def _configure_thresholds(self):
# Each bootstrap sample splits the reference samples into a sub-reference sample (x)
# and an extended test window (y). The extended test window will be treated as W overlapping
# test windows of size W (so 2W-1 test samples in total)
w_size = self.window_size
etw_size = 2*w_size-1 # etw = extended test window
rw_size = self.n - etw_size # rw = sub-ref window
perms = [torch.randperm(self.n) for _ in range(self.n_bootstraps)]
x_inds_all = [perm[:-etw_size] for perm in perms]
y_inds_all = [perm[-etw_size:] for perm in perms]
if self.verbose:
print("Generating permutations of kernel matrix..")
# Need to compute mmd for each bs for each of W overlapping windows
# Most of the computation can be done once however
# We avoid summing the rw_size^2 submatrix for each bootstrap sample by instead computing the full
# sum once and then subtracting the relavent parts (k_xx_sum = k_full_sum - 2*k_xy_sum - k_yy_sum).
# We also reduce computation of k_xy_sum from O(nW) to O(W) by caching column sums
k_full_sum = zero_diag(self.k_xx).sum()
k_xy_col_sums_all = [
self.k_xx[x_inds][:, y_inds].sum(0) for x_inds, y_inds in
(tqdm(zip(x_inds_all, y_inds_all), total=self.n_bootstraps) if self.verbose else
zip(x_inds_all, y_inds_all))
]
k_xx_sums_all = [(
k_full_sum - zero_diag(self.k_xx[y_inds][:, y_inds]).sum() - 2*k_xy_col_sums.sum()
)/(rw_size*(rw_size-1)) for y_inds, k_xy_col_sums in zip(y_inds_all, k_xy_col_sums_all)]
k_xy_col_sums_all = [k_xy_col_sums/(rw_size*w_size) for k_xy_col_sums in k_xy_col_sums_all]
# Now to iterate through the W overlapping windows
thresholds = []
p_bar = tqdm(range(w_size), "Computing thresholds") if self.verbose else range(w_size)
for w in p_bar:
y_inds_all_w = [y_inds[w:w+w_size] for y_inds in y_inds_all] # test windows of size w_size
mmds = [(
k_xx_sum +
zero_diag(self.k_xx[y_inds_w][:, y_inds_w]).sum()/(w_size*(w_size-1)) -
2*k_xy_col_sums[w:w+w_size].sum()
) for k_xx_sum, y_inds_w, k_xy_col_sums in zip(k_xx_sums_all, y_inds_all_w, k_xy_col_sums_all)
]
mmds = torch.tensor(mmds) # an mmd for each bootstrap sample
# Now we discard all bootstrap samples for which mmd is in top (1/ert)% and record the thresholds
thresholds.append(quantile(mmds, 1-self.fpr))
y_inds_all = [y_inds_all[i] for i in range(len(y_inds_all)) if mmds[i] < thresholds[-1]]
k_xx_sums_all = [
k_xx_sums_all[i] for i in range(len(k_xx_sums_all)) if mmds[i] < thresholds[-1]
]
k_xy_col_sums_all = [
k_xy_col_sums_all[i] for i in range(len(k_xy_col_sums_all)) if mmds[i] < thresholds[-1]
]
self.thresholds = thresholds
def score(self, x_t: np.ndarray) -> Union[float, None]:
"""
Compute the test-statistic (squared MMD) between the reference window and test window.
If the test-window is not yet full then a test-statistic of None is returned.
Parameters
----------
x_t
A single instance.
Returns
-------
Squared MMD estimate between reference window and test window
"""
x_t = torch.from_numpy(x_t[None, :]).to(self.device)
kernel_col = self.kernel(self.x_ref[self.ref_inds], x_t)
self.test_window = torch.cat([self.test_window[(1-self.window_size):], x_t], 0)
self.k_xy = torch.cat([self.k_xy[:, (1-self.window_size):], kernel_col], 1)
k_yy = self.kernel(self.test_window, self.test_window)
mmd = (
self.k_xx_sub_sum +
zero_diag(k_yy).sum()/(self.window_size*(self.window_size-1)) -
2*self.k_xy.mean()
)
return float(mmd.detach().cpu())
def _configure_ref_subset(self):
etw_size = 2*self.window_size-1 # etw = extended test window
rw_size = self.n - etw_size # rw = ref-window
# Make split and ensure it doesn't cause an initial detection
mmd_init = None
while mmd_init is None or mmd_init >= self.get_threshold(0):
# Make split
perm = torch.randperm(self.n)
self.ref_inds, self.init_test_inds = perm[:rw_size], perm[-self.window_size:]
self.test_window = self.x_ref[self.init_test_inds]
# Compute initial mmd to check for initial detection
self.k_xx_sub = self.k_xx[self.ref_inds][:, self.ref_inds]
self.k_xx_sub_sum = zero_diag(self.k_xx_sub).sum()/(rw_size*(rw_size-1))
self.k_xy = self.kernel(self.x_ref[self.ref_inds], self.test_window)
k_yy = self.kernel(self.test_window, self.test_window)
mmd_init = (
self.k_xx_sub_sum +
zero_diag(k_yy).sum()/(self.window_size*(self.window_size-1)) -
2*self.k_xy.mean()
)
def score(self, x_t: Union[np.ndarray, Any]) -> float:
"""
Compute the test-statistic (squared MMD) between the reference window and test window.
Parameters
----------
x_t
A single instance to be added to the test-window.
Returns
-------
Squared MMD estimate between reference window and test window.
"""
x_t = super()._preprocess_xt(x_t)
x_t = torch.from_numpy(x_t).to(self.device)
self._update_state(x_t)
k_yy = self.kernel(self.test_window, self.test_window)
mmd = (self.k_xx_sub_sum + zero_diag(k_yy).sum() /
(self.window_size *
(self.window_size - 1)) - 2 * self.k_xy.mean())
return float(mmd.detach().cpu())
def test_zero_diag():
ones = torch.ones(10, 10)
ones_zd = zero_diag(ones)
assert ones_zd.shape == (10, 10)
assert float(ones_zd.trace()) == 0
assert float(ones_zd.sum()) == 90
