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)
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_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 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)