Impedance options - ObjectVision/GeoDMS GitHub Wiki

• the Node_rel, optional for impedance_matrix and impedance_matrix_od64, 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 Euclidean 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 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_table), all Nodes are considered as separate origin zones.

• 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 Euclidean 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.

• 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 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.

• 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.
• 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.
• 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 there but the alt_imp is null there. Both values will not be taken into account in an interaction distribution.
• by itself, this section does not influence which routes are found: routes are chosen on the original Impedance alone, and the alternative impedance is only accumulated along those routes. The pareto section below makes the alternative impedance a second search criterion instead.

• the optional OrgZone_max_imp2 indicates the presence of an additional argument: OrgZone → Impedance2, which can also be a single value (void), bounding the second criterion per origin zone: routes whose alternative impedance reaches this bound are discarded. Analogous to cut(OrgZone_max_imp), which bounds the first criterion.
• this section requires the od result form (impedance_matrix or impedance_matrix_od64; not impedance_table), the alternative(link_imp) argument as the second criterion, and cut(OrgZone_max_imp): a bound on the first criterion is what keeps the set of Pareto-optimal routes bounded.
• the produced od-pair entity holds the Pareto-optimal routes per OrgZone, ordered per OrgZone by increasing impedance and, within one DstZone, decreasing alt_imp. The FIRST row per (OrgZone, DstZone) carries the same impedance as a run without the pareto section: the minimum-impedance route is always part of the front.
• the od-pair products keep their meaning, now per route: impedance and alt_imp are the two criteria of the row's route, and OrgZone_rel, DstZone_rel, StartPoint_rel and EndPoint_rel are attributes of that route -- which zones it connects and which startPoint and endPoint it used. LinkSet is not (yet) supported in this mode, and neither are the interaction section, limit(OrgZone_max_mass,DstZone_mass), alternative(link_attr) and precalculated_NrDstZones. The euclid(maxSqrDist) filter, bidirectional variants and startPoint/endPoint impedances (which apply to the first criterion) combine normally.
• in this mode the alternative link impedances and the startPoint/endPoint impedances must be nonnegative, like the original Impedance: they now drive the route selection.
• performance: the number of Pareto-optimal routes per node is bounded by the number of DISTINCT alternative-impedance totals below the OrgZone_max_imp2 bound. Supply the alternative impedance in coarse units (for instance integer cents or tenths of euros rather than fractional euros) to keep the fronts, and thereby computation time and result size, small; the cut on the first criterion and OrgZone_max_imp2 are the other two controls. The calculation remains exact for the impedances as given.

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).

For the wider theory, from the gravity model through Wilson's family of constrained models to the GTM, see Spatial Interaction Models in the Accessibility Modelling wiki. In Wilson's terms, $α$ spans the unconstrained model ($α = 1$) and the production constrained model ($α = 0$, where $M_{ix} = v_i$); the attraction constrained and doubly constrained models are outside what this option computes, so destinations with hard capacity limits need a competition corrected measure instead.

• $M_{ij} = v_i \cdot w_j \cdot t_{ij} \cdot D_i^{α-1}$

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 $l$.

• 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), 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 avoid 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 $LF_l := \sum\limits_{i,j: l \in (i \rightarrow j)} v_i \cdot w_j \cdot t_{ij} \cdot D_i^{(α-1)}$.

• 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 and impedance_matrix_od64 functions, producing this info is complicated by the fact that a traceback would be available for each origin.

• the optional precalculated_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 StartPoint_rel indicates the production of a subitem StartPoint_rel : OD → startPoint, an attribute with the startPoint from which the shortest route of each od-pair departed. This only adds information beyond OrgZone_rel when startPoint(...,OrgZone_rel) puts several startPoints in one origin zone; it then tells which of them was used. Undefined for od-pairs without an available route. Warning: this option is only correctly implemented since version 20.18.0. Earlier versions do accept StartPoint_rel in the od: list and do produce the sub-item, but never fill it, so the attribute contains uninitialized memory rather than start point references. Results calculated with StartPoint_rel before 20.18.0 are therefore invalid and have to be recalculated.
• the optional EndPoint_rel indicates the production of a subitem EndPoint_rel : OD → endPoint, an attribute with the endPoint at which the shortest route of each od-pair arrived, analogous to StartPoint_rel and similarly informative only when endPoint(...,DstZone_rel) puts several endPoints in one destination zone.
• 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.