diff options
| -rw-r--r-- | README.md | 2 | ||||
| -rw-r--r-- | docs/source/overview.rst | 2 | ||||
| -rw-r--r-- | docs/source/physics.rst | 58 | ||||
| -rw-r--r-- | docs/source/statistics.rst | 14 | ||||
| -rw-r--r-- | examples/inference.ipynb | 6 | ||||
| -rw-r--r-- | examples/tutorial.ipynb | 14 |
6 files changed, 48 insertions, 48 deletions
@@ -49,7 +49,7 @@ For more information on the statistical modeling see * **Distributed and parallel computing**: Scripts included to manage the workload across a CPU cluster using [HTCondor](https://research.cs.wisc.edu/htcondor/). -* **Visualization**: Produce ternary plots of the flavour composition using the +* **Visualization**: Produce ternary plots of the flavor composition using the [python-ternary](https://zenodo.org/badge/latestdoi/19505/marcharper/python-ternary) package and joint posterior plots for analyzing MCMC chains using the [getdist](https://getdist.readthedocs.io/en/latest/) package. diff --git a/docs/source/overview.rst b/docs/source/overview.rst index 65e1871..14e27f8 100644 --- a/docs/source/overview.rst +++ b/docs/source/overview.rst @@ -38,4 +38,4 @@ Features - **MCMC Algorithms**: Affine invariant and nested sampling algorithms provided by `emcee <https://emcee.readthedocs.io/>`_ and `MultiNest <https://doi.org/10.1111/j.1365-2966.2009.14548.x>`_. - **Anarchic Sampling**: Sampling of the neutrino mixing matrix is done under the `*neutrino mixing anarchy* <https://doi.org/10.1016/j.physletb.2003.08.045>`_ hypothesis to ensure an unbiased prior. - **Distributed and parallel computing**: Scripts included to manage the workload across a CPU cluster using `HTCondor <https://research.cs.wisc.edu/htcondor/>`_. -- **Visualization**: Produce ternary plots of the flavour composition using the `python-ternary <https://zenodo.org/badge/latestdoi/19505/marcharper/python-ternary>`_ package and joint posterior plots for analyzing MCMC chains using the `getdist <https://getdist.readthedocs.io/en/latest/>`_ package. +- **Visualization**: Produce ternary plots of the flavor composition using the `python-ternary <https://zenodo.org/badge/latestdoi/19505/marcharper/python-ternary>`_ package and joint posterior plots for analyzing MCMC chains using the `getdist <https://getdist.readthedocs.io/en/latest/>`_ package. diff --git a/docs/source/physics.rst b/docs/source/physics.rst index ed603f2..4589eea 100644 --- a/docs/source/physics.rst +++ b/docs/source/physics.rst @@ -28,13 +28,13 @@ the neutrino it's name as a play on words of *little neutron* in Italian [4]_. It was not until some 20 years later that the discovery of the neutrino was realised. It was eventually understood that neutrinos came in three distinct -*flavours* :math:`\left (\nu_e,\nu_\mu,\nu_\tau\right )` along with their +*flavors* :math:`\left (\nu_e,\nu_\mu,\nu_\tau\right )` along with their associated antiparticles :math:`\left (\bar{\nu}_e,\bar{\nu}_\mu,\bar{\nu}_\tau\right)`. Neutrino Mixing --------------- -For the three massive neutrinos, the flavour eigenstates of the neutrino +For the three massive neutrinos, the flavor eigenstates of the neutrino :math:`\mid{\nu_\alpha}>`, :math:`\alpha\in\{e,\mu,\tau\}`, are related to the mass eigenstates :math:`\mid{\nu_i}>`, :math:`i\in\{1,2,3\}` via a unitary mixing matrix :math:`U_{\alpha i}` known as the PMNS matrix [5]_, [6]_: @@ -49,12 +49,12 @@ This relationship can be seen better in this image: :width: 500px :align: center - Graphical representation of the relationship between the neutrino flavour and + Graphical representation of the relationship between the neutrino flavor and mass eigenstates. The three mass eigenstates are depicted as three boxes, coloured such that the relative area gives the probability of finding the - corresponding flavour neutrino in that given mass state. + corresponding flavor neutrino in that given mass state. -The time evolution of the flavour eigenstate as the neutrino propagates is +The time evolution of the flavor eigenstate as the neutrino propagates is given by: .. math:: @@ -63,7 +63,7 @@ given by: \sum^3_{i=1}U^*_{\alpha i}\mid{\nu_i\left(t\right)}> The oscillation probability gives the probability that a neutrino produced in a -flavour state :math:`\alpha` is then detected in a flavour state :math:`\beta` +flavor state :math:`\alpha` is then detected in a flavor state :math:`\beta` after a propagation distance :math:`L`: .. math:: @@ -98,7 +98,7 @@ where :math:`\Delta m_{ij}^2=m_i^2-m_j^2`. Note that for neutrino oscillations to occur, there must be at least one non-zero :math:`\Delta m_{ij}^2` and therefore there must exist at least one non-zero neutrino mass state. -The mixing matrix can be parameterised using the standard factorisation [8]_: +The mixing matrix can be parameterized using the standard factorization [8]_: .. math:: @@ -126,7 +126,7 @@ where :math:`s_{ij}\equiv\sin\theta_{ij}`, :math:`c_{ij}\equiv\cos\theta_{ij}`, violating phase. Overall phases in the mixing matrix do not affect neutrino oscillations, which only depend on quartic products, and so they have been omitted. Therefore, this gives a total of six independent free parameters -describing neutrino oscillations for three neutrino flavours in a vacuum. This +describing neutrino oscillations for three neutrino flavors in a vacuum. This table outlines the current knowledge of these parameters determined by a fit to global data [9]_: @@ -134,7 +134,7 @@ global data [9]_: :width: 500px :align: center - Three neutrino flavour oscillation parameters from a fit to global data + Three neutrino flavor oscillation parameters from a fit to global data [9]_. This table shows two columns of values, *normal ordering* and *inverted @@ -205,9 +205,9 @@ coincidence with neutrinos coming from a particular source has successfully been able to identify for the very first time, a source of high-energy astrophysical neutrinos [11]_, [12]_. -Of particular interest is the composition of flavours produced at the source. -In the simple pion decay model described above, the *neutrino flavour -composition* (sometimes referred to as the *neutrino flavour ratio*) +Of particular interest is the composition of flavors produced at the source. +In the simple pion decay model described above, the *neutrino flavor +composition* (sometimes referred to as the *neutrino flavor ratio*) produced at the source is: .. math:: @@ -215,12 +215,12 @@ produced at the source is: \pi\text{ decay}\rightarrow \left(f_e:f_\mu:f_\tau\right)_\text{S}=\left(1:2:0\right)_\text{S} -For all discussions on the astrophysical neutrino flavour composition, the +For all discussions on the astrophysical neutrino flavor composition, the neutrino and antineutrino fluxes will been summed over as it is not yet experimentally possible to distinguish between the two. In the case that the muon interacts in the source before it has a chance to decay, e.g.\@ losing energy rapidly in strong magnetic fields or being absorbed in matter, only the -:math:`\nu_\mu` from the initial pion decay escapes and so the source flavour +:math:`\nu_\mu` from the initial pion decay escapes and so the source flavor composition is simply: .. math:: @@ -239,19 +239,19 @@ decay, :math:`n\rightarrow p+e^-+\bar{\nu}_e`, which gives rise to a purely Production of :math:`\nu_\tau` at the source is not expected in standard astrophysics models. However, even in the standard construction, the composition could vary between any of the three idealised models above, which -can be represented as a source flavour composition of :math:`(x:1-x:0)`, where +can be represented as a source flavor composition of :math:`(x:1-x:0)`, where :math:`x` is the fraction of :math:`\nu_e` and can vary between :math:`0\rightarrow1`. Once the neutrinos escape the source, they are free to propagate in the vacuum. -As discussed above, neutrinos can transform from one flavour to another. +As discussed above, neutrinos can transform from one flavor to another. Astrophysical neutrinos have :math:`\mathcal{O}(\text{Mpc})` or higher baselines, large enough that the mass eigenstates completely decouple. The astrophysical neutrinos detected on Earth are decoherent and are propagating in pure mass eigenstates. Taking this assumption greatly simplifies the transition probability as all the interference terms between the three mass eigenstates can be dropped, and all that is left is to convert from the propagating mass -state to the flavour states: +state to the flavor states: .. math:: @@ -259,13 +259,13 @@ state to the flavour states: \phi_{\alpha,\oplus}&=\sum_{i,\beta} \mid{U_{\alpha i}}\mid^2\mid{U_{\beta i}}\mid^2\phi_{\beta,\text{S}} -where :math:`\phi_\alpha` is the flux for a neutrino flavour :math:`\nu_\alpha` +where :math:`\phi_\alpha` is the flux for a neutrino flavor :math:`\nu_\alpha` and :math:`\phi_i` is the flux for a neutrino mass state :math:`\nu_i`. The subscript :math:`\text{S}` denotes the source and :math:`\oplus` denotes as measured on Earth. The same result can be obtained in the plane wave picture of the neutrino mixing equations above and taking the limit :math:`L\rightarrow\infty`, thus this type of decoherent mixing is also known -as oscillation-averaged neutrino mixing. From this, the flavour composition on +as oscillation-averaged neutrino mixing. From this, the flavor composition on Earth is defined as :math:`f_{\alpha,\oplus}=\phi_{\alpha,\oplus}/\sum_\alpha\phi_{\alpha,\oplus}` and this can be calculated using the mixing matrix parameters the table above. @@ -281,15 +281,15 @@ For the three source models discussed above: This can be visualised in a ternary plot, which you can make yourself by checking out the :doc:`examples` section! The axes here are the fraction of -each neutrino flavour as shown below. The coloured circle, square and triangle -show the source flavour compositions. The arrows show the effect of neutrino -mixing on the flavour composition. The unfilled circle, square and triangle -show the corresponding measured flavour composition. Neutrino mixing during -propagation has the effect of averaging out the flavour contributions, which is +each neutrino flavor as shown below. The coloured circle, square and triangle +show the source flavor compositions. The arrows show the effect of neutrino +mixing on the flavor composition. The unfilled circle, square and triangle +show the corresponding measured flavor composition. Neutrino mixing during +propagation has the effect of averaging out the flavor contributions, which is why the arrows point towards the centre of the triangle. This effect is more pronounced for :math:`\nu_\mu\leftrightarrow\nu_\tau` due to the their larger mixings. Also shown on this figure in the hatched *Standard Model* area, is the -region of measured flavour compositions containing all source models of +region of measured flavor compositions containing all source models of :math:`\left(x:1-x:0\right)`, using Gaussian priors on the standard mixing angles. Therefore, this hatched area is the region in which all standard astrophysical models live. @@ -298,11 +298,11 @@ astrophysical models live. :width: 700px :align: center - Astrophysical neutrino flavour composition ternary plot. Axes show the - fraction of each neutrino flavour. Coloured shapes show 3 models for the - source flavour composition. The arrows indicate the effect of neutrino mixing + Astrophysical neutrino flavor composition ternary plot. Axes show the + fraction of each neutrino flavor. Coloured shapes show 3 models for the + source flavor composition. The arrows indicate the effect of neutrino mixing during propagation and the unfilled shapes show the corresponding measured - flavour compositions. The hatched area shows the region in measured flavour + flavor compositions. The hatched area shows the region in measured flavor space in which all standard astrophysical models live. IceCube diff --git a/docs/source/statistics.rst b/docs/source/statistics.rst index e2dd85f..b63c9a0 100644 --- a/docs/source/statistics.rst +++ b/docs/source/statistics.rst @@ -100,7 +100,7 @@ Instead, according to Wilks' theorem [3]_, for sufficiently large :math:`\textbf{x}` and provided certain regularity conditions are met (MLE exists and is unique), :math:`-2\ln\lambda\left(\textbf{x}\right)` can be approximated to follow a :math:`\chi^2` distribution. The :math:`\chi^2` -distribution is parameterised by :math:`k`, the *number of degrees of +distribution is parameterized by :math:`k`, the *number of degrees of freedom*, which is defined as the number of independent normally distributed variables that were summed together. When the profile likelihood is used to account for :math:`n` nuisance parameters, the effective number of degrees of @@ -248,7 +248,7 @@ referred to as the *evidence* of a particular model: \pi_j\left(\mathbf{\theta}_j\right)\text{d}\mathbf{\theta}_j -This was seen before as just a normalisation constant above; however, this +This was seen before as just a normalization constant above; however, this quantity is central in Bayesian model selection, which for two models :math:`\mathcal{M}_0` and :math:`\mathcal{M}_1` is realised through the ratio of the posteriors: @@ -354,7 +354,7 @@ mixing parameters are of concern. These parameters are defined in the mixing matrix :math:`U`, in such a way that any valid combination of the mixing angles can be mapped into a unitary matrix. The ideal and most ignorant choice of prior here is one in which there is no distinction among the three neutrino -flavours, compatible with the hypothesis of *neutrino mixing anarchy*, which is +flavors, compatible with the hypothesis of *neutrino mixing anarchy*, which is the hypothesis that :math:`U` can be described as a result of random draws from an unbiased distribution of unitary :math:`3\times3` matrices [11]_, [12]_, [13]_, [14]_. Simply using a flat prior on the mixing angles however, does @@ -368,7 +368,7 @@ is the central assumption of *basis independence* and from this, distributions over the mixing angles are determined by the integration invariant *Haar measure* [13]_. For the group :math:`U(3)` the Haar measure is given by the volume element :math:`\text{d} U`, which can be written using the mixing angles -parameterisation: +parameterization: .. math:: @@ -385,15 +385,15 @@ chosen according to this Haar measure, i.e. in :math:`\sin^2\theta_{12}`, :math:`\cos^4\theta_{13}`, :math:`\sin^2\theta_{23}` and :math:`\delta`. You can see an example on this in action in the :doc:`examples` notebooks. -This also needs to be considered in the case of a flavour composition +This also needs to be considered in the case of a flavor composition measurement using sampling techniques in a Bayesian approach. In this case, the posterior of the measured composition :math:`f_{\alpha,\oplus}` is sampled over as the parameters of interest. Here, the effective number of parameters can be reduced from three to two due to the requirement :math:`\sum_\alpha f_{\alpha,\oplus}=1`. Therefore, the prior on these two parameters must be -determined by Haar measure of the flavour composition volume element, +determined by Haar measure of the flavor composition volume element, :math:`\text{d} f_{e,\oplus}\wedge\text{d} f_{\mu,\oplus}\wedge\text{d} -f_{\tau,\oplus}`. The following *flavour angles* parameterisation is found to +f_{\tau,\oplus}`. The following *flavor angles* parameterization is found to be sufficient: .. math:: diff --git a/examples/inference.ipynb b/examples/inference.ipynb index 7457f66..129fd86 100644 --- a/examples/inference.ipynb +++ b/examples/inference.ipynb @@ -11,7 +11,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "In this example, we will take the fake data generated in the `tutorial.ipynb` example and use it to make an inference the source flavour composition using Bayesian techniques." + "In this example, we will take the fake data generated in the `tutorial.ipynb` example and use it to make an inference the source flavor composition using Bayesian techniques." ] }, { @@ -324,7 +324,7 @@ " source_angles = llh_paramset.from_tag(ParamTag.SRCANGLES, values=True)\n", " source_composition = angles_to_fr(source_angles)\n", "\n", - " # Calculate the expected measured flavour composition for our sampled values\n", + " # Calculate the expected measured flavor composition for our sampled values\n", " measured_composition = u_to_fr(source_composition, sm_u)\n", "\n", " # Convert flavor angles to flavor compositions for the injected parameters\n", @@ -588,7 +588,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "Great! Looks like our inference of the source flavour composition reflects the injected value $(1:0:0)_S$. Here, the credbility regions include the effect of smearing as well as our uncertainity about the values of the mixing matrix, which is why the values are not exactly at the injected $(1:0:0)_S$ value.\n", + "Great! Looks like our inference of the source flavor composition reflects the injected value $(1:0:0)_S$. Here, the credbility regions include the effect of smearing as well as our uncertainity about the values of the mixing matrix, which is why the values are not exactly at the injected $(1:0:0)_S$ value.\n", "\n", "In a real analysis, an ensemble of nuisance parameters is usually required, related to uncertainties arising from things such as the astrophysical flux, detector calibration and backgrounds from atmospherically produced neutrinos. All these effects come into play when making inferences and careful analysis must be done for each in order to minimize potential biases." ] diff --git a/examples/tutorial.ipynb b/examples/tutorial.ipynb index 6d9bae1..529bb6e 100644 --- a/examples/tutorial.ipynb +++ b/examples/tutorial.ipynb @@ -11,7 +11,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "In this example, we will generate a fake measured flavour composition using a multivariate Gaussian distribution and sample from it using the [emcee](https://emcee.readthedocs.io/) MCMC algorithm." + "In this example, we will generate a fake measured flavor composition using a multivariate Gaussian distribution and sample from it using the [emcee](https://emcee.readthedocs.io/) MCMC algorithm." ] }, { @@ -50,7 +50,7 @@ "\n", "$$\\pi\\:\\text{decay}\\rightarrow\\left(f_e:f_\\mu:f_\\tau\\right)_\\text{S}=\\left(1:2:0\\right)_\\text{S}$$\n", "\n", - "where $f_\\alpha$ is the flavor composition of a neutrino with flavor $\\alpha\\in\\{e,\\mu,\\tau\\}$ and the subscript S represents that this is the flavour composition at the source. In the code below we normalize this to 1 for later calculations." + "where $f_\\alpha$ is the flavor composition of a neutrino with flavor $\\alpha\\in\\{e,\\mu,\\tau\\}$ and the subscript S represents that this is the flavor composition at the source. In the code below we normalize this to 1 for later calculations." ] }, { @@ -84,7 +84,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "For the three massive neutrinos, the flavour eigenstates of the neutrino $|\\nu_\\alpha>$, $\\alpha\\in\\{e,\\mu,\\tau\\}$, are related to the mass eigenstates $|\\nu_i>$, $i\\in\\{1,2,3\\}$ via a unitary mixing matrix $U_{\\alpha i}$ known as the PMNS matrix:\n", + "For the three massive neutrinos, the flavor eigenstates of the neutrino $|\\nu_\\alpha>$, $\\alpha\\in\\{e,\\mu,\\tau\\}$, are related to the mass eigenstates $|\\nu_i>$, $i\\in\\{1,2,3\\}$ via a unitary mixing matrix $U_{\\alpha i}$ known as the PMNS matrix:\n", " \n", "$$ |\\nu_\\alpha>=\\sum^3_{i=1}U^*_{\\alpha i}|\\nu_i> $$\n", "\n", @@ -117,7 +117,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "This mixing matrix says that neutrinos can oscillation from one flavor state $\\alpha\\in\\{e,\\mu,\\tau\\}$ to another $\\beta\\in\\{e,\\mu,\\tau\\}$ as a function of the propagation distance. The oscillation probability gives the probability that a neutrino produced in a flavour state $\\alpha$ is then detected in a flavour state $\\beta$ after a propagation distance $L$:\n", + "This mixing matrix says that neutrinos can oscillation from one flavor state $\\alpha\\in\\{e,\\mu,\\tau\\}$ to another $\\beta\\in\\{e,\\mu,\\tau\\}$ as a function of the propagation distance. The oscillation probability gives the probability that a neutrino produced in a flavor state $\\alpha$ is then detected in a flavor state $\\beta$ after a propagation distance $L$:\n", "\n", "$$\n", "\\begin{align}\n", @@ -295,9 +295,9 @@ " Parameters\n", " ----------\n", " fr : List[float], length 3\n", - " The flavour composition to evaluate at.\n", + " The flavor composition to evaluate at.\n", " fr_bf : List[float], length 3\n", - " The bestfit / injected flavour composition.\n", + " The bestfit / injected flavor composition.\n", " smearing : float\n", " The amount of smearing.\n", " offset : float, optional\n", @@ -350,7 +350,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "Now we have everything we need to do scan over our likelihood, from which we will be able to visualize the effect of this smearing. However, scanning directly in the space of the flavour composition would not be the correct way to do the scan. This particular parameterization has degeneracies, since the total flavor composition must add up to 1, $\\sum_{\\alpha}f_\\alpha=1$, which introduces an unwanted prior dependence." + "Now we have everything we need to do scan over our likelihood, from which we will be able to visualize the effect of this smearing. However, scanning directly in the space of the flavor composition would not be the correct way to do the scan. This particular parameterization has degeneracies, since the total flavor composition must add up to 1, $\\sum_{\\alpha}f_\\alpha=1$, which introduces an unwanted prior dependence." ] }, { |