Had a QuantSci Prof who was fond of asking "Who can name a data collection scenario where the x data has no error?" and then taught Deming regression as a generally preferred analysis [1]

[1] https://en.wikipedia.org/wiki/Deming_regression

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moregrist 3 days ago | parent | next [-]

Most of the time, if you have a sensor that you sample at, say 1 KHz and you’re using a reliable MCU and clock, the noise terms in the sensor will vastly dominate the jitter of sampling.

So for a lot of sensor data, the error in the Y coordinate is orders of magnitude higher than the error in the X coordinate and you can essentially neglect X errors.

	▲	sigmoid10 3 days ago \| parent [-]
		That is actually the case in most fields outside of maybe clinical chemistry and such, where Deming became famous for explaining it (despite not even inventing the method). Ordinary least squares originated in astronomy, where people tried to predict movement of celestial objects. Timing a planet's position was never an issue (in fact time is defined by celestian position), but getting the actual position of a planet was. Total least squares regression also is highly non-trivial because you usually don't measure the same dimension on both axes. So you can't just add up errors, because the fit will be dependent on the scale you chose. Deming skirts around this problem by using the ratio of variances of errors (division also works for different units), but that is rarely known well. Deming works best when the measurement method for both dependent and independent variable is the same (for example when you regress serum levels against one another), meaning the ratio is simply one. Which of course implies that they have the same unit. So you don't run into the scale-invariance issues, which you would in most natural science fields.

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jmpeax 3 days ago | parent | prev | next [-]

From that wikipedia article, delta is the ratio of y variance to x variance. If x variance is tiny compared to y variance (often the case in practice) then will we not get an ill-conditioned model due to the large delta?

	▲	kevmo314 3 days ago \| parent [-]
		If you take the limit of delta -> infinity then you will get beta_1 = s_xy / s_xx which is the OLS estimator. In the wiki page, factor out delta^2 from the sqrt and take delta to infinity and you will get a finite value. Apologies for not detailing the proof here, it's not so easy to type math...

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ghc 2 days ago | parent | prev | next [-]

In my field, the X data error (measurement jitter) is generally <10ns, which might as well be no error.

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Beretta_Vexee 3 days ago | parent | prev | next [-]

For most time series, noise in time measurement is negligible. However, this does not prevent complex coupling phenomena from occurring for other parameters, such as GPS coordinates.

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RA_Fisher 2 days ago | parent | prev [-]

The issue in that case is that OLS is BLUE, the best linear unbiased estimator (best in the sense of minimum variance). This property is what makes OLS exceptional.