def testConsistencyOfBases(self):
m = 6
sa = iisignature.rotinv2dprepare(m, "a")
sk = iisignature.rotinv2dprepare(m, "k")
ca = iisignature.rotinv2dcoeffs(sa)[-1]
ck = iisignature.rotinv2dcoeffs(sk)[-1]
#every row of ck should be in the span of the rows of ca
#i.e. every column of ck.T should be in the span of the columns of
#ca.T
#i.e. there's a matrix b s.t. ca.T b = ck.T
residuals = lstsq(ca.T, ck.T)[1]
self.assertLess(numpy.max(numpy.abs(residuals)), 0.000001)
sq = iisignature.rotinv2dprepare(m, "q")
cq = iisignature.rotinv2dcoeffs(sq)[-1]
ss = iisignature.rotinv2dprepare(m, "s")
cs = iisignature.rotinv2dcoeffs(ss)[-1]
# every row of cs and cq should be in the span of the rows of ca
residuals2 = lstsq(ca.T, cs.T)[1]
self.assertLess(numpy.max(numpy.abs(residuals2)), 0.000001)
residuals2 = lstsq(ca.T, cq.T)[1]
self.assertLess(numpy.max(numpy.abs(residuals2)), 0.000001)
self.assertEqual(cq.shape, cs.shape)
#check that the invariants are linearly independent (not for k)
for c, name in ((cs, "s"), (ca, "a"), (cq, "q")):
self.assertEqual(numpy.linalg.matrix_rank(c), c.shape[0], name)
#check that rows with nonzeros in evil columns are all before
#rows with nonzeros in odious columns
#print ((numpy.abs(ca)>0.00000001).astype("int8"))
for c, name in ((cs, "s"), (ck, "k"), (ca, "a"), (cq, "q")):
evilRows = []
odiousRows = []
for i in range(c.shape[0]):
evil = 0
odious = 0
for j in range(c.shape[1]):
if numpy.abs(c[i, j]) > 0.00001:
if bin(j).count("1") % 2:
odious = odious + 1
else:
evil = evil + 1
if evil > 0:
evilRows.append(i)
if odious > 0:
odiousRows.append(i)
#print (evilRows, odiousRows)
self.assertLess(numpy.max(evil), numpy.min(odious),
"bad order of rows in " + name)
def testConsistencyOfBases(self):
m = 6
sa = iisignature.rotinv2dprepare(m,"a")
sk = iisignature.rotinv2dprepare(m,"k")
ca = iisignature.rotinv2dcoeffs(sa)[-1]
ck = iisignature.rotinv2dcoeffs(sk)[-1]
#every row of ck should be in the span of the rows of ca
#i.e. every column of ck.T should be in the span of the columns of
#ca.T
#i.e. there's a matrix b s.t. ca.T b = ck.T
residuals = lstsq(ca.T,ck.T)[1]
self.assertLess(numpy.max(numpy.abs(residuals)),0.000001)
sq = iisignature.rotinv2dprepare(m, "q")
cq = iisignature.rotinv2dcoeffs(sq)[-1]
ss = iisignature.rotinv2dprepare(m, "s")
cs = iisignature.rotinv2dcoeffs(ss)[-1]
# every row of cs and cq should be in the span of the rows of ca
residuals2 = lstsq(ca.T, cs.T)[1]
self.assertLess(numpy.max(numpy.abs(residuals2)), 0.000001)
residuals2 = lstsq(ca.T, cq.T)[1]
self.assertLess(numpy.max(numpy.abs(residuals2)), 0.000001)
self.assertEqual(cq.shape, cs.shape)
#check that the invariants are linearly independent (not for k)
for c, name in ((cs, "s"), (ca, "a"), (cq, "q")):
self.assertEqual(numpy.linalg.matrix_rank(c),c.shape[0],name)
#check that rows with nonzeros in evil columns are all before
#rows with nonzeros in odious columns
#print ((numpy.abs(ca)>0.00000001).astype("int8"))
for c, name in ((cs, "s"), (ck, "k"), (ca, "a"), (cq, "q")):
evilRows = []
odiousRows = []
for i in range(c.shape[0]):
evil = 0
odious = 0
for j in range(c.shape[1]):
if numpy.abs(c[i, j]) > 0.00001:
if bin(j).count("1") % 2:
odious = odious + 1
else:
evil = evil + 1
if evil > 0:
evilRows.append(i)
if odious > 0:
odiousRows.append(i)
#print (evilRows, odiousRows)
self.assertLess(numpy.max(evil),numpy.min(odious),"bad order of rows in " + name)
def dotest(self,type):
m = 8
nPaths = 95
nAngles = 348
numpy.random.seed(775)
s = iisignature.rotinv2dprepare(m,type)
coeffs = iisignature.rotinv2dcoeffs(s)
angles = numpy.random.uniform(0,math.pi * 2,size=nAngles + 1)
angles[0] = 0
rotationMatrices = [numpy.array([[math.cos(i),math.sin(i)],[-math.sin(i),math.cos(i)]]) for i in angles]
paths = [numpy.random.uniform(-1,1,size=(32,2)) for i in range(nPaths)]
samePathRotInvs = [iisignature.rotinv2d(numpy.dot(paths[0],mtx),s) for mtx in rotationMatrices]
#check the length matches
(length,) = samePathRotInvs[0].shape
self.assertEqual(length,sum(i.shape[0] for i in coeffs))
self.assertEqual(length,iisignature.rotinv2dlength(s))
if type == "a":
self.assertEqual(length,sumCentralBinomialCoefficient(m // 2))
self.assertLess(length,nAngles)#sanity check on the test itself
#check that the invariants are invariant
if 0:
print("\n",numpy.column_stack(samePathRotInvs[0:7]))
for i in range(nAngles):
if 0 and diff(samePathRotInvs[0],samePathRotInvs[1 + i]) > 0.01:
print(i)
print(samePathRotInvs[0] - samePathRotInvs[1 + i])
print(diff(samePathRotInvs[0],samePathRotInvs[1 + i]))
self.assertLess(diff(samePathRotInvs[0],samePathRotInvs[1 + i]),0.01)
#check that the invariants match the coefficients
if 1:
sigLevel=iisignature.sig(paths[0],m)[iisignature.siglength(2,m-1):]
lowerRotinvs = 0 if 2==m else iisignature.rotinv2dlength(iisignature.rotinv2dprepare(m-2,type))
#print("\n",numpy.dot(coeffs[-1],sigLevel),"\n",samePathRotInvs[0][lowerRotinvs:])
#print(numpy.dot(coeffs[-1],sigLevel)-samePathRotInvs[0][lowerRotinvs:])
self.assertTrue(numpy.allclose(numpy.dot(coeffs[-1],sigLevel),samePathRotInvs[0][lowerRotinvs:],atol=0.000001))
#check that we are not missing invariants
if type == "a":
#print("\nrotinvlength=",length,"
#siglength=",iisignature.siglength(2,m))
sigOffsets = []
for path in paths:
samePathSigs = [iisignature.sig(numpy.dot(path,mtx),m) for mtx in rotationMatrices[1:70]]
samePathSigsOffsets = [i - samePathSigs[0] for i in samePathSigs[1:]]
sigOffsets.extend(samePathSigsOffsets)
#print(numpy.linalg.svd(numpy.row_stack(sigOffsets))[1])
def split(a, dim, level):
start = 0
out = []
for m in range(1,level + 1):
levelLength = dim ** m
out.append(a[:,start:(start + levelLength)])
start = start + levelLength
assert(start == a.shape[1])
return out
allOffsets = numpy.row_stack(sigOffsets)
#print (allOffsets.shape)
splits = split(allOffsets,2,m)
#print()
rank_tolerance = 0.01 # this is hackish
#print
#([numpy.linalg.matrix_rank(i.astype("float64"),rank_tolerance)
#for
#i in splits])
#print ([i.shape for i in splits])
#print(numpy.linalg.svd(splits[-1])[1])
#sanity check on the test
self.assertLess(splits[-1].shape[1],splits[0].shape[0])
totalUnspannedDimensions = sum(i.shape[1] - numpy.linalg.matrix_rank(i,rank_tolerance) for i in splits)
self.assertEqual(totalUnspannedDimensions,length)
if 0: #This doesn't work - the rank of the whole thing is less than
#sigLength-totalUnspannedDimensions, which suggests that there are
#inter-level dependencies,
#even though the shuffle product dependencies aren't linear.
#I don't know why this is.
sigLength = iisignature.siglength(2,m)
numNonInvariant = numpy.linalg.matrix_rank(numpy.row_stack(sigOffsets))
predictedNumberInvariant = sigLength - numNonInvariant
print(sigLength,length,numNonInvariant)
self.assertLess(sigLength,nAngles)
self.assertEqual(predictedNumberInvariant,length)
def test_batch(self):
numpy.random.seed(734)
d=2
m=2
n=15
paths = [numpy.random.uniform(-1,1,size=(6,d)) for i in range(n)]
pathArray15=stack(paths)
pathArray1315=numpy.reshape(pathArray15,(1,3,1,5,6,d))
sigs = [iisignature.sig(i,m) for i in paths]
sigArray=stack(sigs)
sigArray15=iisignature.sig(pathArray15,m)
sigArray1315=iisignature.sig(pathArray1315,m)
siglength=iisignature.siglength(d,m)
self.assertEqual(sigArray1315.shape,(1,3,1,5,siglength))
self.assertTrue(numpy.allclose(sigArray1315.reshape(n,siglength),sigs))
self.assertEqual(sigArray15.shape,(15,siglength))
self.assertTrue(numpy.allclose(sigArray15,sigs))
backsigs=[iisignature.sigbackprop(i,j,m) for i,j in zip(sigs,paths)]
backsigArray = stack(backsigs)
backsigs1315=iisignature.sigbackprop(sigArray1315,pathArray1315,m)
self.assertEqual(backsigs1315.shape,(1,3,1,5,6,d))
self.assertTrue(numpy.allclose(backsigs1315.reshape(n,6,2),backsigArray))
data=[numpy.random.uniform(size=(d,)) for i in range(n)]
dataArray1315=stack(data).reshape((1,3,1,5,d))
joined=[iisignature.sigjoin(i,j,m) for i,j in zip(sigs,data)]
joined1315=iisignature.sigjoin(sigArray1315,dataArray1315,m)
self.assertEqual(joined1315.shape,(1,3,1,5,siglength))
self.assertTrue(numpy.allclose(joined1315.reshape(n,-1),stack(joined)))
backjoined=[iisignature.sigjoinbackprop(i,j,k,m) for i,j,k in zip(joined,sigs,data)]
backjoinedArrays=[stack([i[j] for i in backjoined]) for j in range(2)]
backjoined1315=iisignature.sigjoinbackprop(joined1315,sigArray1315,dataArray1315,m)
self.assertEqual(backjoined1315[0].shape,sigArray1315.shape)
self.assertEqual(backjoined1315[1].shape,dataArray1315.shape)
self.assertTrue(numpy.allclose(backjoined1315[0].reshape(n,-1),backjoinedArrays[0]))
self.assertTrue(numpy.allclose(backjoined1315[1].reshape(n,-1),backjoinedArrays[1]))
dataAsSigs=[iisignature.sig(numpy.row_stack([numpy.zeros((d,)),i]),m) for i in data]
dataArray13151=dataArray1315[:,:,:,:,None,:]
dataArray13151=numpy.repeat(dataArray13151,2,4)*[[0.0],[1.0]]
dataArrayAsSigs1315=iisignature.sig(dataArray13151,m)
combined1315=iisignature.sigcombine(sigArray1315,dataArrayAsSigs1315,d,m)
self.assertEqual(joined1315.shape,combined1315.shape)
self.assertTrue(numpy.allclose(joined1315,combined1315))
backcombined1315=iisignature.sigcombinebackprop(joined1315,sigArray1315,dataArrayAsSigs1315,d,m)
backcombined=[iisignature.sigcombinebackprop(i,j,k,d,m) for i,j,k in zip(joined,sigs,dataAsSigs)]
backcombinedArrays=[stack([i[j] for i in backcombined]) for j in range(2)]
self.assertEqual(backcombined1315[0].shape,sigArray1315.shape)
self.assertEqual(backcombined1315[1].shape,sigArray1315.shape)
self.assertTrue(numpy.allclose(backjoined1315[0],backcombined1315[0]))
self.assertTrue(numpy.allclose(backcombined1315[0].reshape(n,-1),backcombinedArrays[0]))
self.assertTrue(numpy.allclose(backcombined1315[1].reshape(n,-1),backcombinedArrays[1]))
scaled=[iisignature.sigscale(i,j,m) for i,j in zip(sigs,data)]
scaled1315=iisignature.sigscale(sigArray1315,dataArray1315,m)
self.assertEqual(scaled1315.shape,(1,3,1,5,siglength))
self.assertTrue(numpy.allclose(scaled1315.reshape(n,-1),stack(scaled)))
backscaled=[iisignature.sigscalebackprop(i,j,k,m) for i,j,k in zip(scaled,sigs,data)]
backscaledArrays=[stack([i[j] for i in backscaled]) for j in range(2)]
backscaled1315=iisignature.sigscalebackprop(scaled1315,sigArray1315,dataArray1315,m)
self.assertEqual(backscaled1315[0].shape,sigArray1315.shape)
self.assertEqual(backscaled1315[1].shape,dataArray1315.shape)
self.assertTrue(numpy.allclose(backscaled1315[0].reshape(n,-1),backscaledArrays[0]))
self.assertTrue(numpy.allclose(backscaled1315[1].reshape(n,-1),backscaledArrays[1]))
s_s=(iisignature.prepare(d,m,"cosax"),iisignature.prepare(d,m,"cosahx"))
for type in ("c","o","s","x","a","ch","oh","sh","ah"):
s=s_s[1 if "h" in type else 0]
logsigs = [iisignature.logsig(i,s,type) for i in paths]
logsigArray=stack(logsigs)
logsigArray1315=iisignature.logsig(pathArray1315,s,type)
self.assertEqual(logsigArray1315.shape,(1,3,1,5,logsigs[0].shape[0]),type)
self.assertTrue(numpy.allclose(logsigArray1315.reshape(n,-1),logsigArray),type)
if type in ("s","x","sh"):
backlogs = stack(iisignature.logsigbackprop(i,j,s,type) for i,j in zip(logsigs,paths))
backlogs1315 = iisignature.logsigbackprop(logsigArray1315,pathArray1315,s,type)
self.assertEqual(backlogs1315.shape,backsigs1315.shape)
self.assertTrue(numpy.allclose(backlogs1315.reshape(n,6,d),backlogs),type)
a=iisignature.rotinv2dprepare(m,"a")
rots=stack([iisignature.rotinv2d(i,a) for i in paths])
rots1315=iisignature.rotinv2d(pathArray1315,a)
self.assertEqual(rots1315.shape,(1,3,1,5,rots.shape[1]))
self.assertTrue(numpy.allclose(rots1315.reshape(n,-1),rots))
mem_limit_mb=30
mem_limit_b = mem_limit_mb * 1024 * 1024
if os.name=="nt":
import ctypes
from ctypes import wintypes
kernel32 = ctypes.WinDLL('kernel32', use_last_error=True)
def errcheck_bool(result, func, args):
if not result:
raise ctypes.WinError(ctypes.get_last_error())
return args
kernel32.GetCurrentProcess.restype = wintypes.HANDLE
kernel32.SetProcessWorkingSetSizeEx.errcheck = errcheck_bool
kernel32.SetProcessWorkingSetSizeEx.argtypes = (wintypes.HANDLE,
ctypes.c_size_t,
ctypes.c_size_t,
wintypes.DWORD)
hProcess = kernel32.GetCurrentProcess()
kernel32.SetProcessWorkingSetSizeEx(hProcess, 204800, mem_limit_b, 6)
elif os.name=="posix":
import resource
resource.setrlimit(resource.RLIMIT_AS,(mem_limit_b,mem_limit_b))
else:
raise RuntimeError("what system")
iisignature.rotinv2dprepare(14,"a")
print ("ok")
#input()
def dotest(self,type):
m = 8
nPaths = 95
nAngles = 348
numpy.random.seed(775)
s = iisignature.rotinv2dprepare(m,type)
coeffs = iisignature.rotinv2dcoeffs(s)
angles = numpy.random.uniform(0,math.pi * 2,size=nAngles + 1)
angles[0] = 0
rotationMatrices = [numpy.array([[math.cos(i),math.sin(i)],[-math.sin(i),math.cos(i)]]) for i in angles]
paths = [numpy.random.uniform(-1,1,size=(32,2)) for i in range(nPaths)]
samePathRotInvs = [iisignature.rotinv2d(numpy.dot(paths[0],mtx),s) for mtx in rotationMatrices]
#check the length matches
(length,) = samePathRotInvs[0].shape
self.assertEqual(length,sum(i.shape[0] for i in coeffs))
self.assertEqual(length,iisignature.rotinv2dlength(s))
if type == "a":
self.assertEqual(length,sumCentralBinomialCoefficient(m // 2))
self.assertLess(length,nAngles)#sanity check on the test itself
#check that the invariants are invariant
if 0:
print("\n",numpy.column_stack(samePathRotInvs[0:7]))
for i in range(nAngles):
if 0 and diff(samePathRotInvs[0],samePathRotInvs[1 + i]) > 0.01:
print(i)
print(samePathRotInvs[0] - samePathRotInvs[1 + i])
print(diff(samePathRotInvs[0],samePathRotInvs[1 + i]))
self.assertLess(diff(samePathRotInvs[0],samePathRotInvs[1 + i]),0.01)
#check that the invariants match the coefficients
if 1:
sigLevel=iisignature.sig(paths[0],m)[iisignature.siglength(2,m-1):]
lowerRotinvs = 0 if 2==m else iisignature.rotinv2dlength(iisignature.rotinv2dprepare(m-2,type))
#print("\n",numpy.dot(coeffs[-1],sigLevel),"\n",samePathRotInvs[0][lowerRotinvs:])
#print(numpy.dot(coeffs[-1],sigLevel)-samePathRotInvs[0][lowerRotinvs:])
self.assertTrue(numpy.allclose(numpy.dot(coeffs[-1],sigLevel),samePathRotInvs[0][lowerRotinvs:],atol=0.000001))
#check that we are not missing invariants
if type == "a":
#print("\nrotinvlength=",length,"
#siglength=",iisignature.siglength(2,m))
sigOffsets = []
for path in paths:
samePathSigs = [iisignature.sig(numpy.dot(path,mtx),m) for mtx in rotationMatrices[1:70]]
samePathSigsOffsets = [i - samePathSigs[0] for i in samePathSigs[1:]]
sigOffsets.extend(samePathSigsOffsets)
#print(numpy.linalg.svd(numpy.row_stack(sigOffsets))[1])
def split(a, dim, level):
start = 0
out = []
for m in range(1,level + 1):
levelLength = dim ** m
out.append(a[:,start:(start + levelLength)])
start = start + levelLength
assert(start == a.shape[1])
return out
allOffsets = numpy.row_stack(sigOffsets)
#print (allOffsets.shape)
splits = split(allOffsets,2,m)
#print()
rank_tolerance = 0.01 # this is hackish
#print
#([numpy.linalg.matrix_rank(i.astype("float64"),rank_tolerance)
#for
#i in splits])
#print ([i.shape for i in splits])
#print(numpy.linalg.svd(splits[-1])[1])
#sanity check on the test
self.assertLess(splits[-1].shape[1],splits[0].shape[0])
totalUnspannedDimensions = sum(i.shape[1] - numpy.linalg.matrix_rank(i,rank_tolerance) for i in splits)
self.assertEqual(totalUnspannedDimensions,length)
if 0: #This doesn't work - the rank of the whole thing is less than
#sigLength-totalUnspannedDimensions, which suggests that there are
#inter-level dependencies,
#even though the shuffle product dependencies aren't linear.
#I don't know why this is.
sigLength = iisignature.siglength(2,m)
numNonInvariant = numpy.linalg.matrix_rank(numpy.row_stack(sigOffsets))
predictedNumberInvariant = sigLength - numNonInvariant
print(sigLength,length,numNonInvariant)
self.assertLess(sigLength,nAngles)
self.assertEqual(predictedNumberInvariant,length)
def test_batch(self):
numpy.random.seed(734)
d=2
m=2
n=15
paths = [numpy.random.uniform(-1,1,size=(6,d)) for i in range(n)]
pathArray15=stack(paths)
pathArray1315=numpy.reshape(pathArray15,(1,3,1,5,6,d))
sigs = [iisignature.sig(i,m) for i in paths]
sigArray=stack(sigs)
sigArray15=iisignature.sig(pathArray15,m)
sigArray1315=iisignature.sig(pathArray1315,m)
siglength=iisignature.siglength(d,m)
self.assertEqual(sigArray1315.shape,(1,3,1,5,siglength))
self.assertTrue(numpy.allclose(sigArray1315.reshape(n,siglength),sigs))
self.assertEqual(sigArray15.shape,(15,siglength))
self.assertTrue(numpy.allclose(sigArray15,sigs))
backsigs=[iisignature.sigbackprop(i,j,m) for i,j in zip(sigs,paths)]
backsigArray = stack(backsigs)
backsigs1315=iisignature.sigbackprop(sigArray1315,pathArray1315,m)
self.assertEqual(backsigs1315.shape,(1,3,1,5,6,d))
self.assertTrue(numpy.allclose(backsigs1315.reshape(n,6,2),backsigArray))
data=[numpy.random.uniform(size=(d,)) for i in range(n)]
dataArray1315=stack(data).reshape((1,3,1,5,d))
joined=[iisignature.sigjoin(i,j,m) for i,j in zip(sigs,data)]
joined1315=iisignature.sigjoin(sigArray1315,dataArray1315,m)
self.assertEqual(joined1315.shape,(1,3,1,5,siglength))
self.assertTrue(numpy.allclose(joined1315.reshape(n,-1),stack(joined)))
backjoined=[iisignature.sigjoinbackprop(i,j,k,m) for i,j,k in zip(joined,sigs,data)]
backjoinedArrays=[stack([i[j] for i in backjoined]) for j in range(2)]
backjoined1315=iisignature.sigjoinbackprop(joined1315,sigArray1315,dataArray1315,m)
self.assertEqual(backjoined1315[0].shape,sigArray1315.shape)
self.assertEqual(backjoined1315[1].shape,dataArray1315.shape)
self.assertTrue(numpy.allclose(backjoined1315[0].reshape(n,-1),backjoinedArrays[0]))
self.assertTrue(numpy.allclose(backjoined1315[1].reshape(n,-1),backjoinedArrays[1]))
scaled=[iisignature.sigscale(i,j,m) for i,j in zip(sigs,data)]
scaled1315=iisignature.sigscale(sigArray1315,dataArray1315,m)
self.assertEqual(scaled1315.shape,(1,3,1,5,siglength))
self.assertTrue(numpy.allclose(scaled1315.reshape(n,-1),stack(scaled)))
backscaled=[iisignature.sigscalebackprop(i,j,k,m) for i,j,k in zip(scaled,sigs,data)]
backscaledArrays=[stack([i[j] for i in backscaled]) for j in range(2)]
backscaled1315=iisignature.sigscalebackprop(scaled1315,sigArray1315,dataArray1315,m)
self.assertEqual(backscaled1315[0].shape,sigArray1315.shape)
self.assertEqual(backscaled1315[1].shape,dataArray1315.shape)
self.assertTrue(numpy.allclose(backscaled1315[0].reshape(n,-1),backscaledArrays[0]))
self.assertTrue(numpy.allclose(backscaled1315[1].reshape(n,-1),backscaledArrays[1]))
s_s=(iisignature.prepare(d,m,"cosax"),iisignature.prepare(d,m,"cosahx"))
for type in ("c","o","s","x","a","ch","oh","sh","ah"):
s=s_s[1 if "h" in type else 0]
logsigs = [iisignature.logsig(i,s,type) for i in paths]
logsigArray=stack(logsigs)
logsigArray1315=iisignature.logsig(pathArray1315,s,type)
self.assertEqual(logsigArray1315.shape,(1,3,1,5,logsigs[0].shape[0]),type)
self.assertTrue(numpy.allclose(logsigArray1315.reshape(n,-1),logsigArray),type)
if type in ("s","x","sh"):
backlogs = stack(iisignature.logsigbackprop(i,j,s,type) for i,j in zip(logsigs,paths))
backlogs1315 = iisignature.logsigbackprop(logsigArray1315,pathArray1315,s,type)
self.assertEqual(backlogs1315.shape,backsigs1315.shape)
self.assertTrue(numpy.allclose(backlogs1315.reshape(n,6,d),backlogs),type)
a=iisignature.rotinv2dprepare(m,"a")
rots=stack([iisignature.rotinv2d(i,a) for i in paths])
rots1315=iisignature.rotinv2d(pathArray1315,a)
self.assertEqual(rots1315.shape,(1,3,1,5,rots.shape[1]))
self.assertTrue(numpy.allclose(rots1315.reshape(n,-1),rots))