<p>An example would be measuring the count rate received from a known source at a known position in the presence of background. A naive estimator can result in a negative count rate.</p>
<p>In my own research I measured the equivalent width of absorption lines. If a chemical element producing the line has a low abundance then it is quite possible, in the presence of noise, to get a negative equivalent width even though an emission line is implausible.</p>

CC BY-SA 4.0
2024-06-08T21:37:22.920Z
Thank you, that is useful.
J. Webb
CC BY-SA 4.0
2024-06-07T20:46:22.310Z
2024-06-07T20:46:22.310Z
ProfRob
<p>I'd assert that no strictly positive quantity that is measured can ever be measured to be negative unless the measurement process is fundamentally flawed.  Instead, there are derived quantities that are <em>calculated</em> to be negative.</p>
<p>This can happen in two ways, firstly if the calculation process involves subtraction.  For example, you can't measure parallax, but you can measure position.  To find the parallax you subtract one position from another, to find the change in position (over a six-month period).  The process of subtraction can (if there are measurement errors) result in a negative value for the parallax.</p>
<p>Alternatively the calculation can use a function may be negative.  For example to find the distance to a galaxy you might measure its redshift. This is valid because redshift correlates with distance (the Hubble Law) but for nearby galaxies the actual motion of the galaxy gives it a blue shift, so naively applying the redshift formula would result in a negative value for the distance. In this case, there is no error in measurement, but the scatter , when fed into a regression formula results in a negative value.  An example would be using the redshift to determine the distance of a galaxy that is nearby and (due to its particular motion) is actually blueshifted. The naive calculation of distance would be found to be negative.</p>
<p>These effect can occur in any situation in which a value is calculated, not directly measured, if the calculation can return a negative value.</p>

CC BY-SA 4.0
2024-06-07T20:53:36.573Z
2024-06-07T20:53:36.573Z
James K
<p>Not exactly what you are apparently looking for, but a similar phenomenon:</p>
<p>Suppose we encounter particles traveling near the speed of light. If we measure their mass/energy and momentum, the ratio of mass/energy to momentum would be greater than the speed of light <span class="math-container">$c$</span> if the velocity is under <span class="math-container">$c$</span>. But measurement errors could make the ratio appear to be less thsn <span class="math-container">$c$</span>, and then we get headlines about possible faster-than-light particles.</p>

CC BY-SA 4.0
2024-06-07T01:01:20.777Z
2024-06-07T01:01:20.777Z
Oscar Lanzi
<p>I am interested in physical quantities which are strictly non-negative, but which, due to errors, may realistically be estimated as negative. The only examples I can think of are interstellar absorption and parallax. Any others?</p>
<p>This is of interest in the construction of statistical tests for whether estimated quantities are zero.</p>

CC BY-SA 4.0
2024-06-03T18:09:24.550Z
I suppose "distance by the naive application of the redshift method" which is erroneously measured as negative for any galaxy whose particular motion towards the Earth exceeds any Hubble Flow, for example the galaxies in the local group (which aren't moving with the Hubble flow, since the universe doesn't expand on that scale) would be measured as having a negative distance if the redshift method were applied naively
James K
CC BY-SA 4.0
2024-06-03T18:09:55.943Z
Speeds, masses (probably), distances (and hence areas and volumes), periods of time, slices of pizza, frequencies, pressure (except dark energy-like stuff), probabilities, entropy (I suppose), refractive indices, fluxes, work, thermal conductivities, magnetic permeabilities, luminosities, …
pela
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2024-06-03T22:03:36.250Z
pela -references to published examples?
J. Webb
CC BY-SA 4.0
2024-06-06T19:33:52.467Z
@pela speeds, masses, distances, pressure, probability, entropy, conductivities, permeabilities, luminosities etc (of/to astronomical objects) are never measured (at best they are inferred from other measurements). Frequencies, periods of time, and fluxes cannot conceivably be measured negatively. Slices of pizza are eaten, not measured. :-)
Walter
CC BY-SA 4.0
2024-06-06T19:36:57.517Z
Strictly speaking, parallaxes and interstellar absorption are not directly measured, but inferred from several measurements (of positions/times and fluxes, respectively). This process of inference can, depending on the measured positions/fluxes, obtain a negative value; it can also give a bias, i.e. a non-zero expectation value if the true parallax/extinction was zero. Interestingly, the such obtained parallaxes for quasars (which are known to be zero) in Gaia follow a normal distribution extremely. well
Walter
CC BY-SA 4.0
2024-06-06T19:37:40.103Z
May I ask what your interest here is? What kind of tests are you thinking of?
Walter
CC BY-SA 4.0
2024-06-07T12:42:40.813Z
what is a direct measurement when a parallax is not directly measured? Where do you draw the line? Does a flux gate magnetometer measure the flux directly? Or is it indirectly as you 'directly' only measure the current to compensate the external field?
planetmaker
CC BY-SA 4.0
2024-06-07T20:16:56.260Z
Walter - I deliberately used the word "estimate" in the question; the way by which the estimate is arrived at is not directly relevant. The application I'm specifically thinking of is correction for interstellar absorption, where estimates may be close to zero or negative. The statistical problem I'm interested in is testing the null hypothesis that E(B-V)=0. This is a so-called "non-standard" problem, since the null hypothesis specifies that the parameter of interest lies on the boundary of the parameter space. I was hoping for suggested other applications.
J. Webb
CC BY-SA 4.0
2024-06-08T07:52:31.207Z
@Walter I agree that examples might sometimes be somewhat conceived, but any time a quantity is smaller than the uncertainty/noise of your measuring device, you can — erroneously — measure the quantity to be negative. It happens often for flux, actually: For a very weak signal, the noise in a spectrum can make it go belo zero. Indeed, you sometimes see negative fluxes reported in published papers.
pela
CC BY-SA 4.0
2024-06-09T08:07:53.763Z
@pela I'm astonished to hear about negative flux measurements. Aren't fluxes essentially measured as counts of electrons in a CCD? Can such counts be negative? Well I suppose, there could be some source of noise in the chip that can actually cause this. But, those negative counts are never significant.
Walter
CC BY-SA 4.0
2024-06-09T08:10:29.727Z
@planetmaker Parallax is definitely not measured. It is estimated from a number of measurements of the objects position taken over a long time, at least 6 months. These positions are then fit to a model, usually assuming a linear motion (i.e. no binary-induced motion), to obtain the parallax and proper motion.
Walter
<observational-astronomy><data-analysis>
Parameters erroneously measured to be negative
J. Webb
# Parameters erroneously measured to be negative

(Asked by: username_0 on 2024-06-03)

I am interested in physical quantities which are strictly non-negative, but which, due to errors, may realistically be estimated as negative. The only examples I can think of are interstellar absorption and parallax. Any others?

This is of interest in the construction of statistical tests for whether estimated quantities are zero.

**Tags:** observational-astronomy, data-analysis
> **Comments:**
> **username_1**: I suppose "distance by the naive application of the redshift method" which is erroneously measured as negative for any galaxy whose particular motion towards the Earth exceeds any Hubble Flow, for example the galaxies in the local group (which aren't moving with the Hubble flow, since the universe doesn't expand on that scale) would be measured as having a negative distance if the redshift method were applied naively  
> **username_2**: Speeds, masses (probably), distances (and hence areas and volumes), periods of time, slices of pizza, frequencies, pressure (except dark energy-like stuff), probabilities, entropy (I suppose), refractive indices, fluxes, work, thermal conductivities, magnetic permeabilities, luminosities, …  
> **username_0**: pela -references to published examples?  
> **username_3**: @pela speeds, masses, distances, pressure, probability, entropy, conductivities, permeabilities, luminosities etc (of/to astronomical objects) are never measured (at best they are inferred from other measurements). Frequencies, periods of time, and fluxes cannot conceivably be measured negatively. Slices of pizza are eaten, not measured. :-)  
> **username_3**: Strictly speaking, parallaxes and interstellar absorption are not directly measured, but inferred from several measurements (of positions/times and fluxes, respectively). This process of inference can, depending on the measured positions/fluxes, obtain a negative value; it can also give a bias, i.e. a non-zero expectation value if the true parallax/extinction was zero. Interestingly, the such obtained parallaxes for quasars (which are known to be zero) in Gaia follow a normal distribution extremely. well  
> **username_3**: May I ask what your interest here is? What kind of tests are you thinking of?  
> **username_4**: what is a direct measurement when a parallax is not directly measured? Where do you draw the line? Does a flux gate magnetometer measure the flux directly? Or is it indirectly as you 'directly' only measure the current to compensate the external field?  
> **username_0**: Walter - I deliberately used the word "estimate" in the question; the way by which the estimate is arrived at is not directly relevant. The application I'm specifically thinking of is correction for interstellar absorption, where estimates may be close to zero or negative. The statistical problem I'm interested in is testing the null hypothesis that E(B-V)=0. This is a so-called "non-standard" problem, since the null hypothesis specifies that the parameter of interest lies on the boundary of the parameter space. I was hoping for suggested other applications.  
> **username_2**: @Walter I agree that examples might sometimes be somewhat conceived, but any time a quantity is smaller than the uncertainty/noise of your measuring device, you can — erroneously — measure the quantity to be negative. It happens often for flux, actually: For a very weak signal, the noise in a spectrum can make it go belo zero. Indeed, you sometimes see negative fluxes reported in published papers.  
> **username_3**: @pela I'm astonished to hear about negative flux measurements. Aren't fluxes essentially measured as counts of electrons in a CCD? Can such counts be negative? Well I suppose, there could be some source of noise in the chip that can actually cause this. But, those negative counts are never significant.  
> **username_3**: @planetmaker Parallax is definitely not measured. It is estimated from a number of measurements of the objects position taken over a long time, at least 6 months. These positions are then fit to a model, usually assuming a linear motion (i.e. no binary-induced motion), to obtain the parallax and proper motion.  

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## 3 answers

### Answer by username_5 on 2024-06-07. Score: 1

An example would be measuring the count rate received from a known source at a known position in the presence of background. A naive estimator can result in a negative count rate.

In my own research I measured the equivalent width of absorption lines. If a chemical element producing the line has a low abundance then it is quite possible, in the presence of noise, to get a negative equivalent width even though an emission line is implausible.

> **Comments:**
> **username_0**: Thank you, that is useful.  

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### Answer by username_1 on 2024-06-07. Score: 0

I'd assert that no strictly positive quantity that is measured can ever be measured to be negative unless the measurement process is fundamentally flawed. Instead, there are derived quantities that are *calculated* to be negative.

This can happen in two ways, firstly if the calculation process involves subtraction. For example, you can't measure parallax, but you can measure position. To find the parallax you subtract one position from another, to find the change in position (over a six-month period). The process of subtraction can (if there are measurement errors) result in a negative value for the parallax.

Alternatively the calculation can use a function may be negative. For example to find the distance to a galaxy you might measure its redshift. This is valid because redshift correlates with distance (the Hubble Law) but for nearby galaxies the actual motion of the galaxy gives it a blue shift, so naively applying the redshift formula would result in a negative value for the distance. In this case, there is no error in measurement, but the scatter , when fed into a regression formula results in a negative value. An example would be using the redshift to determine the distance of a galaxy that is nearby and (due to its particular motion) is actually blueshifted. The naive calculation of distance would be found to be negative.

These effect can occur in any situation in which a value is calculated, not directly measured, if the calculation can return a negative value.


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### Answer by username_6 on 2024-06-07. Score: 0

Not exactly what you are apparently looking for, but a similar phenomenon:

Suppose we encounter particles traveling near the speed of light. If we measure their mass/energy and momentum, the ratio of mass/energy to momentum would be greater than the speed of light $c$ if the velocity is under $c$. But measurement errors could make the ratio appear to be less thsn $c$, and then we get headlines about possible faster-than-light particles.


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