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https://github.com/fabrice-rossi/blvim

Boltzmann–Lotka–Volterra Interaction Model
https://github.com/fabrice-rossi/blvim

boltzmann-lotka-volterra r r-package r-stats spatial-analysis spatial-interaction-modeling

Last synced: 4 months ago
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Boltzmann–Lotka–Volterra Interaction Model

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README 

          
---

output: 

  github_document

---



```{r, include = FALSE}

knitr::opts_chunk$set(

  collapse = TRUE,

  comment = "#>",

  fig.path = "man/figures/README-",

  out.width = "100%"

)

```



# blvim: Boltzmann–Lotka–Volterra Interaction Model blvim website



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`blvim` implements A. Wilson's **Boltzmann–Lotka–Volterra (BLV) interaction model**. 

The model is described in 

[Wilson, A. (2008), "Boltzmann, Lotka and Volterra and spatial structural evolution: an integrated methodology for some dynamical systems", J. R. Soc. Interface, 5:865–871](http://dx.doi.org/10.1098/rsif.2007.1288).



The package's primary goal is to provide a fast implementation of the BLV 

model, complete with a collection of tools designed to explore the results 

through statistical summaries and graphical representations. The secondary goal 

is to facilitate the systematic assessment of how model parameters impact 

the results, again using summaries and graphical representations 

(see `vignette("grid")` for details on this aspect).



## Installation



You can install `blvim` from CRAN with:



``` r

install.packages('blvim')

```



You can install the development  version from 

[R-universe](https://r-universe.dev/) with:



``` r

install.packages('blvim', repos = c('https://fabrice-rossi.r-universe.dev'))

```



## Spatial interaction models



**Spatial interaction models** aim to estimate flows between locations, such as 

workers commuting from residential zones to employment zones. The `blvim` 

package focuses on the maximum entropy models developed by Alan Wilson. 

See `vignette("theory")` for the theoretical background.



In practice, if we have $n$ origin locations and $p$ destination locations, the

goal is to compute a flow matrix $(Y_{ij})_{1\leq i\leq n, 1\leq j\leq p}$, where

$Y_{ij}$ is the flow from origin $i$ to destination $j$. This computation relies 

on characteristics of the origin and destination locations, along with a matrix 

of exchange difficulties, known as a **cost matrix**, 

$(c_{ij})_{1\leq i\leq n, 1\leq j\leq p}$. For example, $c_{ij}$ can represent 

the distance between origin $i$ and destination $j$.



## Usage



The package is loaded in a standard way.

```{r}

library(blvim)

```



### Input data



To compute a spatial interaction model with `blvim`, you need at least a cost 

matrix. The package comes with distance data for a selection of large French

cities. We use the 30 largest ones as origin locations and the 20 smallest ones

as destination locations. The cost matrix is the distance between the cities 

(in meters).



```{r}

## 30 largest cities

origins <- french_cities[1:30, c("th_longitude", "th_latitude")]

## 20 smallest cities

destinations <- french_cities[

  (nrow(french_cities) - 19):nrow(french_cities),

  c("th_longitude", "th_latitude")

]

## cost matrix

cost_matrix <-

  french_cities_distances[

    1:30,

    (nrow(french_cities) - 19):nrow(french_cities)

  ]

rownames(cost_matrix) <- french_cities[1:30, "name"]

colnames(cost_matrix) <-

  french_cities[(nrow(french_cities) - 19):nrow(french_cities), "name"]

```



Additionally, since we focus on **production-constrained models**, we must 

specify the production for each origin location (a vector of positive values 

$(X_i)_{1\leq i\leq n}$). Here, we assume a common unitary production.



```{r}

X <- rep(1, nrow(origins))

```



Finally, the simple **static** model requires an attractiveness value for each 

destination location, a vector of positive values $(Z_j)_{1\leq j\leq p}$. We 

again assume a common unitary attractiveness.



```{r}

Z <- rep(1, nrow(destinations))

```



We could use the population of the cities as production constraints for instance. 



### Static models

In Wilson's production-constrained maximum entropy model, the flows are given

by



$$

Y_{ij} = \frac{X_iZ_j^{\alpha}\exp(-\beta c_{ij})}{\sum_{k=1}^pZ_k^{\alpha}\exp(-\beta c_{ik})},

$$



where $\alpha$ is a return-to-scale parameter and $\beta$ is the inverse of a

cost scale parameter. Note that the flow matrix is **production-constrained**, 

meaning that the total outgoing flow from any origin location is equals 

the production of that location:



$$

\forall i,\quad X_i=\sum_{j=1}^{p}Y_{ij}.

$$



The model is obtained using the `static_blvim()` function:

```{r}

a_model <- static_blvim(cost_matrix, X, alpha = 1.1, beta = 1 / 500000, Z)

a_model

```



Several functions are provided to extract parts of the result. In particular

`flows()` returns the flow matrix $Y$. 

```{r}

a_model_flows <- flows(a_model)

```

which can be displayed using, for instance, the `image()` function.

```{r a_flow}

#| fig.height: 11

#| fig.width: 6.5

#| out.width: 50%

#| fig.align: center

#| fig.alt: >

#|   A matrix representation of the flows. It shows no particular structure

#|   in the flow apart from a dominating flow from Nice to Cagnes-sur-Mer.

par(mar = rep(0.1, 4))

image(t(a_model_flows)[, 30:1],

  col = gray.colors(20, start = 1, end = 0),

  axes = FALSE,

  frame = TRUE

)

```



In this representation, each row shows the flows from one origin location to all

destination locations. The package also provides a `ggplot2::autoplot()` 

function, which can be used as follows:



```{r a_flow_ggplot2}

#| fig.height: 10

#| fig.width: 8

#| out.width: 80%

#| fig.align: center

#| fig.alt: >

#|   A matrix representation of the flows. The color scheme is more contrasted

#|   than in the previous figure, but does not show much more strucutre. The

#|   dominating flow is Nice to Cagnes-sur-Mer, but the figure shows that there

#|   are additional important flows.

library(ggplot2)

autoplot(a_model, "full") +

  scale_fill_gradient(low = "white", high = "black") +

  coord_fixed()

```



```{r}

b_model <- static_blvim(cost_matrix, X, alpha = 1.1, beta = 1 / 100000, Z)

b_model

```



```{r b_flow}

#| fig.height: 10

#| fig.width: 8

#| out.width: 80%

#| fig.align: center

#| fig.alt: >

#|   A matrix representation of the flows. The flows are much more concentrated

#|   than with the previous choice of parameters. Many destination locations

#|   do not receive any flow. Most of the flow goes to Vaulx-en-Vellin, Arles,

#|   Albi and Cagnes-sur-mer.

autoplot(b_model) +

  scale_fill_gradient(low = "white", high = "black") +

  coord_fixed()

```



As the two figures above exemplify, different values of the parameters $\alpha$ 

and $\beta$ result in more or less concentrated flows.



### Dynamic models



A. Wilson's **Boltzmann–Lotka–Volterra (BLV) interaction model** builds upon the

production-constrained maximum entropy model. The core idea is to update the 

attractiveness of the destination locations based on their incoming flows.



Ideally, we aim for the following condition to hold in the limit:



$$

Z_j =\sum_{i=1}^{n}Y_{ij}, 

$$



where the flows are given by the equations above. The model is estimated using

the `blvim()` function as follows.



```{r}

a_blv_model <- blvim(cost_matrix, X, alpha = 1.1, beta = 1 / 500000, Z)

a_blv_model

```



Notice that we start with some initial values of the attractiveness, but the 

final values are different. These final values can be obtained using the 

`attractiveness()` function (and visualised here using a bar plot).



```{r a_blv_Z}

#| fig.height: 4

#| fig.width: 8

#| out.width: 80%

#| fig.align: center

#| fig.alt: >

#|   A bar plot representation of the attractivenesses of the destination

#|   locations. Only one bar is visible: Vaulx-en-Vellin dominates the

#|   destination locations.

par(mar = c(0.1, 4, 1, 0))

a_final_Z <- attractiveness(a_blv_model)

barplot(a_final_Z)

```

In this particular example, one destination location acts as a global attractor 

of all the flows. This pattern is also visible in the final flow matrix.



```{r a_blv_flow}

#| fig.height: 10

#| fig.width: 8

#| out.width: 80%

#| fig.align: center

#| fig.alt: >

#|   A matrix representation of the flows. The matrix has only one black column,

#|   showing that all the flow goes to Vaulx-en-Vellin.

autoplot(a_blv_model) +

  scale_fill_gradient(low = "white", high = "black")

```



The `autoplot()` function can also be used to show the destination flows or the

attractivenesses values:

```{r a_blv_Z_ggplot2}

#| fig.height: 4

#| fig.width: 8

#| fig.alt: >

#|   A bar plot representation of the attractivenesses of the destination

#|   locations. Only one bar is visible: Vaulx-en-Vellin dominates the

#|   destination locations.

autoplot(a_blv_model, "attractiveness", with_names = TRUE) +

  coord_flip()

```



Naturally, the results are strongly influenced by the parameters, as shown in 

this second example.



```{r}

b_blv_model <- blvim(cost_matrix, X, alpha = 1.1, beta = 1 / 50000, Z)

b_blv_model

```



```{r b_blv_Z}

#| fig.height: 4

#| fig.width: 8

#| fig.alt: >

#|   A bar plot representation of the attractivenesses of the destination

#|   locations. Nine out of the twenty destination locations remain

#|   attractive. Vaulx-en-Vellin is still the most attractive location, but

#|   Cholet and Arles are quite attractive also. They are closely followed

#|   by Saint-Ouen-sur-Seine and Saint-Quentin. Then the graph shows a set

#|   of four remaining attractive destinations but with small values (between

#|   0.5 and 1.5, compared to 4 or more for the other locations). The cities

#|   are Albi, Cagnes-sur-Mer, Vannes and Bayonne.

autoplot(b_blv_model, "attractiveness", with_names = TRUE) +

  coord_flip()

```



```{r b_blv_flow}

#| fig.height: 10

#| fig.width: 8

#| out.width: 80%

#| fig.align: center

#| fig.alt: >

#|   A matrix representation of the flows. The figure shows essentially the

#|   same things as the previous bar plot. In most of the cases, the flow

#|   outgoing from a origin location goes to a unique destination location. The

#|   main exception is the pair Saint-Ouen-sur-Seine and Saint-Quentin who

#|   share incoming flows.

autoplot(b_blv_model, with_names = TRUE) +

  scale_fill_gradient(low = "white", high = "black") +

  theme(axis.text.x = element_text(angle = 90))

```



`bvlim` offers a collection of graphical representations that can leverage

the data associated to the origin and destination locations. For instance, 

we can display the full flows using geographical coordinates of the cities.



```{r b_blv_france}

#| fig.height: 8

#| fig.width: 10

#| fig.alt: >

#|   A geographical representation of the flows with dots showing the locations

#|   and arrows between them representation the flows. The figure tells the

#|   same story of the previous ones. It shows in particular star patterns with

#|   a central destination location that receives flows for all the surrounding

#|   origin locations. The figure confirms that flows are organised in a

#|   geographical way as most of the flow goes from one origin to its closests

#|   destination. However, the figure shows also destination locations that

#|   do not receive any flow, emphasizing the concentration effects associated

#|   to the return to scale parameter.

origin_positions(b_blv_model) <- as.matrix(origins)

destination_positions(b_blv_model) <- as.matrix(destinations)

autoplot(b_blv_model,

  with_positions = TRUE,

  show_destination = TRUE,

  show_production = TRUE,

  cut_off = 0

) +

  scale_linewidth(range = c(0, 1.5)) +

  coord_sf(crs = "epsg:4326") +

  scale_color_discrete(type = c("darkorange", "blueviolet"))

```