributions, your standard error might be smaller (*inhibited*) or larger (*clustering*). This is called *dispersion*. --- ### Also ... densities of animals depend on habitat! .pull-left-60[ **Wolf habitat use** ![](images/vikihabitat.png) ] .pull-right-40[ If you look closely: - No locations in lakes - Relatively few in bogs / cultivated areas. - Quite a few in mixed and coniferous forest ] --- ## Imagine a section of forest ... .pull-left-60[ ![](images/Moose1.png)] --- ## ... with observations of moose .pull-left-60[ ![](images/Moose2.png)] .pull-right-40[ **How can we tell what the moose prefers?** Habitat | Area | n | Density ---:|:---:|:---:|:--- open | 100 | 21 | 0.21 mixed | 100 | 43 | 0.43 dense | 200 | 31 | 0.16 **total** | 400 | 95 | 0.24 ] .blue[Knowing how densities differ as a function of **covariates** can be very important for generating estimates of abundances, increasing both **accuracy** and **precision**, and informing **survey design**.] --- ### Sample frames need not be **squares** .pull-left-50[ ![](images/aerial-survey.jpg) ] .pull-right-50[ ## Transects Linear strip, usually from an aerial survey. Efficient way to sample a lot of territory. If "perfect detection", referred to as a **strip transect**. Statistics - essentially - identical to quadrat sampling. ] .footnote[https://media.hhmi.org/biointeractive/click/elephants/survey/survey-aerial-surveys-methods.html] --- ### **Stratified sampling** for more efficient estimation ![](images/stratification1.png) Sample more intensely in those habitats where animals are more likely to be found. Intensely survey .orange[**blocks**] where detection is more difficult. .footnote[https://media.hhmi.org/biointeractive/click/elephants/survey/survey-aerial-surveys-methods.html] --- ### **Stratified sampling** for more efficient estimation ![](images/stratification2.png) Actual elephant flight paths, .footnote[https://media.hhmi.org/biointeractive/click/elephants/survey/survey-aerial-surveys-methods.html] --- ### **Stratified sampling** .pull-left[![](images/stratification1.png)] .pull-right[![](images/stratification2.png)] **Stratification** is used to optimize **effort** and **precision**. Aircraft cost thousands of dollars per hour! (In all of these comprehensize surveys - *design* takes care of **accuracy**). --- ### Sampling strategies .pull-left[![](images/samplingstrategies.png)] .pull-right[ (a) simple random, (b) stratified random, (c) systematic, (d) pseudo-random (systematic unaligned). Each has advantages and disadvantages. See also: *Adaptive Sampling* ] --- ### Detections usually get *worse* with distance! .pull-left-30[ ![](images/DistanceEquations.png) ![](images/DistanceSampling.jpg) ] .pull-right-70[ ## Distance Sampling The statistics of accounting for visibility decreasing with distance ![](images/DistanceCurves.webp) ] --- ## Example reindeer in Svalbard ![](images/DistanceReindeer.png) .large[ **Estimated detection distance**, compared to **total count**, incorporated **vegetation modeling**, computed **standard errors**, concluded that you can get a 15% C.V. for 1/2 the cost.] --- ## Example Ice-Seals ![](images/lobodontini.png) --- ## Example: Flag Counting at Baker ![](images/court.png) --- ## Nice video on counting caribou https://vimeo.com/471257951 ![](images/countingcaribou.png)