Examples
This packages facilitates working with datasets with uncertainties. The core concept of UncertainDatasets is to replace an uncertain data value with a probability distribution describing the point's uncertainty.
There are currently three ways of doing that: by theoretical distributions, theoretical distributions with parameters fitted to empirical data, or kernel density estimations to the distributions of empirical data. Check out the examples below and the documentation in the sidebar!
Uncertain values defined by theoretical distributions¶
First, load the necessary packages:
1 | using UncertainData, Distributions, KernelDensity, Plots |
A uniformly distributed uncertain value¶
Now, consider the following trivial example. We've measure a data value with a poor instrument that tells us that the value lies between -2 and 3. However, we but that we know nothing more about how the value is distributed on that interval. Then it may be reasonable to represent that value as a uniform distribution on [-2, 3].
Define the uncertain value:
1 2 3 4 | u = UncertainValue(Uniform, 1, 2) # Plot the estimated density bar(u, label = "", xlabel = "value", ylabel = "probability density") |
A normally distributed uncertain value¶
Another common example is to use someone else's data from a publication. Usually, these values are reported as the mean or median, with some associated uncertainty. Say we want to use an uncertain value which is normally distributed with mean 2.1 and standard deviation 0.3.
1 2 3 4 | u = UncertainValue(Normal, 2.1, 0.3) # Plot the estimated density bar(u, label = "", xlabel = "value", ylabel = "probability density") |
Uncertain values defined by kernel density estimated distributions¶
One may also be given a a distribution of numbers that's not quite normally distributed. How to represent this uncertainty? Easy: we use a kernel density estimate to the distribution.
In the following example, we generate some random numbers from a mixture of two normal distributions.
1 2 3 4 5 | M = MixtureModel([Normal(-5, 0.5), Normal(0.2)]) u = UncertainValue(UnivariateKDE, rand(M, 250)) # Plot the estimated distribution. plot(u, xlabel = "Value", ylabel = "Probability density") |
Uncertain values defined by theoretical distributions fitted to empirical data¶
One may also be given a dataset whose histogram looks a lot like a theoretical distribution. We may then select a theoretical distribution and fit its parameters to the empirical data.
In the example below, we resample the estimated distribution to obtain a histogram we can compare to the original data.
1 2 3 4 5 6 7 8 9 10 11 12 13 | # Take a sample from a Gamma distribution with parameters α = 1.7 and θ = 5.5 some_sample = rand(Gamma(1.7, 5.5), 2000) uv = UncertainValue(Gamma, some_sample) p1 = histogram(some_sample, normalize = true, fc = :black, lc = :black, label = "", xlabel = "value", ylabel = "density") # Resample the fitted theoretical distribution p2 = histogram(resample(uv, 10000), normalize = true, fc = :blue, lc = :blue, label = "", xlabel = "value", ylabel = "density") plot(p1, p2, layout = (2, 1), link = :x) |
As expected, the histograms closely match (but are not exact because we estimated the distribution using a limited sample).
