# Estimate mobility flows based on different trip distribution models

Source:`R/run_model.R`

`run_model.Rd`

This function estimates mobility flows using different distribution models. As described in Lenormand et al. (2016) , we propose a two-step approach to generate mobility flows by separating the trip distribution law, gravity or intervening opportunities, from the modeling approach used to generate the flows from this law. This function only uses the second step to generate mobility flow based on a matrix of probabilities using different models.

## Usage

```
run_model(
proba,
model = "UM",
nb_trips = 1000,
out_trips = NULL,
in_trips = out_trips,
average = FALSE,
nbrep = 3,
maxiter = 50,
mindiff = 0.01,
check_names = FALSE
)
```

## Arguments

- proba
a squared matrix of probability. The sum of the matrix element must be equal to 1. It will be normalized automatically if it is not the case.

- model
a character indicating which model to use.

- nb_trips
a numeric value indicating the total number of trips. Must be an integer if

`average = FALSE`

(see Details).- out_trips
a numeric vector representing the number of outgoing trips per location. Must be a vector of integers if

`average = FALSE`

(see Details).- in_trips
a numeric vector representing the number of incoming trips per location. Must be a vector of integers if

`average = FALSE`

(see Details).- average
a boolean indicating if the average mobility flow matrix should be generated instead of the

`nbrep`

matrices based on random draws (see Details).- nbrep
an integer indicating the number of replications associated to the model run. Note that

`nbrep = 1`

if`average = TRUE`

(see Details).- maxiter
an integer indicating the maximal number of iterations for adjusting the Doubly Constrained Model (see Details).

- mindiff
a numeric strictly positive value indicating the stopping criterion for adjusting the Doubly Constrained Model (see Details).

- check_names
a boolean indicating if the ID location are used as vector names, matrix rownames and colnames and if they should be checked (see Note).

## Details

We propose four constrained models to generate the flow from the matrix
of probabilities. These models respect different level of
constraints. These constraints can preserve the total number of trips
(argument `nb_trips`

) OR the number of out-going trips
\(O_{i}\) (argument `out_trips`

) AND/OR the number of in-coming
\(D_{j}\) (argument `in_trips`

) according to the model. The sum of
out-going trips \(\sum_{i} O_{i}\) should be equal to the
sum of in-coming trips \(\sum_{j} D_{j}\).

Unconstrained model (

`model = "UM"`

). Only`nb_trips`

will be preserved (arguments`out_trips`

and`in_trips`

will not be used).Production constrained model (

`model = "PCM"`

). Only`out_trips`

will be preserved (arguments`nb_trips`

and`in_trips`

will not be used).Attraction constrained model (

`model = "ACM"`

). Only`in_trips`

will be preserved (arguments`nb_trips`

and`out_trips`

will not be used).Doubly constrained model (

`model = "DCM"`

). Both`out_trips`

and`in_trips`

will be preserved (arguments`nb_trips`

will not be used). The doubly constrained model is based on an Iterative Proportional Fitting process (Deming and Stephan 1940) . The arguments`maxiter`

(50 by default) and`mindiff`

(0.01 by default) can be used to tune the model.`mindiff`

is the minimal tolerated relative error between the simulated and observed marginals.`maxiter`

ensures that the algorithm stops even if it has not converged toward the`mindiff`

wanted value.

By default, when `average = FALSE`

, `nbrep`

matrices are generated from
`proba`

with multinomial random draws that will take different forms
according to the model used. In this case, the models will deal with positive
integers as inputs and outputs. Nevertheless, it is also possible to generate
an average matrix based on a multinomial distribution (based on an infinite
number of drawings). In this case, the models' inputs can be either positive
integer or real numbers and the output (`nbrep = 1`

in this case) will be a
matrix of positive real numbers.

## Note

All the inputs should be based on the same number of
locations sorted in the same order. It is recommended to use the location ID
as vector names, matrix rownames and matrix colnames and to set
`check_names = TRUE`

to verify that everything is in order before running
this function (`check_names = FALSE`

by default). Note that the function
`check_format_names()`

can be used to control the validity of all the inputs
before running the main package's functions.

## References

Lenormand M, Bassolas A, Ramasco JJ (2016).
“Systematic comparison of trip distribution laws and models.”
*Journal of Transport Geography*, **51**, 158-169.

Deming WE, Stephan FF (1940).
“On a Least Squares Adjustment of a Sample Frequency Table When the Expected Marginal Totals Are Known.”
*Annals of Mathematical Statistics*, **11**, 427-444.

## Author

Maxime Lenormand (maxime.lenormand@inrae.fr)

## Examples

```
data(mass)
data(od)
proba <- od / sum(od)
Oi <- as.numeric(mass[, 2])
Dj <- as.numeric(mass[, 3])
res <- run_model(
proba = proba,
model = "DCM", nb_trips = NULL, out_trips = Oi, in_trips = Dj,
average = FALSE, nbrep = 3, maxiter = 50, mindiff = 0.01,
check_names = FALSE
)
# print(res)
```