Ruimtescanner XL in formules

Mathematical description of the Ruimtescanner XL.

= Indices =
 * \( c \) : grid-cell index, aka residence index \( w \).
 * \( t \) : time-period. The states related to the time at the start and end of such period are indicated by the superscript indices /(i/) and /(f/). \(S^i_{t=0}\) indicates a state \(S\) at the start of the base-year.
 * \( j \) : building characteristics, including building type and density, i.e. the number of housing units per ha, to be determined by the current land use (for \( t = 0 \)) or by adaptation by a project developer / land owner.
 * \( r \) : housing zone, as defined by LMS.
 * \( s \) : sub region, relevant for relocation costs of households.
 * \( h \) : household type, including socio-economic characteristics. Center of activity (approximated by a classification of the present housing location) is represented by \( s \).

= Definitions =
 * \( Q_{hrst} \) : demand in [#households]
 * \( j_{ct} \) : building state (including house-densities).
 * \( n(j) \): number of housing-units related to building state j.
 * \( v_{cht} \) : valuation of building \( j_{ct} \) at \(c\) in [EUR / housing-unit ] by household \(h\), which are capitalized yields.
 * \( d_{hsc} \) : relocation costs (= the transition costs for layer 3).
 * \( v_{chst} := v_{cht} - d_{hsc} \) : valuation minus relocation costs.
 * \( e_{ht} < 0 \) : price elasticity, which is assumed to be negative.
 * \( p_{ct} \) : bid-price of building \( j_{ct} \) in [EUR / housing-unit ].
 * \( P_{chst} \): Probability that a single household of type \(hs\) occupies a housing-unit at location \(c\).
 * \( N_{chst} := P_{chst} \times n(j_{ct}) \): Expected number of households at location \(c\) of type \(h\).
 * \( w_{chst} := \exp\left(v_{chst} + e_{ht} \times \log p_{ct}\right) = {p_{ct}} ^ {e_{ht}} \times \exp(v_{chst}) \)
 * \( p_{hst} := \left( \sum\limits_{c} w_{chst} \right)^{-1} \)

= Initial Allocation in layer 3 at \( t = 0 \) = To assess \(P_{ch}\) at \( t = 0 \), we use Tigris output of the initial distribution of Households per type per LMS zone and CBS-Buurt statistics on households per neighbourhood \(b\).

A specific issue here is to estimate a cross table from a series of row- and columntotals.

Known:
 * \( H_{bh} \): number of households of type \(h\) in neighbourhood \(b\), as given by the CBS-Buurt statistics.
 * \( W_{bj} \): number of residences of type \(j\) in neighbourhood \(b\), as derived from the BAG.

We assume that: thus: \( \sum\limits_{h}H_{bh} = \sum\limits_{j}W_{bj} \)
 * for each neighbourhood \(b\), the number of residences and households are (made) equal,
 * \( P_{ch} \) is equal for all \(c\) in the same \(b\) and with the same \(j\), thus depends only on \( P(h|bj) \).
 * \(Q_{bjh}\) has a categorical distribution per row \(j\) and zone \(b\) with \( P(h|bj) := Q_{bjh} / W_{bj} \)
 * \( Q_{bjh} := f_j * f_h * P_{jh} \) such that \( \sum\limits_{h} Q_{bjh} = W_{bj} \) and \( \sum\limits_{j} Q_{bjh} = H_{bh} \), to be determined by Iterative proportional fitting
 * thus: \( \sum\limits_{h} P(h|bj) = 1 \), \( f_j = W_{bj} / \sum\limits_{h} f_h * P_{jh} \), and \( f_h = H_{bh} / \sum\limits_{j} f_j * P_{jh} \).
 * \( P_{jh} \) is to be determined by regression of \( H_{bh} \) by \( W_{bj} \), thus, written in matrix notation: \( H = W*P+\epsilon \) with \(\epsilon\) a \(b \times h \) matrix of independent stochasts with zero expectation.
 * it follows that: \( P := (W^T * W)^{-1}*(W^T * H) \)

ToDo:
 * Consider Alternative: \( Q_{bjh} \) is determined by discrete allocation with the given constraints and suitability \( P_{jh} \)
 * Consider Alternative: \( P := (W^T * H) * (H^T * H)^{-1}\) which follows from regression of \( W_{bj} \) by \( H_{bh} \).
 * Consider Alternative: \( P := (W^T * H) \)
 * Consider Effect of possible heteroscedasticity of \(\epsilon\) with \( H \); i.e. assume \( var(\epsilon) \sim H\), as each element of \(H\) is assumed to be the sum over \(j\) of the results of \( W_{bj} \) trials of categorical distributions with conditional probabilities \( P_{j|h} \).

= Household Allocation in layer 3 at \( t > 0 \) =

Purpose of this layer is to estimate \(N^f_{wh}\), the final number of households per type per residence and the (changed) residence prices.

For each time step \( t\) and zone \( r\), the following is given:
 * \(N^i_{rh}\), and \(N^f_{rh}\), the initial and final number of households per type \(h \).
 * \(R^{t, in}_{rh}\), and \( R^{t, out}_{rh} \), the migration from and to other zones during the time step.

Also, \(N^i_{wh}\), an estimation of the initial number of households per type and per residence \( w \) is given, which is assumed to be or made to be a distribution of \(N^i_{rh}\).

regional balance
The change of \(N^i_{rh}\) to \(N^f_{rh}\) is assumed to reflect the following processes, where \( b^t_{rh}\) reflect balancing factors to keep the endogenous change equal to the given change: \[ \begin{align} T^t_{r f i} &:= {b^t_{rf} \over b^t_{ri} } N^i_{ri} t_{f i} & &\text{Transition of households} \, i \to f \, \text{with rate} \, t_{f i} \\ M^t_{r f i j} &:= { b^t_{r f} \over b^t_{r i} b^t_{r j} } { N^i_{r i} N^i_{r j} \over N^i_{r} }m_{f i j}. & &\text{Merge events (marriage, \( j \) moving in with \( i\) to form type \( f \) )} \, i \times j \to f \, \text{with rate} \, m_{f i j} \\ S^t_{r f g i} &:= {b^t_{r f} b^t_{r g} \over b^t_{r i} } N^i_{ri} s_{f g i} & &\text{Split events (separation, start of \( g \) from \( i \) that becomes \( f \) ) )} \, i \to f \times g \, \text{with rate} \, s_{f g i} \\ R^{t,out}_{rh} & & &\text{Migration from zone \( r \) as determined by Tigris} \\ A^{t,out}_{rh} &:= { 1 \over b^t_{rh} } N^i_{rh} a^{out}_h & &\text{Emigration beyond the Tigris zone scope with rate} \, a^{out}_h \\ D^t_{rh} &:= { 1 \over b^t_{rh} } N^i_{rh} d_h & &\text{Mortality with rate} \, d_h \\ O^t_{rh} &:= S^{t,g}_{rh} + R^{t,in}_{rh} + A^{t,in}_{rh}  & &\text{Occupation events (buy, rent) by residence seekers.} \\ \\ \text{with} \\ \\ N^i_r &:= \sum\limits_h N^i_{rh} & &\text{Initial total house occupation in zone } r\\ S^{t,g}_{rh}  &:= \sum\limits_{f,i} S^t_{r f h i} & &\text{Parting type of household-split} \\ R^{t,in}_{rh} & & &\text{Migration to} \, r \, \text{as projected by Tigris} \\ A^{t,in}_{rh} &:= b^t_{rh} N^i_h a^{t,in}_h & &\text{Immigration from beyond Tigris Zones} \end{align} \]

Other events are assumed to be sequences of the above, such as Split, Relocate and Merge. The housing stock and its change \( \Delta N^{t,max}_{r} := N^{f, max}_r - N^{i, max}_r \) is either occupied of vacant: \( \sum\limits_h {\Delta N^t_{rh}} + \Delta V^t_r = \Delta N^{t,max}_{r} \).

By definition of \( \Delta N^t_{rh} \), for each \(r\), \(t\), and \(h\), the following terms:


 * \( T^{t,f}_{rh} := \sum\limits_i T^t_{r h i} \)
 * \( M^{t,f}_{rh} := \sum\limits_{i, j} M^t_{r h i j} \)
 * \( S^{t,f}_{rh} := \sum\limits_{g,i} S^t_{r h g i} \)
 * \( O^t_{r h} \)

balance


 * \( \Delta N^t_{rh} \)
 * \( T^{t,i}_{rh} := \sum\limits_{f} T^t_{r f h} \)
 * \( M^{t,i}_{rh} := \sum\limits_{f, j} M^t_{r f h j}\)
 * \( M^{t,j}_{rh} := \sum\limits_{f, i} M^t_{r f i h}\)
 * \( S^{t,i}_{rh} := \sum\limits_{f, g} S^t_{r f g h} \)
 * \( R^{t, out}_h \).
 * \( A^{t,out}_{r h} \)
 * \( D^t_{r h} \)

Assuming market clearance (no camping at wintertime) gives that


 * \( O^t_{r h} \)

balances the sum of


 * \( S^{t,g}_{rh} := \sum\limits_{f, i} S^t_{r f h i} \)
 * \( R^{t,in}_{rh} \)
 * \( A^{t,in}_{rh} \).

Substitution of \( O^t_{r h} \), and reordering of terms, gives (indices \( t \) and \( r \) are omitted for simplicity):

\( T^f_h + M^f_h + S^f_h + S^g_h + A^{in}_h - ( T^i_h + M^i_h + M^j_h + S^i_h + A^{out}_h + D_h ) = \Delta N_h + R^{out}_h - R^{in}_h \).

Further substitution, multiplication with \( b_h \), and factor collection gives:

\( A b^2_h + B b_h + C = 0 \)

with

\(A := N_h a^{in}_h + \sum\limits_i { N_i \over b_i} [ t_{hi} + \sum\limits_f b_f ( s_{h f i} + s_{f h i} ) + \sum\limits_j {1 \over b_j} { N_j \over N_t } m_{h i j} ] \),

\( B := \Delta N_h + R^{out}_h - R^{in}_h \), and

\( C := - N_h ( d_h + a^{out}_h + \sum\limits_f [ t_{f h} + \sum\limits_g b_f b_g s_{f g h} + \sum\limits_i { b_f \over b_i } { N_i \over N_t } ( m_{f h i} + m_{f i h} ) ] )\),

which can be solved by Iterative Quadratic Fitting on

\( b_h = { -B \pm \sqrt {B^2-4AC} \over 2A} \).

As \( A > 0 \), \( B > 0 \), and \( C < 0 \), it follows that \( \sqrt {B^2-4AC} > B \) and thus the only positive solution for \( b_h \) is the larger root.

house allocation
The \( T^i, M^i, M^j, S^i, R^{out}, A^{out}, \text{and} \, D \) processes are assumed to equally affect all households in a zone of the same type, thus (index \( t \) is omitted for simplicity) with \( c_{wh} := N^i_{wh} / N^i_{rh}\): \[ \begin{align} T^i_{wh} &:= c_{wh} T^i_{rh} \\ M^i_{wh} &:= c_{wh} M^i_{rh} \\ M^j_{wh} &:= c_{wh} M^j_{rh} \\ S^i_{wh} &:= c_{wh} S^i_{rh} \\ R^{out}_{wh} &:= c_{wh} R^{out}_{rh} \\ A^{out}_{wh} &:= c_{wh} A^{out}_{rh} \\ D_{wh}      &:= c_{wh} D_{rh} \end{align} \]

The resident transitions are defined accordingly: \[ \begin{align} T^f_{wh} &:= \sum\limits_i    c_{wi} T_{rfi} \\ M^f_{wh} &:= \sum\limits_{i,j} c_{wi} M_{rfij} \\ S^f_{wh} &:= \sum\limits_{i,g} c_{wi} S_{rhgi} \end{align} \]

To estimate the final number of households per type per residence, we need a camping with potential buyers and renters:

\( C_{rh} := S^g_{rh} + R^{in}_{rh} + A^{in}_{rh} \)

Resulting House occupation dynamics:

\( N^a_{wh} := N^i_{wh} + T^f_{wh} + M^f_{wh} + S^f_{wh} - T^i_{wh} - M^i_{wh} - M^j_{wh} - S^i_{wh} - R^{out}_{wh} - A^{out}_{wh} - D_{wh} \) Final house occupation without reoccupation.

Note that: \( N^f_{rh} = \sum\limits_{w \in r} N^a_{wh} + C_{rh} \).

\( N^f_{wh} \) should be determined such that:
 * \( \forall r,h: \sum\limits_{w \in r} N^f_{wh} = N^f_{rh} \)
 * \( \forall w,h: N^f_{wh} \ge 0 \)
 * \( \forall w: \sum\limits_h N^f_{wh} \le N^{f,max}_w \)

Vacancy \( V_w := N^{max}_w - \sum\limits_h N_{wh} \), thus \( \Delta V_w = \Delta N^{max}_w - \sum\limits_{h} \Delta N_{wh} \). Changes of \( N^{max}_w \) are modeled in layer 2 as \( n(j_{ct}) \).

In each period, the housing market is cleared by allocating all households on the camping \( C_{rh} \) to vacant (fractions) of residences by a MNL choice model that incorporates the size, price, and possibly other characteristics of the dwellings such as house-density, accessibility, and the proximity of services, and household characteristics, more specifically: price-elasticity.

Resulting House allocation:

\[ \begin{align} P'_{wh} &:= N^a_{wh} / N^{max}_w \\ (P_{wh} - P'_{wh}) &\sim v_{wh} &\text{such that} \, \forall r,h: \sum\limits_{w \in r} P_{wh} = 1, \text{thus}: \end{align} \]


 * \( P_{wh} := p_{rh} * v_{wh} \).
 * \( p_w \) is such that \( \forall w: \sum\limits_{r,h} P_{wh} * C_{rh} = \sum\limits_{hs} p_{rh} * {p_w}^{e_{ht}} * \exp(v_{cht}-d_{cs}) * Q_{hst} \le n(j_{ct}) \space \bot \space p_w \ge 0 \).

Note that:
 * \( C_{rh} \) indicates the demand on time \( t \) for specific \( rh \). The increase/decrease of total demand \( \overset{.}C_{r} := \Delta \sum\limits_{h}C_{rh} / \Delta t \) is communicated to project developers in layer 2. The specific demands are communicated by higher or by higher or lower \( P_{chst} \).
 * if \(e_{ht} \) equals a constant \(e_t\) independent from household type \(h\), then \( {p_{ct}}^{e_t} \) and \( p_{ht} \) can be solved simultaneously by Iterative proportional fitting with \( \exp(v_{chst}) \) as proxy on the following two equations that follow from the definitions:

\( {p_{ct}}^{e_t} = {n(j_{ct}) \over \sum\limits_{hs} p_{hst} * Q_{hst} * \exp(v_{chst})} \) and \( p_{hst} = {1 \over \sum\limits_{c} {p_{ct}}^{e_t} * \exp(v_{chst})}\) Assumptions:
 * \( v_{chst} \) is a linear combination of household type dependent characteristics and location specific characteristics and household center of activity \(s\), but not the price.

= Layer 2 =
 * \( v_{cjhst} \) : valuation of potential building \( j \) at location \( c\) in [EUR / housing-unit ] by household \(hs\), which are capitalized yields minus relocation costs (= the transition costs for layer 3).
 * \( p_{cjt} \) : price of building j at location c and time t in [EUR / housing-unit ].
 * \( P_{cjhst} \): Probability that a single household of type \(hs\) occupies a housing-unit of type j at location \(c\).
 * \( w_{cjhst} := exp\left(v_{cjhst} - e_{ht} * \log p_{ct}\right) \)

Determination of * \( p_{cjt} \) of each alternative \(j\) in layer 2 is similar to the actual price determination for current \(j\) in layer 3; \( p_{hst} \) are assumed fixed.


 * \( s_{cjt} := n(j) * p_{cjt} - T(j_{ct} \rightarrow j) \) : suitability of development \( j \).

Simulation of the building development process \( j_c \) is based on yield maximization \( \sum\limits_{c} s_{cjt} \), under restriction of:
 * building volume \( \sum\limits_{c} T(j_{ct} \rightarrow j_c) \le B_t * \Delta t \space \bot \space \lambda_t \ge 0 \)
 * per region at least built number of housing-units: \( \overset{.}Q_{rt} + Q_{rt} - \sum\limits_{c}n(j_{ct}) \le (n(j_c)-n(j_{ct})) \space \ \bot \space \lambda_{rt} \ge 0 \)

Thus:
 * \( \sum\limits_{c} n(j_c) * p_{cjt} + \lambda_{rt} * (n(j_c)-n(j_{ct}))- (1+\lambda_t)*T(j_{ct} \rightarrow j_c) \ge 0 \space \bot \space \sum\limits_{c} T(j_{ct} \rightarrow j_c) \le B_t * \Delta t \)

Large timesteps, represented by large \( B_t \) result in oscillation of building volume and prices; smaller \( B_t \) result in development yield adjustment based on adjusted demand prices.

= Possible closed form =
 * \( n(j) \uparrow \space \Rightarrow \space v_j*n(j) \uparrow \)
 * \( n(j) \uparrow \space \Rightarrow \space v_j \downarrow \)

. bijvoorbeeld s:= j*S_w met S_w daalt lineair bij stijgende dichtheid. t_cjt(j_ct) : j^ -> t^ en j^ -> (dt/dj)^ bijvoorbeeld: t = (j-j_ct) * T_w en T_w stijgt met j.

v_cjt := s_cjt-t_cjt

j = combinatie van land use type, dichtheid, kwaliteit en verder alles wat mogelijk relevant is.

max j(c): v_cjt subject to #woningen wordt gehaald.

=> v_cjt + lambda_Q*j >= v_cj't + lambda_Q*j' EN ( lambda_Q >= 0 ORTHO SUM(j) >= Q_rt ) => d(v +lambda_Q) / dj = 0

Als v = aj^2 + bj + c met a < 0 Dan d(v +lambda_Q) / dj = 2aj+b+lambda_W Dus j =  (b+lambda_W) / -2a

= Links / bronnen =


 * notitie uit mail van 2013-06-25
 * Ruimtescanner L in formules

= TODO lijst =

Maarten
 * 1) Hierboven documenteren van het samenvoegen in 1 loop van laag 2 (gebouwen) en laag 3 (gebruik van verblijfsobjecten).
 * 2) testen en calibreren rsl_alloc over meerdere perioden (sloop en nieuwbouw integreren in de bestaande voorraad).
 * 3) implementeren http://www.pbl.nl/sites/default/files/cms/publicaties/Dynamiek_stedelijke_milieus_web_PBL.pdf als constructie opties.

Bart
 * 1) rekenregels voor geschiktheden bepalen per combinatie van gebouwtype en actortype.
 * 2) * basisjaar situatie obv NVM, etc.
 * 3) * zichtjaren obv Tigrix woz-indices
 * 4) opschonen van de RSXL config.
 * 5) check zonale mismatches BAG-LMS, zie: /Zichtjaren/Run1/Scenarios/test2013/RunPeriods/Y2010_Y2020/rsl_alloc/SimYears/Y2011/Ratio_LMS_vow @r3592.

Validatie
 * 1) ter vergelijking met de endogene balancing prices:
 * 2) * biedprijzen per m2 of woning per actortype (huishoudenstype en TGL7).
 * 3) * verzamelen prijzen per m2 bvo per gebouwtype per regio (tbv toegevoegde waarde van transities)
 * 4) RO restricties bijwerken.
 * 5) checken geimplementeerde bouw- en sloopkosten.

Onderscheid huur/koop
 * 1) uitwerken of en hoe in RSXL op te nemen. Welke data en classificaties moeten worden uitgesplitst en waar heeft dit onderscheid invloed op de allocatie.

Indicatoren
 * balansprijs per zone, actortype en per gebouwtype
 * conversiepaden, van betaande omgevingen naar constructieopties; flow VBO's per gebouwtype en per actortype.
 * totale sloop- en bouwkosten per region, type, etc.
 * restende schaarste, niet feasible allocaties of onbetaalbare transities (bij controntatie met exogene biedprijzen).
 * soil sealing