Math Bridge: What can no imputation method recover?

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Math BridgeAnalytics & MLStruggle-friendly runway

What can no imputation method recover?

Separate missing values that software may estimate from physical detail the acquisition system never captured.

Data Dora, the guideData Dora guides
The one targetConnect sample interval and ADC bit depth to information limits before imputation.
The chapter case60-second temperature samples and a 12-bit, 0–100% humidity channel.
What it buys youA repair pipeline that does not invent evidence.

A field team faces an unresolved physical question: What can no imputation method recover? They must answer it before changing sample interval on the real device. Predict the direction first.

See the relationship before changing it

The figure reads from left to right. The blue card is sample interval. The middle card applies this page's relationship. The green card is sample rate. Walk the arrows once: set the input, apply the rule, then read the result with its unit.

The retained audit below checks several chapter fixtures. This added model holds every other chapter fixture fixed, so the numeric fixture does not switch without explanation.

Sample interval changes sample rate An input card leads through the page relationship to the sample rate result. SET INPUT ONE CONTROL APPLY RULE predict calculate check units READ RESULT
Walk the arrows. Aliasing changes the apparent time pattern before values go missing. Quantisation limits amplitude detail. Imputation can address neither loss after the fact.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline for sample interval is 60.

  2. 2

    Name the relationship. Ts=60 s: fs=0.01667 Hz; fNyquist=0.00833 Hz; Tmin=120 s N=12 over 100%: q=0.0244%; σq=0.00705%; ideal SNR=74.0 dB

  3. 3

    Substitute the chapter fixture. Set sample interval to 60. The page ledger gives sample rate as 0.01667 Hz.

  4. 4

    Read the result. Keep Hz beside the value. Use it only inside the technical boundary on this page.

Predict, then change sample interval

Try Predict the direction of sample rate. Move one control, calculate, then check your prediction.

60
Chapter baseline
Sample rate

Observe Aliasing changes the apparent time pattern before values go missing. Quantisation limits amplitude detail. Imputation can address neither loss after the fact. Reset the control to 60 and compare sample rate.

Explain Only sample interval moves here. The other chapter fixtures remain fixed.

Check yourself

What should you do before trusting a moved-control result?
Answer: Predict its direction, apply the shown relationship, keep the units, and reset to the worked baseline.
What does this small model leave out?
Answer: Only sample interval moves. Field effects named in the page's technical boundary stay fixed.

Honesty note. This teaching model changes only sample interval. It replays the chapter fixture; it is not a field prediction.

1. Start before the gap appears

Imputation estimates a missing sample from samples that exist. It cannot undo aliasing or recover changes smaller than the acquisition system's quantisation step.

Data Dora: A smooth repaired line can still be a precise picture of the wrong signal.

2. Name every algebra move

1

Invert the intervalfs=1/Ts.

2

Halve the ratefNyquist=fs/2.

3

Invert frequencyThe fastest unaliased period is 1/fNyquist=2Ts.

4

Count ADC levelsL=2^N and q=full scale/L.

5

Convert step to RMSσq=q/√12 and SNR=6.02N+1.76 dB.

3. Reproduce the chapter case

Ts=60 s: fs=0.01667 Hz; fNyquist=0.00833 Hz; Tmin=120 s
N=12 over 100%: q=0.0244%; σq=0.00705%; ideal SNR=74.0 dB

A 90-second physical cycle sampled every 60 seconds aliases into a false 180-second cycle. Filling later gaps cannot restore the original 90-second motion.

4. Try the sample interval

TryChange the interval and watch the fastest defensible cycle and the 90-second alias move.

Sample interval
Sample rate
Nyquist ceiling
Fastest cycle
Test signal period
Apparent alias period
ADC levels
ADC step
RMS floor
Ideal SNR

ObserveAt 60.00 s, the 0.01667 Hz sample rate supports cycles no faster than 120.00 s, and the 90.00 s cycle appears as 180.00 s. The ADC still has 4,096 levels, 0.0244% steps, and 0.0070% RMS noise.

ExplainAliasing changes the apparent time pattern before values go missing. Quantisation limits amplitude detail. Imputation can address neither loss after the fact.

Technical boundaries.

This compact engine isolates acquisition limits; it does not choose an imputation method.

Signal
The 90-second cycle is an illustrative fast HVAC fault inside the chapter's stated 1–3 minute range
Sampling
Nyquist is necessary but an analog anti-alias filter is still required
ADC
The ideal quantisation floor omits sensor noise, drift, nonlinearity, and calibration error

Preserve raw timestamps, quality flags, and the acquisition contract beside repaired values.

5. Classify the missingness

Before filling anything, distinguish dropout, invalid outlier, maintenance, offline device, and intentionally unsampled time. Different causes permit different claims.

6. Keep the repair record

Record sample rate, anti-alias filter, ADC range and bits, missingness cause, validation order, method, maximum gap, quality flag, owner, and the condition that invalidates the repair.

7. Check yourself

Why is the fastest unaliased period 120 seconds?
Answer: A 60-second interval gives fs=1/60 Hz; Nyquist halves that to 1/120 Hz, whose period is 120 seconds.
Where does the 0.0244% ADC step come from?
Answer: A 12-bit converter has 4,096 levels, so 100%/4,096=0.0244% per ideal code.
Can linear interpolation recover a 90-second cycle from these samples?
Answer: No. The cycle has already aliased into a slower pattern before interpolation sees the data.
Honesty boundary.

The fixed acquisition values are the chapter's own examples; the alias signal is explicitly labelled.

60 seconds
The chapter's forward-fill temperature interval
12 bit and 0–100%
The chapter's humidity-channel quiz example
90 seconds
An illustrative short-cycling HVAC period within the chapter's bounded range

Correct, not complete: sensor dynamics and analog filtering must also be measured.

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