From 708b847586a4a7ea7596d957678d45cab7caf687 Mon Sep 17 00:00:00 2001 From: "Martin D. Weinberg" Date: Wed, 24 Dec 2025 13:07:47 -0500 Subject: [PATCH 01/11] Begin documenting the covariance computation --- topics/covariance.rst | 26 ++++++++++++++++++++++++++ 1 file changed, 26 insertions(+) create mode 100644 topics/covariance.rst diff --git a/topics/covariance.rst b/topics/covariance.rst new file mode 100644 index 0000000..179d5ab --- /dev/null +++ b/topics/covariance.rst @@ -0,0 +1,26 @@ +.. role:: python(code) + :language: python + :class: highlight + +.. _units: + +Coefficient covariance estimates +================================ + +.. attention:: + + The `exp` interface described below is under development in the + `OutSample` branch. You will need to ``git checkout devel`` and + recompile to use these features in the `exp` N-body code. The + `pyEXP` interface is currently available in the `main` branch. + + +Overview +-------- + +`exp` and `pyEXP` compute both the empirical covariance matrices and +coefficients in _partitions_ or _batches_ that may be used to estimate +statistical consistence of coefficients. + +.. _unit-schema: + From a587ca545406b02189b9a906c124b6e03c744fa5 Mon Sep 17 00:00:00 2001 From: "Martin D. Weinberg" Date: Wed, 24 Dec 2025 17:23:46 -0500 Subject: [PATCH 02/11] Preliminary implementation of covariance documentation --- conf.py | 4 +- index.rst | 7 +- topics/covariance.rst | 367 +++++++++++++++++++++++++++++++++++++++++- 3 files changed, 367 insertions(+), 11 deletions(-) diff --git a/conf.py b/conf.py index 1a0e8ab..6c46a3a 100644 --- a/conf.py +++ b/conf.py @@ -72,9 +72,9 @@ # -- project information ----------------------------------------------------- # project = 'EXP' -copyright = '2023-2025, EXP-code collaboration' +copyright = '2023-2026, EXP-code collaboration' author = 'EXP-code collaboration' -release = '0.172' +release = '0.173' version = '7.x' # -- General configuration --------------------------------------------------- diff --git a/index.rst b/index.rst index 848fe8e..a36c78c 100644 --- a/index.rst +++ b/index.rst @@ -149,6 +149,7 @@ How to solve specific problems topics/EXP-output topics/flatdisk topics/debug + topics/covariance :doc:`topics/visualizing-bases` Learn how to visualize the EXP bases using pyEXP @@ -156,13 +157,17 @@ How to solve specific problems :doc:`topics/making-coefficients` Get familiar with producing coefficients for particle data in pyEXP +:doc:`topics/covariance + How to think about and compute significance of coefficients using + exp and pyEXP + :doc:`topics/saving-coefficients` Learn how to save and read the coefficients you have produced from your N-body snapshots in pyEXP :doc:`topics/visualizing-fields` Learn how to convert your coefficients to physical fields and - visualize them with Python and pyEXP + visualize them with Python and pyEXP :doc:`topics/make-movies` Make some movies from your fields in Python diff --git a/topics/covariance.rst b/topics/covariance.rst index 179d5ab..534c578 100644 --- a/topics/covariance.rst +++ b/topics/covariance.rst @@ -2,16 +2,16 @@ :language: python :class: highlight -.. _units: +.. _covariance: -Coefficient covariance estimates -================================ +Coefficient covariance and signal-to-noise estimates +==================================================== .. attention:: The `exp` interface described below is under development in the - `OutSample` branch. You will need to ``git checkout devel`` and - recompile to use these features in the `exp` N-body code. The + `OutSample` branch. You will need to ``git checkout OutSample`` + and recompile to use these features in the `exp` N-body code. The `pyEXP` interface is currently available in the `main` branch. @@ -19,8 +19,359 @@ Overview -------- `exp` and `pyEXP` compute both the empirical covariance matrices and -coefficients in _partitions_ or _batches_ that may be used to estimate -statistical consistence of coefficients. +compute coefficients in *partitions* or *batches* that may be used to +estimate statistical consistency of coefficients. -.. _unit-schema: +.. _covariance_theory: + +Theory +------ + +Setup and notation +^^^^^^^^^^^^^^^^^^ + +Let :math:`\{c_i\}_{i=1}^N\subset\mathbb{R}^d` be the contribution to +the coefficient vector from Particle :math:`i`. Specifically, if the +potential basis function is :math:`\Phi_k(\cdot)`, then +:math:`c_i=\{\Phi_1(\mathbf{x}_i), \Phi_1(\mathbf{x}_i), \ldots, +\Phi_d(\mathbf{x}_i)\}` where :math:`\mathbf{x}_i` is the position +vector for Particle :math:`i`. Denote the empirical mean for the +entire ensemble of :math:`N` particles as + +.. math:: + + \bar{c} \;=\; \frac{1}{N}\sum_{i=1}^N c_i + +and the sample covariance as + +.. math:: + + \Sigma \;=\; \frac{1}{N}\sum_{i=1}^N (c_i-\bar{c})(c_i-\bar{c})^{\!\top}. + +Partition the particles into :math:`K` disjoint subsamples or _blocks_ +with sizes :math:`n_k` so that :math:`\sum_{k=1}^K n_k = N`. For block +:math:`k` define + +.. math: + + \bar{c}_k \;=\; \frac{1}{n_k}\sum_{i\in\text{block }k} c_i, + \qquad S_k \;=\; \sum_{i\in\text{block }k} + (c_i-\bar{c}_k)(c_i-\bar{c}_k)^{\!\top}, + +where :math:`S_k` is the within‑block scatter. + +Overall mean and within and between blocks +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ + +The overall mean is the weighted average of block means: + +.. math:: + + \bar{c} \;=\; \frac{1}{N}\sum_{k=1}^K n_k \bar{c}_k. + +The total scatter is + +.. math:: + + T \;=\; \sum_{i=1}^N (x_i-\bar{c})(x_i-\bar{c})^{\!\top}, + +and the exact decomposition is + +.. math:: + + T \;=\; \underbrace{\sum_{k=1}^K S_k}_{W} \;+\; + \underbrace{\sum_{k=1}^K + n_k(\bar{c}_k-\bar{c})(\bar{c}_k-\bar{c})^{\!\top}}_{B}. + +The two terms are the *in block* and *between block* scatter. Hence +the population covariance is + +.. math:: + + \Sigma \;=\; \frac{1}{N}T \;=\; \frac{1}{N}\sum_{k=1}^K S_k \;+\; + \sum_{k=1}^K + \frac{n_k}{N}(\bar{c}_k-\bar{c})(\bar{c}_k-\bar{c})^{\!\top} \;=\; + \frac{W}{N} + \frac{B}{N}. + +The between term :math:`B/N` can be computed from the +:math:`\{\bar{c}_k\}` and :math:`\{n_k\}` alone; the within aggregate +:math:`W` requires the :math:`S_k` or additional assumptions. + +Special case: equal-size blocks +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ + +Assume each block has the same size :math:`m`, so :math:`n_k = m` for +all :math:`k`, and :math:`N = Km`. Define the block‑means covariance +as + +.. math:: + + S_{\bar{c}}^{\mathrm{pop}} \;=\; \frac{1}{K}\sum_{k=1}^K + (\bar{c}_k-\bar{c})(\bar{c}_k-\bar{c})^{\!\top}. + +Then the *between scatter* contribution is + +.. math:: + + B \;=\; \sum_{k=1}^K m(\bar{c}_k-\bar{c})(\bar{c}_k-\bar{c})^{\!\top} + \;=\; m \sum_{k=1}^K (\bar{c}_k-\bar{c})(\bar{c}_k-\bar{c})^{\!\top}, + +and therefore + +.. math:: + + \frac{B}{N} \;=\; \frac{m}{Km}\sum_{k=1}^K + (\bar{c}_k-\bar{c})(\bar{c}_k-\bar{c})^{\!\top} \;=\; + \frac{1}{K}\sum_{k=1}^K + (\bar{c}_k-\bar{c})(\bar{c}_k-\bar{c})^{\!\top} \;=\; + S_{\bar{c}}^{\mathrm{pop}}. + +Thus, deterministically, + +.. math:: + + \Sigma \;=\; \frac{W}{N} + S_{\bar{c}}^{\mathrm{pop}}, + \qquad\text{where } W=\sum_{k=1}^K S_k. + +IID block-sampling model +^^^^^^^^^^^^^^^^^^^^^^^^ + +If additionally each block is formed by :math:`m` independent draws +from the same distribution with mean :math:`\bar{c}` and covariance +:math:`\Sigma` (block independent), then + +.. math:: + + \mathbb{E}[\bar{c}_k] = \bar{c},\qquad + \operatorname{Cov}(\bar{c}_k) = \frac{\Sigma}{m}. + +This is generally true for `exp` simulations and can be made true for +any simulation by selecting particles from the entire particle +ensemble randomly. Consequently, + +.. math:: + + \mathbb{E}\!\big[ S_{\bar{c}}^{\mathrm{pop}} \big] \;=\; + \frac{\Sigma}{m}. + +Therefore an estimator of :math:`\Sigma` based on the +block means is + +.. math:: + + \widehat{\Sigma}_{\bar{c}} \;=\; m\,S_{\bar{c}}, + +where :math:`S_{\bar{c}}` is a chosen empirical covariance of the block +means. Two common choices: + +- Population-style (divide by :math:`K`): + + .. math:: + + S_{\bar{c}}^{(K)} \;=\; \frac{1}{K}\sum_{k=1}^K + (\bar{c}_k-\bar{c})(\bar{c}_k-\bar{c})^{\!\top}, + + for which :math:`\mathbb{E}[S_{\bar{c}}^{(K)}]=\Sigma/m` when the true + mean :math:`\bar{c}` is used (i.e., if :math:`\bar{c}` is known). Then + :math:`\widehat{\Sigma}=m S_{\bar{c}}^{(K)}` is unbiased (when + :math:`\bar{c}` is the true mean). + +- Sample-style (divide by :math:`K-1` using the sample mean + :math:`\bar{\bar{c}}=(1/K)\sum_k \bar{c}_k`): + + .. math:: + + S_{\bar{c}}^{(K-1)} \;=\; \frac{1}{K-1}\sum_{k=1}^K + (\bar{c}_k-\bar{\bar{c}})(\bar{c}_k-\bar{\bar{c}})^{\!\top}. + + This satisfies + + .. math:: + + \mathbb{E}\!\big[S_{\bar{c}}^{(K-1)}\big] \;=\; \frac{\Sigma}{m}, + + so the estimator + + .. math:: + + \widehat{\Sigma} \;=\; m\; S_{\bar{c}}^{(K-1)} + + is unbiased for :math:`\Sigma` under the iid block model. + +Relation of the Finite-sample to the full covariance +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ + +Under the iid model, the expected within aggregate satisfies + +.. math:: + + \mathbb{E}[S_k] = (m-1)\Sigma, \qquad \mathbb{E}[W] = K(m-1)\Sigma, + +hence + +.. math:: + + \mathbb{E} \left[\frac{W}{N}\right] = \frac{K(m-1)\Sigma}{Km} = \frac{m-1}{m}\Sigma, + +and + +.. math:: + + \mathbb{E} \left[\frac{B}{N}\right] = \frac{\Sigma}{m}. + +Adding these gives :math:`\mathbb{E}[\Sigma_{\mathrm{emp}}] = \Sigma`, +verifying unbiasedness of the global empirical covariance. + +Summary +^^^^^^^ + +- If you have only the block means :math:`\{\bar{c}_k\}` and you know that each block is an average of :math:`m` iid draws, then :math:`\widehat{\Sigma}=m\,S_{\bar{c}}^{(K-1)}` is an unbiased estimator of the full covariance :math:`\Sigma`. + +- If blocks are not iid samples (e.g. they are clusters with internal + structure), then multiplying block-mean covariance by $m$ is not + generally valid; additional modeling or within-block information is + required. + +- If you have both the full empirical covariance + :math:`\Sigma_{\mathrm{emp}}` and the block means, you can compute + the aggregate within scatter as + + .. math:: + + W \;=\; N\Sigma_{\mathrm{emp}} - B, + + and therefore recover the sum of the within-block scatters even + without individual :math:`S_k`. + + +.. _coefficient_SN: + +Signal-to-noise estimates +------------------------- + +.. _outsample: + +Using `OutSample` with `exp` +---------------------------- + +Covariance data generation within `exp` is a two step process: + +1. The force method computes the subsample coefficient and covariance + estimates every `nint` steps. The configuration parameter `nint` + is zero by default; no subsamples or sample covariance is computed + by default. + +2. You need to instantiate an `outsamp` output routine in the `Output` + collection with the `name` parameter to match the name of the + component whose force has a non-zero `nint` value. + +The covariance files are written in HDF5 format and take the form: +`coefcovar.{component name}.{runtag}`. + + +.. _pyEXP_covariance: + +Covariance examples in `pyEXP` +------------------------------ + +The `pyEXP` interface is split between the creation and the reading of +HDF5 covariance files as follows: + +1. Similar to the `exp` force methods, the `pyEXP.basis.BiorthBasis` + classes can be configured to produce subsample and covariance data + that get written to an HDF5 file. + +2. The utility class `pyEXP.basis.BasisClasses.SubsampleCovariance` + reads the HDF5 covariance files and returns sample information, + subsample coefficients and covariance information in `numpy` + compatible format. + +The controlling configuration YAML keys are: + +- `pcavar: true` -- This enables the covariance computation + +- `samplesz: 100` -- This sets the number of subsamples to `100`. The + value `1` is also valid. + +- `totalCovar: true` -- Computes a single covariance matrix for the + entire ensemble. + +- `fullCovar: true` -- Computes a covariance matrix for each subsample + +Assume that you have created a basis instance using pyEXP named +`basis`. After reading and generating coefficients for a particular +snapshot in the usual way, the call + +.. code-block:: python + + compname = 'halo' + runtag = 'run1' + time = 3.14 + basis.writeCoefCovariance(compname, runtag, time) + +will write covariance data to file named `coefcovar.halo.run1`, +creating the new file if it does not already exist, and store the +covariance data for time `3.14`. + +You can read and HDF5 covariance file with the following code: + +.. code-block: python + + compname = 'halo' + runtag = 'run1' + covar = pyEXP.basis.CovarianceReader('coefcovar.{}.{}'.format(compname, runtag)) + + +A typical use case might be as follows: + +.. code-block: python + + # This gets the list of available snapshot times + times = covar.Times() + ntimes = len(times) + + # As an example, let's look at the data for the final time + indx = -1 + time = times[indx] + + # Get all the subsample data for time + counts, masses, coefs, covrs = covar['dark'].getCoefCovariance(time) + + # The counts array contains the number of particles in each sample + + # The masses array contains the mass of the particles in each sample + + # The coefs is an ndarray of coefficient vectors indexed by sample + # number and harmonic order + + # The coefs is an ndarray of covariance matrices indexed by sample + # number and harmonic order + + # Following code shows how to visualize the covariance matrices + # using pcolormesh + + + numb = np.sum(counts) + mass = np.sum(masses) + nsamp = counts.shape[0] + + # For pcolormesh below + x = np.arange(0, nmaxh) + y = np.arange(0, nmaxh) + X, Y = np.meshgrid(x, y) + + for l in range(3): + for m in range(l+1): + L = basis['dark'].I(l, m) + sumCof = np.zeros((nmaxh), dtype=np.complex128) + sumVar = np.zeros((nmaxh, nmaxh), dtype=np.complex128) + for T in range(nsamp): + sumCof += coefs[T, L, :]/mass + sumVar += covrs[T, L, :, :]/mass + sumVar -= np.outer(sumCof, sumCof) + # sumVar is not the full normalized covariance + plt.pcolormesh(X, Y, np.log10(np.abs(sumVar)), cmap='viridis') + plt.colorbar() + plt.title('Halo covariance for l={} m={} T={}'.format(l, m, times[-1])) + plt.show() From f31373f575fbcc128b3b90e868af9c6f670d4637 Mon Sep 17 00:00:00 2001 From: "Martin D. Weinberg" Date: Thu, 25 Dec 2025 14:33:03 -0500 Subject: [PATCH 03/11] A complete draft of the covariance documentation --- index.rst | 11 +- topics/covariance.rst | 419 ++++++++++++++++++++++++++--------- topics/images/halo_covar.png | Bin 0 -> 20703 bytes topics/images/halo_sn.png | Bin 0 -> 120377 bytes 4 files changed, 324 insertions(+), 106 deletions(-) create mode 100644 topics/images/halo_covar.png create mode 100644 topics/images/halo_sn.png diff --git a/index.rst b/index.rst index a36c78c..c92aaae 100644 --- a/index.rst +++ b/index.rst @@ -94,6 +94,7 @@ EXP concepts topics/centering topics/yamlconfig topics/howtosim + topics/covariance These topics provide some background detail on the software and mathematics used in EXP. @@ -122,6 +123,11 @@ mathematics used in EXP. :doc:`topics/howtosim` A guide for running EXP simulations +:doc:`topics/covariance + How to think about and compute significance of coefficients using + exp and pyEXP + + .. toctree:: :caption: Frequently Asked Questions (FAQ) @@ -149,7 +155,6 @@ How to solve specific problems topics/EXP-output topics/flatdisk topics/debug - topics/covariance :doc:`topics/visualizing-bases` Learn how to visualize the EXP bases using pyEXP @@ -157,10 +162,6 @@ How to solve specific problems :doc:`topics/making-coefficients` Get familiar with producing coefficients for particle data in pyEXP -:doc:`topics/covariance - How to think about and compute significance of coefficients using - exp and pyEXP - :doc:`topics/saving-coefficients` Learn how to save and read the coefficients you have produced from your N-body snapshots in pyEXP diff --git a/topics/covariance.rst b/topics/covariance.rst index 534c578..c5726fe 100644 --- a/topics/covariance.rst +++ b/topics/covariance.rst @@ -4,8 +4,8 @@ .. _covariance: -Coefficient covariance and signal-to-noise estimates -==================================================== +Coefficient significance analysis +================================= .. attention:: @@ -22,13 +22,36 @@ Overview compute coefficients in *partitions* or *batches* that may be used to estimate statistical consistency of coefficients. +Basis function expansions estimate the underlying fields from the +particles that sample their gravitational fields that generate the +particle orbits. The primary goal of analyses is improving the field +estimates by characterizing their significance. We do this in two +ways: + +1. The variation represented by the BFE is spread across basis + functions. In other words, the independently varying signals not + isolated to specific basis functions. We may use covariance + analysis, such as ideas from Principal Component Analysis (PCA), to + empirically determine a basis which separates the initially + spatially correlated signals. + +2. One may treat the significance of the underlying fields represented + by the expansion as an estimation problem and attempt to analyze + the significance of each coefficient using methods from probability + and statistics. + +The next section introduces the terminology and concepts from sampling +theory necessary to address each of these two goals. We then move on +to practical applications using `exp`-generated coefficients sets with +`pyEXP`. + .. _covariance_theory: Theory ------ -Setup and notation -^^^^^^^^^^^^^^^^^^ +Notation +^^^^^^^^ Let :math:`\{c_i\}_{i=1}^N\subset\mathbb{R}^d` be the contribution to the coefficient vector from Particle :math:`i`. Specifically, if the @@ -40,161 +63,165 @@ entire ensemble of :math:`N` particles as .. math:: - \bar{c} \;=\; \frac{1}{N}\sum_{i=1}^N c_i + \hat{c} \;=\; \frac{1}{N}\sum_{i=1}^N c_i -and the sample covariance as +and the sample covariance is the :math:`d\times d` matrix .. math:: - \Sigma \;=\; \frac{1}{N}\sum_{i=1}^N (c_i-\bar{c})(c_i-\bar{c})^{\!\top}. + \Sigma \;=\; \frac{1}{N}\sum_{i=1}^N (c_i-\hat{c})(c_i-\hat{c})^{\!\top}. -Partition the particles into :math:`K` disjoint subsamples or _blocks_ -with sizes :math:`n_k` so that :math:`\sum_{k=1}^K n_k = N`. For block -:math:`k` define +Partition the particles into :math:`K` disjoint sub samples or +_blocks_ with sizes :math:`n_k` so that :math:`\sum_{k=1}^K n_k = +N`. For each block :math:`k` define -.. math: +.. math:: - \bar{c}_k \;=\; \frac{1}{n_k}\sum_{i\in\text{block }k} c_i, + \hat{c}_k \;=\; \frac{1}{n_k}\sum_{i\in\text{block }k} c_i, \qquad S_k \;=\; \sum_{i\in\text{block }k} - (c_i-\bar{c}_k)(c_i-\bar{c}_k)^{\!\top}, + (c_i-\hat{c}_k)(c_i-\hat{c}_k)^{\!\top}, where :math:`S_k` is the within‑block scatter. Overall mean and within and between blocks ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ -The overall mean is the weighted average of block means: +The overall mean, the coefficient vector estimate itself, is the +weighted average of block means: .. math:: - \bar{c} \;=\; \frac{1}{N}\sum_{k=1}^K n_k \bar{c}_k. + \hat{c} \;=\; \frac{1}{N}\sum_{k=1}^K n_k \hat{c}_k. -The total scatter is +The total scatter, proportional to the covariance matrix, is .. math:: - T \;=\; \sum_{i=1}^N (x_i-\bar{c})(x_i-\bar{c})^{\!\top}, + S \;=\; \sum_{i=1}^N (c_i-\hat{c})(c_i-\hat{c})^{\!\top}, and the exact decomposition is .. math:: - T \;=\; \underbrace{\sum_{k=1}^K S_k}_{W} \;+\; + S \;=\; \underbrace{\sum_{k=1}^K S_k}_{W} \;+\; \underbrace{\sum_{k=1}^K - n_k(\bar{c}_k-\bar{c})(\bar{c}_k-\bar{c})^{\!\top}}_{B}. + n_k(\hat{c}_k-\hat{c})(\hat{c}_k-\hat{c})^{\!\top}}_{B}. -The two terms are the *in block* and *between block* scatter. Hence -the population covariance is +The two terms, :math:`W` and :math:`B`, are the *in block* and +*between block* scatter, respectively. Then, the empirical population +covariance is .. math:: - \Sigma \;=\; \frac{1}{N}T \;=\; \frac{1}{N}\sum_{k=1}^K S_k \;+\; + \Sigma_\textrm{emp} \;=\; \frac{1}{N}S \;=\; + \frac{1}{N}\sum_{k=1}^K S_k \;+\; \sum_{k=1}^K - \frac{n_k}{N}(\bar{c}_k-\bar{c})(\bar{c}_k-\bar{c})^{\!\top} \;=\; + \frac{n_k}{N}(\hat{c}_k-\hat{c})(\hat{c}_k-\hat{c})^{\!\top} \;=\; \frac{W}{N} + \frac{B}{N}. -The between term :math:`B/N` can be computed from the -:math:`\{\bar{c}_k\}` and :math:`\{n_k\}` alone; the within aggregate +The *between block* term :math:`B/N` can be computed from the +:math:`\{\hat{c}_k\}` and :math:`\{n_k\}` alone; the within aggregate :math:`W` requires the :math:`S_k` or additional assumptions. Special case: equal-size blocks ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Assume each block has the same size :math:`m`, so :math:`n_k = m` for -all :math:`k`, and :math:`N = Km`. Define the block‑means covariance +all :math:`k`, and :math:`N = Km`. Define the block mean covariance as .. math:: - S_{\bar{c}}^{\mathrm{pop}} \;=\; \frac{1}{K}\sum_{k=1}^K - (\bar{c}_k-\bar{c})(\bar{c}_k-\bar{c})^{\!\top}. + \Sigma_{\hat{c}} \;=\; \frac{1}{K}\sum_{k=1}^K + (\hat{c}_k-\hat{c})(\hat{c}_k-\hat{c})^{\!\top}. Then the *between scatter* contribution is .. math:: - B \;=\; \sum_{k=1}^K m(\bar{c}_k-\bar{c})(\bar{c}_k-\bar{c})^{\!\top} - \;=\; m \sum_{k=1}^K (\bar{c}_k-\bar{c})(\bar{c}_k-\bar{c})^{\!\top}, + B \;=\; \sum_{k=1}^K m(\hat{c}_k-\hat{c})(\hat{c}_k-\hat{c})^{\!\top} + \;=\; m \sum_{k=1}^K (\hat{c}_k-\hat{c})(\hat{c}_k-\hat{c})^{\!\top}, and therefore .. math:: \frac{B}{N} \;=\; \frac{m}{Km}\sum_{k=1}^K - (\bar{c}_k-\bar{c})(\bar{c}_k-\bar{c})^{\!\top} \;=\; + (\hat{c}_k-\hat{c})(\hat{c}_k-\hat{c})^{\!\top} \;=\; \frac{1}{K}\sum_{k=1}^K - (\bar{c}_k-\bar{c})(\bar{c}_k-\bar{c})^{\!\top} \;=\; - S_{\bar{c}}^{\mathrm{pop}}. + (\hat{c}_k-\hat{c})(\hat{c}_k-\hat{c})^{\!\top} \;=\; + \Sigma_{\hat{c}} -Thus, deterministically, +Putting all of this together, we have .. math:: - \Sigma \;=\; \frac{W}{N} + S_{\bar{c}}^{\mathrm{pop}}, + \Sigma \;=\; \frac{W}{N} + S_{\hat{c}}^{\mathrm{pop}}, \qquad\text{where } W=\sum_{k=1}^K S_k. IID block-sampling model ^^^^^^^^^^^^^^^^^^^^^^^^ If additionally each block is formed by :math:`m` independent draws -from the same distribution with mean :math:`\bar{c}` and covariance +from the same distribution with mean :math:`\hat{c}` and covariance :math:`\Sigma` (block independent), then .. math:: - \mathbb{E}[\bar{c}_k] = \bar{c},\qquad - \operatorname{Cov}(\bar{c}_k) = \frac{\Sigma}{m}. + \mathbb{E}[\hat{c}_k] = \hat{c},\qquad + \operatorname{Cov}(\hat{c}_k) = \frac{\Sigma}{m}. -This is generally true for `exp` simulations and can be made true for -any simulation by selecting particles from the entire particle -ensemble randomly. Consequently, +This is generally true for `exp` simulations which do not sort +particles within their component and can be made true for any +simulation by selecting particles from the entire particle ensemble +randomly. Consequently, .. math:: - \mathbb{E}\!\big[ S_{\bar{c}}^{\mathrm{pop}} \big] \;=\; + \mathbb{E}\!\big[ \Sigma_{\hat{c}} \big] \;=\; \frac{\Sigma}{m}. -Therefore an estimator of :math:`\Sigma` based on the -block means is +Therefore, an estimator of :math:`\Sigma` based on the block means is .. math:: - \widehat{\Sigma}_{\bar{c}} \;=\; m\,S_{\bar{c}}, + \widehat{\Sigma}_{\hat{c}} \;=\; m\,\Sigma_{\hat{c}}, -where :math:`S_{\bar{c}}` is a chosen empirical covariance of the block -means. Two common choices: +where :math:`S_{\hat{c}}` is a chosen empirical covariance of the +block means. One can choose either the natural population-style or +unbiased sample-style form for these estimates: - Population-style (divide by :math:`K`): .. math:: - S_{\bar{c}}^{(K)} \;=\; \frac{1}{K}\sum_{k=1}^K - (\bar{c}_k-\bar{c})(\bar{c}_k-\bar{c})^{\!\top}, + \Sigma_{\hat{c}}^{(K)} \;=\; \frac{1}{K}\sum_{k=1}^K + (\hat{c}_k-\hat{c})(\hat{c}_k-\hat{c})^{\!\top}, - for which :math:`\mathbb{E}[S_{\bar{c}}^{(K)}]=\Sigma/m` when the true - mean :math:`\bar{c}` is used (i.e., if :math:`\bar{c}` is known). Then - :math:`\widehat{\Sigma}=m S_{\bar{c}}^{(K)}` is unbiased (when - :math:`\bar{c}` is the true mean). + for which :math:`\mathbb{E}[S_{\hat{c}}^{(K)}]=\Sigma/m` when the true + mean :math:`\hat{c}` is used (i.e., if :math:`\hat{c}` is known). Then + :math:`\widehat{\Sigma}=m \Sigma_{\hat{c}}^{(K)}` is unbiased (when + :math:`\hat{c}` is the true mean). - Sample-style (divide by :math:`K-1` using the sample mean - :math:`\bar{\bar{c}}=(1/K)\sum_k \bar{c}_k`): + :math:`\hat{c}=(1/K)\sum_k \hat{c}_k`): .. math:: - S_{\bar{c}}^{(K-1)} \;=\; \frac{1}{K-1}\sum_{k=1}^K - (\bar{c}_k-\bar{\bar{c}})(\bar{c}_k-\bar{\bar{c}})^{\!\top}. + \Sigma_{\hat{c}}^{(K-1)} \;=\; \frac{1}{K-1}\sum_{k=1}^K + (\hat{c}_k-\hat{\hat{c}})(\hat{c}_k-\hat{\hat{c}})^{\!\top}. This satisfies .. math:: - \mathbb{E}\!\big[S_{\bar{c}}^{(K-1)}\big] \;=\; \frac{\Sigma}{m}, + \mathbb{E}\!\big[\Sigma_{\hat{c}}^{(K-1)}\big] \;=\; \frac{\Sigma}{m}, so the estimator .. math:: - \widehat{\Sigma} \;=\; m\; S_{\bar{c}}^{(K-1)} + \widehat{\Sigma} \;=\; m\; \Sigma_{\hat{c}}^{(K-1)} is unbiased for :math:`\Sigma` under the iid block model. @@ -205,7 +232,7 @@ Under the iid model, the expected within aggregate satisfies .. math:: - \mathbb{E}[S_k] = (m-1)\Sigma, \qquad \mathbb{E}[W] = K(m-1)\Sigma, + \mathbb{E}[\Sigma_k] = (m-1)\Sigma, \qquad \mathbb{E}[W] = K(m-1)\Sigma, hence @@ -225,7 +252,7 @@ verifying unbiasedness of the global empirical covariance. Summary ^^^^^^^ -- If you have only the block means :math:`\{\bar{c}_k\}` and you know that each block is an average of :math:`m` iid draws, then :math:`\widehat{\Sigma}=m\,S_{\bar{c}}^{(K-1)}` is an unbiased estimator of the full covariance :math:`\Sigma`. +- If you have only the block means :math:`\{\hat{c}_k\}` and you know that each block is an average of :math:`m` iid draws, then :math:`\widehat{\Sigma}=m\,S_{\hat{c}}^{(K-1)}` is an unbiased estimator of the full covariance :math:`\Sigma`. - If blocks are not iid samples (e.g. they are clusters with internal structure), then multiplying block-mean covariance by $m$ is not @@ -244,11 +271,123 @@ Summary without individual :math:`S_k`. +Practical considerations +^^^^^^^^^^^^^^^^^^^^^^^^ + +`exp` and `pyEXP` give you the option of computing equal block samples +:math:`\hat{c}_k`, the full empirical covariance +:math:`\Sigma_{\mathrm{emp}}` and the :math:`\Sigma_k=S_k/n_k` for +each block. + +The main disadvantage to computing all three quantities is disk +space. Let us take a disk basis function expansion with +:math:`M_{max}=4, N_{max}=16` as an example. The full set of +coefficient vectors for double precision is 2048 bytes. The +covariance matrix requires 32768 bytes. If we choose :math:`K=100` +and have 1000 snapshots, our covariance file will be 3.5 GB! 95% of +this is covariance matrices. To save space, a better choice might be +to save the coefficient samples and only the full covariance matrix. +This would reduce our file to 0.2 GB. + +Our currently recommended and default strategy saving :math:`K=100` +samples :math:`\hat{c}_k` and the full covariance +:math:`\Sigma_{\mathrm{emp}}` only. We will described the YAML +configuration for these choices below. + .. _coefficient_SN: Signal-to-noise estimates ------------------------- +Covariance analysis leads to two alternative strategies that serve +different goals, as follows: + +1. The eigenvectors of the covariance matrix :math:`\Sigma` provides a + new basis that decorrelates the variance. Each eigenvalue + :math:`\lambda_j` describes the relative contribution of that new + basis element to the total variance. Let + :math:`F(p)\equiv\sum_{j=1}^p\lambda_j` with :math:`p\le d`. The + cumulative total may be written as :math:`C(p) = F(p)/F(d)`. The + denominator :math:`F(d)` simply provides normalization. One may trim + or truncate on :math:`p` after the cumulative total reaches some + value. A typical strategy is to trim at the value of :math:`p` + that maximizes the curvature of :math:`C(p)`, the so-called *knee* + method. One may then project the back to the original correlated + space using only the first :math:`p` vectors. + +2. A second strategy is the determination of the squared coefficient + magnitude to variance ratio, the *signal-to-noise* ratio, using the + diagonal of the covariance matrix as a proxy of the variance. + Alternatively, one can characterize the shape of the sub sample + coefficient distribution and scale the width by :math:`1/\sqrt{K}`, + following the central limit theorem. Let the variance of the + coefficient of order :math:`n` be + :math:`\mathrm{Var}(\hat{c}_n)`. We may define the signal to noise + ratio for coefficient :math:`n` as: + + .. math:: + S/N \equiv \frac{|\hat{c}_n|^2}{\mathrm{Var}(\hat{c}_n)}. + + We also know that the expectation for the estimate of the + coefficient is: + + .. math:: + + \mathbb{E}[|\hat{c}_n|^2] = |c_n|^2 + \mathrm{Var}(\hat{c}_n). + + This motivates trimming the coefficient series when :math:`S/N\sim + 1`. + + More formally, the functional loss is + + .. math:: + + \text{loss}_N = \sum_{n>N} |c_n|^2. + + For smoothing smooth functions, the run of :math:`\text{loss}_N` + will converge quickly with increasing values of :math:`N`. + However, if applied directly to :math:`|\hat{c}_n|^2`, this + estimator will diverge more slowly or may diverge from the exact + functional loss as :math:`\hat{c}_n` become dominated by noise. + This motivates the need for smoothing the :math:`\hat{c}_n` to zero + as :math:`n\rightarrow\infty`. Applying the trimming argument by + substituting the expression for :math:`\mathbb{E}[|\hat{c}_n|^2]` + the definition of :math:`\text{loss}_N`, we find that the estimator + of the loss from this trimming process is: + + .. math:: + + \widehat{\text{loss}}_N = \sum_{n>N}\left[|\hat{c}_n|^2 - + \mathrm{Var}(\hat{c}_n)\right]_+, + + where the notation :math:`[V]_+` indicates that the negative values + of :math:`V` are truncated to zero. In other words, choosing the + truncation value for :math:`N` to be at the point in the series + where :math:`S/N\sim1` minimizes the functional loss. This + suggests that trimming will produce better results, as expected. + + An alternative smoothing procedure follows from the mean-integrated + square error analysis of the estimator for the series + :math:`\{\hat{c}_n\}` and this leads to a smoothing weight of the + form: + + .. math:: + + \lambda_n = \frac{|\hat{c}_n|^2}{|\hat{c}_n|^2 + + \mathrm{Var}(\hat{c}_n)}. + + This bias--variance trade off is described by Silverman in nice + detail for kernel density estimation. In practice, this + prescription seems to perform worse than hard trimming: + + .. math:: + + \lambda_n = \begin{cases} 1 & \mathrm{if}\,|\hat{c}_n|^2 > + \mathrm{Var}(\hat{c}_n) \\ 0 & \mathrm{otherwise}. \end{cases} + + Regardless, some trimming algorithm desirable to prevent noise + affects from modifying the inferred field estimates. + .. _outsample: Using `OutSample` with `exp` @@ -256,9 +395,9 @@ Using `OutSample` with `exp` Covariance data generation within `exp` is a two step process: -1. The force method computes the subsample coefficient and covariance +1. The force method computes the sub sample coefficient and covariance estimates every `nint` steps. The configuration parameter `nint` - is zero by default; no subsamples or sample covariance is computed + is zero by default; no sub samples or sample covariance is computed by default. 2. You need to instantiate an `outsamp` output routine in the `Output` @@ -278,25 +417,25 @@ The `pyEXP` interface is split between the creation and the reading of HDF5 covariance files as follows: 1. Similar to the `exp` force methods, the `pyEXP.basis.BiorthBasis` - classes can be configured to produce subsample and covariance data + classes can be configured to produce sub sample and covariance data that get written to an HDF5 file. 2. The utility class `pyEXP.basis.BasisClasses.SubsampleCovariance` reads the HDF5 covariance files and returns sample information, - subsample coefficients and covariance information in `numpy` + sub sample coefficients and covariance information in `numpy` compatible format. The controlling configuration YAML keys are: - `pcavar: true` -- This enables the covariance computation -- `samplesz: 100` -- This sets the number of subsamples to `100`. The +- `samplesz: 100` -- This sets the number of sub samples to `100`. The value `1` is also valid. - `totalCovar: true` -- Computes a single covariance matrix for the entire ensemble. -- `fullCovar: true` -- Computes a covariance matrix for each subsample +- `fullCovar: true` -- Computes a covariance matrix for each sub sample Assume that you have created a basis instance using pyEXP named `basis`. After reading and generating coefficients for a particular @@ -315,53 +454,121 @@ covariance data for time `3.14`. You can read and HDF5 covariance file with the following code: -.. code-block: python +.. code-block:: python compname = 'halo' runtag = 'run1' covar = pyEXP.basis.CovarianceReader('coefcovar.{}.{}'.format(compname, runtag)) -A typical use case might be as follows: +A typical use case might be as follows. We want to estimate the +signal to noise for coefficients in a simulation. We read in +the HDF5 covariance file and estimate the empirical sample covariance +using the estimator from block means. Code to do this might look like +the following: + +.. code-block:: python + + times = covar['dark'].Times() + ntimes = len(times) + indx = 0 + time = times[indx] + # + # Get the sample data for this time. In this example, we use indx=0 + which is T=0.0 + # + counts, masses, coefs, covrs = covar['dark'].getCoefCovariance(time) + # + # The basic info for this block sampled data is: + # + numb = np.sum(masses) + mass = np.sum(counts) + nsamp = counts.shape[0] + # + # We now plot the signal to noise ratios for l=0,1,2 + # + for l in range(3): + for m in range(l+1): + L = basis['dark'].I(l, m) + # + sumCof = np.zeros((nmaxh), dtype=np.complex128) + for T in range(nsamp): + sumCof += coefs[T, L, :] + # The sub sample mean + sumCof /= nsamp + # Make the covariance matrix using np.cov. This will automatically normalize + varmat = np.zeros((nmaxh, nsamp), dtype=np.complex128) + for T in range(nsamp): + varmat[:, T] = coefs[T, L, :] + val = np.cov(varmat) + sn = np.zeros(val.shape[0]) + for i in range(val.shape[0]): + sn[i] = np.abs(sumCof[i])**2*nsamp/np.abs(val[i, i]) + plt.semilogy(sn, '-o', label='l={} m={}'.format(l, m)) + plt.legend() + plt.xlabel('n') + plt.ylabel('S/N') + plt.title('S/N with variance estimated from $\hat c_k$ at T={:4.2f}'.format(times[indx])) + plt.savefig('sn_k.png') + +This simulation is an mildly :math:`l=1` unstable truncated NFW-like +DM model. The plot below shows the S/N ratios for the coefficients +for the initial conditions and later, after the instability has developed. + +The result is: + +.. figure:: images/halo_sn.png + :figwidth: 100 % + :width: 100 % + :align: center + + Signal-to-noise ratios for harmonics :math:`l=0,1,2` for all + :math:`m` values at the initial time (left) and after the dipole + instability has developed (:math:`T=20`). At :math:`T=0`, only + the first term with :math:`l=m=n=0` has significance. At later + time, :math:`T=20`, the :math:`l=1` terms in addition to the + :math:`l=1` terms are significant. + +The following example illustrates the visualization of the full +covariance matrix for the same simulation. -.. code-block: python +.. code-block:: python # This gets the list of available snapshot times times = covar.Times() ntimes = len(times) - + # # As an example, let's look at the data for the final time - indx = -1 - time = times[indx] - - # Get all the subsample data for time - counts, masses, coefs, covrs = covar['dark'].getCoefCovariance(time) - - # The counts array contains the number of particles in each sample - - # The masses array contains the mass of the particles in each sample - - # The coefs is an ndarray of coefficient vectors indexed by sample - # number and harmonic order - - # The coefs is an ndarray of covariance matrices indexed by sample - # number and harmonic order - - # Following code shows how to visualize the covariance matrices - # using pcolormesh - - - numb = np.sum(counts) - mass = np.sum(masses) - nsamp = counts.shape[0] - - # For pcolormesh below - x = np.arange(0, nmaxh) - y = np.arange(0, nmaxh) - X, Y = np.meshgrid(x, y) - - for l in range(3): - for m in range(l+1): + indx = -1 + time = times[indx] + # + # Get all the sub sample data for time + counts, masses, coefs, covrs = covar['dark'].getCoefCovariance(time) + # + # The counts array contains the number of particles in each sample + # + # The masses array contains the mass of the particles in each sample + # + # The coefs is an ndarray of coefficient vectors indexed by sample + # number and harmonic order + # + # The coefs is an ndarray of covariance matrices indexed by sample + # number and harmonic order + # + # Following code shows how to visualize the covariance matrices + # using pcolormesh + # + numb = np.sum(counts) + mass = np.sum(masses) + nsamp = counts.shape[0] + # + # For pcolormesh below + x = np.arange(0, nmaxh) + y = np.arange(0, nmaxh) + X, Y = np.meshgrid(x, y) + # + for l in range(3): + for m in range(l+1): L = basis['dark'].I(l, m) sumCof = np.zeros((nmaxh), dtype=np.complex128) sumVar = np.zeros((nmaxh, nmaxh), dtype=np.complex128) @@ -375,3 +582,13 @@ A typical use case might be as follows: plt.title('Halo covariance for l={} m={} T={}'.format(l, m, times[-1])) plt.show() + +.. figure:: images/halo_covar.png + :figwidth: 80 % + :width: 80 % + :align: center + + Example output for :math:`l=m=2` for a simulation of a DM halo with + :math:`N=10^6` particles. The color scale shows the logarithm + base 10 of the absolute value of the covariance elements. 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Weinberg" Date: Thu, 25 Dec 2025 22:38:50 -0500 Subject: [PATCH 04/11] Add warning about bug fix to main --- topics/covariance.rst | 7 +++++++ 1 file changed, 7 insertions(+) diff --git a/topics/covariance.rst b/topics/covariance.rst index c5726fe..e0278dd 100644 --- a/topics/covariance.rst +++ b/topics/covariance.rst @@ -14,6 +14,13 @@ Coefficient significance analysis and recompile to use these features in the `exp` N-body code. The `pyEXP` interface is currently available in the `main` branch. +.. danger:: + + The current `main` branch has an error in the covariance + computation for the `Cylindrical` force in `pyEXP`. The fix for + this is in a recent PR. Users will need to merge that branch + manually or use the `OutSample` branch for correctness. + Overview -------- From ae4e95adf8ad1c7da3e7062c9ee709087e03f72b Mon Sep 17 00:00:00 2001 From: "Martin D. 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zxXICAA9gUjey|8!Z5a;u9>vd3!H|B@rg7e$)n~nVQ~Sy&z;o)<2Ua@F&06dKs3p%Y zZx|gNHQAUbX&D?;5fBs%*_x>ht#(>@3#zGW*L6Pvx2088RTcd0+c)w-A1_5Lg5bFbO8*ysXQhz?nIc)> zRZH&rDVUB1pq<`sUyo}l zJ8H&8M?(N+47Q)CLDjUB*=XgBR_Kf$7Gl~IILOKdpti2IfiINN)Qp{|uuBRK4<`rB zA}=p5cO9*iGdDMH86DLCjytp6Y2xYJQ|v`vTQn5A$Q1VTYs%9txDU`J_7%E(sxOH0 zK&AFiml&*2e{%IGl?J(!f9E_XjB@q(f9Hz-jluleW$sL1$jFR`(F&sYr)rM+@&0#} z1t9Oto&Vy04T#C)m6!$CWelneYZa@zyZfb+Q!9Y*J77gFhVjofjTPula-kaPYW=e- z>h*3r#V+!44(V4}$^U?w5E~KFRftmPoc(}|2Iggk+xBccw<%U2I4crB1U>?A;V8N# zMlRCHtHiiH&_0Y^g#o<*So*KPBaB3W9_G^LD4IM26y)xW3RZ=o^@D%@8^vf#E&s)# zqgwy-%l@(Ff2Gk^;y{gkXETTmgZHf~>Mkp_GyD{{TXD(ry3% From de7f6b9bc8e706fd1bc6593599c6dac8f1b0a8a1 Mon Sep 17 00:00:00 2001 From: Martin Weinberg Date: Sun, 28 Dec 2025 11:30:33 -0500 Subject: [PATCH 06/11] Update index.rst Co-authored-by: Copilot <175728472+Copilot@users.noreply.github.com> --- index.rst | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/index.rst b/index.rst index c92aaae..2f75f62 100644 --- a/index.rst +++ b/index.rst @@ -168,7 +168,7 @@ How to solve specific problems :doc:`topics/visualizing-fields` Learn how to convert your coefficients to physical fields and - visualize them with Python and pyEXP + visualize them with Python and pyEXP :doc:`topics/make-movies` Make some movies from your fields in Python From b8da92dab5c1f5bd37c6304c7a2ad8d2f36bc8b2 Mon Sep 17 00:00:00 2001 From: Martin Weinberg Date: Sun, 28 Dec 2025 11:30:48 -0500 Subject: [PATCH 07/11] Update index.rst Co-authored-by: Copilot <175728472+Copilot@users.noreply.github.com> --- index.rst | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/index.rst b/index.rst index 2f75f62..dcb5611 100644 --- a/index.rst +++ b/index.rst @@ -123,7 +123,7 @@ mathematics used in EXP. :doc:`topics/howtosim` A guide for running EXP simulations -:doc:`topics/covariance +:doc:`topics/covariance` How to think about and compute significance of coefficients using exp and pyEXP From 262aac9e32b865b1045b1e690ed42167c2f2e39e Mon Sep 17 00:00:00 2001 From: "Martin D. Weinberg" Date: Mon, 29 Dec 2025 17:29:02 -0500 Subject: [PATCH 08/11] Some minor copy editing --- topics/covariance.rst | 44 +++++++++++++++++++++++++------------------ 1 file changed, 26 insertions(+), 18 deletions(-) diff --git a/topics/covariance.rst b/topics/covariance.rst index e0278dd..bdd0428 100644 --- a/topics/covariance.rst +++ b/topics/covariance.rst @@ -26,26 +26,26 @@ Overview -------- `exp` and `pyEXP` compute both the empirical covariance matrices and -compute coefficients in *partitions* or *batches* that may be used to +the coefficients in *partitions* or *batches* that may be used to estimate statistical consistency of coefficients. Basis function expansions estimate the underlying fields from the particles that sample their gravitational fields that generate the -particle orbits. The primary goal of analyses is improving the field -estimates by characterizing their significance. We do this in two -ways: +particle orbits. The primary goal of these analyses is the +improvement of field estimates by characterizing their significance. +We do this in two ways: -1. The variation represented by the BFE is spread across basis - functions. In other words, the independently varying signals not - isolated to specific basis functions. We may use covariance - analysis, such as ideas from Principal Component Analysis (PCA), to - empirically determine a basis which separates the initially - spatially correlated signals. +1. The dynamically induced variation in the gravitational field is + spread across basis functions. In other words, the independently + varying signals are not isolated to specific subset of basis + functions. We may use covariance analysis, such as ideas from + Principal Component Analysis (PCA), to empirically determine a + basis which separates the initially spatially correlated signals. 2. One may treat the significance of the underlying fields represented by the expansion as an estimation problem and attempt to analyze - the significance of each coefficient using methods from probability - and statistics. + the significance of each coefficient using established methods from + probability and statistics. The next section introduces the terminology and concepts from sampling theory necessary to address each of these two goals. We then move on @@ -62,7 +62,7 @@ Notation Let :math:`\{c_i\}_{i=1}^N\subset\mathbb{R}^d` be the contribution to the coefficient vector from Particle :math:`i`. Specifically, if the -potential basis function is :math:`\Phi_k(\cdot)`, then +potential basis functions are :math:`\Phi_k(\cdot)`, then :math:`c_i=\{\Phi_1(\mathbf{x}_i), \Phi_1(\mathbf{x}_i), \ldots, \Phi_d(\mathbf{x}_i)\}` where :math:`\mathbf{x}_i` is the position vector for Particle :math:`i`. Denote the empirical mean for the @@ -88,7 +88,15 @@ N`. For each block :math:`k` define \qquad S_k \;=\; \sum_{i\in\text{block }k} (c_i-\hat{c}_k)(c_i-\hat{c}_k)^{\!\top}, -where :math:`S_k` is the within‑block scatter. +where :math:`S_k` is the within‑block scatter. The may also write the +covariance matrix for each block and relate that to :math`S_k` as +follows: + +.. math:: + + \Sigma_k = \frac{1}{n_k}\sum_{i\in\text{block }k} + (c_i - \hat{c}_k) (c_i - \hat{c}_k)^{\!\top} = \frac{S_k}{n_k}. + Overall mean and within and between blocks ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ @@ -232,7 +240,7 @@ unbiased sample-style form for these estimates: is unbiased for :math:`\Sigma` under the iid block model. -Relation of the Finite-sample to the full covariance +Relation of the finite-sample to the full covariance ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Under the iid model, the expected within aggregate satisfies @@ -262,9 +270,9 @@ Summary - If you have only the block means :math:`\{\hat{c}_k\}` and you know that each block is an average of :math:`m` iid draws, then :math:`\widehat{\Sigma}=m\,S_{\hat{c}}^{(K-1)}` is an unbiased estimator of the full covariance :math:`\Sigma`. - If blocks are not iid samples (e.g. they are clusters with internal - structure), then multiplying block-mean covariance by $m$ is not - generally valid; additional modeling or within-block information is - required. + structure), then multiplying block-mean covariance by :math:`m` is + not generally valid; additional modeling or within-block information + is required. - If you have both the full empirical covariance :math:`\Sigma_{\mathrm{emp}}` and the block means, you can compute From 4eff92eacab49347e3ddd5c03d3a82e3363c54db Mon Sep 17 00:00:00 2001 From: "Martin D. Weinberg" Date: Mon, 29 Dec 2025 19:45:09 -0500 Subject: [PATCH 09/11] A few more typo fixes --- topics/covariance.rst | 42 +++++++++++++++++++++++++----------------- 1 file changed, 25 insertions(+), 17 deletions(-) diff --git a/topics/covariance.rst b/topics/covariance.rst index bdd0428..6368cbc 100644 --- a/topics/covariance.rst +++ b/topics/covariance.rst @@ -88,13 +88,13 @@ N`. For each block :math:`k` define \qquad S_k \;=\; \sum_{i\in\text{block }k} (c_i-\hat{c}_k)(c_i-\hat{c}_k)^{\!\top}, -where :math:`S_k` is the within‑block scatter. The may also write the -covariance matrix for each block and relate that to :math`S_k` as +where :math:`S_k` is the within‑block scatter. We may also write the +covariance matrix for each block and relate that to :math:`S_k` as follows: .. math:: - \Sigma_k = \frac{1}{n_k}\sum_{i\in\text{block }k} + \Sigma_k \equiv \frac{1}{n_k}\sum_{i\in\text{block }k} (c_i - \hat{c}_k) (c_i - \hat{c}_k)^{\!\top} = \frac{S_k}{n_k}. @@ -114,7 +114,7 @@ The total scatter, proportional to the covariance matrix, is S \;=\; \sum_{i=1}^N (c_i-\hat{c})(c_i-\hat{c})^{\!\top}, -and the exact decomposition is +and the exact decomposition in block quantities is .. math:: @@ -122,8 +122,8 @@ and the exact decomposition is \underbrace{\sum_{k=1}^K n_k(\hat{c}_k-\hat{c})(\hat{c}_k-\hat{c})^{\!\top}}_{B}. -The two terms, :math:`W` and :math:`B`, are the *in block* and -*between block* scatter, respectively. Then, the empirical population +The two terms, :math:`W` and :math:`B`, are the *in-block* and +*between-block* scatter, respectively. Then, the empirical population covariance is .. math:: @@ -136,21 +136,23 @@ covariance is The *between block* term :math:`B/N` can be computed from the :math:`\{\hat{c}_k\}` and :math:`\{n_k\}` alone; the within aggregate -:math:`W` requires the :math:`S_k` or additional assumptions. +:math:`W` requires the scatter matrices :math:`S_k` or additional +assumptions. Special case: equal-size blocks ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ -Assume each block has the same size :math:`m`, so :math:`n_k = m` for -all :math:`k`, and :math:`N = Km`. Define the block mean covariance -as +`pyEXP` and `exp` partion samples into equal-size blocks by default. +This leads to some additional convenient properties. Assume each +block has the same size :math:`m`, so :math:`n_k = m` for all +:math:`k`, and :math:`N = Km`. Define the block mean covariance as .. math:: \Sigma_{\hat{c}} \;=\; \frac{1}{K}\sum_{k=1}^K (\hat{c}_k-\hat{c})(\hat{c}_k-\hat{c})^{\!\top}. -Then the *between scatter* contribution is +Then, the *between-block* scatter contribution is .. math:: @@ -184,25 +186,24 @@ from the same distribution with mean :math:`\hat{c}` and covariance .. math:: \mathbb{E}[\hat{c}_k] = \hat{c},\qquad - \operatorname{Cov}(\hat{c}_k) = \frac{\Sigma}{m}. + \mathbb{E}\!\big[ \Sigma_{\hat{c}} \big] = + \mathbb{E}\!\big[ \operatorname{Cov}(\hat{c}_k) \big] = \frac{\Sigma}{m}. This is generally true for `exp` simulations which do not sort particles within their component and can be made true for any simulation by selecting particles from the entire particle ensemble -randomly. Consequently, +randomly. .. math:: - \mathbb{E}\!\big[ \Sigma_{\hat{c}} \big] \;=\; - \frac{\Sigma}{m}. Therefore, an estimator of :math:`\Sigma` based on the block means is .. math:: - \widehat{\Sigma}_{\hat{c}} \;=\; m\,\Sigma_{\hat{c}}, + \widehat{\Sigma} = m\,\Sigma_{\hat{c}}, -where :math:`S_{\hat{c}}` is a chosen empirical covariance of the +where :math:`\Sigma_{\hat{c}}` is the empirical covariance of the block means. One can choose either the natural population-style or unbiased sample-style form for these estimates: @@ -285,6 +286,13 @@ Summary and therefore recover the sum of the within-block scatters even without individual :math:`S_k`. +- An analysis of the :math:`K` block covariance :math:`\Sigma_k` + estimates may be used to diagnose the consistency the total + estimate. For example, one might use an eigen analysis to determine + how much of the variance in one block is captured another block, + such as Krzanowski common subspaces analysis. This is not currently + offered directly by `pyEXP`. + Practical considerations ^^^^^^^^^^^^^^^^^^^^^^^^ From 24bd781007b05ae4c3dabd30cc4a25fcaff75c78 Mon Sep 17 00:00:00 2001 From: "Martin D. Weinberg" Date: Mon, 29 Dec 2025 20:24:32 -0500 Subject: [PATCH 10/11] Describe axes in caption --- topics/covariance.rst | 6 ++++-- 1 file changed, 4 insertions(+), 2 deletions(-) diff --git a/topics/covariance.rst b/topics/covariance.rst index 6368cbc..b6e4d82 100644 --- a/topics/covariance.rst +++ b/topics/covariance.rst @@ -611,7 +611,9 @@ covariance matrix for the same simulation. :width: 80 % :align: center - Example output for :math:`l=m=2` for a simulation of a DM halo with - :math:`N=10^6` particles. The color scale shows the logarithm + Example covariance output :math:`\Sigma_{n_1\,n_2}` for + :math:`l=m=2` for a simulation of a DM halo with :math:`N=10^6` + particles. The horizontal and vertical axes show :math:`n_1` and + :math:`n_2`, respectively. The color scale shows the logarithm base 10 of the absolute value of the covariance elements. The diagonal dominance is expected. From d0a09fafe0828a6a3985dbad50607a09ba84c01a Mon Sep 17 00:00:00 2001 From: "Martin D. Weinberg" Date: Tue, 30 Dec 2025 10:01:17 -0500 Subject: [PATCH 11/11] Remove warning message --- topics/covariance.rst | 69 ++++++++++++++++++++++++------------------- 1 file changed, 39 insertions(+), 30 deletions(-) diff --git a/topics/covariance.rst b/topics/covariance.rst index b6e4d82..b95d9a9 100644 --- a/topics/covariance.rst +++ b/topics/covariance.rst @@ -13,13 +13,6 @@ Coefficient significance analysis `OutSample` branch. You will need to ``git checkout OutSample`` and recompile to use these features in the `exp` N-body code. The `pyEXP` interface is currently available in the `main` branch. - -.. danger:: - - The current `main` branch has an error in the covariance - computation for the `Cylindrical` force in `pyEXP`. The fix for - this is in a recent PR. Users will need to merge that branch - manually or use the `OutSample` branch for correctness. Overview @@ -27,25 +20,26 @@ Overview `exp` and `pyEXP` compute both the empirical covariance matrices and the coefficients in *partitions* or *batches* that may be used to -estimate statistical consistency of coefficients. +estimate statistical properties and consistency of coefficients. Basis function expansions estimate the underlying fields from the particles that sample their gravitational fields that generate the particle orbits. The primary goal of these analyses is the -improvement of field estimates by characterizing their significance. -We do this in two ways: +improvement of field estimates by characterizing their physical and +statistical significance. We do this in two ways: 1. The dynamically induced variation in the gravitational field is - spread across basis functions. In other words, the independently - varying signals are not isolated to specific subset of basis - functions. We may use covariance analysis, such as ideas from - Principal Component Analysis (PCA), to empirically determine a - basis which separates the initially spatially correlated signals. + spread across all of the basis functions. In other words, the + independently varying signals induced by gravity are not isolated + to specific subset of basis functions. We may use covariance + analysis, such as ideas from Principal Component Analysis (PCA), to + empirically determine a basis that separates the initially + spatially correlated signals. 2. One may treat the significance of the underlying fields represented by the expansion as an estimation problem and attempt to analyze - the significance of each coefficient using established methods from - probability and statistics. + the statistical significance of each coefficient using established + methods. The next section introduces the terminology and concepts from sampling theory necessary to address each of these two goals. We then move on @@ -142,7 +136,7 @@ assumptions. Special case: equal-size blocks ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ -`pyEXP` and `exp` partion samples into equal-size blocks by default. +`pyEXP` and `exp` partition samples into equal-size blocks by default. This leads to some additional convenient properties. Assume each block has the same size :math:`m`, so :math:`n_k = m` for all :math:`k`, and :math:`N = Km`. Define the block mean covariance as @@ -173,7 +167,7 @@ Putting all of this together, we have .. math:: - \Sigma \;=\; \frac{W}{N} + S_{\hat{c}}^{\mathrm{pop}}, + \Sigma \;=\; \frac{W}{N} + \Sigma_{\hat{c}}, \qquad\text{where } W=\sum_{k=1}^K S_k. IID block-sampling model @@ -181,7 +175,8 @@ IID block-sampling model If additionally each block is formed by :math:`m` independent draws from the same distribution with mean :math:`\hat{c}` and covariance -:math:`\Sigma` (block independent), then +:math:`\Sigma` (the sampling procedure is block independent, in other +words), then .. math:: @@ -190,14 +185,13 @@ from the same distribution with mean :math:`\hat{c}` and covariance \mathbb{E}\!\big[ \operatorname{Cov}(\hat{c}_k) \big] = \frac{\Sigma}{m}. This is generally true for `exp` simulations which do not sort -particles within their component and can be made true for any +particles within their components and can be made true for any simulation by selecting particles from the entire particle ensemble -randomly. - -.. math:: - +randomly. Some n-body codes **do** sort their particles spatially, so +make sure to check. -Therefore, an estimator of :math:`\Sigma` based on the block means is +Assuming independence, an estimator of :math:`\Sigma` based on the +block means is .. math:: @@ -268,7 +262,10 @@ verifying unbiasedness of the global empirical covariance. Summary ^^^^^^^ -- If you have only the block means :math:`\{\hat{c}_k\}` and you know that each block is an average of :math:`m` iid draws, then :math:`\widehat{\Sigma}=m\,S_{\hat{c}}^{(K-1)}` is an unbiased estimator of the full covariance :math:`\Sigma`. +- If you have only the block means :math:`\{\hat{c}_k\}` and you know + that each block is an average of :math:`m` iid draws, then + :math:`\widehat{\Sigma}=m\,S_{\hat{c}}^{(K-1)}` is an unbiased + estimator of the full covariance :math:`\Sigma`. - If blocks are not iid samples (e.g. they are clusters with internal structure), then multiplying block-mean covariance by :math:`m` is @@ -293,6 +290,15 @@ Summary such as Krzanowski common subspaces analysis. This is not currently offered directly by `pyEXP`. +- Similarly, the distribution of :math:`\hat{c}_k` contains more + information about the true variance than the second-moment + covariance matrix. While the covariance matrix is sufficient for + characterization as :math:`N\rightarrow\infty` by the central limit + theorem, some poorly determined high-order coefficients may be far + from that asymptotic regime. In this case, empirical widths + measured from the distribution of :math:`\hat{c}_k` will more + reliably capture the truth. + Practical considerations ^^^^^^^^^^^^^^^^^^^^^^^^ @@ -312,10 +318,13 @@ this is covariance matrices. To save space, a better choice might be to save the coefficient samples and only the full covariance matrix. This would reduce our file to 0.2 GB. -Our currently recommended and default strategy saving :math:`K=100` +Our currently recommended and default strategy saving :math:`K\sim100` samples :math:`\hat{c}_k` and the full covariance -:math:`\Sigma_{\mathrm{emp}}` only. We will described the YAML -configuration for these choices below. +:math:`\Sigma_{\mathrm{emp}}` only. Larger values of :math:`K` may be +valuable in some cases. As described above, quantile estimates based +on the distribution of :math:`\Sigma_k=S_k/n_k` may provide better +information than :math:`\Sigma_{\mathrm{emp}}` alone. We will +described the YAML configuration for these choices below. .. _coefficient_SN: