class: center, middle, inverse, title-slide .title[ # Modelling spatiotemporal point patterns: a new era of point process models ] .subtitle[ ## Charlotte M. Jones-Todd
@cmjonestodd
] .author[ ### NZSA UnConference, November 22nd – 23rd 2022 ] .institute[ ###
University of Auckland ] --- #
I am not a robot <img src="img/recaptcha.jpg" width="40%" style="display: block; margin: auto;" /> --- # Stroma and tumour cells...
Statistics in Medicine (2018)
Identifying prognostic structural features in tissue sections of colon cancer patients using point pattern analysis
Jones-Todd, Charlotte M. et. al.
.center[ <img src="img/cancer_im.png" width="100%" style="display: block; margin: auto;" /> ] --- # The data <br> <br> <br> + 42 patients diagnosed with CRC + At follow up, 19 had died and the remaining were alive + Severity of the cancerous tissue graded by a pathologist as + Dukes A (least severe, all alive at follow up), + Dukes B ( `\(\sim\)` 50% alive at follow up), or + Dukes C (most severe, all dead at follow up). --- # Summarising the structure <br> <br> <br> .center[ <img src="img/cancer_pp.png" width="100%" style="display: block; margin: auto;" /> ] --- # 🌴 The palm likelihood approach + The Palm intensity is a function of distance `\(r\)`, characterised by the parameter vector `\(\theta\)`: `\(\lambda(r; \theta)\)` + It describes how the point density varies as a function of distance `\(r\)` from an arbitrarily chosen point. + It returns the expected intensity of a point process at a distance `\(r\)` from an arbitrarily chosen point. -- .center[ <img src="img/cancer_nn.png" width="80%" style="display: block; margin: auto;" /> ] --- # Results( `\(\thickapprox\)` ) <br> <br> <br> <br> + **Stroma cells**
+ Smaller clusters in patients that died + Fewer voids in patients that died + **Tumour cells**
+ Fewer clusters in patients that died --- #
What about the unknown... -- ## Random fields! <br> <br> <img src="img/rfs_cov.png" width="90%" style="display: block; margin: auto;" /> --- # log-Gaussian Cox process: the maths <br> Intensity surface `\(\rightarrow\)` `\(\{\Lambda(x): x \in {\rm I\!R}^{k+1}\}\)` .small[for *k*D space and 1D time] <br> <br> <br> Overall spatiotemporal intensity `$$\lambda(x) = \int_{{\rm I\!R^{k+1}}} \Lambda(x)dx$$` where `$$\Lambda(x) = \textrm{exp}(\bf{X}\beta + Z(x) + \epsilon)$$` for Gaussian random field, `\(Z(x)\)`, with mean `\(\boldsymbol{\mu}\)`, variance-covariance matrix `\(\boldsymbol{Q}^{-1}\)` --- #
Can I gift wrap that for you? <br> <br> <img width=400 height=400 src="https://encrypted-tbn0.gstatic.com/images?q=tbn:ANd9GcSCgRXxn4qNpSgkI4qHW-1SCWpJH-nHJWwiYg&usqp=CAU"/><img width=400 height=400 src="https://havefunbiking.com/wp-content/uploads/2016/12/hfb007.jpg"/> --- # Voronoi and Delaunay <br> <img src="img/mesh_plots.png" width="100%" style="display: block; margin: auto;" /> --- class: inverse, center, middle # Is there more? --- # Joint likelihood(s) <br> <br> `$$\begin{aligned} \Lambda(s) = \text{exp}(\boldsymbol{X}\boldsymbol{\beta} + G(\boldsymbol{x}) + \epsilon) \\ M_j(s) = f^{-1}((\boldsymbol{X}\boldsymbol{\beta})_{m_j} + G_{m_j}(\boldsymbol{x}) + \alpha_{m_j}\; G(\boldsymbol{x}) + \epsilon_{m_j}) \end{aligned}$$` <br> <br> The `\(\alpha_{m_j}\)` are coefficient(s) linking the point process and the `\(\text{n}_\text{mark}\)` mark(s), `\(m_j\)` `\((j = 1, ..., \text{n}_\text{mark})\)`, and the form of `\(M_j(s)\)` and `\(f^{-1}()\)` depends on the assumed distribution of the marks. --- # Multiple (shared) random fields in a multispecies model
Journal of the Royal Statistical Society: Series C (Applied Statistics) (2017)
A spatiotemporal multispecies model of a semicontinuous response
Jones‐Todd, Charlotte M. et. al.
Let `\(z_{ik}\)` be a binary indicator of the `\(k^{th}\)` species' presence ( `\(k = 1,2,3 =\)` sparrowhawk, collared dove, house sparrow) at site `\(i\)`. Then `\(z_{ik} \sim \text{Bernoulli}(p_{ik})\)`, where `\(p_{ik}\)` is the probability of presence of the `\(k^{th}\)` species at site `\(i\)`. Letting `\(d_{ik}\)` be the density of the `\(k^{th}\)` species at location `\(i\)`, then `\(d_{ik}\)` is given by, `$$d_{ik} = \left\{\begin{array}{ll} \text{Gamma}(a_{ik},b_{ik}) &\text{with probability}\; p_{ik}\\ 0, & \textrm{otherwise,} \end{array}\right.$$` with shape and scale parameters `\((a_{ik},b_{ik})\)` respectively, so that `\(E[d_{k}] = a_k\,b_k = \mu_k\)`. --- # Multiple (shared) random fields in a multispecies model
Journal of the Royal Statistical Society: Series C (Applied Statistics) (2017)
A spatiotemporal multispecies model of a semicontinuous response
Jones‐Todd, Charlotte M. et. al.
The joint model is given by, `$$\begin{aligned} \text{sparrowhawk} &\left\{ \begin{array}{l} \text{logit}(p_{i1}) = \mathbf{x_1}(\mathbf{s}_i,t) \\ \text{log}(\mu_{i1}) = \beta_{1} \mathbf{x_1}(\mathbf{s}_i,t), \\ \end{array} \right.\\ \text{collared dove} &\left\{ \begin{array}{l} \text{logit}(p_{i2}) = \mathbf{x_2}(\mathbf{s}_i,t) \\ \text{log}(\mu_{i2}) = \beta_{2} \mathbf{x_2}(\mathbf{s}_i,t), \\ \end{array} \right.\\ \text{house sparrow} &\left\{ \begin{array}{l} \text{logit}(p_{i3}) = \alpha + \gamma_{1} \mathbf{x_1}(\mathbf{s}_i,t) + \gamma_{2} \mathbf{x_2}(\mathbf{s}_i,t) + \mathbf{x_3}(\mathbf{s}_i,t) \\ \text{log}(\mu_{i3}) = \alpha_{y} + \gamma_{3} \mathbf{x_1}(\mathbf{s}_i,t) + \gamma_{4} \mathbf{x_2}(\mathbf{s}_i,t) + \beta_3\mathbf{x_3}(\mathbf{s}_i,t).\\ \end{array} \right.\\ \end{aligned}$$` Each `\(\mathbf{x_j}(\mathbf{s}_i,t),\: (j=1,2,3)\)` is a spatio-temporal random effect modelled by a SPDE model --- # Results( `\(\thickapprox\)` ) <br> <br> <br> + **Collared doves**
+ areas of high occurrence related to areas of high house sparrow occurrence and abundance + **Sparrowhawks**
+ areas of high occurrence related to areas of low house sparrow occurrence and abundance --- class: inverse, center, middle # Is there more? --- #
.footnote[.tiny[First to yell out the correct answer gets my permission to skip the queue @ morning tea]] <video width="500" height="20" controls> <source src="img/whoop.mp3" type="video/mp4"> </video> -- .center[ <img src="img/sasquatch.png" width ="60%" /> ] --- #
Sasquatch/Bigfoot .pull-left[ <br> <br> ![](https://upload.wikimedia.org/wikipedia/en/thumb/9/99/Patterson%E2%80%93Gimlin_film_frame_352.jpg/230px-Patterson%E2%80%93Gimlin_film_frame_352.jpg)] .pull-right[<img src = https://upload.wikimedia.org/wikipedia/commons/thumb/1/14/Pikes_peak_highway_big_foot.jpg/450px-Pikes_peak_highway_big_foot.jpg width=60%/>] [![Bigfoot Field Researchers Organization](https://www.bfro.net/images/bionics_v1_1213.jpg)](https://www.bfro.net/) --- # Self-exciting point process models
CRAN: R package version 0.0.1. (2022)
stelfi: Hawkes and Log-Gaussian Cox Point Processes Using Template Model Builder
Charlotte M. Jones-Todd; Alec van Helsdingen
<br> <br> .center[ <img src="img/hawked_fitted.png" width ="90%" /> ] --- # Temporal Hawkes process <br> <br> `$$\lambda(t) = \mu + \alpha \Sigma_{i:\tau_i<t}\text{exp}(-\beta * (t-\tau_i)) + \epsilon$$` <br> <br> + `\(n = 972\)` sightings over `\(\text{T} = 2188\)` days + `\(\hat{\mu} \text{T} = 0.12 \times 2188 \sim 263\)` baseline sightings + Expected number of sightings triggered by any one sighting `\(\frac{\hat{\alpha}}{\hat{\beta}} = \frac{0.06}{0.09} = \frac{2}{3}\)`. + Expected number of descendants per sighting `\(\frac{\hat{\beta}}{\hat{\beta} - \hat{\alpha}} = \frac{0.09}{0.09 - 0.06} = 3\)` + Rate of decay for the self-excitement `\(\frac{1}{\hat{\beta}} = \frac{1}{0.09} \sim 11\)` days. --- # Spatiotemporal self-exciting process .center[ <img src="img/stelfi.png" width ="70%" /> ] --- #
<i class="fas fa-image faa-flash animated "></i>
cmjt.github.io/slides/nzsa_2022
Statistics in Medicine (2018)
Identifying prognostic structural features in tissue sections of colon cancer patients using point pattern analysis
Jones-Todd, Charlotte M. et. al.
Journal of the Royal Statistical Society: Series C (Applied Statistics) (2017)
A spatiotemporal multispecies model of a semicontinuous response
Jones‐Todd, Charlotte M. et. al.
CRAN: R package version 0.0.1. (2022)
stelfi: Hawkes and Log-Gaussian Cox Point Processes Using Template Model Builder
Charlotte M. Jones-Todd; Alec van Helsdingen
--- # Diolch am wrando .pull-left[ .animate__animated.animate__bounceInDown[ ``` [1] " ____________" [2] "< Questions? >" [3] " ------------" [4] " \\ / \\ //\\" [5] " \\ |\\___/| / \\// \\\\" [6] " /0 0 \\__ / // | \\ \\ " [7] " / / \\/_/ // | \\ \\ " [8] " @_^_@'/ \\/_ // | \\ \\ " [9] " //_^_/ \\/_ // | \\ \\" [10] " ( //) | \\/// | \\ \\" [11] " ( / /) _|_ / ) // | \\ _\\" [12] " ( // /) '/,_ _ _/ ( ; -. | _ _\\.-~ .-~~~^-." [13] " (( / / )) ,-{ _ `-.|.-~-. .~ `." [14] " (( // / )) '/\\ / ~-. _ .-~ .-~^-. \\" [15] " (( /// )) `. { } / \\ \\" [16] " (( / )) .----~-.\\ \\-' .~ \\ `. \\^-." [17] " ///.----..> \\ _ -~ `. ^-` ^-_" [18] " ///-._ _ _ _ _ _ _}^ - - - - ~ ~-- ,.-~" [19] " /.-~" ``` ] ] .pull-right[ .center[ <img src="https://www.royalsociety.org.nz/assets/Uploads/Marsden-logo-rgb-96dpi.jpg" width ="20%" /> ] .center[ <img src="https://inro.pdn.ac.lk/assets/images/opportunities/AOARD.png" width ="20%" /> ] .center[[
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