Code for self-organizing state-space model (modified version)
Motivation
Since I was able to correct the code I wrote last time, I have posted a correction article. Please see here for the previous contents.
References
Same as last time. I will post it just in case.
- [Basics of Statistical Modeling for Prediction-From Introduction to Bayesian Statistics to Applications](http://www.amazon.co.jp/gp/product/4061557955/ref=as_li_qf_sp_asin_tl?ie=UTF8&camp=247&creative=1211&creativeASIN=4061557955&linkCode= as2 & tag = shimashimao06-22) <img src = "http://ir-jp.amazon-adsystem.com/e/ir?t=shimashimao06-22&l=as2&o=9&a=4061557955" width="1" height="1 "border =" 0 "alt =" "style =" border: none! Important; margin: 0px! Important; "/>
- Introduction to Time Series Analysis
- [Bayes Statistical Data Analysis (Data Science Learned in R)](http://www.amazon.co.jp/gp/product/4320019237/ref=as_li_qf_sp_asin_tl?ie=UTF8&camp=247&creative=1211&creativeASIN=4320019237&linkCode=as2&tag= shimashimao06-22)
- Knowledge discovery and self-organization statistical model
- Filtering of motion trajectory using self-organized state space model
- [Forefront of financial time series analysis by self-organized state space model with initial distribution search: Application to stochastic volatility fluctuation model with t distribution](http://www.fsa.go.jp/frtc/nenpou/ 2006a / 05.pdf)
Reference code
The cause of the suspicious behavior was that scipy.stats.cauchy.rvs did not give size. If you don't give the number of elements in the vector, it will return an incomprehensible value without warning. .. .. After that, I modified it so that it can be used for general purposes. However, if the model changes, The three functions (1) get_system_noise, (2) calc_pred_particles, and (3) calc_particles_weight will need to be modified. Then, I will describe the modified code below.
python
# coding: utf-8
from math import log, pow, sqrt
import numpy as np
from scipy.stats import norm, cauchy
from numpy.random import uniform, multivariate_normal
from multiprocessing import Pool
import matplotlib.pyplot as plt
class ParticleFilter:
log_likelihood = 0.0 #Log likelihood
TIME = 1
PR=8 # unmber of processing
def __init__(self, PARTICLES_NUM, k=1, ydim=1, sys_pdim=1, ob_pdim=1, sh_parameters=[0.01, 0.35]):
self.nois_sh_parameters = sh_parameters # nu:System noise position ultra-super parameter, xi:Observation noise position ultra-super parameter
pdim = sys_pdim+ob_pdim
self.PARTICLES_NUM = PARTICLES_NUM #Number of particles
self.TEETH_OF_COMB = np.arange(0, 1, float(1.0)/self.PARTICLES_NUM)
self.weights = np.zeros((ydim, self.PARTICLES_NUM), dtype=np.float64)
self.particles = np.zeros((k*ydim+pdim ,self.PARTICLES_NUM), dtype=np.float64)
self.predicted_particles = np.zeros((k*ydim+pdim , self.PARTICLES_NUM), dtype=np.float64)
np.random.seed(555)
self.predicted_value = []
self.filtered_value = []
self.sys_nois = []
self.ob_nois = []
self.LSM = np.zeros(ydim) #Square error
self.F, self.G, self.H= self.FGHset(k, ydim, pdim)
self.k = k
self.ydim = ydim
self.pdim = pdim
self.sys_pdim = sys_pdim
self.ob_pdim = ob_pdim
def init_praticles_distribution(self, P, r):
"""initialize particles
x_0|0
tau_0|0
sigma_0|0
"""
data_particles = multivariate_normal([1]*self.ydim*self.k,
np.eye(self.ydim*self.k)*10, self.PARTICLES_NUM).T
param_particles = np.zeros((self.pdim, self.PARTICLES_NUM))
for i in xrange(self.pdim):
param_particles[i,:] = uniform(P-r, P+r, self.PARTICLES_NUM)
self.particles = np.vstack((data_particles, param_particles))
def get_system_noise(self):
"""v_t vector"""
data_noise = np.zeros((self.ydim, self.PARTICLES_NUM), dtype=np.float64)
for i in xrange(self.ydim):
data_noise[i,:] = cauchy.rvs(loc=[0]*self.PARTICLES_NUM, scale=np.power(10,self.particles[self.ydim]),
size=self.PARTICLES_NUM)
data_noise[data_noise==float("-inf")] = -1e308
data_noise[data_noise==float("inf")] = 1e308
parameter_noises = np.zeros((self.pdim, self.PARTICLES_NUM), dtype=np.float64)
for i in xrange(self.pdim):
parameter_noises[i,:] = cauchy.rvs(loc=0, scale=self.nois_sh_parameters[i], size=self.PARTICLES_NUM)
return np.vstack((data_noise, parameter_noises))
def calc_pred_particles(self):
"""calculate system function
x_t|t-1 = F*x_t-1 + Gv_t
"""
return self.F.dot(self.particles) + self.G.dot(self.get_system_noise()) # linear non-Gaussian
def calc_particles_weight(self,y):
"""calculate fitness probabilities between observation value and predicted value
w_t
"""
locs = self.calc_pred_particles()
self.predicted_particles = locs
scale=np.power(10,locs[-1])
scale[scale==0] = 1e-308
#Need to be modified in case of multivariate
self.weights = cauchy.pdf( np.array([y]*self.PARTICLES_NUM) - self.H.dot(locs), loc=[0]*self.PARTICLES_NUM,
scale=scale, size=self.PARTICLES_NUM).flatten()
def calc_likelihood(self):
"""calculate likelihood at that point
p(y_t|y_1:t-1)
"""
res = np.sum(self.weights)/self.PARTICLES_NUM
self.log_likelihood += log(res)
def normalize_weights(self):
"""wtilda_t"""
self.weights = self.weights/np.sum(self.weights)
def resample(self,y):
"""x_t|t"""
self.normalize_weights()
self.memorize_predicted_value()
# accumulate weight
cum = np.cumsum(self.weights)
# create roulette pointer
base = uniform(0,float(1.0)/self.PARTICLES_NUM)
pointers = self.TEETH_OF_COMB + base
# select particles
selected_idx = [np.where(cum>=p)[0][0] for p in pointers]
"""
pool = Pool(processes=self.PR)
selected_idx = pool.map(get_slected_particles_idx, ((cum,p) for p in pointers))
pool.close()
pool.join()
"""
self.particles = self.predicted_particles[:,selected_idx]
self.memorize_filtered_value(selected_idx, y)
def memorize_predicted_value(self):
predicted_value = np.sum(self.predicted_particles*self.weights, axis=1)[0]
self.predicted_value.append(predicted_value)
def memorize_filtered_value(self, selected_idx, y):
filtered_value = np.sum(self.particles*self.weights[selected_idx] , axis=1) \
/np.sum(self.weights[selected_idx])
self.filtered_value.append(filtered_value[:self.ydim])
self.sys_nois.append(np.power(10,filtered_value[self.ydim:self.ydim+self.sys_pdim]))
self.ob_nois.append(np.power(10,filtered_value[self.ydim+self.sys_pdim:]))
self.calculate_LSM(y,filtered_value[:self.ydim])
def calculate_LSM(self,y,filterd_value):
self.LSM += pow(y-filterd_value,2)
def forward(self,y):
"""compute system model and observation model"""
print 'calculating time at %d' % self.TIME
self.calc_pred_particles()
self.calc_particles_weight(y)
self.calc_likelihood()
self.resample(y)
self.TIME += 1
def FGHset(self, k, vn_y, n_h_parameters):
"""Matrix setting for state-space representation
vn_y:Input vector dimensions
n_h_parameters:Number of hyper parameters
k: Difference
"""
G_upper_block = np.zeros((k*vn_y, vn_y+n_h_parameters))
G_lower_block = np.zeros((n_h_parameters, vn_y+n_h_parameters))
G_lower_block[-n_h_parameters:, -n_h_parameters:] = np.eye(n_h_parameters)
G_upper_block[:vn_y, :vn_y] = np.eye(vn_y)
G = np.vstack( (G_upper_block, G_lower_block) )
H = np.hstack( (np.eye(vn_y),
np.zeros((vn_y, vn_y*(k-1)+n_h_parameters))
) )
#Construction of block matrix of trend model
F_upper_block = np.zeros((k*vn_y, k*vn_y+n_h_parameters))
F_lower_block = np.zeros((n_h_parameters, k*vn_y+n_h_parameters))
F_lower_block[-n_h_parameters:, -n_h_parameters:] = np.eye(n_h_parameters)
if k==1:
F_upper_block[:vn_y, :vn_y] = np.eye(vn_y)
elif k==2:
F_upper_block[:vn_y, :vn_y] = np.eye(vn_y)*2
F_upper_block[:vn_y, vn_y:k*vn_y] = np.eye(vn_y)*-1
F_upper_block[vn_y:k*vn_y, :vn_y] = np.eye(vn_y)
F = np.vstack((F_upper_block, F_lower_block))
return F, G, H
def get_slected_particles_idx((cum,p)):
"""multiprocessing function"""
try:
return np.where(cum>=p)[0][0]
except Exception, e:
import sys
import traceback
sys.stderr.write(traceback.format_exc())
if __name__=='__main__':
n_particle = 1000
nu=0.01
xi=0.35
pf = ParticleFilter(n_particle, k=1, ydim=1, sys_pdim=1, ob_pdim=1, sh_parameters=[nu, xi])
pf.init_praticles_distribution(0, 8) # P, r
data = np.hstack((norm.rvs(0,1,size=20),norm.rvs(10,1,size=60),norm.rvs(-30,0.5,size=20)))
for d in data:
pf.forward(d)
print 'log likelihood:', pf.log_likelihood
print 'LSM:', pf.LSM
rng = range(100)
plt.plot(rng,data,label=u"training data")
plt.plot(rng,pf.predicted_value,label=u"predicted data")
plt.plot(rng,pf.filtered_value,label=u"filtered data")
# plt.plot(rng,pf.sys_nois,label=u"system noise hyper parameter")
# plt.plot(rng,pf.ob_nois,label=u"observation noise hyper parameter")
plt.xlabel('TIME',fontsize=18)
plt.ylabel('Value',fontsize=18)
plt.legend(loc = 'upper left')
plt.show()
Experimental result
I experimented several times by changing the value of the ultra-super parameter and the number of particles. Below are the experimental figures.
Number of particles 100, nu = 0.1, xi = 0.35
Number of particles 1000, nu = 0.1, xi = 0.35
Number of particles 10000, nu = 0.1, xi = 0.35
Number of particles 10000, nu = 0.00001, xi = 0.00001
Noise parameters (number of particles 1000, nu = 0.0008, xi = 0.35)
comment
You can see that the accuracy increases as the number of particles increases. Also, due to capacity limitation, I could only show two figures with 10,000 particles, but when the number of particles exceeds 10,000, the influence of the ultra-super parameter value is reduced, and I think that a good tendency was seen. The observed noise parameter $ σ ^ 2 $ has a large value in the part where the value changes rapidly. If you look at this, you can see the advantages of the self-organizing model.
Summary
I'm glad I managed to make something that works properly w
We apologize for the inconvenience, but if you make a mistake, we would appreciate it if you could point it out.