def _next_time_step(self, prev_xa, prev_Pa, xo_in_AW):
N = prev_xa.size
xa = prev_xa
Pa = prev_Pa
for j in range(self.J):
H = ffa.get_H(xo_in_AW[j])
the_xo = ffa.left_aligned(xo_in_AW[j])
the_xf = self.l96.run(xa, days=self.assim_interval_days * (j + 1))
M = self.l96.jacobian(xa, days=self.assim_interval_days * (j + 1))
PaMTHT = Pa @ M.T @ H.T
K = PaMTHT @ np.linalg.inv(H @ M @ PaMTHT + self.R)
the_xa = xa + K @ (the_xo - (H @ the_xf))
the_Pa = (np.identity(N) - K @ H @ M) @ Pa
# xa, Paをfor loopの中でupdateする
xa = the_xa
Pa = the_Pa
xa_in_AW = self.l96.get_xf_in_AW(xa,
self.J,
days=self.assim_interval_days)
M = self.l96.jacobian(xa, days=self.assim_interval_days * self.J)
the_Pa = (1 + self.delta) * M @ Pa @ M.T
# xaは、時刻t=-1に於ける解析値
# xa_in_AWは、時刻t=0~J-1に於ける解析値
# the_Paは、時刻t=J-1に於ける解析誤差共分散行列
return xa_in_AW, xa, the_Pa
def _next_time_step(self, prev_Xa, the_xo):
H = ffa.get_H(the_xo)
Xf = self.l96.ensemble_run(prev_Xa, days=self.assim_interval_days)
Xf_bar = np.average(Xf, axis=1)
# broadcasting
dXf = np.sqrt(1 + self.delta) * (Xf - Xf_bar[:, None])
dYf = H.dot(dXf)
# broadcasting
K = dXf @ dYf.T @ np.linalg.inv(dYf @ dYf.T + (self.m - 1) * self.R)
# K = dXf.dot(dYf.T.dot(np.linalg.inv(dYf.dot(dYf.T) + (self.m - 1) * self.R)))
the_xo = ffa.left_aligned(the_xo)
# Xa = np.zeros((self.N, self.m))
# for k in range(self.m):
# # Perturbation。理論上はalpha = 1
# e = self.alpha * np.random.randn(self.p)
# Kxoxf = K @ (the_xo + e - H @ Xf[:, k])
# # Kxoxf = K @ (the_xo + e - H @ Xf[:, k].reshape[0]).reshape(K.shape[0])
# # Kxoxf = K.dot(
# # the_xo + e - H.dot(Xf[:, k]).reshape(H.shape[0])).reshape(K.shape[0])
# Xa[:, k] = Xf[:, k] + Kxoxf
# 上記のコードを最適化すると以下のとおりとなる
e_mat = self.alpha * np.random.randn(self.p, self.m)
Xa = Xf + K @ (the_xo[:, None] + e_mat - H @ Xf)
return Xa
def _next_time_step_estimating_F(prev_xa, prev_F, prev_Pa, the_xo, R, delta, delta_of_F,
dt=0.01, days=0.25, KSC=1e+5):
N = prev_xa.size # 解析値の地点数
the_xf = Lorenz96.run(prev_xa, F=prev_F, dt=dt, days=days)
M = Lorenz96.jacobian_for_s(prev_xa, dt=dt, days=days, F=prev_F)
# Fの部分だけ別の値でinflationする。
the_Pf = (1 + delta) * M @ prev_Pa @ M.T
the_Pf[N, :] = (1 + delta_of_F) * the_Pf[N, :] / (1 + delta)
# the_Pf[:, N] = (1 + delta_of_F) * the_Pf[:, N] / (1 + delta)
# the_Pf[N, N] = (1 + delta_of_F) * the_Pf[N, N] / (1 + delta)
# ↑このコメントアウトを外すと発散してしまう。
# あくまでクロスタームに対応する部分だけをFに関する部分と見なす。
H = ffa.get_H(the_xo, KSC=KSC)
H = ffa.get_H_for_s(H, 1)
K = the_Pf @ H.T @ np.linalg.inv(H @ the_Pf @ H.T + R)
the_sf = np.hstack((the_xf, prev_F))
the_xo = ffa.left_aligned(the_xo, KSC=KSC)
the_sa = the_sf + K @ (the_xo - H @ the_sf)
the_xa = the_sa[:N]
the_F = the_sa[the_sa.size - 1]
the_Pa = (np.identity(K.shape[0]) - K @ H) @ the_Pf
return the_xa, the_Pa, the_F
def _next_time_step(self, prev_xa, prev_Pa, the_xo):
# 予報
the_xf = self.l96.run(prev_xa, days=self.assim_interval_days)
# カルマンゲイン計算
M = self.l96.jacobian(prev_xa, days=self.assim_interval_days)
MPM = M @ prev_Pa @ M.T
the_Pf = (1 + self.delta) * MPM
H = ffa.get_H(the_xo)
K = the_Pf @ H.T @ np.linalg.inv(H @ the_Pf @ H.T + self.R)
# 解析値算出
the_xo = ffa.left_aligned(the_xo)
d_ob = the_xo - H @ the_xf
the_xa = the_xf + K @ d_ob
the_Pa = (np.identity(self.N) - K @ H) @ the_Pf
# inflation deltaの推定
d_ab = the_xa - the_xf
Cov_for_est_Pf1 = using_jit.cal_covmat(H @ d_ab, d_ob)
Cov_for_est_Pf2 = using_jit.cal_covmat(d_ob, d_ob) - self.R
# Rの推定
d_oa = the_xo - H @ the_xa
Cov_for_est_R = using_jit.cal_covmat(d_oa, d_ob)
# 推定値保存
self.Cov_for_est_Pf[self.l] = Cov_for_est_Pf1
self.Cov_for_est_Pf2[self.l] = Cov_for_est_Pf2
self.Cov_for_est_R[self.l] = Cov_for_est_R
self.Pf[self.l] = the_Pf
return the_xa, the_Pa
def _next_time_step(self, prev_Enxa, the_xo):
# 予報
Xf = self.l96.ensemble_run(prev_Enxa, days=self.assim_interval_days)
# 予測にノイズを加える(発散対策)
Xf += self.delta * np.random.randn(*Xf.shape)
# 観測行列作成・xoを欠損値抜きにして寄せる。
H = ffa.get_H(the_xo)
# 観測があるグリッドのインデックス
indices_not_ms = np.arange(self.N)[~np.isnan(the_xo)]
the_xo = ffa.left_aligned(the_xo)
# 各アンサンブル(粒子)に対して尤度を求める
lkhs = self.likelihood(Xf, the_xo, H)
# 重みwの計算
w = get_w(lkhs)
# Xf[:, i]から、重みwによって「重み付きサンプリング」することで解析値を求める
Xa = resampling(Xf, w, indices_not_ms)
# メンバ変数に保存
self.w_timeseries[self.l] = w
self.Enxf[self.l] = Xf
self.Xf[self.l] = np.average(Xf, axis=1)
return Xa
def _next_time_step(self, prev_xa, xo_in_AW):
# jedit_flg = 0の時は何も表示させない。 = 1の時は評価関数Jが小さくなっている様子を表示させる。
days = self.assim_interval_days
J = self.J
N = self.N
p = self.p
H_in_AW = np.zeros((J, p, N))
xo_in_AW_aligned = np.zeros((J, p))
for j in range(J):
xo_in_AW_aligned[j] = ffa.left_aligned(xo_in_AW[j])
H_in_AW[j] = ffa.get_H(xo_in_AW[j])
# 参考：http://org-technology.com/posts/scipy-unconstrained-minimization-of-multivariate-scalar-functions.html
def ObjectiveFunction(x):
xf_in_AW = self.l96.get_xf_in_AW(x, J, days)
obs_term = 0.0
for j in range(J):
d = H_in_AW[j] @ xf_in_AW[j] - xo_in_AW_aligned[j]
obs_term += 0.5 * d @ self.Rinv @ d
return 0.5 * (x - prev_xa) @ self.Binv @ (x - prev_xa) + obs_term
def gradient(x):
xf_in_AW = self.l96.get_xf_in_AW(x, J, days)
M_in_AW = self.l96.get_M_in_AW(x, J, days, analysis=self.analysis)
obs_term = 0.0
for j in range(J):
Mj = M_in_AW[j]
Hj = H_in_AW[j]
# 線形アジョイントモデル
obs_term += Mj.T @ Hj.T @ self.Rinv @ (Hj @ xf_in_AW[j] -
xo_in_AW_aligned[j])
return self.Binv @ (x - prev_xa) + obs_term
# 初期値はランダム。ただし収束を速くするために、prev_xaとする
x0 = prev_xa
if self.optimization_method == 'l-bfgs':
# scipyを用いて最適化し、ObjectiveFunctionを最小にするxを求める。それがx0になる
res = minimize(ObjectiveFunction,
x0,
jac=gradient,
method='l-bfgs-b')
x0 = res.x
elif self.optimization_method == 'steepest':
# 最急降下法を自分で実装してやってみた。
x0 = self.steepest_descent_method(gradient, x0)
else:
raise ValueError("your optimization method is not valid!")
xa_in_AW = self.l96.get_xf_in_AW(x0, J, days)
if abs(xa_in_AW[J - 1, 0] - self.l96.run(x0, J * days)[0]) > 1e-14:
raise ValueError('your calculation is not valid!')
return xa_in_AW, x0
def _next_time_step(self, prev_Xa, xo_in_AW):
Xa = prev_Xa
for j in range(self.J):
# このfor loopの中で同化ウィンドウ内の観測を取り込んで、Xaをアップデートしていく。
# 観測が存在するモデルグリッド
Not_ms = np.arange(self.N)[~np.isnan(xo_in_AW[j])]
Xf = self.l96.ensemble_run(Xa,
days=self.assim_interval_days * (j + 1))
Xf_bar = np.average(Xf, axis=1)
dXf = Xf - Xf_bar[:, None]
Pf = dXf @ dXf.T / (self.m - 1)
the_xo = ffa.left_aligned(xo_in_AW[j])
for i in range(self.p):
# for loop Obs start
obs_point = Not_ms[i]
RHO = self.k_localizer.get_rho(obs_point)
Xa_bar = np.average(Xa, axis=1)
if i == 0:
dXa = np.sqrt(1 + self.delta) * (Xa - Xa_bar[:, None])
else:
dXa = Xa - Xa_bar[:, None]
# 観測点1点に対応するHの計算
x_for_making_H = np.full(self.N, np.nan)
x_for_making_H[obs_point] = 0.0 # 使う観測点1点だけをKSC以外の値に
H = ffa.get_H(x_for_making_H) # 観測1点に対応するH
# 観測点1点に対応するHの計算
dYf = H @ dXf
localR = self.R[i, i] # スカラー
localPf = Pf[obs_point, obs_point] # スカラー
K = RHO[:, None] * dXa @ dYf.T / ((self.m - 1) *
(localPf + localR))
# アンサンブルアップデート(第一推定値の置き換え)
Xa_bar_new = Xa_bar + K @ (the_xo[i] - H @ Xf_bar)
alpha = 1.0 / (1.0 + np.sqrt(localR / (localR + localPf)))
K_childa = alpha * K
dXa_new = dXa - K_childa @ H @ dXf
# アンサンブルアップデート
Xa = Xa_bar_new[:, None] + dXa_new
# for loop Obs end
xa = np.average(Xa, axis=1)
xa_in_AW = self.l96.get_xf_in_AW(xa,
self.J,
days=self.assim_interval_days)
the_Xa = self.l96.ensemble_run(Xa,
days=self.assim_interval_days * self.J)
return xa_in_AW, xa, the_Xa
def _next_time_step(self, prev_xa, prev_Pa, next_xo):
# 時刻t=Tに於ける観測を同化して、時刻t=0に於ける解析値を計算する。
# 解析値を算出するために、次のステップにおける予報値も計算する必要がある。
the_xf = self.l96.run(prev_xa, days=self.assim_interval_days)
next_xf = self.l96.run(the_xf, days=self.assim_interval_days)
M = self.l96.jacobian(prev_xa, days=self.assim_interval_days)
the_Pf = (1 + self.delta) * M @ prev_Pa @ M.T
H = ffa.get_H(next_xo)
PfMTHT = the_Pf @ M.T @ H.T
K = PfMTHT @ np.linalg.inv(H @ M @ PfMTHT + self.R)
next_xo = ffa.left_aligned(next_xo)
the_xa = the_xf + K @ (next_xo - H @ next_xf)
the_Pa = (np.identity(self.N) - K @ H @ M) @ the_Pf
return the_xa, the_Pa
def _next_time_step_estimating_dynamicF(prev_xa, prev_F, prev_Pa, the_xo, R, dt=0.01, days=0.25,
delta=0.0, delta_of_F=1e-2, KSC=1e+5):
N = prev_xa.size # 解析値の地点数
the_xf = Lorenz96.run_dynamic_F(prev_xa, F=prev_F, dt=dt, days=days)
M = Lorenz96.jacobian_dynamic_F(prev_xa, dt=dt, days=days, F=prev_F)
the_Pf = (1 + delta) * M @ prev_Pa @ M.T
the_Pf[N:, :] = (1 + delta_of_F) * the_Pf[N:, :] / (1 + delta)
H = ffa.get_H(the_xo, KSC=KSC)
H = ffa.get_H_for_s(H, prev_F.size)
K = the_Pf @ H.T @ np.linalg.inv(H @ the_Pf @ H.T + R)
the_sf = np.hstack((the_xf, prev_F))
the_xo = ffa.left_aligned(the_xo, KSC=KSC)
the_sa = the_sf + K @ (the_xo - H @ the_sf)
the_xa = the_sa[:N]
the_F = the_sa[N:]
the_Pa = (np.identity(K.shape[0]) - K @ H) @ the_Pf
return the_xa, the_F, the_Pa, the_xf, the_Pf
def _next_time_step(self, prev_Xa, next_xo):
# 時刻6hの観測データを同化して時刻0hにおける解析値を計算する
Not_ms = np.arange(self.N)[~np.isnan(next_xo)] # 観測が存在するモデルグリッド
the_Xf = self.l96.ensemble_run(prev_Xa, days=self.assim_interval_days)
next_Xf = self.l96.ensemble_run(the_Xf, days=self.assim_interval_days)
next_xo = ffa.left_aligned(next_xo)
next_Xf_bar = np.average(next_Xf, axis=1)
next_dXf = next_Xf - next_Xf_bar[:, None]
next_Pf = next_dXf @ next_dXf.T / (self.m - 1)
for i in range(self.p):
obs_point = Not_ms[i]
RHO = self.k_localizer.get_rho(obs_point)
the_Xf_bar = np.average(the_Xf, axis=1)
if i == 0:
the_dXf = np.sqrt(1 + self.delta) * (the_Xf - the_Xf_bar[:, None])
else:
the_dXf = the_Xf - the_Xf_bar[:, None]
# 観測点1点に対応するHの計算
x_for_making_H = np.full(self.N, np.nan)
x_for_making_H[obs_point] = 0.0 # 使う観測点1点だけをKSC以外の値に
H = ffa.get_H(x_for_making_H) # 観測1点に対応するH
# 観測点1点に対応するHの計算
next_dYf = H @ next_dXf
localR = self.R[i, i] # スカラー
localPf = next_Pf[obs_point, obs_point] # スカラー
K = RHO[:, None] * the_dXf @ next_dYf.T / ((self.m - 1) * (localPf + localR))
# アンサンブルアップデート(第一推定値の置き換え)
Xa_bar = the_Xf_bar + K @ (next_xo[i] - H @ next_Xf_bar)
alpha = 1.0 / (1.0 + np.sqrt(localR / (localR + localPf)))
K_childa = alpha * K
dXa = the_dXf - K_childa @ H @ next_dXf
# アンサンブルアップデート
the_Xf = Xa_bar[:, None] + dXa
Xa = Xa_bar[:, None] + dXa
return Xa
def _next_time_step(self, prev_xa, the_xo):
the_xf = self.l96.run(prev_xa, days=self.assim_interval_days)
H = ffa.get_H(the_xo)
K = self.B @ H.T @ (np.linalg.inv(H @ self.B @ H.T + self.R))
the_xo = ffa.left_aligned(the_xo)
d_ob = the_xo - H @ the_xf
the_xa = the_xf + K @ d_ob
# Bの推定
# なお、BそのものではなくHBH.Tを推定する。
d_ab = the_xa - the_xf
est_B1 = using_jit.cal_covmat(H @ d_ab, d_ob)
est_B2 = using_jit.cal_covmat(d_ob, d_ob) - self.R
# Rの推定
d_oa = the_xo - H @ the_xa
Cov_for_est_R = using_jit.cal_covmat(d_oa, d_ob)
return the_xa, est_B1, est_B2, Cov_for_est_R
def _next_time_step(self, prev_Xa, xo_in_AW):
Xa = prev_Xa
for j in range(self.J):
# このfor loopの中で同化ウィンドウ内の観測を取り込んで、Xaをアップデートしていく。
H = ffa.get_H(xo_in_AW[j])
the_xo = ffa.left_aligned(xo_in_AW[j])
Xa_bar = np.average(Xa, axis=1)
dXa = Xa - Xa_bar[:, None]
Xf = self.l96.ensemble_run(Xa, days=self.assim_interval_days * (j + 1))
Xf_bar = np.average(Xf, axis=1)
dXf = Xf - Xf_bar[:, None]
dYf = H @ dXf
A = (self.m - 1) / self.rho * np.identity(self.m) + dYf.T @ self.Rinv @ dYf
U, D = ffa.Eigenvalue_decomp(A)
Dinv = np.diag(1. / np.diag(D)) # Dは対角行列より、np.linalg.invよりこっちのほうが高速
sqrtDinv = np.sqrt(Dinv) # Dinvは対角行列より、np.sqrtでMatrix square rootが求まる
K = dXa @ U @ Dinv @ U.T @ dYf.T @ self.Rinv
T = np.sqrt(self.m - 1) * U @ sqrtDinv @ U.T
Kxoxf = K @ (the_xo - H @ Xf_bar)
# すべての列に同じ配列を格納している。(N, m)の行列になる。
# 以下のコードの最適化。
# Xa_bar_MAT = np.zeros((self.N, self.m))
# Kxoxf_MAT = np.zeros((self.N, self.m))
# for k in range(self.m):
# Xa_bar_MAT[:, k] = Xa_bar[:]
# Kxoxf_MAT[:, k] = Kxoxf[:]
Xa_bar_MAT = np.repeat(Xa_bar[:, None], self.m, axis=1)
Kxoxf_MAT = np.repeat(Kxoxf[:, None], self.m, axis=1)
Xa = Xa_bar_MAT + Kxoxf_MAT + dXa @ T
xa = np.average(Xa, axis=1)
xa_in_AW = self.l96.get_xf_in_AW(xa, self.J, days=self.assim_interval_days)
the_Xa = self.l96.ensemble_run(Xa, days=self.assim_interval_days * self.J)
return xa_in_AW, xa, the_Xa
def _next_time_step(self, prev_Xa, next_xo):
H = ffa.get_H(next_xo)
next_xo = ffa.left_aligned(next_xo)
the_Xf = self.l96.ensemble_run(prev_Xa, days=self.assim_interval_days)
the_Xf_bar = np.average(the_Xf, axis=1)
the_dXf = the_Xf - the_Xf_bar[:, None]
next_Xf = self.l96.ensemble_run(the_Xf, days=self.assim_interval_days)
next_Xf_bar = np.average(next_Xf, axis=1)
next_dXf = next_Xf - next_Xf_bar[:, None]
next_dYf = H @ next_dXf
A = (self.m - 1) / self.rho * np.identity(
self.m) + next_dYf.T @ self.Rinv @ next_dYf
U, D = ffa.Eigenvalue_decomp(A)
Dinv = np.diag(1. / np.diag(D)) # Dは対角行列より、np.linalg.invよりこっちのほうが高速
sqrtDinv = np.sqrt(Dinv) # Dinvは対角行列より、np.sqrtでMatrix square rootが求まる
K = the_dXf @ U @ Dinv @ U.T @ next_dYf.T @ self.Rinv
T = np.sqrt(self.m - 1) * U @ sqrtDinv @ U.T
Kxoxf = K @ (next_xo - H @ next_Xf_bar)
# すべての列に同じ配列を格納している。(N, m)の行列になる。
# 以下のコードの最適化。
# the_Xf_bar_MAT = np.zeros((self.N, self.m))
# Kxoxf_MAT = np.zeros((self.N, self.m))
# for k in range(self.m):
# the_Xf_bar_MAT[:, k] = the_Xf_bar[:]
# Kxoxf_MAT[:, k] = Kxoxf[:]
the_Xf_bar_MAT = np.repeat(the_Xf_bar[:, None], self.m, axis=1)
Kxoxf_MAT = np.repeat(Kxoxf[:, None], self.m, axis=1)
Xa = the_Xf_bar_MAT + Kxoxf_MAT + the_dXf @ T
return Xa