/
quadratic_estimate.jl
198 lines (155 loc) · 8.6 KB
/
quadratic_estimate.jl
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
export quadratic_estimate
"""
quadratic_estimate(ds::DataSet, which; wiener_filtered=true)
quadratic_estimate((ds1::DataSet,ds2::DataSet), which; wiener_filtered=true)
Compute quadratic estimate of ϕ given data.
The `ds` or `(ds1,ds2)` tuple contain the DataSet object(s) which houses the
data and covariances used in the estimate. Note that only the Fourier-diagonal
approximations for the beam, mask, and noise,, i.e. `ds.B̂`, `ds.M̂`, and
`ds.Cn̂`, are accounted for. To account full operators (if they are not actually
Fourier-diagonal), you should compute the impact using Monte Carlo.
If a tuple is passed in, the result will come from correlating the data from
`ds1` with that from `ds2`, which can be useful for debugging / isolating
various noise terms.
An optional keyword argument `AL` can be passed in in case the QE normalization
was already computed, in which case it won't be recomputed during the
calculation.
Returns a NamedTuple `(ϕqe, AL, Nϕ)` where `ϕqe` is the (possibly Wiener
filtered, depending on `wiener_filtered` option) quadratic estimate, `AL` is the
normalization (which is already applied to ϕqe, it does not need to be applied
again), and `Nϕ` is the analytic N0 noise bias (Nϕ==AL if using unlensed
weights, currently only Nϕ==AL is always returned, no matter the weights)
"""
function quadratic_estimate((ds1,ds2)::NTuple{2,DataSet}, which; wiener_filtered=true, AL=nothing, weights=:unlensed) where {F1,F2}
@assert weights in (:lensed, :unlensed) "weights should be :lensed or :unlensed"
@assert (which in [:TT, :EE, :EB]) "which='$which' not implemented"
@assert (ds1.Cf===ds2.Cf && ds1.Cf̃===ds2.Cf̃ && ds1.Cn̂===ds2.Cn̂ && ds1.Cϕ===ds2.Cϕ && ds1.B̂===ds2.B̂) "operators in `ds1` and `ds2` should be the same"
@assert spin(ds1.d)==spin(ds2.d)
check_hat_operators(ds1)
@unpack Cf, Cf̃, Cn̂, Cϕ, B̂, M̂ = ds1()
(quadratic_estimate_func, pol) = @match which begin
:TT => (quadratic_estimate_TT, :I)
:EE => (quadratic_estimate_EE, :P)
:EB => (quadratic_estimate_EB, :P)
_ => error("`which` argument to `quadratic_estimate` should be one of (:TT, :EE, :EB)")
end
quadratic_estimate_func((ds1.d[pol], ds2.d[pol]), Cf[pol], Cf̃[pol], Cn̂[pol], Cϕ, (M̂*B̂)[pol], wiener_filtered, weights, AL)
end
quadratic_estimate(ds::DataSet, which; kwargs...) = quadratic_estimate((ds,ds), which; kwargs...)
quadratic_estimate(ds::DataSet; kwargs...) = quadratic_estimate(ds, ds.d isa FlatS0 ? :TT : :EB; kwargs...) # somewhat arbitraritly make default P estimate be EB
quadratic_estimate_TT(d::Field, args...) = quadratic_estimate_TT((d,d), args...)
quadratic_estimate_EB(d::Field, args...) = quadratic_estimate_EB((d,d), args...)
quadratic_estimate_EE(d::Field, args...) = quadratic_estimate_EE((d,d), args...)
@doc doc"""
All of the terms in the quadratic estimate and normalization expressions look like
C * l[i] * l̂[j] * l̂[k] * ...
where C is some field or diagonal covariance. For example, there's a term in the EB
estimator that looks like:
(CE * (CẼ+Cn) \ d[:E])) * l[i] * l̂[j] * l̂[k]
(where note that `l̂[j]` and `l̂[k]` are unit vectors, but `l[i]` is not). The
function `get_term_memoizer` returns a function `term` which could be called in
the following way to compute this term:
term((CE * (CẼ+Cn) \ d[:E])), [i], j, k)
(note that the fact that `l[i]` is not a unit vector is specified by putting the
`[i]` index in brackets).
Additionally, all of these terms are symmetric in their indices, i.e. in
`(i,j,k)` in this case. The `term` function is smart about this, and is memoized
so that each unique set of indices is only computed once. This leads to a pretty
drastic speedup for terms with many indices like those that arize in the EE and
EB normalizations, and lets us write code which is both clear and fast without
having to think too hard about these symmetries.
"""
function get_term_memoizer(f)
Ðf = Ð(f)
Lfactors() = 1
Lfactors(inds...) = broadcast!(*, similar(Ðf), getproperty.(getindex.(Ref(∇),inds),:diag)...)
term(C::Diagonal, inds...) = term(C.diag, inds...)
term(C::Field, inds...) = term(count((x->x isa Int),inds)/2f0, C, first.(inds)...)
@sym_memo term(n, C::Field, @sym(inds...)) = Map(nan2zero.(C .* Lfactors(inds...) ./ real.(∇².diag).^n))
term
end
ϵ(x...) = levicivita([x...])
inds(n) = collect(product(repeated(1:2,n)...))[:]
function quadratic_estimate_TT((d1,d2)::NTuple{2,FlatS0}, Cf, Cf̃, Cn, Cϕ, TF, wiener_filtered, weights, AL=nothing)
term = get_term_memoizer(d1)
ΣTtot = TF^2 * Cf̃ + Cn
CT = (weights==:unlensed) ? Cf : Cf̃
# unnormalized estimate
ϕqe_unnormalized = @subst -sum(∇[i] * Fourier(term($(ΣTtot\(TF*d1))) * term($(CT*(ΣTtot\(TF*d2))), [i])) for i=1:2)
# normalization
if AL == nothing
AL = @subst begin
A(i,j) = (
term($(@. TF^2 * CT^2 / ΣTtot), [i], [j]) * term($(@. TF^2 / ΣTtot) )
+ term($(@. TF^2 * CT / ΣTtot), [i] ) * term($(@. TF^2 * CT / ΣTtot), [j])
)
pinv(Diagonal(sum(∇[i].diag .* ∇[j].diag .* Fourier(A(i,j)) for (i,j) in inds(2))))
end
end
Nϕ = AL # true only for unlensed weights
ϕqe = (wiener_filtered ? (Cϕ*pinv(Cϕ+Nϕ)) : 1) * (AL*ϕqe_unnormalized)
@namedtuple ϕqe AL Nϕ
end
function quadratic_estimate_EE((d1,d2)::NTuple{2,FlatS2}, Cf, Cf̃, Cn, Cϕ, TF, wiener_filtered, weights, AL=nothing)
term = get_term_memoizer(d1[:E])
TF² = TF[:E]^2
ΣEtot = TF² * Cf̃[:E] + Cn[:E]
CE = ((weights==:unlensed) ? Cf : Cf̃)[:E]
# unnormalized estimate
ϕqe_unnormalized = @subst begin
I(i) = -(
2sum(term($(CE * (ΣEtot \ (TF*d1)[:E])), [i], j, k) * term($((ΣEtot \ (TF*d2)[:E])), j, k) for (j,k) in inds(2))
- term($(CE * (ΣEtot \ (TF*d1)[:E])), [i] ) * term($((ΣEtot \ (TF*d2)[:E])) )
)
sum(∇[i] * Fourier(I(i)) for i=1:2)
end
# normalization
if AL == nothing
AL = @subst begin
A1(i,j) = -4 * sum( ϵ(m,p,3) * ϵ(n,q,3) * (
term($(@. TF² * CE^2 / ΣEtot), [i], [j], k, l, m, n) * term($(@. TF² / ΣEtot), k, l, p, q)
+ term($(@. TF² * CE / ΣEtot), [i], k, l, m, n) * term($(@. TF² * CE / ΣEtot), [j], k, l, p, q))
for (k,l,m,n,p,q) in inds(6)
)
A2(i,j) = (
term($(@. TF² * CE^2 / ΣEtot), [i], [j]) * term($(@. TF² / ΣEtot) )
+ term($(@. TF² * CE / ΣEtot), [i] ) * term($(@. TF² * CE / ΣEtot), [j])
)
pinv(Diagonal(sum(∇[i].diag .* ∇[j].diag .* Fourier(A1(i,j) + A2(i,j)) for i=1:2,j=1:2)))
end
end
Nϕ = AL # true only for unlensed weights
ϕqe = (wiener_filtered ? (Cϕ*pinv(Cϕ+Nϕ)) : 1) * (AL*ϕqe_unnormalized)
@namedtuple ϕqe AL Nϕ
end
function quadratic_estimate_EB((d1,d2)::NTuple{2,FlatS2}, Cf, Cf̃, Cn, Cϕ, TF, wiener_filtered, weights, AL=nothing; zeroB=false)
term = get_term_memoizer(d1[:E])
CE, CB = getindex.(Ref((weights==:unlensed) ? Cf : Cf̃),(:E,:B))
TF²E, TF²B = TF[:E]^2, TF[:B]^2
ΣEtot = TF²E * Cf̃[:E] + Cn[:E]
ΣBtot = TF²B * Cf̃[:B] + Cn[:B]
# unnormalized estimate
ϕqe_unnormalized = @subst begin
I(i) = 2 * sum( ϵ(k,l,3) * (
term($(CE * (ΣEtot \ (TF*d1)[:E])), [i], j, k) * term($( (ΣBtot \ (TF*d2)[:B])), j, l)
- (zeroB ? 0 : term($( ΣEtot \ (TF*d1)[:E]), j, k) * term($(CB * (ΣBtot \ (TF*d2)[:B])), [i], j, l)))
for (j,k,l) in inds(3)
)
sum(∇[i] * Fourier(I(i)) for i=1:2)
end
# normalization
if AL == nothing
AL = @subst begin
@sym_memo A(@sym(i,j)) = 4 * sum( ϵ(m,p,3) * ϵ(n,q,3) * (
term($(@. TF²E * CE^2 / ΣEtot), [i], [j], k, l, m, n) * term($(@. TF²B / ΣBtot), k, l, p, q)
+ (zeroB ? 0 : -2term($(@. TF²E * CE / ΣEtot), [i], k, l, m, n) * term($(@. TF²B * CB / ΣBtot), [j], k, l, p, q))
+ (zeroB ? 0 : term($(@. TF²E / ΣEtot), k, l, m, n) * term($(@. TF²B * CB^2 / ΣBtot), [i], [j], k, l, p, q)))
for (k,l,m,n,p,q) in inds(6)
)
pinv(Diagonal(sum(∇[i].diag .* ∇[j].diag .* Fourier(A(i,j)) for i=1:2,j=1:2)))
end
end
Nϕ = AL # true only for unlensed weights
ϕqe = (wiener_filtered ? (Cϕ*pinv(Cϕ+Nϕ)) : 1) * (AL * ϕqe_unnormalized)
@namedtuple ϕqe AL Nϕ
end