def elementwise_divide(a: MatrixVector2D, b: matrix) -> MatrixVector2D:
if a.shape == b.shape:
x = divide(a.x, b)
y = divide(a.y, b)
return MatrixVector2D(x, y)
else:
raise ValueError('MatrixVector2D and matrix shapes not the same.')
def elementwise_divide(a: MatrixVector, b: matrix) -> MatrixVector:
if a.shape == b.shape:
x = divide(a.x, b)
y = divide(a.y, b)
z = divide(a.z, b)
return MatrixVector(x, y, z)
else:
raise ValueError('The shape of a and b need to be the same.')
def elementwise_divide(a: MatrixVector, b: matrix) -> MatrixVector:
if a.shape == b.shape:
x = divide(a.x, b)
y = divide(a.y, b)
z = divide(a.z, b)
return MatrixVector(x, y, z)
else:
raise ValueError()
def phi_doublet_matrix(vecs: MatrixVector, rls: MatrixVector, sgnz: matrix):
mags = vecs.return_magnitude()
ms = divide(vecs.x, rls.y)
ths = arctan(ms)
ths[rls.y == 0.0] = piby2
gs = multiply(ms, divide(rls.z, mags))
Js = arctan(gs)
Js[rls.y == 0.0] = piby2
phids = Js - multiply(sgnz, ths)
return phids, mags
def phi_trailing_doublet_matrix(rls: MatrixVector, sgnz: matrix):
ths = zeros(rls.shape, dtype=float)
chky = absolute(rls.y) < tol
ths[rls.y > 0.0] = piby2
ths[rls.y < 0.0] = -piby2
ths[chky] = -piby2
gs = zeros(rls.shape, dtype=float)
divide(rls.z, rls.y, where=logical_not(chky), out=gs)
Js = arctan(gs)
Js[rls.y == 0.0] = -piby2
phids = Js - multiply(sgnz, ths)
return phids
def vel_doublet_matrix(av, am, bv, bm):
adb = elementwise_dot_product(av, bv)
abm = multiply(am, bm)
dm = multiply(abm, abm+adb)
axb = elementwise_cross_product(av, bv)
axbm = axb.return_magnitude()
chki = (axbm == 0.0)
chki = logical_and(axbm >= -tol, axbm <= tol)
chkd = absolute(dm) < tol
fac = zeros(axbm.shape, dtype=float)
divide(am+bm, dm, where=logical_not(chkd), out=fac)
velvl = elementwise_multiply(axb, fac)
velvl.x[chki] = 0.0
velvl.y[chki] = 0.0
velvl.z[chki] = 0.0
return velvl
def phi_doublet_matrix(vecs: MatrixVector, sgnz: matrix):
mags = vecs.return_magnitude()
chkm = mags < tol
chky = absolute(vecs.y) < tol
vecs.y[chky] = 0.0
ms = zeros(mags.shape, dtype=float)
divide(vecs.x, vecs.y, where=logical_not(chky), out=ms)
ths = arctan(ms)
ths[chky] = piby2
ts = zeros(mags.shape, dtype=float)
divide(vecs.z, mags, where=logical_not(chkm), out=ts)
gs = multiply(ms, ts)
Js = arctan(gs)
Js[chky] = piby2
phids = Js - multiply(sgnz, ths)
return phids, mags
def __init__(self,order,cutoff,Fs,filtertype):
self.order = order
self.Fs = Fs
self.Wn = self.Fs*0.5 # Time 0.5 to create float type
self.type = filtertype;
self.num, self.den = butter(self.order,np.divide(cutoff,self.Wn),self.type)
self.A, self.B, self.C, self.D = tf2ss(self.num,self.den)
self.Xnn = np.zeros((np.size(self.A,1),1))
def velocity_matrix(ra: MatrixVector,
rb: MatrixVector,
rc: MatrixVector,
betm: float = 1.0,
tol: float = 1e-12):
if ra.shape != rb.shape:
return ValueError()
numi = rc.shape[0]
numj = ra.shape[1]
ra = ra.repeat(numi, axis=0)
rb = rb.repeat(numi, axis=0)
rc = rc.repeat(numj, axis=1)
a = rc - ra
b = rc - rb
a.x = a.x / betm
b.x = b.x / betm
am = a.return_magnitude()
bm = b.return_magnitude()
# Velocity from Bound Vortex
adb = elementwise_dot_product(a, b)
abm = multiply(am, bm)
dm = multiply(abm, abm + adb)
axb = elementwise_cross_product(a, b)
axbm = axb.return_magnitude()
chki = (axbm == 0.0)
chki = logical_and(axbm >= -tol, axbm <= tol)
veli = elementwise_multiply(axb, divide(am + bm, dm))
veli.x[chki] = 0.0
veli.y[chki] = 0.0
veli.z[chki] = 0.0
# Velocity from Trailing Vortex A
axx = MatrixVector(zeros(a.shape, dtype=float), a.z, -a.y)
axxm = axx.return_magnitude()
chka = (axxm == 0.0)
vela = elementwise_divide(axx, multiply(am, am - a.x))
vela.x[chka] = 0.0
vela.y[chka] = 0.0
vela.z[chka] = 0.0
# Velocity from Trailing Vortex B
bxx = MatrixVector(zeros(b.shape, dtype=float), b.z, -b.y)
bxxm = bxx.return_magnitude()
chkb = (bxxm == 0.0)
velb = elementwise_divide(bxx, multiply(bm, bm - b.x))
velb.x[chkb] = 0.0
velb.y[chkb] = 0.0
velb.z[chkb] = 0.0
return veli, vela, velb
def phi_source_matrix(am, bm, dab, rl, phid):
numrab = am + bm + dab
denrab = am + bm - dab
Pab = divide(numrab, denrab)
Pab[denrab == 0.0] = 1.0
Qab = log(Pab)
tmps = multiply(rl.y, Qab)
phis = -multiply(rl.z, phid) - tmps
return phis, Qab
def phi_source_matrix(am, bm, dab, rl, phid):
numrab = am+bm+dab
denrab = am+bm-dab
Pab = divide(numrab, denrab)
chkd = absolute(denrab) < tol
Pab[chkd] = 1.0
Qab = log(Pab)
tmps = multiply(rl.y, Qab)
phis = -multiply(rl.z, phid) - tmps
return phis, Qab
def phi_trailing_doublet_matrix(rls: MatrixVector, sgnz: matrix, faco: float):
ths = zeros(rls.shape, dtype=float)
ths[rls.y > 0.0] = piby2
ths[rls.y == 0.0] = -piby2 * faco
ths[rls.y < 0.0] = -piby2
gs = divide(rls.z, rls.y)
Js = arctan(gs)
Js[rls.y == 0.0] = -piby2 * faco
phids = Js - multiply(sgnz, ths)
return phids
def betas(prices_m):
"""
Count beta parameters for all stocks, described in 'prices_m' matrix,
according to 'index' benchmark.
:param prices_m: matrix of prices. Each column represents a stock.
Each row represents price at successive time stamp
:return: matrix with betas. Each column represents a stock. Each row
represents beta at successive time period
"""
returns_m = ml.divide(ml.subtract(prices_m[1:], prices_m[:-1]),
prices_m[:-1])
index_m = ml.matrix(index).T
index_returns_m = ml.divide(ml.subtract(index_m[1:], index_m[:-1]),
index_m[:-1])
result = ml.empty((k, prices_m.shape[1]))
for i in range(k):
for j in range(stock_amount):
x = returns_m[:, j]
y = index_returns_m[:, i]
result[i, j] = np.cov(x, y, rowvar=0)[0][1]/np.var(y)
return result
def vel_doublet_matrix(av, am, bv, bm):
adb = elementwise_dot_product(av, bv)
abm = multiply(am, bm)
dm = multiply(abm, abm + adb)
axb = elementwise_cross_product(av, bv)
axbm = axb.return_magnitude()
chki = (axbm == 0.0)
chki = logical_and(axbm >= -tol, axbm <= tol)
velvl = elementwise_multiply(axb, divide(am + bm, dm))
velvl.x[chki] = 0.0
velvl.y[chki] = 0.0
velvl.z[chki] = 0.0
return velvl
def signal(prices, index):
"""
Signals to buy/sell stocks.
Returns tuple (X1, X2, ..., Xn), where n is amount of stocks,
Xi is one of the ('sell', 'buy', None).
:param prices: list of price lists of stocks, in time ascending order
:param index: list of k(!) price lists of benchmark values for k last
periods, in time ascending order, with same time frame as stocks.
"""
def betas(prices_m):
"""
Count beta parameters for all stocks, described in 'prices_m' matrix,
according to 'index' benchmark.
:param prices_m: matrix of prices. Each column represents a stock.
Each row represents price at successive time stamp
:return: matrix with betas. Each column represents a stock. Each row
represents beta at successive time period
"""
returns_m = ml.divide(ml.subtract(prices_m[1:], prices_m[:-1]),
prices_m[:-1])
index_m = ml.matrix(index).T
index_returns_m = ml.divide(ml.subtract(index_m[1:], index_m[:-1]),
index_m[:-1])
result = ml.empty((k, prices_m.shape[1]))
for i in range(k):
for j in range(stock_amount):
x = returns_m[:, j]
y = index_returns_m[:, i]
result[i, j] = np.cov(x, y, rowvar=0)[0][1]/np.var(y)
return result
def regime(reduced_returns_m):
"""
Make a regime switch based on PCA standard deviation acceleration.
:param reduced_returns_m: matrix of PCA. Each column represents a
stock. Each row represents price at successive time stamp
:return: one of the strings, 'momentum' - if trend is ment to
continue its movement, 'mean_reversion' - otherwise
"""
cross_sect_vol = np.std(reduced_returns_m, axis=1)
changes = cross_sect_vol[1:] - cross_sect_vol[:-1]
squared_changes = np.square(changes)
distance_times = reduced_returns_m.shape[0] - 1 # because there is
# T - 1 changes
distance = np.zeros(distance_times)
for t in range(distance_times):
sum_amount = min(t + 1, H)
for i in range(sum_amount):
distance[t] += squared_changes[t - i, 0]
distance[t] = np.sqrt(distance[t])
signal = distance[1:] - distance[:-1]
if np.max(signal) > 0:
return 'momentum'
else:
return 'mean_reversion'
prices_m = ml.matrix(prices).T
# Preparing main matrices for further computations
try:
log_returns_m = np.log(ml.divide(prices_m[1:], prices_m[:-1]))
except TypeError as e:
raise WrongPricesError(prices_m)
time_period, stock_amount = log_returns_m.shape
mean_log_returns_m = ml.average(log_returns_m, axis=0)
demeaned_log_returns_m = log_returns_m - mean_log_returns_m
covariation_m = demeaned_log_returns_m.T * demeaned_log_returns_m
# Count eigenvectors of covariation matrix and compose PCA matrix from them
e_values, e_vectors = eig(covariation_m)
abs_e_values = np.absolute(e_values)
# TODO: np.absolute(e_vectors) or something like that
indexed_abs_e_values = [(i, v) for i, v in enumerate(abs_e_values)]
w = sorted(indexed_abs_e_values, reverse=False,
key=lambda x: x[1])
e_vectors_m = ml.empty((stock_amount, k))
for j in range(k):
e_vectors_m[:, j] = e_vectors[:, w[j][0]]
# Main part: project returns on PCA universe
reduced_returns_m = (e_vectors_m.T * demeaned_log_returns_m.T).T
# Count beta parameters with respect to given benchmark index
betas_m = betas(prices_m)
time = time_period - time_shift
if time < H:
raise WrongParameterException("time_shift should be less than H")
# Collect data from returns in one vector
accumulated_reduced_returns = ml.zeros((1, k))
for i in range(H):
accumulated_reduced_returns += reduced_returns_m[time - 1 - i]
# Make a prediction about further returns behaviour
estimation = accumulated_reduced_returns * betas_m + \
mean_log_returns_m
if regime_switcher:
regime = regime(reduced_returns_m)
else:
regime = 'mean_reversion'
# Finally, decide for each stock, whether we need to sell it as
# overvalued or buy as undervalued. Other way around for momentum switch
max_recent_log_returns = log_returns_m[-H:].max(0)
result = []
for i in range(stock_amount):
if max_recent_log_returns[0, i] > estimation[0, i] + epsilon:
if regime == 'mean_reversion':
result.append('sell')
else:
result.append('buy')
elif max_recent_log_returns[0, i] < estimation[0, i] - epsilon:
if regime == 'mean_reversion':
result.append('buy')
else:
result.append('sell')
else:
result.append(None)
return result
def derivative(mat: matrix, var: matrix, axis: int = 0):
return divide(diff(mat, axis), diff(var, axis))
