I'm reading about how to fit a straight line with measurement errors in both coordinates ($x$ and $y$).
Let the true unobserved variables be $x_{t,i}$ and $y_{t,i}$ and the observed variables be $x_i$ and $y_i$.
The relationship between the observed and the unobserved data is:
$$ x_i = x_{t,i} + e_{x,i} $$
where $e_{x,i}$ represents unknown error component. And it is assumed that $e_{x,i}$ ~$ N(0, \sigma_{x,i}^2) $.
Unfortunately, this is where I get lost. Can someone please explain to me what is the difference between:
1. Error $e_{x,i}$
2. Variance $\sigma_{x,i}^2$
and
3. Measurement errors (also known as measurement uncertainties?)
My data is obtained by fitting models to the observed light curve. An MCMC approach is used to calculate the uncertainties (measurement errors). I didn't do the model, I simply downloaded the available data in its final form from an online database.
For example, my data look something like the following: dip in the light curve = $ 3 \pm 0.03$
Obviously, my observed parameter: $x_i = 3 $
Measurement error (also known as uncertainty) = $ \pm 0.03 $
Let's suppose that the true unobserved parameter $x_t = 8 $
Is the error $ e = x - x_t = 3 - 8 = -5$?
What about the variance $\sigma_{x}^2$? How can I calculate it?
Finally, how can I assume that my errors are normally distributed?
I know I'm asking basic statistical question, unfortunately I read a lot of references, none where clear about the difference of these 3 different parameters. All the tutorials on regression simply outlined the above and went on with explaining the rest of the problem. Sorry if this is a stupid question.
1 Answer
When talking about measurements, a measurement error is the difference between the measured value of your parameter and its true value (you may or may not know the true value, this depends on the parameter). The wording is really important. What you call Error is measurement error in my understanding. And your term Measurement errors (also known as measurement uncertainties?) should solely be named as measurement uncertainty.
There are several sources of errors resulting from the measurement process (measurment bias, random errors, operator bias, environmental errors,…). Once you know your error sources, you make can estimate an uncertainty in the error of the measurement. The usual lack of knowledge about the sign and magnitude of measurement errors is called measurement uncertainty. You will have uncertainties in each error source. The uncertainty analysis uses the fact that measurement errors can be characterized by statistical distributions (often normal distribution for technical measurements). And this is where the variance comes into play. If a quantity $x$ is a random variable representing a population of measurements, then the variance in $x$ is the variance in the error in $x$.
The measurement uncertainty $u$ is the square root of the variance in the measurement error.
There are different ways to estimate variance and uncertainties from measurements ("Type A, Type B"). One way is to determine a sample mean and standard deviation and calculate limits that bound the sample mean with a certain level of confidence, aka confidence limits. Confidence limits can be expressed as multiples of the sample standard deviation, e.g. the Student t-statistic in case of normal distributions.
