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Assessing partial observability of link flow

inference problems at large scale networksFrancesco VITI & Marco Rinaldi - University of Luxembourg

ERC SCALE 2018 – 10-11 September 2018, Grenoble

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Outline

Introduction

Network Sensor Location Problems

Full and partial observability concepts

A new metric for assessing partial observability

NSP metric

Ranking partial observability solutions

Examples

The impact of route set generation in large scale applications

Hypergraph

Exact and approximate solutions

Examples

Conclusions & future research directions

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Introduction

Information from traffic sensors crucial to various applications, e.g.

State estimation

OD flow estimation

Model calibration

Traffic management

…

• Problem: Obtaining full information not realistic in real-sized networks

need to strategically position the (limited) amount of sensors

The quality of traffic information depends on number and type of

sensors and where they are positioned

network sensor location problem, NSLP

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Network Sensor Location

Problem (NSLP)

Formal definition:

“Find an optimal number and location of sensors to

maximize the available information on a network”

Analogy with a sudoku game: different initial states (positions

of numbers) determine difficulty (estimation reliability) and

uniqueness (solution determinacy)

Need rules to solve them

Link-Route-OD relationships

Local consistency (e.g., flows at nodes)

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The NSLP formulation

Determining link/route information based on simple algebraic relations

Fundamental relationships used

OD flows f,

route flows h σ𝑟∈ℛ 𝛿𝑎𝑟ℎ𝑟 = 𝑣𝑎 , ∀𝑙 ∈ ℒ

σ𝑟∈ℛ 𝜌𝑤𝑟ℎ𝑟 = 𝑓𝑤 , ∀𝑤 ∈ 𝒲

𝒗𝒇 =

∙ 𝒉

link flows v

Location rules

Mostly based on widely accepted statements, i.e.

Select those variables that can tell something about unknown variables

(information coverage)

Try and get as much flow information as possible (information capturing)

Don’t waste sensors to get information you have already (information

independence)

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Location rules

Heuristic rules in literature (Yang and Zhou’s, 1998, Larsson et al., 2010, Cipriani et al.,

2006, Yang et al., 2006, Gentili and Mirchandani, 2012, Castillo et al., 2014)

1. OD/route/link coverage

2. Maximum (link/route/OD) flow fraction

3. Maximum intercepting/net (link/route/OD) flow

4. Link/route independence

No general methodology exists that applies all rules at once

Two families of NSLP approaches (Gentili& Mirchandani, 2012)

Observability problems – exploiting topological supply relations

Flow estimation problems – exploting demand flow relations

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Observability vs. Flow

estimation

Observability problems

Need only relations between link,

route and OD variables (topological

characteristics)

Specifies coverage requirements

Set the positions of the known

numbers, in relation to the unknown

ones

Assess solution determinacy

Flow-estimation problems

Optimal solutions related to estimation

model adopted

Prior information necessary

Different rules to determine quality of

information a priori

Set the values of numbers in relation to

those unknown

Assess solution reliability

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Full observability problem

Determine the minimum amount/combination of sensors to be placed

such that all flow information is known

Obvious solution: equip all links with sensors

Better solution: exploit topological relationships to reduce number of variables

necessary

8

3 1 2

4 3 5

v v v

v v v

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Full observability problem

Full observability: find set of linearly independent variables:

𝒙𝒅 = 𝐏𝒙𝒊

Different solution methods:

Pivoting (Castillo et al., 2008);

Gaussian Elimination (Hu et al., 2009);

Node-based (Ng, 2012);

Topological tree (He, 2013);

Non-planar Holes (Castillo et al., 2014).

Derive new relations 𝐏 using matrix transformations

𝑣1𝑣2𝑣3𝑣4𝑣5

=

1 0 1 00 1 0 11 1 1 11 1 0 00 0 1 1

ℎ1ℎ2ℎ3ℎ4

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Full observability: Pivoting


=

1 0 1 00 1 0 11 1 1 11 1 0 00 0 1 1

ℎ1ℎ2ℎ3ℎ4

ℎ1𝑣2𝑣3𝑣4𝑣5

=

1 0 −1 00 1 0 11 1 0 11 1 −1 00 0 1 1

𝒗1ℎ2ℎ3ℎ4

ℎ1ℎ2𝒗3𝑣4𝑣5

=

1 0 −1 00 1 0 −11 1 0 01 1 −1 −10 0 1 1

𝒗1𝒗2ℎ3ℎ4

ℎ1ℎ2𝒗3ℎ3𝒗5

=

0 −1 1 10 1 0 −11 1 0 01 1 −1 −11 1 −1 0

𝒗1𝒗2𝒗4ℎ4

Problem 1 : Full observability solutions only theoretical in real-sized networks

(~60% of links need to be measured)

Problem 2: Exact solutions, not unique permutation-dependent

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Research questions

Research questions:

Is there a way of assessing partial observability solutions?

What is the added value of placing or removing an extra sensor?

What if only a limited number of sensors is available for budget reasons?

What would be the # of sensors needed to guarantee an acceptable level of

under-determinedness?

Are full observability solutions all the same in terms of potential partial

observability solutions?


=

0 −1 1 10 1 0 −11 1 0 01 1 −1 −11 1 −1 0

𝒗1𝒗2𝒗4ℎ4


=

1 0 1 00 1 0 11 1 1 11 1 0 00 0 1 1

ℎ1ℎ2ℎ3ℎ4

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Partial observability

Goal Find an optimal* set of locations in full observability

solution(s) in terms of partial observability

* minimizing the magnitude of missing information, while

* respecting given budget constraints (e.g. maximum number of sensors)

Why starting from the full observability solution?

1. We can assure that links with sensors are linearly independent

2. Solutions provide a smallest set of linearly independent variables

3. The search space of solutions is reduced significantly


=

0 −1 1 10 1 0 −11 1 0 01 1 −1 −11 1 −1 0

𝒗1𝒗2𝒗4ℎ4

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Towards a new metric

Rewrite 𝒙𝒅 = 𝐏𝒙𝒊 into

𝒙𝑖𝑛𝑑𝑒𝑝𝒙𝑑𝑒𝑝

=𝐈 𝟎𝐏 𝟎


Subdivide variables in those observed and those unobserved𝒙𝑖𝑛𝑑𝑒𝑝 &𝑜𝑏𝑠

𝒙𝑖𝑛𝑑𝑒𝑝 &𝑢𝑛𝑜𝑏𝑠

𝒙𝑑𝑒𝑝=

𝐈 𝟎𝐏 𝟎


=𝐈′𝟎𝐏′

𝟎𝐈′′𝐏′′

𝟎𝟎𝟎

𝒙𝑖𝑛𝑑𝑒𝑝 &𝑜𝑏𝑠

𝒙𝑖𝑛𝑑𝑒𝑝 &𝑢𝑛𝑜𝑏𝑠

𝒙𝑑𝑒𝑝

Focus on the ‘size’ of the solution space of the reduced pivoted matrix and on its basic

dimensions knowing that

The reduced matrix P’ will have infinite possible solutions due to under-

determinedness

The basis B’ provides the ‘basic’ vectors characterizing the infinite solutions, i.e. any

combination 𝜶𝐁′ belongs to the Null space

We want now

A metric that gives a scalar value to the degree of under-determinedness

It is normalized, i.e. it can be seen as ‘fraction of full information lost’13

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Null-Space Metric of P, NSP

• Characterize the Null Space, i.e. set of vectors mapping to zero in the given

(sub)space measuring the degrees of freedom in solution space

Compare extent of the basis of the Null space of the reduced matrix wrt basis

of full matrix normalization of the solution spaces

New metric for assessing partial observability solutions based on the trace

function:

𝑁𝑆𝑃 =

𝐈 𝟎𝐏 𝟎

𝑇

𝐁′𝐹

𝐈 𝟎𝐏 𝟎 𝐹

Frobenius norm:

related to the trace of the matrix, i.e. the maximum error extent in the null-space

related to the singular values of the reduced matrix

Basis of the Partial

Observability matrixProjection onto full set of

independent variables

Extent of the full matrix

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../Videos/Null-space Impedance Control with Disturbance Observer.mp4

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Geometric interpretation

NSP metric related to the

maximum size of the prism

derived from the basis

A

B

C

D

v3v1

v2

v4

v5

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Greedy algorithm(s)

Questions

1. How to optimally identify sensor placements?

2. What is the added value of placing or removing an extra sensor?

3. Given n sensors, where should these be installed to get maximum

information?

Two local search algorithms introduced:

Add

starts from an empty set of observed variables, and adds as variable

to be observed, the one that results in the biggest decrease of error

Remove

starts from the set observing all variables, and chooses the single

variable that causes the least decrease in information

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Proof of concept (1)

Test on simplified ‘candy’

network

Easier to illustrate properties

Has only two ‘pivot’ families to

evaluate

‘Center’ family, i.e. l3 is

observed

‘Star’ family, where solution

does not contain I3

A

B

C

D

v3v1

v2

v4

v5

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Star

A

B

C

D

v3v1

v2

v4

v5 Max. Estimation Error

Observed: l1 & l4 Observed: l1 & l5 Observed: l4 & l5l1 0 0 −∞,∞

l2 0 + −∞,∞ − 0 −∞,∞ + 0 − 0 0 + 0 − −∞,∞

l3 0 + −∞,∞ −∞,∞ + 0 0

l4 0 −∞,∞ 0

l5 −∞,∞ 0 0

NSP 0.44 0.44 0.44

Max. Estimation Error

Observed: l1 Observed: l4 Observed: l5

l10 −∞,∞ −∞,∞

l2 −∞,∞ + −∞,∞ − 0 −∞,∞ + 0 − [−∞,∞] −∞,∞ + 0 − [−∞,∞]

l3 −∞,∞ + [−∞,∞] −∞,∞ + 0 0 + −∞,∞

l4−∞,∞ 0 −∞,∞

l5−∞,∞ −∞,∞ 0

NSP 0.79 0.64 0.64

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Center

A

B

C

D

v3v1

v2

v4

v5Max. Estimation Error

Observed: l1 & l5 Observed: l1 & l3 Observed: l3 & l5l1 0 0 −∞,∞

l2 0 − −∞,∞ 0 0 − −∞,∞

l3 −∞,∞ 0 0

l4 0 − −∞,∞ 0 − −∞,∞ 0

l5 0 −∞,∞ 0

NSP 0.53 0.47 0.47

Max. Estimation Error

Observed: l1 Observed: l3 Observed: l5l1 0 −∞,∞ −∞,∞

l2 −∞,∞ − 0 0 − −∞,∞ −∞,∞ − [−∞,∞]

l3 −∞,∞ 0 −∞,∞

l4 −∞,∞ − [−∞,∞] 0 − −∞,∞ −∞,∞ − 0

l5 −∞,∞ −∞,∞ 0

NSP 0.8 0.69 0.8

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Pivot “Families”: Ranking

Question: are pivots all the same to find optimal locations?

Answer: NO

And our NSP is able to provide different results for the same number of sensors

Singular Value Decomposition can be employed to rank pivots a-priori

σ𝑖σ𝑗 𝑠𝑖𝑗 as a measure of pivot’s information content

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• “Star” Family: 2.7979

• “Center” Family: 2.7321

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Pivot “Families”: different

information

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More informative

Faster error decrease

Rank: 2.7321

Rank: 2.7979

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Observations

Tested different network sizes and it seems that

Ranking is consistently independent on the number of sensors, i.e. better ranked

families remain better ranked till full observability solution

Best families can be found already from first sensors placed!

‘Parallel’ network (14 links; 4 ODs)

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Test on large networks:

Rotterdam

Test on real-sized networks shows that

Most informative links are near nodes and centroids (good news!)

Nice spatial distribution (even if no such rule was imposed)

Not all full observability solutions contain the minimum number of sensors

(consistent with recent findings of Castillo, 2013)

Rank:

266.27

Rank:

342.74

!!!

Links 476

Nodes 243

ODs 1890

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Conclusions (intermezzo)

A new definition of partial observability problems that accounts for

full and partially observed variables was introduced

A novel metric able to assess information quantity based on

analysis of the extent of the Null-Space (NSP)

Local search algorithms developed to explore the solution space,

to determine optimal sensor locations

The methodology is generic as requires only the fundamental

topological relations between links, routes and eventually OD flows

different applications and extensions possible

Solutions on large networks strongly depend on route set

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Node vs. Route based

approaches

Node-based relations

• Flow conservation at nodes

• Local, simple laws

• No route enumeration necessary

• Misses complete information on link-path-OD relations

solutions require systematically more sensors

Route-based relations• Flow conservation on routes

• Connects links across whole network

• Needs some kind of route enumeration

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Observability and route

information (1)

Different route set compositions yield different full observability

solutions

Routes may be selected because of behavioral/flow capturing rules

(e.g. shortest paths)

| |

1 0 1 0

1 0 0 1

l RA

{ }odR r | | | |dep indep

Hypothesis: more efficient full (and partial) observability solutions can

be found with efficient route set generation.

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Observability and route

information (2)

Three different sorts of information

Non Redundant (linearly independent)

Redundant + Informative (linearly dependent, but useful to derive

independence relationships)

Pure Redundant (linearly dependent)

Strongly dependent on the chosen route set

| |

1 0 1 0

[ ]

1 0 0 1

l RA NR RI PR

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Example: parallel highway

network

Best full obs. route set: Best* partial obs. route set:

«Parallel highway» network

4 routes, fully diverse 9 routes, highly overlapping

* Non-unique solution

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Route set generation

Main idea:

Enumerate routes so to obtain the maximum linearly independent route set

Evaluate resulting Partial Observability Solutions

Challenges:

High dimensionality, combinatorial problem

Non-uniqueness of solutions

Non-convex condition (linear independence)

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Hypergraph approach (1)

Hypergraph

Express combinations of routes as vertices

Edges capture linear independence through bitwise logical operations

Example:

2 3 4

1 3 4

2 3 5

{ , , }

{ , }

{ ,

,

, }

v

A v v v

B v v

C v v v

(0,1,1,1,0)

(1,0,1,1,0)

(0,1,1,0,1)

A

B

C

Route combination A + B is independent iif:

,A B A B

(1,1,1,1,0)A B

A

B

C

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Hypergraph approach (2)

B CA B A C

A B C

A B C

Independence condition:

A

B

C

AA B C

B

B C

A C

A

B

C

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3 equivalent solutions

A B

B C

B C A C

BA A C

A

B

C

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Finding exact solutions (1)

Problem to be solved: constrained maximum clique in

the hypergraph|| ||

| |

{0,1}

(

(

A {

)

}

)

h h

h

V V

g ij

g V g

HG g

A J

A

a

I A

Q I

min

. .0

{0,1}

T

x HG

HG

x Q x f x

xPs t

x

Quadratic component,

captures adjacency

| |

max(|:

| ) 1

ii

i h

v

vf f

V

Linear component,

Captures importance

1 | |[ , ] { }

1:

| |

,...,hHG i iV

ik k i

ii i

P i

v

pp

vp v

p

Parenthood

constraints

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Finding exact solutions (2)

2R

2 (2 1)

2

R R

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Exact solutions based on solving the max clique problem

computationally intractable even for small networks!

Real challenge: building the hypergraph

nodes to be generated

edges to be checked for eligibility

Quickly unfeasible with realistic network sizes, akin to brute

forcing.

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Full ‘candy’ example (1)

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Exact solutions based on solving the max clique problem

computationally intractable even for small networks!

8

| | 256

| | 32385

h

h

R

V

L

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Full ‘candy’ example (2)

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Hypergraph generation:

heuristics

Vertex culling rules:

(Cul-1): Remove vertices if no new information is introduced w.r.t. parents

(Cul-2): Remove vertices if expected information is lower than best bound

Metaheuristics used to find efficient solutions (e.g. GA)

A B

BA

A B

BA

B CA B A C

A B C

D

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Impact of culling rules

Cul-1

Solution: « BCH »Cul-1 + Cul-2

Solution: « BC »

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Parallel Highways example

Hypergraph Statistics Solution statistics

# Vertices # Arcs# Vertices Max

Clique

Final route set size

[ind, full coverage]

Tot Memory usage (MB)

[RAM + Nodefiles]

Comp. Time

(s)% Gap

3583 5991142 405 6, 6 217.77 3600+278.7 4.11

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Test on large scale networks:

Route set generation methods

Yen’s K-Shortest path

Free parameter: k, how many paths per O/D

Intuition: higher k -> higher information content

Intuition 2: an upper bound to k must exist, after which adding routes to the

routeset will only bring redundant information*

Enumerate according to Yen’s K-Shortest paths

Extra check: if new path added is NOT independent, discard

Stop when k is met or no more independent paths can be found

Independency check: performed through matrix rank (can be computationally

cumbersome)

Castillo’s (2015) Independent paths

Enumerates routes such that all routes (and combinations thereof) are

independent from one another

Higher degree of information wrt. randomly chosen k-shortest paths

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Sioux Falls example

Generated Solution Statistics Solution statistics

# VerticesTot. GA

GenerationsFinal route set size


[RAM]Comp. Time (s)

37 165 - 37 0.72 281.9

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Rotterdam example

Generated Solution Statistics Solution statistics

# VerticesTot. GA

GenerationsFinal route set size


[RAM]Comp. Time (s)

144 541 - 150 98.26 12259.1

Links 476

Nodes 243

ODs 1890

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Vulnerability analysis

43

KSP

KISP

C1

MI

PivotingNetwork

Topology

MI

KSP

KISP

C

'

'

'

'

MI

KSP

KISP

C

ℝ 𝑖𝑛𝑑𝑒𝑝 −|fail|×|𝑑𝑒𝑝|ℝ|𝑖𝑛𝑑𝑒𝑝|×|𝑑𝑒𝑝|

MI

KSP

KISP

C

A

A

A

A

ℝ|𝐿|×|𝑅|

( , )G N L

Random Sensor

FailureLoss of

Information

Hypothesis: different full (and partial) observability solutions bear

different levels of resilience to sensor failure.

'||

||

||

||

T

F

F

NSB

P

')(B null

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Test Results (1)

Information level for different route set generation policies at 50 sensors

44

( ' )KISPNSP

( ' )KSPNSP

1)( 'CNSP

)( 'MINSP

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Test Results (2)

Loss of information upon 10% sensor failure (100 draws, 5x ~U(0,50))

45

( )NSP

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Conclusions

Different full observability solutions indeed bear different resilience to

sensor failure

Max. Independent Route Set policy

Achieves highest information content and density

Interestingly, also yields most resilient solution upon sensor failure

Future research topics:

Mixing different types of sensors (scanned links, FCDs,…)

Topology to data: impact on flow estimation techniques, comparison

between topology prediction and data validation

Potential for combination with flow capturing methods to be explored,

especially for state estimation

From observability to controllability…

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References

Viti F., Rinaldi M., Corman F., Tampére C.M.J. (2014). Assessing Partial

Observability in Network Sensor Location Problems. Transportation

Research Part B: Methodological, Vol. 70, pp. 65-89.

Rinaldi M., Viti F. (2017). Exact and Approximate Route Set Generation for

Resilient Partial Observability in Sensor Location Problems. Transportation

Research Part B: Methodological, Vol. 105, pp. 86-119.

Thank you!

{francesco.viti,marco.rinaldi}@uni.lu

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