def test_arma_acov_compare_theoretical_arma_acov():
# Test against the older version of this function, which used a different
# approach that nicely shows the theoretical relationship
# See GH:5324 when this was removed for full version of the function
# including documentation and inline comments
def arma_acovf_historical(ar, ma, nobs=10):
if np.abs(np.sum(ar) - 1) > 0.9:
nobs_ir = max(1000, 2 * nobs)
else:
nobs_ir = max(100, 2 * nobs)
ir = arma_impulse_response(ar, ma, leads=nobs_ir)
while ir[-1] > 5 * 1e-5:
nobs_ir *= 10
ir = arma_impulse_response(ar, ma, leads=nobs_ir)
if nobs_ir > 50000 and nobs < 1001:
end = len(ir)
acovf = np.array([np.dot(ir[:end-nobs-t], ir[t:end-nobs])
for t in range(nobs)])
else:
acovf = np.correlate(ir, ir, 'full')[len(ir) - 1:]
return acovf[:nobs]
assert_allclose(arma_acovf([1, -0.5], [1, 0.2]),
arma_acovf_historical([1, -0.5], [1, 0.2]))
assert_allclose(arma_acovf([1, -0.99], [1, 0.2]),
arma_acovf_historical([1, -0.99], [1, 0.2]))
def test_arma_acov_compare_theoretical_arma_acov():
# Test against the older version of this function, which used a different
# approach that nicely shows the theoretical relationship
# See GH:5324 when this was removed for full version of the function
# including documentation and inline comments
def arma_acovf_historical(ar, ma, nobs=10):
if np.abs(np.sum(ar) - 1) > 0.9:
nobs_ir = max(1000, 2 * nobs)
else:
nobs_ir = max(100, 2 * nobs)
ir = arma_impulse_response(ar, ma, leads=nobs_ir)
while ir[-1] > 5 * 1e-5:
nobs_ir *= 10
ir = arma_impulse_response(ar, ma, leads=nobs_ir)
if nobs_ir > 50000 and nobs < 1001:
end = len(ir)
acovf = np.array(
[
np.dot(ir[: end - nobs - t], ir[t : end - nobs])
for t in range(nobs)
]
)
else:
acovf = np.correlate(ir, ir, "full")[len(ir) - 1 :]
return acovf[:nobs]
assert_allclose(
arma_acovf([1, -0.5], [1, 0.2]),
arma_acovf_historical([1, -0.5], [1, 0.2]),
)
assert_allclose(
arma_acovf([1, -0.99], [1, 0.2]),
arma_acovf_historical([1, -0.99], [1, 0.2]),
)
def test_innovations_algo_filter_kalman_filter(reset_randomstate):
# Test the innovations algorithm and filter against the Kalman filter
# for exact likelihood evaluation of an ARMA process
ar_params = np.array([0.5])
ma_params = np.array([0.2])
# TODO could generalize to sigma2 != 1, if desired, after #5324 is merged
# and there is a sigma2 argument to arma_acovf
# (but maybe this is not really necessary for the point of this test)
sigma2 = 1
endog = np.random.normal(size=10)
# Innovations algorithm approach
acovf = arma_acovf(np.r_[1, -ar_params], np.r_[1, ma_params],
nobs=len(endog))
theta, v = innovations_algo(acovf)
u = innovations_filter(endog, theta)
llf_obs = -0.5 * u ** 2 / (sigma2 * v) - 0.5 * np.log(2 * np.pi * v)
# Kalman filter apparoach
mod = SARIMAX(endog, order=(len(ar_params), 0, len(ma_params)))
res = mod.filter(np.r_[ar_params, ma_params, sigma2])
# Test that the two approaches are identical
atol = 1e-6 if PLATFORM_WIN else 0.0
assert_allclose(u, res.forecasts_error[0], rtol=1e-6, atol=atol)
assert_allclose(theta[1:, 0], res.filter_results.kalman_gain[0, 0, :-1],
atol=atol)
assert_allclose(llf_obs, res.llf_obs, atol=atol)
def test_innovations_algo_filter_kalman_filter(reset_randomstate):
# Test the innovations algorithm and filter against the Kalman filter
# for exact likelihood evaluation of an ARMA process
ar_params = np.array([0.5])
ma_params = np.array([0.2])
# TODO could generalize to sigma2 != 1, if desired, after #5324 is merged
# and there is a sigma2 argument to arma_acovf
# (but maybe this is not really necessary for the point of this test)
sigma2 = 1
endog = np.random.normal(size=10)
# Innovations algorithm approach
acovf = arma_acovf(np.r_[1, -ar_params], np.r_[1, ma_params],
nobs=len(endog))
theta, v = innovations_algo(acovf)
u = innovations_filter(endog, theta)
llf_obs = -0.5 * u**2 / (sigma2 * v) - 0.5 * np.log(2 * np.pi * v)
# Kalman filter apparoach
mod = SARIMAX(endog, order=(len(ar_params), 0, len(ma_params)))
res = mod.filter(np.r_[ar_params, ma_params, sigma2])
# Test that the two approaches are identical
atol = 1e-6 if PLATFORM_WIN else 0.0
assert_allclose(u, res.forecasts_error[0], atol=atol)
assert_allclose(theta[1:, 0], res.filter_results.kalman_gain[0, 0, :-1],
atol=atol)
assert_allclose(llf_obs, res.llf_obs, atol=atol)
def test_arma_acovf():
# Check for specific AR(1)
N = 20
phi = 0.9
sigma = 1
# rep 1: from module function
rep1 = arma_acovf([1, -phi], [1], N)
# rep 2: manually
rep2 = [1.0 * sigma * phi ** i / (1 - phi ** 2) for i in range(N)]
assert_allclose(rep1, rep2)
def test_arma_acovf():
# Check for specific AR(1)
N = 20
phi = 0.9
sigma = 1
# rep 1: from module function
rep1 = arma_acovf([1, -phi], [1], N)
# rep 2: manually
rep2 = [1. * sigma * phi ** i / (1 - phi ** 2) for i in range(N)]
assert_almost_equal(rep1, rep2, 7) # 7 is max precision here
def test_arma_acovf():
# Check for specific AR(1)
N = 20
phi = 0.9
sigma = 1
# rep 1: from module function
rep1 = arma_acovf([1, -phi], [1], N)
# rep 2: manually
rep2 = [1. * sigma * phi ** i / (1 - phi ** 2) for i in range(N)]
assert_almost_equal(rep1, rep2, 7) # 7 is max precision here
def _params2cov(self, params, nobs):
'''get autocovariance matrix from ARMA regression parameter
ar parameters are assumed to have rhs parameterization
'''
ar = np.r_[[1], -params[:self.nar]]
ma = np.r_[[1], params[-self.nma:]]
#print('ar', ar
#print('ma', ma
#print('nobs', nobs
autocov = arma_acovf(ar, ma, nobs=nobs)
#print('arma_acovf(%r, %r, nobs=%d)' % (ar, ma, nobs)
#print(autocov.shape
#something is strange fixed in aram_acovf
autocov = autocov[:nobs]
sigma = toeplitz(autocov)
return sigma
def _params2cov(self, params, nobs):
'''get autocovariance matrix from ARMA regression parameter
ar parameters are assumed to have rhs parameterization
'''
ar = np.r_[[1], -params[:self.nar]]
ma = np.r_[[1], params[-self.nma:]]
#print('ar', ar
#print('ma', ma
#print('nobs', nobs
autocov = arma_acovf(ar, ma, nobs=nobs)
#print('arma_acovf(%r, %r, nobs=%d)' % (ar, ma, nobs)
#print(autocov.shape
#something is strange fixed in aram_acovf
autocov = autocov[:nobs]
sigma = toeplitz(autocov)
return sigma
def test_innovations_algo_filter_kalman_filter(ar_params, ma_params, sigma2):
# Test the innovations algorithm and filter against the Kalman filter
# for exact likelihood evaluation of an ARMA process
ar = np.r_[1, -ar_params]
ma = np.r_[1, ma_params]
endog = np.random.normal(size=10)
nobs = len(endog)
# Innovations algorithm approach
arma_process_acovf = arma_acovf(ar, ma, nobs=nobs, sigma2=sigma2)
transformed_acov = _arma_innovations.darma_transformed_acovf_fast(
ar, ma, arma_process_acovf / sigma2)
acovf, acovf2 = (np.array(mv) for mv in transformed_acov)
theta, r = _arma_innovations.darma_innovations_algo_fast(
nobs, ar_params, ma_params, acovf, acovf2)
u = _arma_innovations.darma_innovations_filter(endog, ar_params, ma_params,
theta)
v = np.array(r) * sigma2
u = np.array(u)
llf_obs = -0.5 * u**2 / v - 0.5 * np.log(2 * np.pi * v)
# Kalman filter apparoach
mod = SARIMAX(endog, order=(len(ar_params), 0, len(ma_params)))
res = mod.filter(np.r_[ar_params, ma_params, sigma2])
# Test that the two approaches are identical
assert_allclose(u, res.forecasts_error[0])
# assert_allclose(theta[1:, 0], res.filter_results.kalman_gain[0, 0, :-1])
assert_allclose(llf_obs, res.llf_obs)
# Get llf_obs directly
llf_obs2 = _arma_innovations.darma_loglikeobs_fast(
endog, ar_params, ma_params, sigma2)
assert_allclose(llf_obs2, res.llf_obs)
def test_innovations_algo_filter_kalman_filter(ar_params, ma_params, sigma2):
# Test the innovations algorithm and filter against the Kalman filter
# for exact likelihood evaluation of an ARMA process
ar = np.r_[1, -ar_params]
ma = np.r_[1, ma_params]
endog = np.random.normal(size=10)
nobs = len(endog)
# Innovations algorithm approach
arma_process_acovf = arma_acovf(ar, ma, nobs=nobs, sigma2=sigma2)
acovf, acovf2 = np.array(_arma_innovations.darma_transformed_acovf_fast(
ar, ma, arma_process_acovf / sigma2))
theta, r = _arma_innovations.darma_innovations_algo_fast(
nobs, ar_params, ma_params, acovf, acovf2)
u = _arma_innovations.darma_innovations_filter(endog, ar_params, ma_params,
theta)
v = np.array(r) * sigma2
u = np.array(u)
llf_obs = -0.5 * u**2 / v - 0.5 * np.log(2 * np.pi * v)
# Kalman filter apparoach
mod = SARIMAX(endog, order=(len(ar_params), 0, len(ma_params)))
res = mod.filter(np.r_[ar_params, ma_params, sigma2])
# Test that the two approaches are identical
assert_allclose(u, res.forecasts_error[0])
# assert_allclose(theta[1:, 0], res.filter_results.kalman_gain[0, 0, :-1])
assert_allclose(llf_obs, res.llf_obs)
# Get llf_obs directly
llf_obs2 = _arma_innovations.darma_loglikeobs_fast(
endog, ar_params, ma_params, sigma2)
assert_allclose(llf_obs2, res.llf_obs)
[('ma1', acovf_ma1),
('ma2', acovf_ma2),
('arma11', acovf_arma11),
('ar1', acovf_arma11)])
cases = [('ma1', (ar0, ma1)),
('ma2', (ar0, ma2)),
('arma11', (ar1, ma1)),
('ar1', (ar1, ma0))]
for c, args in cases:
ar, ma = args
print('')
print(c, ar, ma)
myacovf = arma_acovf(ar, ma, nobs=10)
myacf = arma_acf(ar, ma, nobs=10)
if c[:2]=='ma':
othacovf = comparefn[c](ma)
else:
othacovf = comparefn[c](ar, ma)
print(myacovf[:5])
print(othacovf[:5])
#something broke again,
#for high persistence case eg ar=0.99, nobs of IR has to be large
#made changes to arma_acovf
assert_array_almost_equal(myacovf, othacovf,10)
assert_array_almost_equal(myacf, othacovf/othacovf[0],10)
#from nitime.utils
def test_brockwell_davis_ex533():
# See Brockwell and Davis (2009) - Time Series Theory and Methods
# Example 5.3.3: ARMA(1, 1) process, p.g. 177
nobs = 10
ar_params = np.array([0.2])
ma_params = np.array([0.4])
sigma2 = 8.92
p = len(ar_params)
q = len(ma_params)
m = max(p, q)
ar = np.r_[1, -ar_params]
ma = np.r_[1, ma_params]
# First, get the autocovariance of the process
arma_process_acovf = arma_acovf(ar, ma, nobs=nobs, sigma2=sigma2)
unconditional_variance = (
sigma2 * (1 + 2 * ar_params[0] * ma_params[0] + ma_params[0]**2) /
(1 - ar_params[0]**2))
assert_allclose(arma_process_acovf[0], unconditional_variance)
# Next, get the autocovariance of the transformed process
# Note: as required by {{prefix}}arma_transformed_acovf, we first divide
# through by sigma^2
arma_process_acovf /= sigma2
unconditional_variance /= sigma2
out = np.array(_arma_innovations.darma_transformed_acovf_fast(
ar, ma, arma_process_acovf))
acovf = np.array(out[0])
acovf2 = np.array(out[1])
# `acovf` is an m^2 x m^2 matrix, where m = max(p, q)
# but it is only valid for the autocovariances of the first m observations
# (this means in particular that the block `acovf[m:, m:]` should *not* be
# used)
# `acovf2` then contains the (time invariant) autocovariance terms for
# the observations m + 1, ..., nobs - since the autocovariance is the same
# for these terms, to save space we do not construct the autocovariance
# matrix as we did for the first m terms. Thus `acovf2[0]` is the variance,
# `acovf2[1]` is the first autocovariance, etc.
# Test the autocovariance function for observations m + 1, ..., nobs
# (it is time invariant here)
assert_equal(acovf2.shape, (nobs - m,))
assert_allclose(acovf2[0], 1 + ma_params[0]**2)
assert_allclose(acovf2[1], ma_params[0])
assert_allclose(acovf2[2:], 0)
# Test the autocovariance function for observations 1, ..., m
# (it is time varying here)
assert_equal(acovf.shape, (m * 2, m * 2))
# (we need to check `acovf[:m * 2, :m]`, i.e. `acovf[:2, :1])`
ix = np.diag_indices_from(acovf)
ix_lower = (ix[0][:-1] + 1, ix[1][:-1])
# acovf[ix] is the diagonal, and we want to check the first m
# elements of the diagonal
assert_allclose(acovf[ix][:m], unconditional_variance)
# acovf[ix_lower] is the first lower off-diagonal
assert_allclose(acovf[ix_lower][:m], ma_params[0])
# Now, check that we compute the moving average coefficients and the
# associated variances correctly
out = _arma_innovations.darma_innovations_algo_fast(
nobs, ar_params, ma_params, acovf, acovf2)
theta = np.array(out[0])
v = np.array(out[1])
# Test v (see eq. 5.3.13)
desired_v = np.zeros(nobs)
desired_v[0] = unconditional_variance
for i in range(1, nobs):
desired_v[i] = 1 + (1 - 1 / desired_v[i - 1]) * ma_params[0]**2
assert_allclose(v, desired_v)
# Test theta (see eq. 5.3.13)
# Note that they will have shape (nobs, m + 1) here, not (nobs, nobs - 1)
# as in the original (non-fast) version
assert_equal(theta.shape, (nobs, m + 1))
desired_theta = np.zeros(nobs)
for i in range(1, nobs):
desired_theta[i] = ma_params[0] / desired_v[i - 1]
assert_allclose(theta[:, 0], desired_theta)
assert_allclose(theta[:, 1:], 0)
# Test against Table 5.3.1
endog = np.array([
-1.1, 0.514, 0.116, -0.845, 0.872, -0.467, -0.977, -1.699, -1.228,
-1.093])
u = _arma_innovations.darma_innovations_filter(endog, ar_params, ma_params,
theta)
# Note: Table 5.3.1 has \hat X_n+1 = -0.5340 for n = 1, but this seems to
# be a typo, since equation 5.3.12 gives the form of the prediction
# equation as \hat X_n+1 = \phi X_n + \theta_n1 (X_n - \hat X_n)
# Then for n = 1 we have:
# \hat X_n+1 = 0.2 (-1.1) + (0.2909) (-1.1 - 0) = -0.5399
# And for n = 2 if we use what we have computed, then we get:
# \hat X_n+1 = 0.2 (0.514) + (0.3833) (0.514 - (-0.54)) = 0.5068
# as desired, whereas if we used the book's number for n=1 we would get:
# \hat X_n+1 = 0.2 (0.514) + (0.3833) (0.514 - (-0.534)) = 0.5045
# which is not what Table 5.3.1 shows.
desired_hat = np.array([
0, -0.540, 0.5068, -0.1321, -0.4539, 0.7046, -0.5620, -0.3614,
-0.8748, -0.3869])
desired_u = endog - desired_hat
assert_allclose(u, desired_u, atol=1e-4)
def test_brockwell_davis_ex534():
# See Brockwell and Davis (2009) - Time Series Theory and Methods
# Example 5.3.4: ARMA(1, 1) process, p.g. 178
nobs = 10
ar_params = np.array([1, -0.24])
ma_params = np.array([0.4, 0.2, 0.1])
sigma2 = 1
p = len(ar_params)
q = len(ma_params)
m = max(p, q)
ar = np.r_[1, -ar_params]
ma = np.r_[1, ma_params]
# First, get the autocovariance of the process
arma_process_acovf = arma_acovf(ar, ma, nobs=nobs, sigma2=sigma2)
assert_allclose(arma_process_acovf[:3],
[7.17133, 6.44139, 5.06027], atol=1e-5)
# Next, get the autocovariance of the transformed process
out = np.array(_arma_innovations.darma_transformed_acovf_fast(
ar, ma, arma_process_acovf))
acovf = np.array(out[0])
acovf2 = np.array(out[1])
# See test_brockwell_davis_ex533 for details on acovf vs acovf2
# Test acovf
assert_equal(acovf.shape, (m * 2, m * 2))
ix = np.diag_indices_from(acovf)
ix_lower1 = (ix[0][:-1] + 1, ix[1][:-1])
ix_lower2 = (ix[0][:-2] + 2, ix[1][:-2])
ix_lower3 = (ix[0][:-3] + 3, ix[1][:-3])
ix_lower4 = (ix[0][:-4] + 4, ix[1][:-4])
assert_allclose(acovf[ix][:m], 7.17133, atol=1e-5)
desired = [6.44139, 6.44139, 0.816]
assert_allclose(acovf[ix_lower1][:m], desired, atol=1e-5)
assert_allclose(acovf[ix_lower2][0], 5.06027, atol=1e-5)
assert_allclose(acovf[ix_lower2][1:m], 0.34, atol=1e-5)
assert_allclose(acovf[ix_lower3][:m], 0.1, atol=1e-5)
assert_allclose(acovf[ix_lower4][:m], 0, atol=1e-5)
# Test acovf2
assert_equal(acovf2.shape, (nobs - m,))
assert_allclose(acovf2[:4], [1.21, 0.5, 0.24, 0.1])
assert_allclose(acovf2[4:], 0)
# Test innovations algorithm output
out = _arma_innovations.darma_innovations_algo_fast(
nobs, ar_params, ma_params, acovf, acovf2)
theta = np.array(out[0])
v = np.array(out[1])
# Test v (see Table 5.3.2)
desired_v = [7.1713, 1.3856, 1.0057, 1.0019, 1.0016, 1.0005, 1.0000,
1.0000, 1.0000, 1.0000]
assert_allclose(v, desired_v, atol=1e-4)
# Test theta (see Table 5.3.2)
assert_equal(theta.shape, (nobs, m + 1))
desired_theta = np.array([
[0, 0.8982, 1.3685, 0.4008, 0.3998, 0.3992, 0.4000, 0.4000, 0.4000,
0.4000],
[0, 0, 0.7056, 0.1806, 0.2020, 0.1995, 0.1997, 0.2000, 0.2000, 0.2000],
[0, 0, 0, 0.0139, 0.0722, 0.0994, 0.0998, 0.0998, 0.0999, 0.1]]).T
assert_allclose(theta[:, :m], desired_theta, atol=1e-4)
assert_allclose(theta[:, m:], 0)
# Test innovations filter output
endog = np.array([1.704, 0.527, 1.041, 0.942, 0.555, -1.002, -0.585, 0.010,
-0.638, 0.525])
u = _arma_innovations.darma_innovations_filter(endog, ar_params, ma_params,
theta)
desired_hat = np.array([
0, 1.5305, -0.1710, 1.2428, 0.7443, 0.3138, -1.7293, -0.1688,
0.3193, -0.8731])
desired_u = endog - desired_hat
assert_allclose(u, desired_u, atol=1e-4)
def test_brockwell_davis_ex534():
# See Brockwell and Davis (2009) - Time Series Theory and Methods
# Example 5.3.4: ARMA(1, 1) process, p.g. 178
nobs = 10
ar_params = np.array([1, -0.24])
ma_params = np.array([0.4, 0.2, 0.1])
sigma2 = 1
p = len(ar_params)
q = len(ma_params)
m = max(p, q)
ar = np.r_[1, -ar_params]
ma = np.r_[1, ma_params]
# First, get the autocovariance of the process
arma_process_acovf = arma_acovf(ar, ma, nobs=nobs, sigma2=sigma2)
assert_allclose(arma_process_acovf[:3],
[7.17133, 6.44139, 5.06027], atol=1e-5)
# Next, get the autocovariance of the transformed process
transformed_acovf = _arma_innovations.darma_transformed_acovf_fast(
ar, ma, arma_process_acovf)
acovf, acovf2 = (np.array(arr) for arr in transformed_acovf)
# See test_brockwell_davis_ex533 for details on acovf vs acovf2
# Test acovf
assert_equal(acovf.shape, (m * 2, m * 2))
ix = np.diag_indices_from(acovf)
ix_lower1 = (ix[0][:-1] + 1, ix[1][:-1])
ix_lower2 = (ix[0][:-2] + 2, ix[1][:-2])
ix_lower3 = (ix[0][:-3] + 3, ix[1][:-3])
ix_lower4 = (ix[0][:-4] + 4, ix[1][:-4])
assert_allclose(acovf[ix][:m], 7.17133, atol=1e-5)
desired = [6.44139, 6.44139, 0.816]
assert_allclose(acovf[ix_lower1][:m], desired, atol=1e-5)
assert_allclose(acovf[ix_lower2][0], 5.06027, atol=1e-5)
assert_allclose(acovf[ix_lower2][1:m], 0.34, atol=1e-5)
assert_allclose(acovf[ix_lower3][:m], 0.1, atol=1e-5)
assert_allclose(acovf[ix_lower4][:m], 0, atol=1e-5)
# Test acovf2
assert_equal(acovf2.shape, (nobs - m,))
assert_allclose(acovf2[:4], [1.21, 0.5, 0.24, 0.1])
assert_allclose(acovf2[4:], 0)
# Test innovations algorithm output
out = _arma_innovations.darma_innovations_algo_fast(
nobs, ar_params, ma_params, acovf, acovf2)
theta = np.array(out[0])
v = np.array(out[1])
# Test v (see Table 5.3.2)
desired_v = [7.1713, 1.3856, 1.0057, 1.0019, 1.0016, 1.0005, 1.0000,
1.0000, 1.0000, 1.0000]
assert_allclose(v, desired_v, atol=1e-4)
# Test theta (see Table 5.3.2)
assert_equal(theta.shape, (nobs, m + 1))
desired_theta = np.array([
[0, 0.8982, 1.3685, 0.4008, 0.3998, 0.3992, 0.4000, 0.4000, 0.4000,
0.4000],
[0, 0, 0.7056, 0.1806, 0.2020, 0.1995, 0.1997, 0.2000, 0.2000, 0.2000],
[0, 0, 0, 0.0139, 0.0722, 0.0994, 0.0998, 0.0998, 0.0999, 0.1]]).T
assert_allclose(theta[:, :m], desired_theta, atol=1e-4)
assert_allclose(theta[:, m:], 0)
# Test innovations filter output
endog = np.array([1.704, 0.527, 1.041, 0.942, 0.555, -1.002, -0.585, 0.010,
-0.638, 0.525])
u = _arma_innovations.darma_innovations_filter(endog, ar_params, ma_params,
theta)
desired_hat = np.array([
0, 1.5305, -0.1710, 1.2428, 0.7443, 0.3138, -1.7293, -0.1688,
0.3193, -0.8731])
desired_u = endog - desired_hat
assert_allclose(u, desired_u, atol=1e-4)
'''
print('\n*********************\n chapter2: 1.5\n*********************')
p, q, acovf = 0, 2, (12.4168, -4.7520, 5.2)
maPara = armaME(p, q, acovf)
print('\n*********************\n chapter2: 2.4\n*********************')
p, q, acovf = 2, 2, (5.61, -1.1, 0.23, 0.43, -0.1)
armaPara = armaME(p, q, acovf)
'''
### chapter 3: 2.5
use function arima_process.arma_acovf() in statsmodels.tsa
for computing theoretical autocovariance function of ARMA process
'''
ar, ma, sigma2 = (1, -0.0894, 0.6265), (1, -0.3334, 0.8158), 4.0119
armaAcovf = arima_process.arma_acovf(ar, ma, nobs=11, sigma2=sigma2)
print('\n*********************\n chapter2: 2.5\n*********************')
for i, value in enumerate(armaAcovf):
if i > 4:
print(' γ%d = %.4f' % (i, value))
print('\n')
'''
### chapter 4: 3.4
use custom function _rejRate()
for computing rejection rate of White Noise hypothesis test
with different c2 for fixed c1, d1 and d2
'''
print('\n*********************\n chapter4: 3.4\n*********************')
print('\nCannot get all availabe (c1,c2,d1,d2)'
ma1 = [1., 0.4]
ma2 = [1., 0.4, 0.6]
ma0 = [1., 0.]
comparefn = dict([('ma1', acovf_ma1), ('ma2', acovf_ma2),
('arma11', acovf_arma11), ('ar1', acovf_arma11)])
cases = [('ma1', (ar0, ma1)), ('ma2', (ar0, ma2)), ('arma11', (ar1, ma1)),
('ar1', (ar1, ma0))]
for c, args in cases:
ar, ma = args
print('')
print(c, ar, ma)
myacovf = arma_acovf(ar, ma, nobs=10)
myacf = arma_acf(ar, ma, nobs=10)
if c[:2] == 'ma':
othacovf = comparefn[c](ma)
else:
othacovf = comparefn[c](ar, ma)
print(myacovf[:5])
print(othacovf[:5])
#something broke again,
#for high persistence case eg ar=0.99, nobs of IR has to be large
#made changes to arma_acovf
assert_array_almost_equal(myacovf, othacovf, 10)
assert_array_almost_equal(myacf, othacovf / othacovf[0], 10)
#from nitime.utils
def arma_innovations(endog,
ar_params=None,
ma_params=None,
sigma2=1,
normalize=False,
prefix=None):
"""
Compute innovations using a given ARMA process
Parameters
----------
endog : ndarray
The observed time-series process, may be univariate or multivariate.
ar_params : ndarray, optional
Autoregressive parameters.
ma_params : ndarray, optional
Moving average parameters.
sigma2 : ndarray, optional
The ARMA innovation variance. Default is 1.
normalize : boolean, optional
Whether or not to normalize the returned innovations. Default is False.
prefix : str, optional
The BLAS prefix associated with the datatype. Default is to find the
best datatype based on given input. This argument is typically only
used internally.
Returns
-------
innovations : ndarray
Innovations (one-step-ahead prediction errors) for the given `endog`
series with predictions based on the given ARMA process. If
`normalize=True`, then the returned innovations have been "whitened" by
dividing through by the square root of the mean square error.
innovations_mse : ndarray
Mean square error for the innovations.
"""
# Parameters
endog = np.array(endog)
squeezed = endog.ndim == 1
if squeezed:
endog = endog[:, None]
ar_params = np.atleast_1d([] if ar_params is None else ar_params)
ma_params = np.atleast_1d([] if ma_params is None else ma_params)
nobs, k_endog = endog.shape
ar = np.r_[1, -ar_params]
ma = np.r_[1, ma_params]
# Get BLAS prefix
if prefix is None:
prefix, dtype, _ = find_best_blas_type(
[endog, ar_params, ma_params,
np.array(sigma2)])
dtype = prefix_dtype_map[prefix]
# Make arrays contiguous for BLAS calls
endog = np.asfortranarray(endog, dtype=dtype)
ar_params = np.asfortranarray(ar_params, dtype=dtype)
ma_params = np.asfortranarray(ma_params, dtype=dtype)
sigma2 = dtype(sigma2).item()
# Get the appropriate functions
arma_transformed_acovf_fast = getattr(
_arma_innovations, prefix + 'arma_transformed_acovf_fast')
arma_innovations_algo_fast = getattr(_arma_innovations,
prefix + 'arma_innovations_algo_fast')
arma_innovations_filter = getattr(_arma_innovations,
prefix + 'arma_innovations_filter')
# Run the innovations algorithm for ARMA coefficients
arma_acovf = arima_process.arma_acovf(ar, ma, sigma2=sigma2,
nobs=nobs) / sigma2
acovf, acovf2 = arma_transformed_acovf_fast(ar, ma, arma_acovf)
theta, v = arma_innovations_algo_fast(nobs, ar_params, ma_params, acovf,
acovf2)
v = np.array(v)
if normalize:
v05 = v**0.5
# Run the innovations filter across each series
u = []
for i in range(k_endog):
u_i = np.array(
arma_innovations_filter(endog[:, i], ar_params, ma_params, theta))
u.append(u_i / v05 if normalize else u_i)
u = np.vstack(u).T
# Post-processing
if squeezed:
u = u.squeeze()
return u, v
def test_brockwell_davis_ex533():
# See Brockwell and Davis (2009) - Time Series Theory and Methods
# Example 5.3.3: ARMA(1, 1) process, p.g. 177
nobs = 10
ar_params = np.array([0.2])
ma_params = np.array([0.4])
sigma2 = 8.92
p = len(ar_params)
q = len(ma_params)
m = max(p, q)
ar = np.r_[1, -ar_params]
ma = np.r_[1, ma_params]
# First, get the autocovariance of the process
arma_process_acovf = arma_acovf(ar, ma, nobs=nobs, sigma2=sigma2)
unconditional_variance = (
sigma2 * (1 + 2 * ar_params[0] * ma_params[0] + ma_params[0]**2) /
(1 - ar_params[0]**2))
assert_allclose(arma_process_acovf[0], unconditional_variance)
# Next, get the autocovariance of the transformed process
# Note: as required by {{prefix}}arma_transformed_acovf, we first divide
# through by sigma^2
arma_process_acovf /= sigma2
unconditional_variance /= sigma2
transformed_acovf = _arma_innovations.darma_transformed_acovf_fast(
ar, ma, arma_process_acovf)
acovf, acovf2 = (np.array(arr) for arr in transformed_acovf)
# `acovf` is an m^2 x m^2 matrix, where m = max(p, q)
# but it is only valid for the autocovariances of the first m observations
# (this means in particular that the block `acovf[m:, m:]` should *not* be
# used)
# `acovf2` then contains the (time invariant) autocovariance terms for
# the observations m + 1, ..., nobs - since the autocovariance is the same
# for these terms, to save space we do not construct the autocovariance
# matrix as we did for the first m terms. Thus `acovf2[0]` is the variance,
# `acovf2[1]` is the first autocovariance, etc.
# Test the autocovariance function for observations m + 1, ..., nobs
# (it is time invariant here)
assert_equal(acovf2.shape, (nobs - m,))
assert_allclose(acovf2[0], 1 + ma_params[0]**2)
assert_allclose(acovf2[1], ma_params[0])
assert_allclose(acovf2[2:], 0)
# Test the autocovariance function for observations 1, ..., m
# (it is time varying here)
assert_equal(acovf.shape, (m * 2, m * 2))
# (we need to check `acovf[:m * 2, :m]`, i.e. `acovf[:2, :1])`
ix = np.diag_indices_from(acovf)
ix_lower = (ix[0][:-1] + 1, ix[1][:-1])
# acovf[ix] is the diagonal, and we want to check the first m
# elements of the diagonal
assert_allclose(acovf[ix][:m], unconditional_variance)
# acovf[ix_lower] is the first lower off-diagonal
assert_allclose(acovf[ix_lower][:m], ma_params[0])
# Now, check that we compute the moving average coefficients and the
# associated variances correctly
out = _arma_innovations.darma_innovations_algo_fast(
nobs, ar_params, ma_params, acovf, acovf2)
theta = np.array(out[0])
v = np.array(out[1])
# Test v (see eq. 5.3.13)
desired_v = np.zeros(nobs)
desired_v[0] = unconditional_variance
for i in range(1, nobs):
desired_v[i] = 1 + (1 - 1 / desired_v[i - 1]) * ma_params[0]**2
assert_allclose(v, desired_v)
# Test theta (see eq. 5.3.13)
# Note that they will have shape (nobs, m + 1) here, not (nobs, nobs - 1)
# as in the original (non-fast) version
assert_equal(theta.shape, (nobs, m + 1))
desired_theta = np.zeros(nobs)
for i in range(1, nobs):
desired_theta[i] = ma_params[0] / desired_v[i - 1]
assert_allclose(theta[:, 0], desired_theta)
assert_allclose(theta[:, 1:], 0)
# Test against Table 5.3.1
endog = np.array([
-1.1, 0.514, 0.116, -0.845, 0.872, -0.467, -0.977, -1.699, -1.228,
-1.093])
u = _arma_innovations.darma_innovations_filter(endog, ar_params, ma_params,
theta)
# Note: Table 5.3.1 has \hat X_n+1 = -0.5340 for n = 1, but this seems to
# be a typo, since equation 5.3.12 gives the form of the prediction
# equation as \hat X_n+1 = \phi X_n + \theta_n1 (X_n - \hat X_n)
# Then for n = 1 we have:
# \hat X_n+1 = 0.2 (-1.1) + (0.2909) (-1.1 - 0) = -0.5399
# And for n = 2 if we use what we have computed, then we get:
# \hat X_n+1 = 0.2 (0.514) + (0.3833) (0.514 - (-0.54)) = 0.5068
# as desired, whereas if we used the book's number for n=1 we would get:
# \hat X_n+1 = 0.2 (0.514) + (0.3833) (0.514 - (-0.534)) = 0.5045
# which is not what Table 5.3.1 shows.
desired_hat = np.array([
0, -0.540, 0.5068, -0.1321, -0.4539, 0.7046, -0.5620, -0.3614,
-0.8748, -0.3869])
desired_u = endog - desired_hat
assert_allclose(u, desired_u, atol=1e-4)
def arma_innovations(endog, ar_params=None, ma_params=None, sigma2=1,
normalize=False, prefix=None):
"""
Compute innovations using a given ARMA process
Parameters
----------
endog : ndarray
The observed time-series process, may be univariate or multivariate.
ar_params : ndarray, optional
Autoregressive parameters.
ma_params : ndarray, optional
Moving average parameters.
sigma2 : ndarray, optional
The ARMA innovation variance. Default is 1.
normalize : boolean, optional
Whether or not to normalize the returned innovations. Default is False.
prefix : str, optional
The BLAS prefix associated with the datatype. Default is to find the
best datatype based on given input. This argument is typically only
used internally.
Returns
-------
innovations : ndarray
Innovations (one-step-ahead prediction errors) for the given `endog`
series with predictions based on the given ARMA process. If
`normalize=True`, then the returned innovations have been "whitened" by
dividing through by the square root of the mean square error.
innovations_mse : ndarray
Mean square error for the innovations.
"""
# Parameters
endog = np.array(endog)
squeezed = endog.ndim == 1
if squeezed:
endog = endog[:, None]
ar_params = np.atleast_1d([] if ar_params is None else ar_params)
ma_params = np.atleast_1d([] if ma_params is None else ma_params)
nobs, k_endog = endog.shape
ar = np.r_[1, -ar_params]
ma = np.r_[1, ma_params]
# Get BLAS prefix
if prefix is None:
prefix, dtype, _ = find_best_blas_type(
[endog, ar_params, ma_params, np.array(sigma2)])
dtype = prefix_dtype_map[prefix]
# Make arrays contiguous for BLAS calls
endog = np.asfortranarray(endog, dtype=dtype)
ar_params = np.asfortranarray(ar_params, dtype=dtype)
ma_params = np.asfortranarray(ma_params, dtype=dtype)
sigma2 = dtype(sigma2).item()
# Get the appropriate functions
arma_transformed_acovf_fast = getattr(
_arma_innovations, prefix + 'arma_transformed_acovf_fast')
arma_innovations_algo_fast = getattr(
_arma_innovations, prefix + 'arma_innovations_algo_fast')
arma_innovations_filter = getattr(
_arma_innovations, prefix + 'arma_innovations_filter')
# Run the innovations algorithm for ARMA coefficients
arma_acovf = arima_process.arma_acovf(ar, ma,
sigma2=sigma2, nobs=nobs) / sigma2
acovf, acovf2 = arma_transformed_acovf_fast(ar, ma, arma_acovf)
theta, v = arma_innovations_algo_fast(nobs, ar_params, ma_params,
acovf, acovf2)
v = np.array(v)
if normalize:
v05 = v**0.5
# Run the innovations filter across each series
u = []
for i in range(k_endog):
u_i = np.array(arma_innovations_filter(endog[:, i], ar_params,
ma_params, theta))
u.append(u_i / v05 if normalize else u_i)
u = np.vstack(u).T
# Post-processing
if squeezed:
u = u.squeeze()
return u, v