# **Impedance options**

*network-functions Impedance options*

## options specification

The options parameter indicates how to calculate routes, which additional arguments are given, and which results will be produced. Each section is separated by a ‘;’, starts with a label and indicates comma-separated extra arguments between ‘(‘ and ‘)’. Optional results are specified by name after a ‘:’. All sections, extra arguments and products must be specified in the order of the following description. All labels and names are case-sensitive, and no intermediate spaces are allowed.

## link direction

The first section is obligatory and indicates the allowed traversal direction of each link (directed versus indirected graphs).

It can be:

**directed**: only routes from a start point to an endpoint that traverses each link forward from node f1 to node f2 are considered and not vice versa.**bidirectional**: also, backward traversal is considered through each link, from node f2 to node f1.**bidirectional(link_flag)**: indicates an additional argument**Link → Bool**, a boolean attribute of**Link**, that indicates for each link if it is to be considered as bidirectional. False indicates forward traversal only.

## startPoints

The optional **startPoint(Node_rel,impedance,OrgZone_rel,OrgZone_loc):max_imp** specifies a set of origin zones.

## options:

- the **Node_rel**, optional for **impedance_matrix_od**, indicates the presence of an additional argument:**startPoint → Node**, a Node attribute of startPoint in order to select a subset of nodes as startPoints. Required for the **impedance_table** function. Default: when omitted with the **impedance_matrix** function, all Nodes are considered as startPoints and **startPoint** := **Node**. - the optional **impedance** indicates the presence of an additional argument:

**startPoint → Impedance**, as departure impedance attribute of startPoint, useful mainly when origin zones contain multiple startPoints with specific departure impedances. Default value when omitted: zero. - the **OrgZone_rel**, optional for **impedance_matrix** and not allowed for **impedance_table**, indicates the presence of an additional argument: startPoint → OrgZone, an origin zone attribute of startPoint, in order to define multiple startPoints per origin zone. The default for the **impedance_matrix** function when omitted: all startPoints are considered as separate origin zones. This option is not allowed with the **impedance_table**, where all startPoints are assumed to belong to the same origin zone. - the **OrgZone_loc** is only allowed and required in combination with the **euclid(maxSqrDist)** option and indicates the presence of an additional argument: OrgZone → SPoint, indicating the location of each OrgZone for which the euclidian distance to DstZones is calculated in order to filter out od-pairs that have a squared Euclidian distance of more than maxSqrDist. - the optional **max_imp**, indicates the production of a [subitem](subitem.html)

**OrgZone → Imp**, an Imp attribute of OrgZone, named **MaxImp** that contains the maximum impedance for each OrgZone for all connected DstZones. This is especially useful in combination with the **limit(OrgZone_max_mass,DstZone_mass)** option with which one can produce the distance to the n-th unit of a DstZone specific quantity. When all Nodes are considered as separate OrgZones, one can specify **startPoints:max_imp**. This option is available in GeoDMS 7.168 and later versions. - when no additional arguments are indicated, this section should be omitted; **startPoint()** is a non-allowed syntax. - when no startPoints are specified (not allowed with **impedance_matrix_s**), all Nodes are considered as separate origin zones.

## endPoints

The optional **endPoint(Node_rel,impedance,DstZone_rel, DstZone_loc)** specifies a set of destination zones.

## options:

- the optional **Node_rel** indicates the presence of an additional argument **endPoint → Node**, a Node attribute of endPoint in order to select a subset of nodes as endPoints. Default: all Nodes are considered as separate endPoints - the optional **impedance** indicates the presence of an additional argument **endPoint → MeasureType**, an arrival impedance attribute of endPoint, useful mainly for when Destination Zones contain multiple endPoints. - the optional **DstZone_rel** indicates the presence of an additional argument **endPoint → DstZone**, a destination zone per endPoint, in order to define multiple endPoints per DstZone. Default: each endPoint is considered as a separate destination. - the **DstZone_loc** is only allowed and required in combination with the **euclid(maxSqrDist)** option and indicates the presence of an additional argument: DstZone → SPoint, indicating the location of each DstZone for which the euclidian distance from each OrgZone is calculated in order to filter out od-pairs that have a squared Euclidian distance of more than maxSqrDist. - when no additional arguments are indicated, this section should be omitted; **endPoint()** is a non-allowed syntax. - when no endPoints are specified, all Nodes are considered as separate destination zones.## filters

Optional **filters** can be applied to limit the resulting set.

## options:

- The optional **cut(OrgZone_max_imp)** specifies that a maximum route impedance will limit the route search, which requires an additional argument: OrgZone → Impedance, an impedance attribute of the origin zones or a single impedance limit parameter that is applied for all origin zones. - The optional **limit(OrgZone_max_mass,DstZone_mass)** specifies that a maximum amount of destinations per origin zone will limit the route development, which requires two additional arguments: - **OrgZone → Mass or {∅} → Mass** (must have the same [value-type](value-type.html) as **Impedance**), a Mass attribute of OrgZone or a single Mass parameter, that sets a maximum on the amount of Mass to be reached at the DstZones. - **DstZone → Mass or {∅} → Mass** (must have the same value type as **Impedance**), a Mass attribute of DstZone or a single Mass parameter, that indicates a Mass for each DstZone, which is accumulated until the limit is reached for each OrgZone.## euclidean distance filter

The optional **euclid(maxSqrDist)** specifies the maximum Euclean search distance for destinations. The specified parameter is the maximum squared distance for an od-pair to be admittable. This option requires **startPoint(OrgZone_loc)** and **endPoint(DstZone_loc)**.

## alternative impedance

The optional **alternative(link_imp, link_attr):alt_imp,link_ttr** specifies that an alternative link impedance is to be used to calculate an impedance per od-pair for further use in the interaction potential calculation and/or an alternative link_attr to be used as $e_l$ in interaction aggregation per origin.

## options:

- the optional **link_imp** requires an additional argument: **Links → Impedance2**, an alternative impedance per link, where Impedance2 must have the same value type as the original impedance, but may have a different [metric](metric.html). - the optional **link_attr** requires an additional argument: **Links → Impedance3**, an alternative impedance per link, where Impedance3 must have the same value type as the original impedance, but may have a different [metric](metric.html). - the optional product **alt_imp** indicates the production of an Impedance2 attribute of the resulting od-pair entity named **alt_imp** with the total alternative impedance of the found route for each od-pair. - the optional product **link_attr** indicates the production of an Impedance3 attribute of the resulting od-pair entity named **link_attr** with the total $e_l$ of the found route for each od-pair. - when this section is omitted, the original Impedance is used in the interaction calculation and the unit Impedance2 is set to be equal to Impedance. - when no filtering applies, some od-pairs may represent combinations without connecting routes. The impedance is MAX_VALUE<ImpType> there but the alt_imp is [null](null.html) there. Both values will not be taken into account in an interaction distribution.## interaction

The optional **interaction(OrgZone_min,DstZone_min,v_i,w_j,dist_decay,dist_logit(alpha,beta,gamma),OrgZone_alpha):NrDstZones,D_i,M_ix,SumImp,SumLinkAttr,C_j,M_xj,Link_flow** indicates the application of a free or origin constrained interaction model following the thoughts and notation of Alonso’s General Theory of Movement (GTM) with a fixed supply elasticity per destination of 1 (thus *β*_{j} = 1.0; no constraint on the destination side). Based on De Vries, Nijkamp & Rietveld (2000).

## theory:

This describes the theory of a free or origin-constrained interaction model following the thoughts and notation of Alonso's General Theory of Movement (GTM) with a fixed supply elasticity per destination of 1 (thus $β_j = 1.0$; no constraint on the destination side). - $M_{ij} = v_i \\cdot w_j \\cdot t_{ij} \\cdot D_i^{α-1}$ in which: - $M_{ij}$ = potential interaction between i and j - $v_i$ = trip generation weight per origin - $w_j$ = trip distribution weight per destination - $t_{ij}$ = facility of movement between i and j - $t_{ij} = d_{ij}^{-γ}$, in case of a normal distance decay specification. - $t_{ij} = (1+exp(alpha)⋅d_{ij}^{beta}⋅exp(d_{ij})^{gamma})^{−1}$, in case of a log-logistic distance decay specification. - $d_{ij}$ = impedance (e.g. distance or travel time) from i to j - $γ$ = distance decay factor, and in the case of the log-logistic specification there are the $alpha$, $beta$, and $gamma$ parameters. Summations: - $D_i := \\sum\\limits_{j} w_j \\cdot t_{ij}$ - $M_{ix} := \\sum\\limits_{j} M_{ij} = \\sum\\limits_{j} v_i \\cdot w_j \\cdot t_{ij} \\cdot D_i^{α-1} = v_i \\cdot \\sum\\limits_{j} w_j \\cdot t_{ij} \\cdot D_i^{α-1} = v_i \\cdot D_i \\cdot D_i^{α-1} = v_i \\cdot\\ D_i^α$ = summation of decayed impedances from origin $i$. - $SumImp_i := \\sum\\limits_{j} d_{ij} \\cdot M_{ij}$ - $SumLinkAttr_i := \\sum\\limits_{j} e_{ij} \\cdot M_{ij}$ with $e_{ij}$ an aggregation of $e_l$ over each route $i \\rightarrow j$. $e_l$ must be provided as alternative(link_attr). - $C_j := \\sum\\limits_{i} v_i \\cdot t_{ij} \\cdot D_i^{α-1}$ - $M_{xj} := \\sum\\limits_{i} M_{ij} = \\sum\\limits_{i} v_i \\cdot w_j \\cdot t_{ij} \\cdot D_i^{α-1} = w_j \\cdot C_j$ = summation of decayed impedances to destination $j$. - $LF_l := \\sum\\limits_{i,j: l \in (i \\rightarrow j)} M_{ij} = \\sum\\limits_{i,j: l \in (i \\rightarrow j)} v_i \\cdot w_j \\cdot t_{ij} \\cdot D_i^{α-1}$ = link flow over link $l$.## options:

- Note that if alt_imp is defined. The impedance within the interaction is the **alt_imp**. The first impedance is used for route decisions, while the alt impedance will be used to aggregate the interaction calculations. - the optional **OrgZone_min** indicates an additional argument: $OrgZone → Impedance2$, which can also be a single value ([void](void.html)), indicating a minimum (alternative) impedance to be used for each destination to avoid infinite auto interaction potential for each origin. The default value is 0. - the optional **DstZone_min** indicates an additional argument: $DstZone → Impedance2$, which can also be a single value, indicating a minimum (alternative) impedance to be used for each origin to a void infinite auto interaction potential for each destination. The default value is 0. - From this we define the distance measure to be $d_{ij} := max(impedance_{ij}, OrgZone min_i, DstZone min_j)$. - the obligatory **v_i** indicates an additional argument: $OrgZone → Mass$, which can also be a single value, indicating a trip generation weight per origin, aka $v_i$. The value type should match the value type of the impedance. - the obligatory **w_j** indicates an additional argument: $DstZone → Mass$, which can also be a single value, indicating a trip distribution weight per destination, aka $w_j$. The value type should match the value type of the impedance. - the optional **dist_decay** indicates an additional argument $gamma$ : Float64 that indicates the rate of distance decay. The used interaction potential $t_{ij}$ is calculated as $t_{ij} := d_{ij}^{−gamma}$. - $gamma = 1.0$ thus indicates $t_{ij} = 1/d_{ij}$ - If no route exists from *i* to *j* (which can be visible in the results when no filtering was applied), $t_{ij}$ is set to 0. - $gamma = 0.0$ indicates $t_{ij} = 1$, even when $d_{ij} ≤ 0$, except when no route exists. - $gamma = − 1.0$ indicates $t_{ij} = d_{ij}$ - when $d_{ij} ≤ 0$ and gamma ≠ 0.0, $t_{ij}$ is set to zero to avoid incorporating infinite potentials when no minimum impedances nor departure or arrival impedances were specified. Auto-interaction is then excluded but included when gamma = 0.0. - the optional **dist_logit(alpha,beta,gamma)** indicates three additional arguments that can be used to alternatively define $t_{ij}$ as a log-logistic distance decay function of $d_{ij}$: - $t_{ij} := (1+exp(alpha+beta⋅ln(d_{ij})+gamma⋅d_{ij}))^{−1} = (1+exp(alpha)⋅d_{ij}^{beta}⋅exp(d_{ij})^{gamma})^{−1}$ - The value type of all parameters should be float64. - note that either $dist_decay$ or $dist_logit(alpha,beta,gamma)$ must be specified in an interaction section. - note that we name the parameters in Latin characters instead of Greek characters. Since this alternative definition of $t_{ij}$ uses the same parameter names as in the other interaction definitions. So, to avoid confusion, we use the Latin characters for the distance decay parameters. - the optional **OrgZone_alpha** indicates an additional argument: $OrgZone −> Float64$, which can also be a single value, indicating the elasticity of the origin's supply for the number of demand alternatives. $α = 1.0$ indicates an elastic model; $α = 0.0$ indicates a push model with fixed supply $v_i$ per origin. The default value is 0. Calculated products (at least one should be specified): - the optional **NrDstZones**, indicates the production of a subitem **OrgZone → UInt32**, an attribute representing the number of reachable destination zones for each origin zone. - the optional **D_i**, indicates the production of a subitem **OrgZone → Mass**, a Mass attribute of OrgZone, named D_i, defined as $D_i:= \\sum\\limits_{j} w_j \\cdot t_{ij}$. - the optional **M_ix**, indicates the production of a subitem **OrgZone → Mass**, a Mass attribute of OrgZone, named M_ix, defined as $M_{ix} := v_i \\cdot\\ D_i^α = \\sum\\limits_{j} v_i \\cdot w_j \\cdot t_{ij} \\cdot D_i^{(α-1)}$ - Note that when $D_i = 0$, $D_i^α$ and $M_{ix}$ are assumed to be zero, even if $α = 0$, in order to respect the fact that if there is no demand potential at all, it cannot be raised to a set demand whatever the elasticity. - the optional **SumImp**, indicates the production of a subitem, called **OrgZone_SumImp**, representing $SumImp_i$ - the optional **SumLinkAttr**, indicates the production of a subitem, called **OrgZone_SumLinkAttr**, representing $SumLinkAttr_i$ - the optional **C_j**, indicates the production of a subitem **DstZone → Mass**, a Mass attribute of DstZone, named C_j, defined as $C_j := \\sum\\limits_{i} v_i \\cdot t_{ij} \\cdot D_i^{(α-1)}$. - the optional **M_xj**, indicates the production of a subitem **DstZone → Mass**, a Mass attribute of DstZone, named M_xj, defined as $M_{xj}:= w_j \\cdot C_j = \\sum\\limits_{i} v_i \\cdot w_j \\cdot t_{ij} \\cdot D_i^{(α-1)}$. - the optional **Link_flow**, indicates the production of a traffic flow network assignment, a subitem **Link → Mass**, a Mass attribute of Link, named **Link_flow**, defined as **$LF_l := \\sum\\limits_{i,j: l \in (i \\rightarrow j)} v_i \\cdot w_j \\cdot t_{ij} \\cdot D_i^{(α-1)}$**.## traceback production

## options:

- The optional **node:TraceBack** (only available for **impedance_table**) produces a subitem**TraceBack : Node → Link**, a Link attribute of Node, indicating for each node which Link of the grown tree(s) traces back to the origin of startPoint Node(s). For the **impedance_matrix_od** function, producing this info is complicated by the fact that a traceback would be available for each origin.

## od-pair specification and attribute production

the optional **od(precalculated_NrDstZones):impedance,OrgZone_rel,DstZone_rel,LinkSet**, indicates the production of a specific kind of od-paid entity and related attributes.

## options:

- the optional **precalculateted_NrDstZones** indicates the usage of an earlier calculated number of destination zones per origin zone to avoid (repeated) counting of the number of OD-pairs. This precalculation will determine the resulting domain range. If, for each origin zone, the number of destination zones is less than or equal to the precalculated number, the remaining rows will remain undefined. If, for some origin zones, the number of destination zones is greater than the precalculated number for that origin zone, an error message will be provided, and no calculation result will be available. - the optional **impedance** indicates the production of a subitem **impedance : OD → Impedance**, an attribute with the lowest route impedance per found od-pair. This impedance includes startPoint(impedance) and endPoint(impedance) but not OrgZone_min nor DstZone_min. - the optional **OrgZone_rel** indicates the production of a subitem **OrgZone_rel : OD → OrgZone**, an attribute with the OrgZone of each od-pair. - the optional **DstZone_rel** indicates the production of a subitem **DstZone_rel : OD → DstZone**, an attribute with the DstZone of each od-pair. - the optional **LinkSet** indicates the production of a subitem **LinkSet : OD → Link-sequence**, an attribute with the sequence of Links of each od-pair route. When no od-pair-related products are requested (including no **alt_imp**), the memory footprint of the operation will be greatly reduced as intermediate potentials and route flows per od-pair are produced per OrgZone and directly aggregated to the OrgZone, DstZone, and/or Link level and no storage is required per od-pair simultaneously for different OrgZones.## production of t_ij and M_ij

In order to get $t_{ij}$ and $M_{ij}$, the calculated interaction potential and trip flow per od-pair, one can recalculate them from the results available from **od:impedance,OrgZone_rel,DstZone_rel**

- When no alternative link impedance is given, defined as an extra subitem of the resulting od-pair unit:

```
attribute<float32> t_ij := impedance >= 1e+38f ? 0.0f : distDecay == 0.0f ? 1.0f : impedance^-distDecay;
```

Note that $t_{ij} >= max_dist$ only for od-pairs without available route, which only appears in results when no filtering option was used. If filtering was used and it is known that distDecay is non-zero, the above can be simplified to

```
attribute<float32> t_ij := impedance^-distDecay;
```

- When alternative(link_imp):alt_imp is used, use:

```
attribute<float32> t_ij := !IsDefined(alt_imp) ? 0.0f : distDecay == 0.0f ? 1.0f : alt_imp^-distDecay;
```

- When
**OrgZone_min**or**DstZone_min**were specified , replace the last impedance measure (**impedance**or**alt_imp**) by**max_elem(impedance, min_imp[OrgZone_rel])**when**min_imp**is given per origin zone,**max_elem(impedance, min_imp[DstZone_rel])**when**min_imp**is given per destination zone, or just**max_elem(impedance, min_imp)**when it is a single value parameter, applied for all od-pairs.

- The </B>
*M*_{ij}</B> can then be calculated with

```
attribute<float32> M_ij := D_i[OrgZone_rel] <= 0.0f ? 0.0f : t_ij*D_i[OrgZone_rel]^(demand_alpha-1.0f);
```

- if the demand for each zone i is assumed to be inelastic, i.e. demand_alpha == 0f, this can be simplified to

```
attribute<float32> M_ij := D_i[OrgZone_rel] <= 0.0f ? 0.0f : t_ij*(1.0f / D_i)[OrgZone_rel];
```

An example of this can be found in the operator test configuration at **/Network/UnTiled/dijkstra_all_interaction**.