Cross-Validation and Spatial Sampling

1. From closed loop to cross-validation

In SHM-02 verification was by closed loop: we generated a signal with known properties and measured whether the engine recovered them. That is a strong test, but it has a limit — it only shows the engine is internally consistent.

Cross-validation is a different kind of test: two mutually independent routes are compared. Here the routes are the FDD mode shapes extracted from the raw acceleration records, and a model of the same building calibrated separately.

2. The missing piece: the channel map

In SHM-03 the analysis was deliberately confined to frequency, because which channel sat on which floor and in which direction was not known. The sensor layout drawing was obtained later:

3. The inferred assignment was tested

Before the map arrived, the direction assignment of the channels had been inferred from the data itself: channels sharing a common peak frequency were grouped into the same direction. When the map arrived that inference could be checked.

4. A third independent check: published frequencies

The building has been studied in the literature. Comparing the frequencies published there against the ones we identified gives a third route:

Agreement is close in the low modes. That matters: it means the disagreement found in the next section is specific to the mode shapes, not a general problem with the identification.

5. Comparing the mode shapes

MAC is computed between the FDD shapes of each event and the corresponding model shape, and averaged over the events:

6. Agreement collapses with mode order

The pattern is regular: high in the first mode, halved in the second, essentially absent in the third. And it repeats in both directions and in every event. A regular pattern is not noise — it has a cause.

7. Three candidate explanations

1. The model is wrong in the high modes. Possible: a calibrated model is usually tuned on the low modes and its high-mode behaviour is not verified.
2. Identification is poor in the high modes. Also possible: high modes have lower energy and a worse signal-to-noise ratio.
3. The measurement layout cannot resolve the high modes. This is the one examined below, and the data supports it.

The three are not mutually exclusive. But the third is testable with the data in hand, and if it holds the other two become secondary.

8. Spatial sampling: Nyquist, but in space

In SHM-01 we saw that at least two points per cycle are needed to recognise a cycle in time. The same logic applies in space.

Now look at the sensor layout. Sensors sit on adjacent floor pairs: 1–2, 7–8, 13–14, 19–20. Two floors adjacent in a twenty-storey building are almost the same height as far as a mode shape is concerned. So although there are eight sensor levels, the number of effectively distinct heights is four.

Four distinct heights: comfortable for mode 1, borderline for mode 2, exactly at the limit for mode 3. That ordering matches the MAC table one-to-one.

9. Evidence: the pairs carry no independent information

The claim is testable. If two sensors in a pair really sit at effectively the same height, the difference between the model's mode shape values at those two floors should be small — and should grow with mode order.

In the first mode the within-pair difference is around 5% — the two sensors of a pair are reading almost the same thing. It rises in the second and third modes, but never enough to make the pair into two independent points. This is the quantitative form of "eight sensors, four heights".

10. The layout is not wrong — it is right for another purpose

So this is a design choice, not an error. The instrumentation was laid out to measure torsion and the fundamental translational modes — and it does that well, as SHM-05 shows. What it cannot do is resolve the third mode shape, and expecting it to would be to misread the layout's intent.

11. What to do in practice

1. Decide the sensor layout from the modes you want. For mode n, at least n+1 distinct heights.
2. Count effectively distinct heights, not sensors. Sensors on adjacent floors are one height as far as the shape is concerned.
3. State the purpose of the layout. A torsion-oriented layout and a high-mode-oriented layout are different; one instrumentation cannot serve both without more channels.
4. Do not read low MAC as damage straight away. Insufficient spatial sampling produces the same symptom.
5. Cross-validate. Data, model and published values are three independent routes; where they agree the conclusion is strong, and where they disagree the disagreement itself is informative.

12. Test yourself

1. How many distinct measurement heights are needed to define the shape of mode 4?
2. Why do eight sensors give only four effective heights?
3. Low MAC in the third mode — list three candidate explanations and say how you would test each.
4. Why does the layout put two sensors on the same floor instead of spreading them out?
5. The frequencies agree with the published values but the mode shapes do not. What does that tell you?
6. How does cross-validation differ from closed-loop verification? What does each show that the other cannot?