def ridge_corr(Rstim, Pstim, Rresp, Presp, alphas, normalpha=False, dtype=np.single, corrmin=0.2,
singcutoff=1e-10, use_corr=True, logger=logging.getLogger("ridge_corr")):
"""Uses ridge regression to find a linear transformation of [Rstim] that approximates [Rresp].
Then tests by comparing the transformation of [Pstim] to [Presp]. This procedure is repeated
for each regularization parameter alpha in [alphas]. The correlation between each prediction and
each response for each alpha is returned. Note that the regression weights are NOT returned.
Parameters
----------
Rstim : array_like, shape (TR, N)
Training stimuli with TR time points and N features. Each feature should be Z-scored across time.
Pstim : array_like, shape (TP, N)
Test stimuli with TP time points and N features. Each feature should be Z-scored across time.
Rresp : array_like, shape (TR, M)
Training responses with TR time points and M responses (voxels, neurons, what-have-you).
Each response should be Z-scored across time.
Presp : array_like, shape (TP, M)
Test responses with TP time points and M responses.
alphas : list or array_like, shape (A,)
Ridge parameters to be tested. Should probably be log-spaced. np.logspace(0, 3, 20) works well.
normalpha : boolean
Whether ridge parameters should be normalized by the Frobenius norm of Rstim. Good for
comparing models with different numbers of parameters.
dtype : np.dtype
All data will be cast as this dtype for computation. np.single is used by default for memory
efficiency.
corrmin : float in [0..1]
Purely for display purposes. After each alpha is tested, the number of responses with correlation
greater than corrmin minus the number of responses with correlation less than negative corrmin
will be printed. For long-running regressions this vague metric of non-centered skewness can
give you a rough sense of how well the model is working before it's done.
singcutoff : float
The first step in ridge regression is computing the singular value decomposition (SVD) of the
stimulus Rstim. If Rstim is not full rank, some singular values will be approximately equal
to zero and the corresponding singular vectors will be noise. These singular values/vectors
should be removed both for speed (the fewer multiplications the better!) and accuracy. Any
singular values less than singcutoff will be removed.
use_corr : boolean
If True, this function will use correlation as its metric of model fit. If False, this function
will instead use variance explained (R-squared) as its metric of model fit. For ridge regression
this can make a big difference -- highly regularized solutions will have very small norms and
will thus explain very little variance while still leading to high correlations, as correlation
is scale-free while R**2 is not.
Returns
-------
Rcorrs : array_like, shape (A, M)
The correlation between each predicted response and each column of Presp for each alpha.
"""
## Calculate SVD of stimulus matrix
logger.info("Doing SVD...")
try:
U,S,Vh = np.linalg.svd(Rstim, full_matrices=False)
except np.linalg.LinAlgError, e:
logger.info("NORMAL SVD FAILED, trying more robust dgesvd..")
from text.regression.svd_dgesvd import svd_dgesvd
U,S,Vh = svd_dgesvd(Rstim, full_matrices=False)
def ridge(stim,
resp,
alpha,
singcutoff=1e-10,
normalpha=False,
logger=ridge_logger):
"""Uses ridge regression to find a linear transformation of [stim] that approximates
[resp]. The regularization parameter is [alpha].
Parameters
----------
stim : array_like, shape (T, N)
Stimuli with T time points and N features.
resp : array_like, shape (T, M)
Responses with T time points and M separate responses.
alpha : float or array_like, shape (M,)
Regularization parameter. Can be given as a single value (which is applied to
all M responses) or separate values for each response.
normalpha : boolean
Whether ridge parameters should be normalized by the largest singular value of stim. Good for
comparing models with different numbers of parameters.
Returns
-------
wt : array_like, shape (N, M)
Linear regression weights.
"""
try:
U, S, Vh = np.linalg.svd(stim, full_matrices=False)
except np.linalg.LinAlgError:
logger.info("NORMAL SVD FAILED, trying more robust dgesvd..")
from text.regression.svd_dgesvd import svd_dgesvd
U, S, Vh = svd_dgesvd(stim, full_matrices=False)
UR = np.dot(U.T, np.nan_to_num(resp))
# Expand alpha to a collection if it's just a single value
if isinstance(alpha, (float, int)):
alpha = np.ones(resp.shape[1]) * alpha
# Normalize alpha by the LSV norm
norm = S[0]
if normalpha:
nalphas = alpha * norm
else:
nalphas = alpha
# Compute weights for each alpha
ualphas = np.unique(nalphas)
wt = np.zeros((stim.shape[1], resp.shape[1]))
for ua in ualphas:
selvox = np.nonzero(nalphas == ua)[0]
#awt = reduce(np.dot, [Vh.T, np.diag(S/(S**2+ua**2)), UR[:,selvox]])
awt = Vh.T.dot(np.diag(S / (S**2 + ua**2))).dot(UR[:, selvox])
wt[:, selvox] = awt
return wt
def precompute_svd(Rstim, singcutoff=1e-10):
#logger.info("Doing SVD...")
try:
U, S, Vh = np.linalg.svd(Rstim, full_matrices=False)
except np.linalg.LinAlgError, e:
print "NORMAL SVD FAILED, trying more robust dgesvd.."
#logger.info("NORMAL SVD FAILED, trying more robust dgesvd..")
from text.regression.svd_dgesvd import svd_dgesvd
U, S, Vh = svd_dgesvd(Rstim, full_matrices=False)
def ridge(stim, resp, alpha, singcutoff=1e-10, normalpha=False, logger=ridge_logger):
"""Uses ridge regression to find a linear transformation of [stim] that approximates
[resp]. The regularization parameter is [alpha].
Parameters
----------
stim : array_like, shape (T, N)
Stimuli with T time points and N features.
resp : array_like, shape (T, M)
Responses with T time points and M separate responses.
alpha : float or array_like, shape (M,)
Regularization parameter. Can be given as a single value (which is applied to
all M responses) or separate values for each response.
normalpha : boolean
Whether ridge parameters should be normalized by the largest singular value of stim. Good for
comparing models with different numbers of parameters.
Returns
-------
wt : array_like, shape (N, M)
Linear regression weights.
"""
try:
U,S,Vh = np.linalg.svd(stim, full_matrices=False)
except np.linalg.LinAlgError:
logger.info("NORMAL SVD FAILED, trying more robust dgesvd..")
from text.regression.svd_dgesvd import svd_dgesvd
U,S,Vh = svd_dgesvd(stim, full_matrices=False)
UR = np.dot(U.T, np.nan_to_num(resp))
# Expand alpha to a collection if it's just a single value
if isinstance(alpha, (float,int)):
alpha = np.ones(resp.shape[1]) * alpha
# Normalize alpha by the LSV norm
norm = S[0]
if normalpha:
nalphas = alpha * norm
else:
nalphas = alpha
# Compute weights for each alpha
ualphas = np.unique(nalphas)
wt = np.zeros((stim.shape[1], resp.shape[1]))
for ua in ualphas:
selvox = np.nonzero(nalphas==ua)[0]
#awt = reduce(np.dot, [Vh.T, np.diag(S/(S**2+ua**2)), UR[:,selvox]])
awt = Vh.T.dot(np.diag(S/(S**2+ua**2))).dot(UR[:,selvox])
wt[:,selvox] = awt
return wt
def ridge_corr(Rstim, Pstim, Rresp, Presp, alphas, normalpha=False, dtype=np.single, corrmin=0.2, singcutoff=1e-10, use_corr=True): """Uses ridge regression to find a linear transformation of [Rstim] that approximates [Rresp]. Then tests by comparing the transformation of [Pstim] to [Presp]. This procedure is repeated for each regularization parameter alpha in [alphas]. The correlation between each prediction and each response for each alpha is returned. Note that the regression weights are NOT returned. Parameters ---------- Rstim : array_like, shape (TR, N) Training stimuli with TR time points and N features. Each feature should be Z-scored across time. Pstim : array_like, shape (TP, N) Test stimuli with TP time points and N features. Each feature should be Z-scored across time. Rresp : array_like, shape (TR, M) Training responses with TR time points and M responses (voxels, neurons, what-have-you).Each response should be Z-scored across time. Presp : array_like, shape (TP, M) Test responses with TP time points and M responses. alphas : list or array_like, shape (A,) Ridge parameters to be tested. Should probably be log-spaced. np.logspace(0, 3, 20) works well. normalpha : boolean Whether ridge parameters should be normalized by the Frobenius norm of Rstim. Good forcomparing models with different numbers of parameters. dtype : np.dtype All data will be cast as this dtype for computation. np.single is used by default for memory efficiency. corrmin : float in [0..1] Purely for display purposes. After each alpha is tested, the number of responses with correlation greater than corrmin minus the number of responses with correlation less than negative corrmin will be printed. For long-running regressions this vague metric of non-centered skewness can give you a rough sense of how well the model is working before it's done. singcutoff : float The first step in ridge regression is computing the singular value decomposition (SVD) of the stimulus Rstim. If Rstim is not full rank, some singular values will be approximately equal to zero and the corresponding singular vectors will be noise. These singular values/vectors should be removed both for speed (the fewer multiplications the better!) and accuracy. Any singular values less than singcutoff will be removed. use_corr : boolean If True, this function will use correlation as its metric of model fit. If False, this function will instead use variance explained (R-squared) as its metric of model fit. For ridge regression this can make a big difference -- highly regularized solutions will have very small norms and will thus explain very little variance while still leading to high correlations, as correlation is scale-free while R**2 is not. Returns ------- Rcorrs : array_like, shape (A, M) The correlation between each predicted response and each column of Presp for each alpha. """ ## Calculate SVD of stimulus matrix print "Doing SVD..." try: U,S,Vh = np.linalg.svd(Rstim, full_matrices=False) except np.linalg.LinAlgError, e: print "NORMAL SVD FAILED, trying more robust dgesvd.." from text.regression.svd_dgesvd import svd_dgesvd U,S,Vh = svd_dgesvd(Rstim, full_matrices=False)
def ridge_corr_pred(Rstim, Pstim, Rresp, Presp, valphas, normalpha=False,
singcutoff=1e-10, use_corr=True, logger=ridge_logger):
"""Uses ridge regression to find a linear transformation of [Rstim] that approximates [Rresp],
then tests by comparing the transformation of [Pstim] to [Presp]. Returns the correlation
between predicted and actual [Presp], without ever computing the regression weights.
This function assumes that each voxel is assigned a separate alpha in [valphas].
Parameters
----------
Rstim : array_like, shape (TR, N)
Training stimuli with TR time points and N features. Each feature should be Z-scored across time.
Pstim : array_like, shape (TP, N)
Test stimuli with TP time points and N features. Each feature should be Z-scored across time.
Rresp : array_like, shape (TR, M)
Training responses with TR time points and M responses (voxels, neurons, what-have-you).
Each response should be Z-scored across time.
Presp : array_like, shape (TP, M)
Test responses with TP time points and M responses.
valphas : list or array_like, shape (M,)
Ridge parameter for each voxel.
normalpha : boolean
Whether ridge parameters should be normalized by the largest singular value (LSV) norm of
Rstim. Good for comparing models with different numbers of parameters.
corrmin : float in [0..1]
Purely for display purposes. After each alpha is tested, the number of responses with correlation
greater than corrmin minus the number of responses with correlation less than negative corrmin
will be printed. For long-running regressions this vague metric of non-centered skewness can
give you a rough sense of how well the model is working before it's done.
singcutoff : float
The first step in ridge regression is computing the singular value decomposition (SVD) of the
stimulus Rstim. If Rstim is not full rank, some singular values will be approximately equal
to zero and the corresponding singular vectors will be noise. These singular values/vectors
should be removed both for speed (the fewer multiplications the better!) and accuracy. Any
singular values less than singcutoff will be removed.
use_corr : boolean
If True, this function will use correlation as its metric of model fit. If False, this function
will instead use variance explained (R-squared) as its metric of model fit. For ridge regression
this can make a big difference -- highly regularized solutions will have very small norms and
will thus explain very little variance while still leading to high correlations, as correlation
is scale-free while R**2 is not.
Returns
-------
corr : array_like, shape (M,)
The correlation between each predicted response and each column of Presp.
"""
## Calculate SVD of stimulus matrix
logger.info("Doing SVD...")
try:
U,S,Vh = np.linalg.svd(Rstim, full_matrices=False)
except np.linalg.LinAlgError:
logger.info("NORMAL SVD FAILED, trying more robust dgesvd..")
from text.regression.svd_dgesvd import svd_dgesvd
U,S,Vh = svd_dgesvd(Rstim, full_matrices=False)
## Truncate tiny singular values for speed
origsize = S.shape[0]
ngoodS = np.sum(S > singcutoff)
nbad = origsize-ngoodS
U = U[:,:ngoodS]
S = S[:ngoodS]
Vh = Vh[:ngoodS]
logger.info("Dropped %d tiny singular values.. (U is now %s)"%(nbad, str(U.shape)))
## Normalize alpha by the LSV norm
norm = S[0]
logger.info("Training stimulus has LSV norm: %0.03f"%norm)
if normalpha:
nalphas = valphas * norm
else:
nalphas = valphas
## Precompute some products for speed
UR = np.dot(U.T, Rresp) ## Precompute this matrix product for speed
PVh = np.dot(Pstim, Vh.T) ## Precompute this matrix product for speed
#Prespnorms = np.apply_along_axis(np.linalg.norm, 0, Presp) ## Precompute test response norms
zPresp = zs(Presp)
#Prespvar = Presp.var(0)
Prespvar_actual = Presp.var(0)
Prespvar = (np.ones_like(Prespvar_actual) + Prespvar_actual) / 2.0
logger.info("Average difference between actual & assumed Prespvar: %0.3f" % (Prespvar_actual - Prespvar).mean())
ualphas = np.unique(nalphas)
corr = np.zeros((Rresp.shape[1],))
for ua in ualphas:
selvox = np.nonzero(nalphas==ua)[0]
alpha_pred = PVh.dot(np.diag(S/(S**2+ua**2))).dot(UR[:,selvox])
if use_corr:
corr[selvox] = (zPresp[:,selvox] * zs(alpha_pred)).mean(0)
else:
resvar = (Presp[:,selvox] - alpha_pred).var(0)
Rsq = 1 - (resvar / Prespvar)
corr[selvox] = np.sqrt(np.abs(Rsq)) * np.sign(Rsq)
return corr
def ridge_corr(Rstim, Pstim, Rresp, Presp, alphas, normalpha=False, corrmin=0.2,
singcutoff=1e-10, use_corr=True, logger=ridge_logger):
"""Uses ridge regression to find a linear transformation of [Rstim] that approximates [Rresp],
then tests by comparing the transformation of [Pstim] to [Presp]. This procedure is repeated
for each regularization parameter alpha in [alphas]. The correlation between each prediction and
each response for each alpha is returned. The regression weights are NOT returned, because
computing the correlations without computing regression weights is much, MUCH faster.
Parameters
----------
Rstim : array_like, shape (TR, N)
Training stimuli with TR time points and N features. Each feature should be Z-scored across time.
Pstim : array_like, shape (TP, N)
Test stimuli with TP time points and N features. Each feature should be Z-scored across time.
Rresp : array_like, shape (TR, M)
Training responses with TR time points and M responses (voxels, neurons, what-have-you).
Each response should be Z-scored across time.
Presp : array_like, shape (TP, M)
Test responses with TP time points and M responses.
alphas : list or array_like, shape (A,)
Ridge parameters to be tested. Should probably be log-spaced. np.logspace(0, 3, 20) works well.
normalpha : boolean
Whether ridge parameters should be normalized by the largest singular value (LSV) norm of
Rstim. Good for comparing models with different numbers of parameters.
corrmin : float in [0..1]
Purely for display purposes. After each alpha is tested, the number of responses with correlation
greater than corrmin minus the number of responses with correlation less than negative corrmin
will be printed. For long-running regressions this vague metric of non-centered skewness can
give you a rough sense of how well the model is working before it's done.
singcutoff : float
The first step in ridge regression is computing the singular value decomposition (SVD) of the
stimulus Rstim. If Rstim is not full rank, some singular values will be approximately equal
to zero and the corresponding singular vectors will be noise. These singular values/vectors
should be removed both for speed (the fewer multiplications the better!) and accuracy. Any
singular values less than singcutoff will be removed.
use_corr : boolean
If True, this function will use correlation as its metric of model fit. If False, this function
will instead use variance explained (R-squared) as its metric of model fit. For ridge regression
this can make a big difference -- highly regularized solutions will have very small norms and
will thus explain very little variance while still leading to high correlations, as correlation
is scale-free while R**2 is not.
Returns
-------
Rcorrs : array_like, shape (A, M)
The correlation between each predicted response and each column of Presp for each alpha.
"""
## Calculate SVD of stimulus matrix
logger.info("Doing SVD...")
try:
U,S,Vh = np.linalg.svd(Rstim, full_matrices=False)
except np.linalg.LinAlgError:
logger.info("NORMAL SVD FAILED, trying more robust dgesvd..")
from text.regression.svd_dgesvd import svd_dgesvd
U,S,Vh = svd_dgesvd(Rstim, full_matrices=False)
## Truncate tiny singular values for speed
origsize = S.shape[0]
ngoodS = np.sum(S > singcutoff)
nbad = origsize-ngoodS
U = U[:,:ngoodS]
S = S[:ngoodS]
Vh = Vh[:ngoodS]
logger.info("Dropped %d tiny singular values.. (U is now %s)"%(nbad, str(U.shape)))
## Normalize alpha by the LSV norm
norm = S[0]
logger.info("Training stimulus has LSV norm: %0.03f"%norm)
if normalpha:
nalphas = alphas * norm
else:
nalphas = alphas
## Precompute some products for speed
UR = np.dot(U.T, Rresp) ## Precompute this matrix product for speed
PVh = np.dot(Pstim, Vh.T) ## Precompute this matrix product for speed
#Prespnorms = np.apply_along_axis(np.linalg.norm, 0, Presp) ## Precompute test response norms
zPresp = zs(Presp)
#Prespvar = Presp.var(0)
Prespvar_actual = Presp.var(0)
Prespvar = (np.ones_like(Prespvar_actual) + Prespvar_actual) / 2.0
logger.info("Average difference between actual & assumed Prespvar: %0.3f" % (Prespvar_actual - Prespvar).mean())
Rcorrs = [] ## Holds training correlations for each alpha
for na, a in zip(nalphas, alphas):
#D = np.diag(S/(S**2+a**2)) ## Reweight singular vectors by the ridge parameter
D = S / (S ** 2 + na ** 2) ## Reweight singular vectors by the (normalized?) ridge parameter
pred = np.dot(mult_diag(D, PVh, left=False), UR) ## Best (1.75 seconds to prediction in test)
# pred = np.dot(mult_diag(D, np.dot(Pstim, Vh.T), left=False), UR) ## Better (2.0 seconds to prediction in test)
# pvhd = reduce(np.dot, [Pstim, Vh.T, D]) ## Pretty good (2.4 seconds to prediction in test)
# pred = np.dot(pvhd, UR)
# wt = reduce(np.dot, [Vh.T, D, UR]).astype(dtype) ## Bad (14.2 seconds to prediction in test)
# wt = reduce(np.dot, [Vh.T, D, U.T, Rresp]).astype(dtype) ## Worst
# pred = np.dot(Pstim, wt) ## Predict test responses
if use_corr:
#prednorms = np.apply_along_axis(np.linalg.norm, 0, pred) ## Compute predicted test response norms
#Rcorr = np.array([np.corrcoef(Presp[:,ii], pred[:,ii].ravel())[0,1] for ii in range(Presp.shape[1])]) ## Slowly compute correlations
#Rcorr = np.array(np.sum(np.multiply(Presp, pred), 0)).squeeze()/(prednorms*Prespnorms) ## Efficiently compute correlations
Rcorr = (zPresp * zs(pred)).mean(0)
else:
## Compute variance explained
resvar = (Presp - pred).var(0)
Rsq = 1 - (resvar / Prespvar)
Rcorr = np.sqrt(np.abs(Rsq)) * np.sign(Rsq)
Rcorr[np.isnan(Rcorr)] = 0
Rcorrs.append(Rcorr)
log_template = "Training: alpha=%0.3f, mean corr=%0.5f, max corr=%0.5f, over-under(%0.2f)=%d"
log_msg = log_template % (a,
np.mean(Rcorr),
np.max(Rcorr),
corrmin,
(Rcorr>corrmin).sum()-(-Rcorr>corrmin).sum())
logger.info(log_msg)
return Rcorrs
def ridge_corr_pred(Rstim,
Pstim,
Rresp,
Presp,
valphas,
normalpha=False,
singcutoff=1e-10,
use_corr=True,
logger=ridge_logger):
"""Uses ridge regression to find a linear transformation of [Rstim] that approximates [Rresp],
then tests by comparing the transformation of [Pstim] to [Presp]. Returns the correlation
between predicted and actual [Presp], without ever computing the regression weights.
This function assumes that each voxel is assigned a separate alpha in [valphas].
Parameters
----------
Rstim : array_like, shape (TR, N)
Training stimuli with TR time points and N features. Each feature should be Z-scored across time.
Pstim : array_like, shape (TP, N)
Test stimuli with TP time points and N features. Each feature should be Z-scored across time.
Rresp : array_like, shape (TR, M)
Training responses with TR time points and M responses (voxels, neurons, what-have-you).
Each response should be Z-scored across time.
Presp : array_like, shape (TP, M)
Test responses with TP time points and M responses.
valphas : list or array_like, shape (M,)
Ridge parameter for each voxel.
normalpha : boolean
Whether ridge parameters should be normalized by the largest singular value (LSV) norm of
Rstim. Good for comparing models with different numbers of parameters.
corrmin : float in [0..1]
Purely for display purposes. After each alpha is tested, the number of responses with correlation
greater than corrmin minus the number of responses with correlation less than negative corrmin
will be printed. For long-running regressions this vague metric of non-centered skewness can
give you a rough sense of how well the model is working before it's done.
singcutoff : float
The first step in ridge regression is computing the singular value decomposition (SVD) of the
stimulus Rstim. If Rstim is not full rank, some singular values will be approximately equal
to zero and the corresponding singular vectors will be noise. These singular values/vectors
should be removed both for speed (the fewer multiplications the better!) and accuracy. Any
singular values less than singcutoff will be removed.
use_corr : boolean
If True, this function will use correlation as its metric of model fit. If False, this function
will instead use variance explained (R-squared) as its metric of model fit. For ridge regression
this can make a big difference -- highly regularized solutions will have very small norms and
will thus explain very little variance while still leading to high correlations, as correlation
is scale-free while R**2 is not.
Returns
-------
corr : array_like, shape (M,)
The correlation between each predicted response and each column of Presp.
"""
## Calculate SVD of stimulus matrix
logger.info("Doing SVD...")
try:
U, S, Vh = np.linalg.svd(Rstim, full_matrices=False)
except np.linalg.LinAlgError:
logger.info("NORMAL SVD FAILED, trying more robust dgesvd..")
from text.regression.svd_dgesvd import svd_dgesvd
U, S, Vh = svd_dgesvd(Rstim, full_matrices=False)
## Truncate tiny singular values for speed
origsize = S.shape[0]
ngoodS = np.sum(S > singcutoff)
nbad = origsize - ngoodS
U = U[:, :ngoodS]
S = S[:ngoodS]
Vh = Vh[:ngoodS]
logger.info("Dropped %d tiny singular values.. (U is now %s)" %
(nbad, str(U.shape)))
## Normalize alpha by the LSV norm
norm = S[0]
logger.info("Training stimulus has LSV norm: %0.03f" % norm)
if normalpha:
nalphas = valphas * norm
else:
nalphas = valphas
## Precompute some products for speed
UR = np.dot(U.T, Rresp) ## Precompute this matrix product for speed
PVh = np.dot(Pstim, Vh.T) ## Precompute this matrix product for speed
#Prespnorms = np.apply_along_axis(np.linalg.norm, 0, Presp) ## Precompute test response norms
zPresp = zs(Presp)
#Prespvar = Presp.var(0)
Prespvar_actual = Presp.var(0)
Prespvar = (np.ones_like(Prespvar_actual) + Prespvar_actual) / 2.0
logger.info("Average difference between actual & assumed Prespvar: %0.3f" %
(Prespvar_actual - Prespvar).mean())
ualphas = np.unique(nalphas)
corr = np.zeros((Rresp.shape[1], ))
for ua in ualphas:
selvox = np.nonzero(nalphas == ua)[0]
alpha_pred = PVh.dot(np.diag(S / (S**2 + ua**2))).dot(UR[:, selvox])
if use_corr:
corr[selvox] = (zPresp[:, selvox] * zs(alpha_pred)).mean(0)
else:
resvar = (Presp[:, selvox] - alpha_pred).var(0)
Rsq = 1 - (resvar / Prespvar)
corr[selvox] = np.sqrt(np.abs(Rsq)) * np.sign(Rsq)
return corr
def ridge_corr(Rstim,
Pstim,
Rresp,
Presp,
alphas,
normalpha=False,
corrmin=0.2,
singcutoff=1e-10,
use_corr=True,
logger=ridge_logger):
"""Uses ridge regression to find a linear transformation of [Rstim] that approximates [Rresp],
then tests by comparing the transformation of [Pstim] to [Presp]. This procedure is repeated
for each regularization parameter alpha in [alphas]. The correlation between each prediction and
each response for each alpha is returned. The regression weights are NOT returned, because
computing the correlations without computing regression weights is much, MUCH faster.
Parameters
----------
Rstim : array_like, shape (TR, N)
Training stimuli with TR time points and N features. Each feature should be Z-scored across time.
Pstim : array_like, shape (TP, N)
Test stimuli with TP time points and N features. Each feature should be Z-scored across time.
Rresp : array_like, shape (TR, M)
Training responses with TR time points and M responses (voxels, neurons, what-have-you).
Each response should be Z-scored across time.
Presp : array_like, shape (TP, M)
Test responses with TP time points and M responses.
alphas : list or array_like, shape (A,)
Ridge parameters to be tested. Should probably be log-spaced. np.logspace(0, 3, 20) works well.
normalpha : boolean
Whether ridge parameters should be normalized by the largest singular value (LSV) norm of
Rstim. Good for comparing models with different numbers of parameters.
corrmin : float in [0..1]
Purely for display purposes. After each alpha is tested, the number of responses with correlation
greater than corrmin minus the number of responses with correlation less than negative corrmin
will be printed. For long-running regressions this vague metric of non-centered skewness can
give you a rough sense of how well the model is working before it's done.
singcutoff : float
The first step in ridge regression is computing the singular value decomposition (SVD) of the
stimulus Rstim. If Rstim is not full rank, some singular values will be approximately equal
to zero and the corresponding singular vectors will be noise. These singular values/vectors
should be removed both for speed (the fewer multiplications the better!) and accuracy. Any
singular values less than singcutoff will be removed.
use_corr : boolean
If True, this function will use correlation as its metric of model fit. If False, this function
will instead use variance explained (R-squared) as its metric of model fit. For ridge regression
this can make a big difference -- highly regularized solutions will have very small norms and
will thus explain very little variance while still leading to high correlations, as correlation
is scale-free while R**2 is not.
Returns
-------
Rcorrs : array_like, shape (A, M)
The correlation between each predicted response and each column of Presp for each alpha.
"""
## Calculate SVD of stimulus matrix
logger.info("Doing SVD...")
try:
U, S, Vh = np.linalg.svd(Rstim, full_matrices=False)
except np.linalg.LinAlgError:
logger.info("NORMAL SVD FAILED, trying more robust dgesvd..")
from text.regression.svd_dgesvd import svd_dgesvd
U, S, Vh = svd_dgesvd(Rstim, full_matrices=False)
## Truncate tiny singular values for speed
origsize = S.shape[0]
ngoodS = np.sum(S > singcutoff)
nbad = origsize - ngoodS
U = U[:, :ngoodS]
S = S[:ngoodS]
Vh = Vh[:ngoodS]
logger.info("Dropped %d tiny singular values.. (U is now %s)" %
(nbad, str(U.shape)))
## Normalize alpha by the LSV norm
norm = S[0]
logger.info("Training stimulus has LSV norm: %0.03f" % norm)
if normalpha:
nalphas = alphas * norm
else:
nalphas = alphas
## Precompute some products for speed
UR = np.dot(U.T, Rresp) ## Precompute this matrix product for speed
PVh = np.dot(Pstim, Vh.T) ## Precompute this matrix product for speed
#Prespnorms = np.apply_along_axis(np.linalg.norm, 0, Presp) ## Precompute test response norms
zPresp = zs(Presp)
#Prespvar = Presp.var(0)
Prespvar_actual = Presp.var(0)
Prespvar = (np.ones_like(Prespvar_actual) + Prespvar_actual) / 2.0
logger.info("Average difference between actual & assumed Prespvar: %0.3f" %
(Prespvar_actual - Prespvar).mean())
Rcorrs = [] ## Holds training correlations for each alpha
for na, a in zip(nalphas, alphas):
#D = np.diag(S/(S**2+a**2)) ## Reweight singular vectors by the ridge parameter
D = S / (
S**2 + na**2
) ## Reweight singular vectors by the (normalized?) ridge parameter
pred = np.dot(mult_diag(D, PVh, left=False),
UR) ## Best (1.75 seconds to prediction in test)
# pred = np.dot(mult_diag(D, np.dot(Pstim, Vh.T), left=False), UR) ## Better (2.0 seconds to prediction in test)
# pvhd = reduce(np.dot, [Pstim, Vh.T, D]) ## Pretty good (2.4 seconds to prediction in test)
# pred = np.dot(pvhd, UR)
# wt = reduce(np.dot, [Vh.T, D, UR]).astype(dtype) ## Bad (14.2 seconds to prediction in test)
# wt = reduce(np.dot, [Vh.T, D, U.T, Rresp]).astype(dtype) ## Worst
# pred = np.dot(Pstim, wt) ## Predict test responses
if use_corr:
#prednorms = np.apply_along_axis(np.linalg.norm, 0, pred) ## Compute predicted test response norms
#Rcorr = np.array([np.corrcoef(Presp[:,ii], pred[:,ii].ravel())[0,1] for ii in range(Presp.shape[1])]) ## Slowly compute correlations
#Rcorr = np.array(np.sum(np.multiply(Presp, pred), 0)).squeeze()/(prednorms*Prespnorms) ## Efficiently compute correlations
Rcorr = (zPresp * zs(pred)).mean(0)
else:
## Compute variance explained
resvar = (Presp - pred).var(0)
Rsq = 1 - (resvar / Prespvar)
Rcorr = np.sqrt(np.abs(Rsq)) * np.sign(Rsq)
Rcorr[np.isnan(Rcorr)] = 0
Rcorrs.append(Rcorr)
log_template = "Training: alpha=%0.3f, mean corr=%0.5f, max corr=%0.5f, over-under(%0.2f)=%d"
log_msg = log_template % (a, np.mean(Rcorr), np.max(Rcorr), corrmin,
(Rcorr > corrmin).sum() -
(-Rcorr > corrmin).sum())
logger.info(log_msg)
return Rcorrs
def bootstrap_ridge(Rstim, Rresp, Pstim, Presp, alphas, nboots, chunklen, nchunks, dtype=np.single,
corrmin=0.2, joined=None, singcutoff=1e-10, normalpha=False, single_alpha=False,
use_corr=True, logger=logging.getLogger("ridge_corr")):
"""Uses ridge regression with a bootstrapped held-out set to get optimal alpha values for each response.
[nchunks] random chunks of length [chunklen] will be taken from [Rstim] and [Rresp] for each regression
run. [nboots] total regression runs will be performed. The best alpha value for each response will be
averaged across the bootstraps to estimate the best alpha for that response.
If [joined] is given, it should be a list of lists where the STRFs for all the voxels in each sublist
will be given the same regularization parameter (the one that is the best on average).
Parameters
----------
Rstim : array_like, shape (TR, N)
Training stimuli with TR time points and N features. Each feature should be Z-scored across time.
Rresp : array_like, shape (TR, M)
Training responses with TR time points and M different responses (voxels, neurons, what-have-you).
Each response should be Z-scored across time.
Pstim : array_like, shape (TP, N)
Test stimuli with TP time points and N features. Each feature should be Z-scored across time.
Presp : array_like, shape (TP, M)
Test responses with TP time points and M different responses. Each response should be Z-scored across
time.
alphas : list or array_like, shape (A,)
Ridge parameters that will be tested. Should probably be log-spaced. np.logspace(0, 3, 20) works well.
nboots : int
The number of bootstrap samples to run. 15 to 30 works well.
chunklen : int
On each sample, the training data is broken into chunks of this length. This should be a few times
longer than your delay/STRF. e.g. for a STRF with 3 delays, I use chunks of length 10.
nchunks : int
The number of training chunks held out to test ridge parameters for each bootstrap sample. The product
of nchunks and chunklen is the total number of training samples held out for each sample, and this
product should be about 20 percent of the total length of the training data.
dtype : np.dtype
All data will be cast as this dtype for computation. np.single is used by default for memory efficiency,
as using np.double will thrash most machines on a big problem. If you want to do regression on
complex variables, this should be changed to np.complex128.
corrmin : float in [0..1]
Purely for display purposes. After each alpha is tested for each bootstrap sample, the number of
responses with correlation greater than this value will be printed. For long-running regressions this
can give a rough sense of how well the model works before it's done.
joined : None or list of array_like indices
If you want the STRFs for two (or more) responses to be directly comparable, you need to ensure that
the regularization parameter that they use is the same. To do that, supply a list of the response sets
that should use the same ridge parameter here. For example, if you have four responses, joined could
be [np.array([0,1]), np.array([2,3])], in which case responses 0 and 1 will use the same ridge parameter
(which will be parameter that is best on average for those two), and likewise for responses 2 and 3.
singcutoff : float
The first step in ridge regression is computing the singular value decomposition (SVD) of the
stimulus Rstim. If Rstim is not full rank, some singular values will be approximately equal
to zero and the corresponding singular vectors will be noise. These singular values/vectors
should be removed both for speed (the fewer multiplications the better!) and accuracy. Any
singular values less than singcutoff will be removed.
normalpha : boolean
Whether ridge parameters (alphas) should be normalized by the Frobenius norm of Rstim. Good for rigorously
comparing models with different numbers of parameters.
single_alpha : boolean
Whether to use a single alpha for all responses. Good for identification/decoding.
use_corr : boolean
If True, this function will use correlation as its metric of model fit. If False, this function
will instead use variance explained (R-squared) as its metric of model fit. For ridge regression
this can make a big difference -- highly regularized solutions will have very small norms and
will thus explain very little variance while still leading to high correlations, as correlation
is scale-free while R**2 is not.
Returns
-------
wt : array_like, shape (N, M)
Regression weights for N features and M responses.
corrs : array_like, shape (M,)
Validation set correlations. Predicted responses for the validation set are obtained using the regression
weights: pred = np.dot(Pstim, wt), and then the correlation between each predicted response and each
column in Presp is found.
alphas : array_like, shape (M,)
The regularization coefficient (alpha) selected for each voxel using bootstrap cross-validation.
bootstrap_corrs : array_like, shape (A, M, B)
Correlation between predicted and actual responses on randomly held out portions of the training set,
for each of A alphas, M voxels, and B bootstrap samples.
valinds : array_like, shape (TH, B)
The indices of the training data that were used as "validation" for each bootstrap sample.
"""
nresp, nvox = Rresp.shape
bestalphas = np.zeros((nboots, nvox)) ## Will hold the best alphas for each voxel
valinds = [] ## Will hold the indices into the validation data for each bootstrap
Rcmats = []
for bi in counter(range(nboots), countevery=1, total=nboots):
logger.info("Selecting held-out test set..")
allinds = range(nresp)
indchunks = zip(*[iter(allinds)]*chunklen)
random.shuffle(indchunks)
heldinds = list(itools.chain(*indchunks[:nchunks]))
notheldinds = list(set(allinds)-set(heldinds))
valinds.append(heldinds)
RRstim = Rstim[notheldinds,:]
PRstim = Rstim[heldinds,:]
RRresp = Rresp[notheldinds,:]
PRresp = Rresp[heldinds,:]
## Run ridge regression using this test set
Rcmat = ridge_corr(RRstim, PRstim, RRresp, PRresp, alphas,
dtype=dtype, corrmin=corrmin, singcutoff=singcutoff,
normalpha=normalpha, use_corr=use_corr)
Rcmats.append(Rcmat)
## Find weights for each voxel
try:
U,S,Vh = np.linalg.svd(Rstim, full_matrices=False)
except np.linalg.LinAlgError, e:
logger.info("NORMAL SVD FAILED, trying more robust dgesvd..")
from text.regression.svd_dgesvd import svd_dgesvd
U,S,Vh = svd_dgesvd(Rstim, full_matrices=False)
def bootstrap_ridge(Rstim, Rresp, Pstim, Presp, alphas, nboots, chunklen, nchunks, dtype=np.single, corrmin=0.2, joined=None, singcutoff=1e-10, normalpha=False, single_alpha=False, use_corr=True): """Uses ridge regression with a bootstrapped held-out set to get optimal alpha values for each response. [nchunks] random chunks of length [chunklen] will be taken from [Rstim] and [Rresp] for each regression run. [nboots] total regression runs will be performed. The best alpha value for each response will be averaged across the bootstraps to estimate the best alpha for that response. If [joined] is given, it should be a list of lists where the STRFs for all the voxels in each sublist will be given the same regularization parameter (the one that is the best on average). Parameters ---------- Rstim : array_like, shape (TR, N) Training stimuli with TR time points and N features. Each feature should be Z-scored across time. Rresp : array_like, shape (TR, M) Training responses with TR time points and M different responses (voxels, neurons, what-have-you). Each response should be Z-scored across time. Pstim : array_like, shape (TP, N) Test stimuli with TP time points and N features. Each feature should be Z-scored across time. Presp : array_like, shape (TP, M) Test responses with TP time points and M different responses. Each response should be Z-scored across time. alphas : list or array_like, shape (A,) Ridge parameters that will be tested. Should probably be log-spaced. np.logspace(0, 3, 20) works well. nboots : int The number of bootstrap samples to run. 15 to 30 works well. chunklen : int On each sample, the training data is broken into chunks of this length. This should be a few times longer than your delay/STRF. e.g. for a STRF with 3 delays, I use chunks of length 10. nchunks : int The number of training chunks held out to test ridge parameters for each bootstrap sample. The product of nchunks and chunklen is the total number of training samples held out for each sample, and this product should be about 20 percent of the total length of the training data. dtype : np.dtype All data will be cast as this dtype for computation. np.single is used by default for memory efficiency, as using np.double will thrash most machines on a big problem. If you want to do regression on complex variables, this should be changed to np.complex128. corrmin : float in [0..1] Purely for display purposes. After each alpha is tested for each bootstrap sample, the number of responses with correlation greater than this value will be printed. For long-running regressions this can give a rough sense of how well the model works before it's done. joined : None or list of array_like indices If you want the STRFs for two (or more) responses to be directly comparable, you need to ensure that the regularization parameter that they use is the same. To do that, supply a list of the response sets that should use the same ridge parameter here. For example, if you have four responses, joined could be [np.array([0,1]), np.array([2,3])], in which case responses 0 and 1 will use the same ridge parameter (which will be parameter that is best on average for those two), and likewise for responses 2 and 3. singcutoff : float The first step in ridge regression is computing the singular value decomposition (SVD) of the stimulus Rstim. If Rstim is not full rank, some singular values will be approximately equal to zero and the corresponding singular vectors will be noise. These singular values/vectors should be removed both for speed (the fewer multiplications the better!) and accuracy. Any singular values less than singcutoff will be removed. normalpha : boolean Whether ridge parameters (alphas) should be normalized by the Frobenius norm of Rstim. Good for rigorously comparing models with different numbers of parameters. single_alpha : boolean Whether to use a single alpha for all responses. Good for identification/decoding. use_corr : boolean If True, this function will use correlation as its metric of model fit. If False, this function will instead use variance explained (R-squared) as its metric of model fit. For ridge regression this can make a big difference -- highly regularized solutions will have very small norms and will thus explain very little variance while still leading to high correlations, as correlation is scale-free while R**2 is not. Returns ------- wt : array_like, shape (N, M) Regression weights for N features and M responses. corrs : array_like, shape (M,) Validation set correlations. Predicted responses for the validation set are obtained using the regression weights: pred = np.dot(Pstim, wt), and then the correlation between each predicted response and each column in Presp is found. alphas : array_like, shape (M,) The regularization coefficient (alpha) selected for each voxel using bootstrap cross-validation. bootstrap_corrs : array_like, shape (A, M, B) Correlation between predicted and actual responses on randomly held out portions of the training set, for each of A alphas, M voxels, and B bootstrap samples. valinds : array_like, shape (TH, B) The indices of the training data that were used as "validation" for each bootstrap sample. """ nresp, nvox = Rresp.shape bestalphas = np.zeros((nboots, nvox)) ## Will hold the best alphas for each voxel valinds = [] ## Will hold the indices into the validation data for each bootstrap Rcmats = [] for bi in counter(range(nboots), countevery=1, total=nboots): print "Selecting held-out test set.." allinds = range(nresp) indchunks = zip(*[iter(allinds)]*chunklen) random.shuffle(indchunks) heldinds = list(itools.chain(*indchunks[:nchunks])) notheldinds = list(set(allinds)-set(heldinds)) valinds.append(heldinds) RRstim = Rstim[notheldinds,:] PRstim = Rstim[heldinds,:] RRresp = Rresp[notheldinds,:] PRresp = Rresp[heldinds,:] ## Run ridge regression using this test set Rcmat = ridge_corr(RRstim, PRstim, RRresp, PRresp, alphas, dtype=dtype, corrmin=corrmin, singcutoff=singcutoff, normalpha=normalpha, use_corr=use_corr) Rcmats.append(Rcmat) ## Find weights for each voxel try: U,S,Vh = np.linalg.svd(Rstim, full_matrices=False) except np.linalg.LinAlgError, e: print "NORMAL SVD FAILED, trying more robust dgesvd.." from text.regression.svd_dgesvd import svd_dgesvd U,S,Vh = svd_dgesvd(Rstim, full_matrices=False)
