randomized svd draft #3008

hanbin973 · 2024-10-03T18:32:08Z

Description

A draft of randomized principal component analysis (PCA) using the TreeSequence.genetic_relatedness_vector. The implementation contains spicy.sparse which should eventually be removed.
This part of the code is only used when collapsing a #sample * #sample GRM into a #individual * #individual matrix.
Therefore, it will not be difficult to replace with pure numpy.

The API was partially taken from scikit-learn.

To add some details, iterated_power is the number of power iterations in the range finder in the randomized algorithm. The error of SVD decreases exponentially as a function of this number.
The effect of power iteration is profound when the eigen spectrum of the matrix decays slowly, which seems to be the case of tree sequence GRMs in my experience.

indices specifies the individuals to be included in the PCA, although decreasing the number of individuals does not meaningfully reduce the amount of computation.

hanbin973 · 2024-10-03T18:34:34Z

@petrelharp Here's the code.

codecov · 2024-10-03T18:35:31Z

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All modified and coverable lines are covered by tests ✅

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python/tskit/trees.py

petrelharp · 2024-10-04T01:45:27Z

This looks great! Very elegant. I think probably we ought to include a samples argument, though? For consistency, but also since the tree sequence represents phased data, and so it's actually informative to look at the PCs of maternally- and paternally-inherited chromosomes separately.

So, how about the signature is like

def pca(samples=None, individuals=None, ...)

and:

the default is equivalent to samples=ts.samples(), individuals=None
you can't have both samples and individuals specified
if individuals is a list of individual IDs then it does as in the code currently
otherwise, is just skips the "sum over individuals" step

Note that we could be getting PCs for non-sample nodes (since individual's nodes need not be samples); I haven't thought through whether the values you get are correct or informative. My guess is that maybe they are? But we need a "user beware" note for this?

python/tskit/trees.py

petrelharp · 2024-10-07T15:41:25Z

Ah, sorry - one more thing - does this work with windows? (It looks like not?)

I think the way to do the windows would be something like

drop_windows = windows is None
if drop_windows:
    windows = [0, self.sequence_length]

# then do stuff; with these windows genetic_relatedness will always return an array where the first dimension is "window";
# so you can operate on each slice separately

if drop_windows:
    # get rid of the first dimension in the output

Basically - get it to work in the case where windows are specified (ie not None) and then we can get it to have the right behavior.

hanbin973 · 2024-10-08T04:00:03Z

A simple test case for the windows feature.

demography = msprime.Demography()
demography.add_population(name="A", initial_size=5_000)
demography.add_population(name="B", initial_size=5_000)
demography.add_population(name="C", initial_size=1_000)
demography.add_population_split(time=1000, derived=["A", "B"], ancestral="C")
ts = msprime.sim_ancestry(
    samples={"A": 500, "B": 500},
    sequence_length=1e6,
    recombination_rate=3e-8,
    demography=demography, 
    random_seed=12)
seq_length = ts.sequence_length

U, _ = ts.pca(individuals=np.asarray([i.id for i in ts.individuals()]), iterated_power=5, random_seed=1, windows=[0, seq_length/2, seq_length])
U0, _ = ts.pca(individuals=np.asarray([i.id for i in ts.individuals()]), iterated_power=5, random_seed=1, windows=[0, seq_length/2])
U1, _ = ts.pca(individuals=np.asarray([i.id for i in ts.individuals()]), iterated_power=5, random_seed=1, windows=[seq_length/2, seq_length])

idx = 0 # idx is the idx-th principal component
# correlation instead of allclose because PCA is rotation symmetric
np.corrcoef(U[0][:,idx], U0[:,idx]), np.corrcoef(U[1][:,idx], U1[:,idx])

Because of the randomness of the algo, the correlation is not exactly 1, although it's nearly 1 like 0.99995623-ish.

hanbin973 · 2024-10-11T18:33:00Z

I just noticed that centre doesn't work with nodes option. The new commit fixed this problem.

hanbin973 · 2024-10-11T18:49:51Z

Check results for two windows.

demography = msprime.Demography()
demography.add_population(name="A", initial_size=5_000)
demography.add_population(name="B", initial_size=5_000)
demography.add_population(name="C", initial_size=1_000)
demography.add_population_split(time=1000, derived=["A", "B"], ancestral="C")
seq_length =1e6
ts = msprime.sim_ancestry(
    samples={"A": 500, "B": 500},
    sequence_length=seq_length,
    recombination_rate=3e-8,
    demography=demography, 
    random_seed=12)

# for individuals
U, _ = ts.pca(individuals=np.asarray([i.id for i in ts.individuals()]), iterated_power=5, random_seed=1, windows=[0, seq_length/2, seq_length])
U0, _ = ts.pca(individuals=np.asarray([i.id for i in ts.individuals()]), iterated_power=5, random_seed=1, windows=[0, seq_length/2])
U1, _ = ts.pca(individuals=np.asarray([i.id for i in ts.individuals()]), iterated_power=5, random_seed=1, windows=[seq_length/2, seq_length])

idx = 0 # idx is the idx-th principal component
# correlation instead of allclose because PCA is rotation symmetric
np.corrcoef(U[0][:,idx], U0[0][:,idx]), np.corrcoef(U[1][:,idx], U1[0][:,idx])

# for nodes
U, _ = ts.pca(iterated_power=5, random_seed=1, windows=[0, seq_length/2, seq_length])
U0, _ = ts.pca(iterated_power=5, random_seed=1, windows=[0, seq_length/2])
U1, _ = ts.pca(iterated_power=5, random_seed=1, windows=[seq_length/2, seq_length])

idx = 0 # idx is the idx-th principal component
# correlation instead of allclose because PCA is rotation symmetric
np.corrcoef(U[0][:,idx], U0[0][:,idx]), np.corrcoef(U[1][:,idx], U1[0][:,idx])

python/tskit/trees.py

petrelharp · 2024-11-16T17:30:15Z

I made a pass through the docs. We need to add time_windows to the tests still, and see what's going on with the CI.

Co-authored-by: Peter Ralph <[email protected]>

jeromekelleher

Looks good to me. I think we need to tidy up the lint and get tests passing next so we can see how coverage is doing?

jeromekelleher · 2024-11-18T13:29:22Z

python/tskit/trees.py

+        samples, sample_individuals = (
+            ij[:, 0],
+            ij[:, 1],
+        )  # sample node index, individual of those nodes


Putting comments at the end of lines is causing them to get broken by Black. Better to put the comments on the line immediately above.

jeromekelleher · 2024-11-18T13:35:58Z

python/tskit/trees.py

+    The principal component factors. Columns are orthogonal, with one entry per sample
+    or individual (see :meth:`pca <.TreeSequence.pca>`).
+    """
+    eigen_values: np.ndarray


eigenvalues is one word, isn't it?

python/tests/test_relatedness_vector.py

petrelharp · 2025-01-08T23:02:56Z

python/tests/test_relatedness_vector.py

+        if np.allclose(x, 0):
+            r = 1.0
+        else:
+            r = np.mean(x / y)


This is not right, as here we want r to be +/-1, I think?

petrelharp · 2025-01-08T23:07:46Z

It looks like the things to do here are:

get the tests working (right now they fail with FAILED tests/test_relatedness_vector.py::TestPCA::test_bad_windows - TypeError: pca() got an unexpected keyword argument 'n_components'
either remove for now the time_windows argument or write tests for it
write tests that exercise the individuals argument (or remove it)
write a test that uses range_sketch
write tests that exercise iterated_power and num_oversamples: probably, just something that checks whether setting these to bigger numbers still gets us (nearly) the same answer

jeromekelleher · 2025-03-03T15:10:51Z

Bumping this one - we want to get ts.pca implemented and released as soon as we can. What's left to do here @hanbin973? Can we help with anything to get it over the line?

hanbin973 · 2025-03-03T15:30:54Z

Bumping this one - we want to get ts.pca implemented and released as soon as we can. What's left to do here @hanbin973? Can we help with anything to get it over the line?

It's just the test codes that are missing. We have to make the tests pass. I told @petrelharp that I will work on it, but well it didn't go as planned :(

jeromekelleher · 2025-03-07T12:39:52Z

I don't think this works if we use it on a tree sequence where the samples aren't 0,...,n. Trying this out on the SARS-CoV-2 data, I got:

/tmp/ipykernel_620501/3297030094.py in ?()
----> 1 pca = ts.pca(4, samples=ts.samples())

~/.local/lib/python3.10/site-packages/tskit/trees.py in ?(self, num_components, windows, samples, individuals, time_window, mode, centre, iterated_power, num_oversamples, random_seed, range_sketch)
   8856                 else:
   8857                     low = _f_low(arr=x, indices=indices, mode=mode, centre=centre, windows=this_window)
   8858                     return high - low
   8859 
-> 8860             U[i], D[i], _, Q[i], E[i] = _rand_svd(
   8861                 operator=_G,
   8862                 operator_dim=dim,
   8863                 rank=num_components,

~/.local/lib/python3.10/site-packages/tskit/trees.py in ?(operator, operator_dim, rank, depth, num_vectors, rng, range_sketch)
   8803             """
   8804             Algorithm 8 in https://arxiv.org/pdf/2002.01387
   8805             """
   8806             assert num_vectors >= rank > 0
-> 8807             Q = _rand_pow_range_finder(
   8808                 operator, operator_dim, num_vectors, depth, num_vectors, rng, range_sketch
   8809             )
   8810             C = operator(Q).T

~/.local/lib/python3.10/site-packages/tskit/trees.py in ?(operator, operator_dim, rank, depth, num_vectors, rng, range_sketch)
   8786             else:
   8787                 Q = range_sketch
   8788             for _ in range(depth):
   8789                 Q = np.linalg.qr(Q).Q
-> 8790                 Q = operator(Q)
   8791             Q = np.linalg.qr(Q).Q
   8792             return Q[:, :rank]

~/.local/lib/python3.10/site-packages/tskit/trees.py in ?(x)
   8852             def _G(x):
-> 8853                 high = _f_high(arr=x, indices=indices, mode=mode, centre=centre, windows=this_window)
   8854                 if time_window is None:
   8855                     return high
   8856                 else:

~/.local/lib/python3.10/site-packages/tskit/trees.py in ?(self, arr, indices, mode, centre, windows)
   8610         centre: bool = True,
   8611         windows = None,
   8612         ) -> np.ndarray:
   8613         x = arr - arr.mean(axis=0) if centre else arr
-> 8614         x = self._expand_indices(x, indices)
   8615         x = self.genetic_relatedness_vector(
   8616             W=x, windows=windows, mode=mode, centre=False, nodes=indices,
   8617         )[0]

~/.local/lib/python3.10/site-packages/tskit/trees.py in ?(self, x, indices)
   8596         x: np.ndarray,
   8597         indices: np.ndarray
   8598         ) -> np.ndarray:
   8599         y = np.zeros((self.num_samples, x.shape[1]))
-> 8600         y[indices] = x
   8601 
   8602         return y

IndexError: index 2482157 is out of bounds for axis 0 with size 2482157
[6]:

but if I first simplify (ts = ts.simplify()) so that the samples are 0 to n it runs fine. Just needs a simple test for this case I'd imageine.

jeromekelleher

Few minor comments on return value interface

jeromekelleher · 2025-03-07T13:20:05Z

python/tskit/trees.py

+        :param np.ndarray range_sketch: Sketch matrix for each window. Default is None.
+        :return: A :class:`.PCAResult` object, containing estimated principal components,
+            eigenvalues, and other information.
+            The principal component loadings are in U


This last bit is out of date now.

I'm always confused by what's a loading and what's a loading score (or factor). U is should be the score. Will work on it.

jeromekelleher · 2025-03-07T13:22:01Z

python/tskit/trees.py

+
+
+    """
+    factors: np.ndarray


We need to define the dimensions here. Currently the first dimension is samples/individuals and the second is the num_componnents. Is there a strong reason for doing it this way? I expected it to be (num_components, num_samples) as you usually want to access all the values for a given component together?

Aha, it's following scikit learn. That's an excellent guide to follow - let's just do what scikit learn does and make our API as compatible with them as possible? We should document this as a stated goal.

jeromekelleher · 2025-03-07T14:06:47Z

I successfully ran this on a 2.5M sample SARS-CoV-2 ARG. Took about 30 seconds, and seemed to converge OK (at least from my cursory check using the recommended approach). A summary of the results is here: jeromekelleher/sc2ts-paper#372

petrelharp · 2025-03-10T14:00:35Z

python/tskit/trees.py

@@ -8779,17 +8804,16 @@ def _rand_pow_range_finder(
            """
            Algorithm 9 in https://arxiv.org/pdf/2002.01387
            """
-            assert num_vectors >= rank > 0, "num_vectors should be larger than rank"
+            assert num_vectors >= rank > 0, "num_vectors should not be smaller than rank"


I changed the words to match the math; is this right?

Good point. geq/leq are tricky.

petrelharp · 2025-03-10T14:01:32Z

python/tskit/trees.py

            else:
                Q = range_sketch
            for _ in range(depth):
                Q = np.linalg.qr(Q).Q
                Q = operator(Q)
            Q = np.linalg.qr(Q).Q
-            return Q[:, :rank]
+            return Q


To pass Q back into the method (losslessly) we need to pass the whole thing, not just the top rank columns.

petrelharp · 2025-03-10T14:10:55Z

Okay - I've put in a bunch of the testing code for correct arguments, etcetera. Still TODO:

testing for the error bounds
figure out why individuals is not passing tests (see below)

One issue I've turned up along the way is that the individuals mode I think was not agreeing with our definition of genetic_relatedness: as described in the relatedness paper, relatedness between two sample sets is the average of the relatedness between pairwise combinations of samples. This is just a factor of 4 for diploids - but that's a discrepancy that will confuse people - and actually makes a difference for mixed ploidy (eg sex chromosomes). I think I've sorted this out correctly.

Have we convinced ourselves that the default iterated_power is a good value? I have a bit of a hard time figuring out what "good" means - like, what tolerance do we need these eigenvectors to?

I also refactored the code to randomly generate range_sketch outside of the loop over windows, because this makes (a) the testing-for-correct-inputs code up top cleaner, and (b) the _rand_pow_range_finder and _rand_svd functions simpler. This has a memory downside of having to have range_sketch for all windows, not just one -- but, this is something we're returning anyhow, so is not a big deal.

I added an individuals argument to the testing implementation, but this isn't matching; I've stared at it a bunch and don't know why. Oddly, ts.pca( samples=x) and ts.pca( individuals=x) match for haploid individuals, but these don't match to the testing code.

petrelharp · 2025-03-10T14:16:53Z

Also - I wonder if a better name for iterated_power would be power_iterations?

Oh, I see that scikit-learn says iterated_power (and also n_components instead of num_components). Hm. I think that being compatible with them is good, but copying their bad API choices is not requred? (Well, n_components is a fine API, but it isn't consistent with the rest of our API?, which is num_X for things?)

jeromekelleher · 2025-03-10T15:02:44Z

I'd vote for compatible with scikit second, compatible with our other APIs first.

hanbin973 · 2025-03-10T17:30:26Z

Okay - I've put in a bunch of the testing code for correct arguments, etcetera. Still TODO:

testing for the error bounds

figure out why individuals is not passing tests (see below)

One issue I've turned up along the way is that the individuals mode I think was not agreeing with our definition of genetic_relatedness: as described in the relatedness paper, relatedness between two sample sets is the average of the relatedness between pairwise combinations of samples. This is just a factor of 4 for diploids - but that's a discrepancy that will confuse people - and actually makes a difference for mixed ploidy (eg sex chromosomes). I think I've sorted this out correctly.

Have we convinced ourselves that the default iterated_power is a good value? I have a bit of a hard time figuring out what "good" means - like, what tolerance do we need these eigenvectors to?

I also refactored the code to randomly generate range_sketch outside of the loop over windows, because this makes (a) the testing-for-correct-inputs code up top cleaner, and (b) the _rand_pow_range_finder and _rand_svd functions simpler. This has a memory downside of having to have range_sketch for all windows, not just one -- but, this is something we're returning anyhow, so is not a big deal.

I added an individuals argument to the testing implementation, but this isn't matching; I've stared at it a bunch and don't know why. Oddly, ts.pca( samples=x) and ts.pca( individuals=x) match for haploid individuals, but these don't match to the testing code.

What is the exact failure about? Is it the eigenvalues not matching or the factor scores not matching? In terms of the factor scores, I think it's better to compare U.T @ U to each other so that we avoid rotation and sign problems. Shall I change the code?

petrelharp · 2025-03-12T00:20:23Z

The failure is that the two answers are Definitely Not the Same. Clicking on the "Tests / ..." link above, or running locally, we get

FAILED tests/test_relatedness_vector.py::TestPCA::test_individuals[1-0-True] - AssertionError: 
Not equal to tolerance rtol=1e-07, atol=1e-08

Mismatched elements: 5 / 5 (100%)
Max absolute difference among violations: 5733.50077407
Max relative difference among violations: 0.60127089
 ACTUAL: array([3802.135878, 3157.318008, 2779.806231, 2273.337602, 1637.086583])
 DESIRED: array([9535.636652, 5611.718265, 4424.8184  , 4081.358172, 2913.365737])

To run this locally, do

python -m pytest tests/test_relatedness_vector.py::TestPCA::test_individuals[1-0-True]

I hear what you're saying about comparing U.T @ U, but I think we do want to actually test the individual eigenvectors, not just that the low dimensional approx is right. (Or, maybe I don't understand your suggestion?) The testing code is robust to ordering of eigen-things and sign switches; however, if two eigenvalues are very close, then you're right, we could have problems with nonidentifiabilty that way. Please do have a look at the testing code, though, to see if you have a better suggestion?

But - I don't think that's the problem - inserting a print into the testing code shows that these are the eigenvalues:

[3837.56623916, 3364.17632569, 2843.38691979, 2419.60371452, 655.31546225]

randomized svd draft

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petrelharp marked this pull request as draft October 4, 2024 01:46

modified api remove scipy

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output shape change when windows=wholegnome

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Co-authored-by: Peter Ralph <[email protected]>

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remove comments; eigen_values -> eigenvalues

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randomized svd draft #3008

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