Assignment result + observed counts → calibrated OD matrixSTAGE 5 · FTT BACKWARD PASS · TOY NETWORK
1. The setup
Network
nodeslinks (with sensors)
OD pairs & paths
A → B1 path · uses link 1
A → C (via B)uses link 1 + link 2
A → C (express)uses link 3
B → C1 path · uses link 2
Path-link matrix A (from assignment)
link
p1: A→B
p2: A→C·B
p3: A→C·exp
p4: B→C
link 1
1
1
0
0
link 2
0
1
0
1
link 3
0
0
1
0
Initial OD prior T₀
OD
T0
A → B
100
A → C
80
B → C
50
A → C splits 60% via-B / 40% express (a route-choice proportion from a prior assignment).
2. The ODME problem
T (OD)
↓
B (OD → path)
↓
path flow f
↓
A (path → link)
↓
link flow v = ABT
↑
backward: minimise ‖v − ŷ‖²
minT ‖ABT − ŷ‖²
+ λ ‖T − T0‖²
Observed counts ŷ
link
observed ŷ
initial v0 = ABT0
gap
link 1
150
148
+2
link 2
120
98
+22
link 3
40
32
+8
Counts say more traffic is moving than the prior OD predicts — especially on link 2. ODME will find a T that fits the counts without straying too far from T0.
Tuning
small λ → trust counts · large λ → trust prior
Residual fit
counts
—
prior
—
3. Calibrated OD T*
100
A→B
80
A→C
50
B→C
Calibrated link flows v* vs observations
link
v*
ŷ
gap
OD change from prior
OD
T0
T*
Δ
Verification
—
RMSE counts
—
‖T*−T₀‖
FTT tie-in: this is exactly the backward pass of the Flow-Through-Tensor. The same A matrix used here to assign T → v is also used (transposed) to push the count residual back into corrections on T. On a real network, this is the same loop, just with bigger tensors and more paths.