def pv(imList, p, median, j):
''' calculate -2log(R_ell,j) and return P-value '''
imList = ee.List(imList)
p = ee.Number(p)
j = ee.Number(j)
f = p
one = ee.Number(1.0)
# 1 - (1. + 1./(j*(j-1)))/(6.*p*n)
rhoj = one.subtract(
one.add(one.divide(j.multiply(j.subtract(one)))).divide(6 * 4.9))
# -(f/4.)*(1.-1./rhoj)**2'
omega2j = one.subtract(one.divide(rhoj)).pow(2.0).multiply(f.divide(-4.0))
Z = ee.Image(ee.Image(log_det_sum(imList,j.subtract(1)))).multiply(j.subtract(1)) \
.add(log_det(imList,j)) \
.add(p.multiply(j).multiply(ee.Number(j).log())) \
.subtract(p.multiply(j.subtract(1)).multiply(j.subtract(1).log())) \
.subtract(ee.Image(log_det_sum(imList,j)).multiply(j)) \
.multiply(rhoj) \
.multiply(-2*4.9)
# (1.-omega2j)*stats.chi2.cdf(Z,[f])+omega2j*stats.chi2.cdf(Z,[f+4])
P = ee.Image(
chi2cdf(Z, f).multiply(one.subtract(omega2j)).add(
chi2cdf(Z, f.add(4)).multiply(omega2j)))
# 3x3 median filter
return ee.Algorithms.If(median, P.focal_median(), P)
def ells_iter(current, prev):
ell = ee.Number(current)
prev = ee.Dictionary(prev)
pv_arr = ee.List(prev.get('pv_arr'))
k = ee.Number(prev.get('k'))
enl = ee.Number(prev.get('enl'))
median = prev.get('median')
imList = ee.List(prev.get('imList'))
# number of bands (degrees of freedom)
p2 = ee.Image(imList.get(0)).bandNames().length()
imList_ell = imList.slice(ell.subtract(1))
js = ee.List.sequence(2, k.subtract(ell).add(1))
first = ee.Dictionary({
'median': median,
'imList': imList_ell,
'enl': enl,
'pvs': ee.List([]),
'Z': ee.Image.constant(0.0)
})
result = ee.Dictionary(js.iterate(js_iter, first))
# list of P-values for R_ell,j, j = ell+1 ... k
pvs = ee.List(result.get('pvs'))
# omnibus test statistic -2lnQ_ell = sum_j(-2lnR_j)
Z = ee.Image(result.get('Z'))
# degrees of freedom
f = k.subtract(ell).multiply(p2)
p = p2.sqrt()
one = ee.Number(1)
# rho, w2 values
rho = ee.Number(ee.Algorithms.If( p2.eq(2),
one,
one.subtract(
p2.multiply(2).subtract(one) \
.multiply(k.subtract(one.divide(k))) \
.divide(k.subtract(one).multiply(p).multiply(6).multiply(enl)) )))
w2 = ee.Number(ee.Algorithms.If( p2.eq(2),
0,
p2.multiply(p2.subtract(one)) \
.multiply(k.subtract(one.divide(k.pow(2)))) \
.divide(rho.pow(2).multiply(24).multiply(enl.pow(2))) \
.subtract(p2.multiply(k.subtract(one)).multiply(one.subtract(one.divide(rho)).pow(2).divide(4))) ))
Z = Z.multiply(rho)
PvQ = ee.Image.constant(1.0).subtract(
chi2cdf(Z, f).multiply(ee.Number(1).subtract(w2))).subtract(
chi2cdf(Z, f.add(4)).multiply(w2))
PvQ = ee.Algorithms.If(median, PvQ.focal_median(), PvQ)
# put at end of current sequence
pvs = pvs.add(PvQ)
return ee.Dictionary({
'k': k,
'median': median,
'enl': enl,
'imList': imList,
'pv_arr': pv_arr.add(pvs)
})
def pv(imList, p, median, j):
''' calculate -2log(R_ell,j) and return P-value '''
imList = ee.List(imList).slice(0, j)
p = ee.Number(p)
Z = ee.Image(ee.Image(log_det_sum(imList,j.subtract(1)))).multiply(j.subtract(1)) \
.add(log_det(imList,j)) \
.add(p.multiply(j).multiply(ee.Number(j).log())) \
.subtract(p.multiply(j.subtract(1)).multiply(j.subtract(1).log())) \
.subtract(ee.Image(log_det_sum(imList,j)).multiply(j)) \
.multiply(-2*4.9)
P = ee.Image(ee.Algorithms.If(p.eq(2), chi2cdf(Z, 2), chi2cdf(Z, 1)))
return ee.Algorithms.If(median, P.focal_median(1.5, 'square', 'pixels', 1),
P)
def pv(imList, median, j, enl):
''' calculate -2log(R_ell,j) and return P-value'''
imList = ee.List(imList)
p2 = ee.Image(imList.get(0)).bandNames().length()
p = ee.Number(ee.Algorithms.If(p2.eq(2).Or(p2.eq(3)), p2, p2.sqrt()))
j = ee.Number(j)
f = p2
one = ee.Number(1.0)
rhoj = ee.Number(ee.Algorithms.If( p2.eq(2),
one,
# 1 - (2*p2-1.)*(1.+1./(j*(j-1.))/6*p*n
one.subtract( p2.multiply(2).subtract(one) \
.multiply(one.add(one.divide(j.multiply(j.subtract(one))))) \
.divide(p.multiply(6).multiply(enl))
) ))
omega2j = ee.Number(ee.Algorithms.If( p2.eq(2),
0,
# (f/4)*(1-1./rhoj)**2 + (1./(24.*n**2))*p2(p2-1.)*(1.+(2.*j-1.)/(j**2*(j-1.)**2))/rh0j**2
f.multiply(one.subtract(one.divide(rhoj))).divide(4). \
add( one.divide(enl.pow(2).multiply(24)) \
.multiply(p2.multiply(p2.subtract(one))) \
.multiply(one.add(j.multiply(2)).subtract(one)) \
.divide(j.pow(2).multiply(j.subtract(one)).pow(2)) \
.divide(rhoj.pow(2)) ) ))
# Zj = -2*lnRj
Zj = ee.Image(log_det_sum(imList,j.subtract(1))) \
.multiply(j.subtract(1)) \
.add(log_det(imList,j)) \
.add(p.multiply(j).multiply(ee.Number(j).log())) \
.subtract(p.multiply(j.subtract(1)).multiply(j.subtract(1).log())) \
.subtract(ee.Image(log_det_sum(imList,j)).multiply(j)) \
.multiply(-2).multiply(enl)
# (1.-omega2j)*stats.chi2.cdf(rhoj*Zj,[f])+omega2j*stats.chi2.cdf(rhoj*Zj,[f+4])
P = chi2cdf(Zj.multiply(rhoj),f).multiply(one.subtract(omega2j)) \
.add(chi2cdf(Zj.multiply(rhoj),f.add(4)).multiply(omega2j))
PV = ee.Image.constant(1.0).subtract(P)
# 3x3 median filter
return (ee.Algorithms.If(median, PV.focal_median(), PV), Zj)
def ells_iter(current,prev):
ell = ee.Number(current)
prev = ee.Dictionary(prev)
pv_arr = ee.List(prev.get('pv_arr'))
k = ee.Number(prev.get('k'))
median = prev.get('median')
p = prev.get('p')
imList = ee.List(prev.get('imList'))
imList_ell = imList.slice(ell.subtract(1))
js = ee.List.sequence(2,k.subtract(ell).add(1))
first = ee.Dictionary({'median':median,'p':p,'imList':imList_ell,'pvs':ee.List([]),'Z':ee.Image.constant(0.0)})
result = ee.Dictionary(js.iterate(js_iter,first))
# list of P-values for R_ell,j, j = ell+1 ... k
pvs = ee.List(result.get('pvs'))
# omnibus test statistic -2lnQ_ell = sum_j(-2lnR_j)
Z = ee.Image(result.get('Z'))
f = k.subtract(ell).multiply(ee.Number(p))
PvQ = ee.Image.constant(1.0).subtract(chi2cdf(Z,f))
PvQ = ee.Algorithms.If(median, PvQ.focal_median(),PvQ)
pvs = pvs.add(PvQ)
return ee.Dictionary({'k':k,'p':p,'median':median,'imList':imList,'pv_arr':pv_arr.add(pvs)})
from scipy.stats import chi2
import numpy as np
ee.Initialize()
x = 2
df = 5
print 'scipy: ', chi2.cdf(x, df)
point = ee.Geometry.Point([50, 6])
img = ee.Image.constant(x)
dfi = ee.Image.constant(df)
#result = chi2cdf_old(img,df)
result1 = chi2cdf(img, df)
print 'chi2cdf: ', result1.reduceRegion(ee.Reducer.first(),
geometry=point,
scale=4000).getInfo()
arr = np.mat(np.random.randn(100, 4))
cov = ee.Array((arr.T * arr).tolist())
print np.array(cov.getInfo())
L = ee.Array(cov.matrixCholeskyDecomposition().get('L'))
print np.array(L.getInfo())
S = L.matrixMultiply(L.matrixTranspose())
print np.array(S.getInfo())