text:
### A Pluto.jl notebook ###
# v0.18.1

using Markdown
using InteractiveUtils

# ╔═╡ 17b325dc-c77c-40be-9ad9-cf85b2ee6dee
begin
	using CSV
	using DataFrames
	using Metrics
	using NearestNeighborModels
	using MLJ
	using MLJFlux
	using LinearAlgebra
	using StatsPlots
end

# ╔═╡ 8db59250-904f-11ec-0f09-bb7059e674a7
md"""
# HW 5

- Name: Eric Nguyen

- TUID: 915786865
"""

# ╔═╡ e768446a-5118-42e3-8606-8f79040fc45a
md"""
## Problem 1
"""

# ╔═╡ 0880650c-80d6-43ef-98f1-b6a0b9d7909c
md"""
### Read data into CSV
"""

# ╔═╡ 0ca91499-40f4-45ed-85c0-3e24eb9fcdb6
dfheaders = [
	"age",
	"workclass",
	"fnlwgt",
	"education",
	"education-num",
	"marital-status",
	"occupation",
	"relationship",
	"race",
	"sex",
	"capital-gain",
	"capital-loss",
	"hours-per-week",
	"native-country",
	"class"
]

# ╔═╡ 51f15a61-f170-4acc-abb8-75d1735ada91
incomedf = CSV.read("data/adult.data", DataFrame, delim=", ", header=dfheaders)

# ╔═╡ c64a4895-3d2f-4229-a33e-e2303bfd5ba8
md"""
### Process data for data modeling
"""

# ╔═╡ 260bf856-7059-4692-9c56-9a3c2f35cf53
md"""
Here, we take only numerical columns from the data.
"""

# ╔═╡ d08b2b4b-83ca-4a57-9b33-3439d3b0b29f
knncolumns(df) = let
	# Take only numerical columns from DataFrame
	df = df[!,eltype.(collect(df[1,:])) .<: Number]

	# Ignore fnlwgt, capital-gain, and capital-loss columns
	df = select!(df, Not(:fnlwgt))
	df = select!(df, Not("capital-gain"))
	df = select!(df, Not("capital-loss"))
end

# ╔═╡ 0b59d895-f80a-4a7f-991d-a0e8d5ca485c
md"""
Here, we normalize each column of the data and then add the class column back.
"""

# ╔═╡ d0065a6c-23dd-45c5-b999-d8021288c9e2
normknndf = let
	df = knncolumns(incomedf)

	# Normalize matrix using min-max normalization
	for x ∈ 1:size(df,2)
		df[!,x] = (normalize(df[:,x]) .- minimum(normalize(df[:,x]))) / (maximum(normalize(df[:,x])) - minimum(normalize(df[:,x])))
	end

	# Coerce class column into Multiclass type
	df[!,:class] = coerce(incomedf[!,:class] .== ">50K", OrderedFactor{2})

	df
end

# ╔═╡ 292a232f-76f0-4461-9bc6-84cf1d0b2f20
md"""
See [https://alan-turing-institute.github.io/MLJ.jl/dev/working\_with\_categorical_data/](https://alan-turing-institute.github.io/MLJ.jl/dev/working_with_categorical_data/) for more information on working with categorical data in MLJ.
"""

# ╔═╡ 5c5fc600-5682-4778-aa24-7bf8f62bd7b3
md"""
The schema of the processed data is as follows:
"""

# ╔═╡ 98919d06-cc8a-4675-a125-4c32a35a3730
schema(normknndf)

# ╔═╡ 66479914-d71f-40aa-9827-99a6320af613
md"""
Separate the features and the target.
"""

# ╔═╡ b88b08a1-cb65-4a99-ac80-35ab2cf4f120
y, X = unpack(normknndf, ==(Symbol("class")); rng=123);

# ╔═╡ e4876122-a269-41ff-81ed-7bff7ec98ed1
md"""
### Choose a model
"""

# ╔═╡ 7c67bb05-55f9-47f0-8ade-93850f86ab73
md"""
Here is a list of possible models we can use for the data we have:
"""

# ╔═╡ ab881e2d-712d-480c-af06-40a647fc9556
models(matching(X,y))

# ╔═╡ 06091f16-e782-41b5-96d3-f31afed32fae
md"""
We choose the KNNClassifer in this case:
"""

# ╔═╡ 670cc398-83ae-4461-bdc2-8695a5282f34
Model = @load KNNClassifier pkg=NearestNeighborModels

# ╔═╡ 6e2f20bb-e836-46f0-89e2-7610563037ff
md"""
### Report F1 score for chosen models
"""

# ╔═╡ a54b6b50-99f3-440a-9bfb-ed93634c752c
md"""
The F1 scores for the KNNClassifer with K = 3 and K = 10 are reported accordingly:
"""

# ╔═╡ 13a54499-b2f2-4e4c-a353-778777f04f03
md"""
**K = 3:**
"""

# ╔═╡ 61c81f71-5e8d-4c26-926f-0395d928dfa7
evaluate(Model(K = 3), X, y, resampling=CV(nfolds=3), measure=f1score, verbosity=0, check_measure=false)

# ╔═╡ 083adea7-cb6e-4de6-acdb-e44c67bc30a7
md"""
$\text{F-score} = 0.456 \qquad\text{for}\qquad K = 3$
"""

# ╔═╡ 8bbd3b33-cc0d-4391-b39b-248da032f118
md"""
**K = 10:**
"""

# ╔═╡ 6a3806dc-624e-424a-b666-6d972547870b
evaluate(Model(K = 10), X, y, resampling=CV(nfolds=3), measure=f1score, verbosity=0, check_measure=false)

# ╔═╡ aa07b8d4-0981-43b3-8f97-5fd1d9f86055
md"""
$\text{F-score} = 0.44 \qquad \text{for} \qquad K = 10$
"""

# ╔═╡ 3762067e-811c-4ffe-83dc-737ba82f21c4
md"""
## Problem 2
"""

# ╔═╡ d9d22117-bb35-4338-b20d-9c53532f6ebb
md"""
### Choose model
"""

# ╔═╡ 8aa33ccc-121a-47eb-bc72-c8f7892a5260
NNModel = @load NeuralNetworkClassifier pkg=MLJFlux

# ╔═╡ be5bbf98-8de7-420d-82b8-9eb12658de65
md"""
We use `epochs=1` to reduce training time significantly at the cost of fitting performance.
"""

# ╔═╡ c2b47571-e081-4d2a-bbc4-9d8745133bea
clf = NNModel(epochs=1)

# ╔═╡ 599c17ab-e44f-4192-89ab-cc472363b34b
md"""
Separate the training and testing data.
"""

# ╔═╡ 55693488-4507-491a-9036-e4847e03ae76
train, test = partition(eachindex(y), 0.66, shuffle=true, rng=123)

# ╔═╡ 48106ca5-c04f-4073-92d3-1a82369d2198
md"""
### Train the model
"""

# ╔═╡ 5b28c4c8-fc89-406f-bc67-0fc595dd3efb
mach = let
	mach = machine(clf, X, y)
	fit!(mach, rows=train)
end

# ╔═╡ 38c0201e-6c9b-4257-a2d9-7474c11909b5
md"""
### Plot the ROC curve
"""

# ╔═╡ f39df55f-9e96-4994-95e8-8a2c4207bb04
let
	train, test = partition(eachindex(y), 0.7)
	ŷ = predict(mach, X[test,:])

	fprs, tprs, ts = roc(ŷ, y[test])
	plot(fprs, tprs, title="ROC of Neural Network", label="NeuralNetworkClassifier(epoch=1)", legend=:bottomright, xlabel="False positive rate", ylabel="True positive rate")
end

# ╔═╡ 72aee613-7f54-446a-a5d1-d2af011f20ab
md"""
## Problem 3

The data is linearly separable only if there exists a separating hyperplane between the data.
In this case we have two dimensions, so the separating hyperplane is defined as

$𝐰^T 𝐱 + b = w_1 x_1 + w_2 x_2 + b$

We construct the following perceptron,

$\hat{2}y = \begin{cases}
1, &\text{if } 𝐰^T 𝐱 + b > 0 \\
-1, &\text{otherwise.}
\end{cases}$

Suppose there exists a hyperplane separating the data.
Then putting the data into the perceptron, we expect:

$\begin{align*}
(x_1 = 1, x_2 = 0) &= 1 &&\implies w_1 + b > 0 &&\implies w_1 > -b \\
(x_1 = 0, x_2 = 1) &= 1 &&\implies w_2 + b > 0 &&\implies w_2 > -b \\
(x_1 = 0, x_2 = 0) &= -1 &&\implies b ≤ 0 \\
(x_1 = 1, x_2 = 1) &= -1 &&\implies w_1 + w_2 + b ≤ 0 &&\implies w_1 + w_2 ≤ -b \\
\end{align*}$

The inequalities contradict each other.
Thus, there exists no separating hyperplane and therefore the data is **not linearly separable**.
"""

# ╔═╡ 0aec950d-36df-4079-8504-01e8ca07bb24
md"""
## Problem 4
"""

# ╔═╡ 9aeed7b3-3dcb-48e2-b28f-89e24aa02b69
md"""
**4(a).**
Let ``U`` represent the condition that a student is an undergraduate, ``G`` represent the condition that a student is a graduate, ``S^+`` represent the condition that a student smokes, and ``S^-`` represent the condition that a student does not smoke.
We are given

$\begin{align*}
P(S^+ ∣ U) &= 0.15 \\
P(S^+ ∣ G) &= 0.23 \\
P(G) &= 0.2 \\
P(U) &= 0.8
\end{align*}$

The probability of a student who smokes being a graduate student is

$\begin{align*}
P(G ∣ S^+) &= \frac{P(S^+ ∣ G) P(G)}{P(S^+ ∣ G) P(G) + P(S^+ ∣ U) P(U)} \\
&= \frac{(0.23)(0.2)}{(0.23)(0.2) + (0.15)(0.8)} \\
&≈ 28\%
\end{align*}$
"""

# ╔═╡ a737ced3-e8fa-4199-b249-62b6a227dea7
let
	PS⁺G = 0.23
	PG = 0.2
	PS⁺U = 0.15
	PU = 0.8

	PGS⁺ = (PS⁺G * PG) / (PS⁺G * PG + PS⁺U * PU)
	PUS⁺ = 1 - PGS⁺

	:PGS⁺ => PGS⁺, :PUS⁺ => PUS⁺
end

# ╔═╡ 9a7b7cf2-a7e4-4f2c-8cc2-2dc8d3cd4563
md"""
**4(b).**
A randomly chosen college student is more likely to be an undergraduate student since ``P(U) > P(G)``.
"""

# ╔═╡ cb63aae4-5b91-484a-9dcb-067255ed3cfd
md"""
**4(c).**
A randomly chosen college student, assuming they are a smoker, is more likely to be an undergraduate student (from ``P(G ∣ S^+) = 28\% < P(U ∣ S^+) = 72\%``).
"""

# ╔═╡ 291e03d6-bb34-4d85-89ba-cb66db93c71e
md"""
**4(d).**
Let ``D`` represent the event that a given student lives in a dorm.
Then we're given the following probabilities,

$\begin{align*}
P(D ∣ G) &= 0.3 \\
P(D ∣ U) &= 0.1
\end{align*}$

The probability that a given student lives in a dorm is

$P(D) = P(U) P(D ∣ U) + P(G) P(D ∣ G) = (0.8) (0.1) + (0.2) (0.3) = 0.14$

and the probability that a given student smokes is

$P(S^+) = P(U) P(S^+ ∣ U) + P(G) P(S^+ ∣ G) = (0.8) (0.15) + (0.2) (0.23) = 0.166$

Assuming the events that students live in a dorm and that students smoke are independent, the probability that a student both smokes and lives in a dorm given that they are a graduate student is

$P((D ∩ S^+) ∣ G) = P(D ∣ G) P(S^+ ∣ G) = (0.3) (0.23) = 0.069$

and the same probability given they are an undergraduate student is

$P((D ∩ S^+) ∣ U) = P(D ∣ U) P(S^+ ∣ U) = (0.1) (0.15) = 0.015$

The probability that a student is a graduate student given they both live in the dorm and smoke is

$P(G ∣ (D ∩ S)) = \frac{P((D ∩ S^+) ∣ G) P(G)}{P(D ∩ S^+)} = \frac{(0.069)(0.2)}{(0.14)(0.166)} ≈ 0.59$

and the probability that they are an undergraduate is

$P(U ∣ (D ∩ S)) = \frac{P((D ∩ S^+) ∣ U) P(U)}{P(D ∩ S^+)} = \frac{(0.015)(0.8)}{(0.14)(0.166)} ≈ 0.52$

Since ``P(G ∣ (D ∩ S)) = 0.59 > P(U ∣ (D ∩ S)) = 0.52``, if a student smokes and lives in the dorm, he/she is more likely to be a graduate student.
"""

# ╔═╡ 680490ac-48ef-44b6-b282-7a0ed90262b4
let
	PU = 0.8
	PG = 0.2
	PDU = 0.1
	PDG = 0.3
	PSU = 0.15
	PSG = 0.23

	PD = PU * PDU + PG * PDG
	PS = PU * PSU + PG * PSG
	PDnSG = PDG * PSG
	PDnSU = PDU * PSU
	PGDnS = PDnSG * PG / (PD * PS)
	PUDnS = PDnSU * PU / (PD * PS)

	"P(D)" => PD, "P(S)" => PS, "P(D ∩ S | G)" => PDnSG, "P(D ∩ S | U)" => PDnSU, "P(G | D ∩ S)" => PGDnS, "P(U ∣ D ∩ S)" => PUDnS
end

# ╔═╡ fca35db7-f3ac-46cf-badb-5a2a0a6eda1d
md"""
### Acknowledgements

- MLJ.jl package documentation: [https://alan-turing-institute.github.io/MLJ.jl/dev/](https://alan-turing-institute.github.io/MLJ.jl/dev/)

- XOR classification problem: [https://computing.dcu.ie/~humphrys/Notes/Neural/single.neural.html](https://computing.dcu.ie/~humphrys/Notes/Neural/single.neural.html)
"""

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version = "1.1.0+4"

[[deps.Xorg_xcb_util_image_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "0fab0a40349ba1cba2c1da699243396ff8e94b97"
uuid = "12413925-8142-5f55-bb0e-6d7ca50bb09b"
version = "0.4.0+1"

[[deps.Xorg_xcb_util_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxcb_jll"]
git-tree-sha1 = "e7fd7b2881fa2eaa72717420894d3938177862d1"
uuid = "2def613f-5ad1-5310-b15b-b15d46f528f5"
version = "0.4.0+1"

[[deps.Xorg_xcb_util_keysyms_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "d1151e2c45a544f32441a567d1690e701ec89b00"
uuid = "975044d2-76e6-5fbe-bf08-97ce7c6574c7"
version = "0.4.0+1"

[[deps.Xorg_xcb_util_renderutil_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "dfd7a8f38d4613b6a575253b3174dd991ca6183e"
uuid = "0d47668e-0667-5a69-a72c-f761630bfb7e"
version = "0.3.9+1"

[[deps.Xorg_xcb_util_wm_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "e78d10aab01a4a154142c5006ed44fd9e8e31b67"
uuid = "c22f9ab0-d5fe-5066-847c-f4bb1cd4e361"
version = "0.4.1+1"

[[deps.Xorg_xkbcomp_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxkbfile_jll"]
git-tree-sha1 = "4bcbf660f6c2e714f87e960a171b119d06ee163b"
uuid = "35661453-b289-5fab-8a00-3d9160c6a3a4"
version = "1.4.2+4"

[[deps.Xorg_xkeyboard_config_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xkbcomp_jll"]
git-tree-sha1 = "5c8424f8a67c3f2209646d4425f3d415fee5931d"
uuid = "33bec58e-1273-512f-9401-5d533626f822"
version = "2.27.0+4"

[[deps.Xorg_xtrans_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "79c31e7844f6ecf779705fbc12146eb190b7d845"
uuid = "c5fb5394-a638-5e4d-96e5-b29de1b5cf10"
version = "1.4.0+3"

[[deps.ZipFile]]
deps = ["Libdl", "Printf", "Zlib_jll"]
git-tree-sha1 = "3593e69e469d2111389a9bd06bac1f3d730ac6de"
uuid = "a5390f91-8eb1-5f08-bee0-b1d1ffed6cea"
version = "0.9.4"

[[deps.Zlib_jll]]
deps = ["Libdl"]
uuid = "83775a58-1f1d-513f-b197-d71354ab007a"

[[deps.Zstd_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "e45044cd873ded54b6a5bac0eb5c971392cf1927"
uuid = "3161d3a3-bdf6-5164-811a-617609db77b4"
version = "1.5.2+0"

[[deps.Zygote]]
deps = ["AbstractFFTs", "ChainRules", "ChainRulesCore", "DiffRules", "Distributed", "FillArrays", "ForwardDiff", "IRTools", "InteractiveUtils", "LinearAlgebra", "MacroTools", "NaNMath", "Random", "Requires", "SparseArrays", "SpecialFunctions", "Statistics", "ZygoteRules"]
git-tree-sha1 = "93285d2877f1f1b09b2a2b029f90e9db10127022"
uuid = "e88e6eb3-aa80-5325-afca-941959d7151f"
version = "0.6.35"

[[deps.ZygoteRules]]
deps = ["MacroTools"]
git-tree-sha1 = "8c1a8e4dfacb1fd631745552c8db35d0deb09ea0"
uuid = "700de1a5-db45-46bc-99cf-38207098b444"
version = "0.2.2"

[[deps.libass_jll]]
deps = ["Artifacts", "Bzip2_jll", "FreeType2_jll", "FriBidi_jll", "HarfBuzz_jll", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "5982a94fcba20f02f42ace44b9894ee2b140fe47"
uuid = "0ac62f75-1d6f-5e53-bd7c-93b484bb37c0"
version = "0.15.1+0"

[[deps.libblastrampoline_jll]]
deps = ["Artifacts", "Libdl", "OpenBLAS_jll"]
uuid = "8e850b90-86db-534c-a0d3-1478176c7d93"

[[deps.libfdk_aac_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "daacc84a041563f965be61859a36e17c4e4fcd55"
uuid = "f638f0a6-7fb0-5443-88ba-1cc74229b280"
version = "2.0.2+0"

[[deps.libpng_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "94d180a6d2b5e55e447e2d27a29ed04fe79eb30c"
uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f"
version = "1.6.38+0"

[[deps.libvorbis_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Ogg_jll", "Pkg"]
git-tree-sha1 = "b910cb81ef3fe6e78bf6acee440bda86fd6ae00c"
uuid = "f27f6e37-5d2b-51aa-960f-b287f2bc3b7a"
version = "1.3.7+1"

[[deps.nghttp2_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d"

[[deps.p7zip_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0"

[[deps.x264_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4fea590b89e6ec504593146bf8b988b2c00922b2"
uuid = "1270edf5-f2f9-52d2-97e9-ab00b5d0237a"
version = "2021.5.5+0"

[[deps.x265_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "ee567a171cce03570d77ad3a43e90218e38937a9"
uuid = "dfaa095f-4041-5dcd-9319-2fabd8486b76"
version = "3.5.0+0"

[[deps.xkbcommon_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Wayland_jll", "Wayland_protocols_jll", "Xorg_libxcb_jll", "Xorg_xkeyboard_config_jll"]
git-tree-sha1 = "ece2350174195bb31de1a63bea3a41ae1aa593b6"
uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd"
version = "0.9.1+5"
"""

# ╔═╡ Cell order:
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text:
module cash # Cash

"""
number  ::domain -> Abstract Space :: codomain
Float64 ::domain -> Application Space :: codomain

Quantity  ::Integer
"""
Quantity :: Integer #number of a bankNote # frequency of cash #Integer
Quality ::  Float64     # how much of a bankNote

#TODO: add bankNotes (functionality)

# Dr cash

end

ext: jl

max_stars_repo_name: adamwillisXanax/SolvencyPredictor

max_issues_repo_name: adamwillisXanax/SolvencyPredictor

max_forks_repo_name: adamwillisXanax/SolvencyPredictor


text:
mutable struct GAMCTune <: MCTunerState
  smmalatune::BasicMCTune
  amtune::BasicMCTune
  totaltune::BasicMCTune
  smmalafrequency::Real
  amfrequency::Real
end

GAMCTune(smmalatune::BasicMCTune, amtune::BasicMCTune, totaltune::BasicMCTune) =
  GAMCTune(smmalatune, amtune, totaltune, NaN, NaN)

GAMCTune(smmalastep::Real=1., malastep::Real=1.) =
  GAMCTune(BasicMCTune(smmalastep), BasicMCTune(NaN), BasicMCTune(1.), NaN, NaN)

struct GAMCTuner <: MCTuner
  smmalatuner::VanillaMCTuner
  amtuner::VanillaMCTuner
  totaltuner::Union{VanillaMCTuner, AcceptanceRateMCTuner}
  verbose::Bool
end

GAMCTuner(smmalatuner::VanillaMCTuner, amtuner::VanillaMCTuner, totaltuner::Union{VanillaMCTuner, AcceptanceRateMCTuner}) =
  GAMCTuner(smmalatuner, amtuner, totaltuner, (totaltuner.verbose || smmalatuner.verbose || amtuner.verbose) ? true : false)

show(io::IO, tuner::GAMCTuner) = print(io, "GAMCTuner")


text:
include("parameters.jl")

############################################################################
# Initial arrays
############################################################################

global TO=zeros(vlevo); global TO1=zeros(vlevo); global TOst=zeros(vlevo);
global UO=zeros(vlevo); global UO1=1zeros(vlevo); global UOst=zeros(vlevo);
global ρO=zeros(vleva);

global UA=zeros(vleva); global UA1=zeros(vleva); global UAst=zeros(vleva);
global qA=zeros(vleva); global qA1=zeros(vleva); global qAst=zeros(vleva);
global ΘA=zeros(vleva); global ΘA1=zeros(vleva); global ΘAst=zeros(vleva);
global ρA=zeros(vleva);
global TA=zeros(vleva);
global PA=zeros(vleva);

############################################################################
# Initial conditions
############################################################################
# atmospheric temperature in DegC
TA=25 .-0.007.*ZAC;
PA[1]=101300;
PA[2:vleva]=PA[1].*exp.(-cumsum(ΔZA./(Rgas.*(TA[2:end].+d2k)./gravity_mks)));
ρA=PA./(Rgas.*(TA.+d2k));
# atmospheric potential temperature in kelvin
ΘA[:]=(TA.+d2k).*(PA[1]./PA).^(2/7);
#ΘA[1:20].=ΘA[20];
ΘA0=ΘA[end].+1;
T0=ones(vlevo).*30;#repmat((16+(vlev-1:-1:0)/vlev*8),n+1,1);
TO=T0[1,1].+15 .*([vlevo:-1:1;]/vlevo.-1)/5;
SST=TO[1];
ssq = saltsat*cvapor_fac*exp(-cvapor_exp./(TA[1]+d2k))./ρA[1];
RH=0.8;
q0=0.5*ssq;
qA=saltsat*cvapor_fac*exp.(-cvapor_exp./(TA.+d2k))./ρA.*RH;
UA=(0.2/karman).*(log.(ZAC./1e-4));
SO=sref

UO=(vlevo:-1:1)/vlevo/10; VO=zeros(size(UO));
atemp=TA[1]+d2k; aqh=qA[1];
speed=abs(UA[1]); sst=SST[1];
tmp0=bulk(atemp,aqh,speed,sst,ZAC[1],ZAC[1],ZAC[1],ZAC[1],ρA[1]);
huol0=tmp0[8];

############################################################################

global LH=zeros(1); global SH=zeros(1); global TAU=zeros(1); TAUO=zeros(1);
global LW=zeros(1); global E=zeros(1); ssq=zeros(1);
xi=zeros(1); ce=zeros(1); ch=zeros(1);
global ustar=zeros(1); qstar=zeros(1); tstar=zeros(1);
psimh=zeros(1); psixh=zeros(1); Ri=zeros(vleva);
rd=zeros(1); rh=zeros(1); re=zeros(1);
global KAm=zeros(vleva); global KAt=zeros(vleva);
global KOm=zeros(vlevo); KOt=zeros(vlevo);
global Qnet=zeros(1);
FWflux=zeros(size(Qnet));


text:
using Luxor

srand(42) # Only the colors will change

N = 240
paper_dim = [16 10]
scaling = 150

xy = (rand(Float64, (N, 2)).*paper_dim.-paper_dim./2).*scaling

# Points
points = [Point(xy[i,1], xy[i,2]) for i in 1:N]

# Distances
dx = xy[:,1] .- xy[:,1]'
dy = xy[:,2] .- xy[:,2]'
dxy = sqrt.(dx.^2 .+ dy.^2)

# Lines
lines = []
for i in 1:(N-1)
	dxi = dxy[i,:]
	cutoff = sort(dxi)[rand(0:4)+1]
	connect = find(x -> dxi[x] <= cutoff, 1:N)
	for c in connect
		push!(lines, (points[i], points[c]))
	end
end

color_teal = ["#00796B", "#00897B", "#009688"]
col_indigo = ["#303F9F", "#3949AB", "#3F51B5"]
col_red = ["#D32F2F", "#E53935", "#F44336"]
col_purple = ["#7B1FA2", "#8E24AA", "#9C27B0"]
col_green = ["#388E3C", "#43A047", "#4CAF50"]

col = col_green

Drawing(paper_dim[1]*scaling, paper_dim[2]*scaling, "background.png")
origin()
background(col[1])
sethue(col[3])
circle.(points, 6.+rand(N).*10, :fill)
sethue(col[2])
circle.(points, 6.*rand(N).+2.0, :fill)
for l in lines
	line(l[1], l[2], :stroke)
end
finish()

max_stars_repo_name: atredennick/asa-envr-talk

max_issues_repo_name: atredennick/asa-envr-talk

max_forks_repo_name: atredennick/asa-envr-talk


text:
module SparseProbVec

using LinearAlgebra
using SpecialFunctions
using Distributions
using StatsBase

include("generalTools.jl")
include("sparseDirichletMixture.jl")
include("stickBreakingMixture.jl")

end


text:
mutable struct ACOPF_fromulation ACOPF_fromulation() = new() end

export PBRAPVmodel, PBRARVmodel, CBRARVmodel, CBRAWVmodel, PNPAPVmodel, PNRAPVmodel, PNRARVmodel

PBRAPVmodel =  ACOPF_fromulation();     #Power Branch Flow Rectangular Admittance Polar Voltage
PBRARVmodel =  ACOPF_fromulation();     #Power Branch Flow Rectangular Admittance Rectangular Voltage
PBRAWVmodel =  ACOPF_fromulation();     #Power Branch Flow Rectangular Admittance W-matrix Voltage
CBRARVmodel =  ACOPF_fromulation();     #Current Branch Flow Rectangular Admittance Rectangular Voltage
CBRAWVmodel =  ACOPF_fromulation();     #Current Branch Flow Rectangular Admittance W-matrix Voltage
PNPAPVmodel =  ACOPF_fromulation();     #Power Nodal Flow Polar Admittance Polar Voltage
PNRAPVmodel =  ACOPF_fromulation();     #Power Nodal Flow Rectangular Admittance Polar Voltage
PNRARVmodel =  ACOPF_fromulation();     #Power Nodal Flow Rectangular Admittance Rectangular Voltage
PNRAWVmodel =  ACOPF_fromulation();     #Power Nodal Flow Rectangular Admittance W-matrix Voltage


text:
# Spinnaker.jl: wrapper for FLIR/Point Grey Spinnaker SDK
# Copyright (C) 2019 Samuel Powell

# build.jl: locate library and write file
using Libdl

function find_spinnaker()

  if Sys.iswindows()
    path = joinpath(ENV["ProgramFiles"], "Point Grey Research", "Spinnaker", "bin", "vs2015")
    libspinnaker = "SpinnakerC_v140.dll"
    libspinvideo = ""
  elseif Sys.islinux()
    path = "/usr/lib"
    libspinnaker = "libSpinnaker_C.so"
    libspinvideo = "libSpinVideo_C.so"
  elseif Sys.isapple()
    path = "/usr/local/lib"
    libspinnaker = "libSpinnaker_C.dylib"
    libspinvideo = "libSpinVideo_C.dylib"
  else
    @error "Spinnaker SDK only supported on Linux, Windows and MacOS platforms"
  end

  return path, libspinnaker, libspinvideo

end

function configure()

  path, libspinnaker, libspinvideo = find_spinnaker()
  libSpinnaker_C = find_library(libspinnaker,[path])
  libSpinVideo_C = find_library(libspinvideo,[path])

  libSpinnaker_C != "" || @error "Spinnaker SDK cannot be found"

  open(joinpath(@__DIR__, "deps.jl"), "w") do f
    write(f, """
      const libSpinnaker_C = \"$(libSpinnaker_C)\"
      const libSpinVideo_C = \"$(libSpinVideo_C)\"
    """)
  end

end

configure()


text:
export DoubleLearner

using Random

"""
    DoubleLearner(;L1, L2, rng=MersenneTwister())

This is a meta-learner, it will randomly select one learner and update another learner.
The estimation of an observation is the sum of result from two learners.
"""
Base.@kwdef struct DoubleLearner{T1<:AbstractLearner,T2<:AbstractLearner,R<:AbstractRNG} <:
                   AbstractLearner
    L1::T1
    L2::T2
    rng::R = MersenneTwister()
end

"""
    DoubleLearner(l1, l2; seed = nothing)
"""
DoubleLearner(l1, l2; seed = nothing) = DoubleLearner(l1, l2, MersenneTwister(seed))

(learner::DoubleLearner)(obs) = learner.L1(obs) .+ learner.L2(obs)

RLBase.extract_experience(t::AbstractTrajectory, learner::DoubleLearner) =
    extract_experience(t, learner.L1)

update!(learner::DoubleLearner, experience) =
    rand(learner.rng, Bool) ? update!(learner.L1, experience) :
    update!(learner.L2, experience)


text:
# Creates a operator that permutes rows, in blocks of size
# length(perm)
immutable PermutationOperator{T} <: Operator{T}
    perm::Vector{Int}
end
PermutationOperator(prm)=PermutationOperator{Int}(prm)

for OP in (:domainspace,:rangespace)
    @eval $OP(T::PermutationOperator) = ℓ⁰
end

Base.convert{T}(::Type{Operator{T}},P::PermutationOperator) =
    PermutationOperator{T}(P.perm)

function bandinds(P::PermutationOperator)
    dfs=P.perm-(1:length(P.perm))
    minimum(dfs),maximum(dfs)
end

function getindex{T}(P::PermutationOperator{T},k::Integer,j::Integer)
    n=length(P.perm)
    if (k-1)÷n == (j-1)÷n  # diagonal blocks
        k=mod(k-1,n)+1
        j=mod(j-1,n)+1
        T(P.perm[k]==j)
    else
        zero(T)
    end
end

multiplyperm(b,a)=Int[a[bk] for bk in b]
perm(a::Vector,b::Vector)=multiplyperm(invperm(sortperm(b)),sortperm(a))
perm(a::Tuple,b::Tuple)=perm([a...],[b...])

# the operator that rearranges a to be b
function PermutationOperator{T}(::Type{T},a::Vector,b::Vector)
    @assert sort(a)==sort(b)
    PermutationOperator{T}(perm(a,b))
end
PermutationOperator{T}(::Type{T},a::Tuple,b::Tuple)=PermutationOperator(T,[a...],[b...])


text:
### A Pluto.jl notebook ###
# v0.16.4

using Markdown
using InteractiveUtils

# ╔═╡ 6541fa72-1528-11eb-157a-a3e1f9aa82a0
md"""# Simulation of a Computer Network
## Router as a Queuing System

Delay/Jitter:
- Transmission
- Service
- Processing
- Waiting

Quality of Service:
- Realtime
- Lossless

Representation:
- ``A\left(t\right)``
- ``D\left(t\right)``
- ``W\left(t\right)``
- ``W_Q\left(t\right)``
- ``T_n``
- ``S_n``
- ``L_n``
- ``L_{Qn}``

## Little's Law

```math
\mathbb E\left[L\right]=\lambda\mathbb E\left[W\right]
```
```math
\mathbb E\left[L_Q\right]=\lambda\mathbb E\left[W_Q\right]
```

## Steady State Probabilities

```math
P_n=\lim_{t\rightarrow\infty}\mathbb P\left\{L(t)=n\right\}
```

Stability
```math
a_n=d_n
```

Poisson arrivals always see time averages (PASTA):
```math
P_n=a_n
```

## MM1

Exponential Arrivals / Exponential Services (``\lambda`` / ``\mu``)

Markov Chain

A technical remark: if one event occurs at an exponential rate ``\lambda``, and another independent event at an exponential rate ``\mu``, then together they occur at an exponential rate ``\lambda+\mu``.

Steady State

```math
\begin{cases}
P_0=1-\frac{\lambda}{\mu}\\
P_n=\left(1-\frac{\lambda}{\mu}\right)\left(\frac{\lambda}{\mu}\right)^n,\quad n>0
\end{cases}
```

```math
\mathbb E\left[L\right]=\frac{\lambda}{\mu-\lambda}
```

```math
\mathbb E\left[W\right]=\frac{1}{\lambda}\mathbb E\left[L\right]=\frac{1}{\mu-\lambda}
```

```math
\mathbb E\left[W_Q\right]=\mathbb E\left[W\right]-\mathbb E\left[S\right]=\frac{\lambda}{\mu\left(\mu-\lambda\right)}
```

```math
\mathbb E\left[L_Q\right]=\lambda\mathbb E\left[W_Q\right]=\frac{\lambda^2}{\mu\left(\mu-\lambda\right)}
```

Finite Capacity?

Exponential Arrivals?

Exponential Services?

Network of Queues?

## GG1

Virtual Waiting Time: ``V\left(t\right)``
```math
\mathbb E\left[V\right]=\lambda\mathbb E\left[S\right]E\left[W_Q\right]+\frac{\lambda}{2}\mathbb E\left[S^2\right]
```

## MG1

Pollaczek–Khintchine formula
```math
\mathbb E\left[W_Q\right] = \frac{\lambda\mathbb E\left[S^2\right]}{2\left(1-\lambda\mathbb E\left[S\right]\right)}
```

## Statistical Processing

Sample Mean and Sample Variance

```math
\bar X = \sum_{i=1}^n X_i
```

```math
\mathbb E\left[\bar X\right]= \mu
```

To determine the “worth” of ``\bar X`` as an estimator of the population mean ``\mu``, we consider its mean square error:
```math
\mathbb E\left[\left(\bar X - \mu\right)^2\right]= \frac{\sigma^2}{n}
```

```math
S^2=\frac{\sum_{i=1}^n\left(X-\bar x\right)^2}{n-1}
```

```math
\mathbb E\left[S^2\right]=\sigma^2
```

``n`` sufficient large: central limit theorema
```math
\mathbb P\left\{\left|X − \mu\right|>c\frac{\sigma}{\sqrt n}\right\} \approx \mathbb P\left\{\left|Z\right|>c\right\}
```
where ``Z`` is a standard normal

Runtime

Number of Runs

The Regenerative Approach

Stationnarity vs Ergodicity
"""

# ╔═╡ Cell order:
# ╟─6541fa72-1528-11eb-157a-a3e1f9aa82a0


text:
using KernelFunctions
using MLDataUtils
using Zygote
using Flux
using Distributions, LinearAlgebra
using Plots

N = 100 #Number of samples
μ = randn(2,2) # Random Centers
xgrid = range(-3,3,length=100) # Create a grid
Xgrid = hcat(collect.(Iterators.product(xgrid,xgrid))...)' #Combine into a 2D grid
y = rand([-1,1],N) # Select randomly between the two classes
X = zeros(N,2)
X[y.==1,:] = rand(MvNormal(μ[:,1],I),count(y.==1))' #Attribute samples from class 1
X[y.==-1,:] = rand(MvNormal(μ[:,2],I),count(y.==-1))' # Attribute samples from class 2

k = SqExponentialKernel(2.0) # Create kernel function
f(x,k,λ) = kernelmatrix(k,x,X,obsdim=1)*inv(kernelmatrix(k,X,obsdim=1)+exp(λ[1])*I)*y # Optimal prediction f
svmloss(y,ŷ)= f(X,k,λ) |> ŷ -> sum(maximum.(0.0,1-y*ŷ)) - λ*norm(ŷ) # Total svm loss with regularisation
pred = f(Xgrid,k,λ) #Compute prediction on a grid
contourf(xgrid,xgrid,pred)
scatter!(eachcol(X)...,color=y,lab="data")


text:
# this file implements direct difference estimation for Δ = Ωx - Ωy

####################################
#
# loss function evaluation
#
####################################

"""
  Computes vecnorm(Σx⋅Δ⋅Σy - Σy + Σx, p)
"""
function diffLoss(
  Σx::Symmetric{T},
  Δ::SymmetricSparseIterate{T},
  Σy::Symmetric{T},
  p::Real) where {T<:AbstractFloat}

  d = size(Σx, 1)
  if p == 2.
    v = zero(T)
    for c=1:d, r=1:d
      v += abs2( A_mul_X_mul_B_rc(Σx, Δ, Σy, r, c) - Σy[r, c] + Σy[r, c] )
    end
    return sqrt(v)
  elseif p==Inf
    v = zero(T)
    for c=1:d, r=1:d
      t = abs( A_mul_X_mul_B_rc(Σx, Δ, Σy, r, c) - Σy[r, c] + Σy[r, c] )
      if t > v
        v = t
      end
    end
    return v
  else
    throw(ArgumentError("p should be 2 or Inf"))
  end
end

"""
  Computes vecnorm(Σx⋅(Δ+UU')⋅Σy - Σy + Σx, p)
"""
function diffLoss(
  Σx::Symmetric{T},
  Δ::SymmetricSparseIterate{T},
  U::StridedMatrix{T},
  Σy::Symmetric{T},
  p::Real) where {T<:AbstractFloat}

  d = size(Σx, 1)
  if p == 2.
    v = zero(T)
    for c=1:d, r=1:d
      v += abs2( A_mul_X_mul_B_rc(Σx, Δ, Σy, r, c) + A_mul_UUt_mul_B_rc(Σx, U, Σy, r, c) - Σy[r, c] + Σy[r, c] )
    end
    return sqrt(v)
  elseif p==Inf
    v = zero(T)
    for c=1:d, r=1:d
      t = abs( A_mul_X_mul_B_rc(Σx, Δ, Σy, r, c) + A_mul_UUt_mul_B_rc(Σx, U, Σy, r, c) - Σy[r, c] + Σy[r, c] )
      if t > v
        v = t
      end
    end
  else
    throw(ArgumentError("p should be 2 or Inf"))
  end
end

####################################
#
# loss tr(Σx⋅Δ⋅Σy⋅Δ)/2 - tr(Δ(Σy-Σx))
#
####################################
struct CDDirectDifferenceLoss{T<:AbstractFloat, S} <: CoordinateDifferentiableFunction
  Σx::Symmetric{T, S}
  Σy::Symmetric{T, S}
  A::Matrix{T}    # stores Σx⋅Δ⋅Σy
  p::Int64
end

function CDDirectDifferenceLoss(Σx::Symmetric{T,S}, Σy::Symmetric{T,S}) where {T<:AbstractFloat} where S
  (issymmetric(Σx) && issymmetric(Σy)) || throw(DimensionMismatch())
  (p = size(Σx, 1)) == size(Σy, 1) || throw(DimensionMismatch())
  CDDirectDifferenceLoss{T,S}(Σx, Σy, zeros(T, p, p), p)
end

CoordinateDescent.numCoordinates(f::CDDirectDifferenceLoss) = div(f.p * (f.p + 1), 2)

function CoordinateDescent.initialize!(f::CDDirectDifferenceLoss{T}, x::SymmetricSparseIterate{T}) where {T<:AbstractFloat}
  # compute residuals for the loss

  Σx = f.Σx
  Σy = f.Σy
  A = f.A
  p = f.p

  for ac=1:p, ar=1:p
      @inbounds A[ar,ac] = A_mul_X_mul_B_rc(Σx, x, Σy, ar, ac)
  end

  nothing
end

function CoordinateDescent.gradient(
  f::CDDirectDifferenceLoss{T},
  x::SymmetricSparseIterate{T},
  j::Int64) where {T<:AbstractFloat}

  Σx = f.Σx
  Σy = f.Σy
  A = f.A
  p = f.p

  ri, ci = ind2subLowerTriangular(p, j)
  @inbounds return (A[ri,ci] + A[ci,ri]) / 2. - (Σy[ri, ci] - Σx[ri, ci])
end

function CoordinateDescent.descendCoordinate!(
  f::CDDirectDifferenceLoss{T},
  g::ProxL1{T},
  x::SymmetricSparseIterate{T},
  j::Int64) where {T<:AbstractFloat}

  Σx = f.Σx
  Σy = f.Σy
  A = f.A
  p = f.p

  ri, ci = ind2subLowerTriangular(p, j)

  a = zero(T)
  b = zero(T)
  if ri == ci
    @inbounds a = Σx[ri,ri] * Σy[ri,ri]
    @inbounds b = Σy[ri,ri] - Σx[ri,ri] - A[ri,ri]
  else
    @inbounds a = (Σx[ri,ri]*Σy[ci,ci] + Σx[ci,ci]*Σy[ri,ri]) + 2. * Σx[ri,ci] * Σy[ri,ci]
    @inbounds b = 2. * (Σy[ri,ci] - Σx[ri,ci]) - A[ri,ci] - A[ci,ri]
  end
  oldVal = x[ri, ci]
  x[ri, ci] += b / a
  newVal = cdprox!(g, x, j, ri == ci ? 1. / a : 2. / a)
  h = newVal - oldVal

  # update internals
  for ac=1:p, ar=1:p
    @inbounds A[ar, ac] += (ri == ci) ? (h * Σx[ar, ri] * Σy[ac, ri]) :
          (h * (Σx[ar, ri] * Σy[ci, ac] + Σx[ar, ci] * Σy[ri, ac]))
  end
  h
end

#####################################################
#
# Direct difference estimation using Active Shooting
#
#####################################################

differencePrecisionActiveShooting!(
  x::SymmetricSparseIterate,
  Σx::Symmetric,
  Σy::Symmetric,
  g::ProxL1,
  options=CDOptions()) =
  coordinateDescent!(x, CDDirectDifferenceLoss(Σx, Σy), g, options)

@inline function differencePrecisionActiveShooting(
  Σx::Symmetric,
  Σy::Symmetric,
  g::ProxL1,
  options=CDOptions())

  f = CDDirectDifferenceLoss(Σx, Σy)
  coordinateDescent!(SymmetricSparseIterate(f.p), f, g, options)
end

function differencePrecisionRefit(
  Σx::Symmetric{T},
  Σy::Symmetric{T},
  S,
  options=CDOptions()
  ) where T

  f = CDDirectDifferenceLoss(Σx, Σy)
  t = CoordinateDescent.numCoordinates(f)
  ω = ones(T, t) * 1e10
  for i=S
    # r,c = ind2sub(Σx, i)
    r, c = Tuple(CartesianIndices(Σx)[i])
    if r >= c
      ind = sub2indLowerTriangular(f.p, r, c)
      ω[ind] = 0.
    end
  end
  g = ProxL1(one(T), ω)
  x = SymmetricSparseIterate(f.p)
  coordinateDescent!(x, f, g, options)
end

########################################################################
#
# Direct difference estimation using iterative hard thresholding
#
########################################################################

# A = Σx⋅(Δ + (UV' + V'U) / 2.)⋅Σy                 --- this is ensured by the algorithm
function differencePrecision_objective(
  A::StridedMatrix{T},
  Σx::Symmetric{T}, Σy::Symmetric{T},
  Δ::SymmetricSparseIterate{T}, U::StridedMatrix{T}, V::StridedMatrix{T}) where {T<:AbstractFloat}

  p, r = size(U)
  v = zero(T)
  # add sparse part
  for c=1:p
    for j=1:p
      t = Δ[j, c]
      v += (A[j,c]/2. + Σx[j,c] - Σy[j,c])*t
    end
  end

  # add low rank part
  for c=1:r
    for a=1:p, b=1:p
      v += (A[a,b] / 2. + Σx[a,b] - Σy[a,b])*U[a,c]*V[b,c] / 2.
      v += (A[a,b] / 2. + Σx[a,b] - Σy[a,b])*U[b,c]*V[a,c] / 2.
    end
  end

  v
end

differencePrecision_objective(
  Σx::Symmetric{T}, Σy::Symmetric{T},
  Δ::SymmetricSparseIterate{T}) where {T<:AbstractFloat} =
    trace((A_mul_X_mul_B(Σx, Δ, Σy) / 2 - (Σy - Σx)) * Δ)

function diffPrecision_grad_delta!(
  grad_out::StridedMatrix{T},
  A::StridedMatrix{T},
  Σx::Symmetric{T}, Σy::Symmetric{T}) where {T<:AbstractFloat}

  p = size(Σx, 1)

  # (a, b) indices of updated grad_out[a,b] element
  for a=1:p, b=a:p
    grad_out[a,b] = (A[a,b] + A[b,a])/2. + Σx[a,b] - Σy[a,b]
    # update the other symmetric element
    if a != b
      grad_out[b,a] = grad_out[a,b]
    end
  end

  grad_out
end

function differencePrecisionIHT_grad_u!(grad_U, grad_Δ, A, Σx, Σy, U)
  diffPrecision_grad_delta!(grad_Δ, A, Σx, Σy)
  scale!(grad_Δ, 2.)
  A_mul_B!(grad_U, grad_Δ, U)
end

# η is the step size
# s is the target sparsity
function differencePrecisionIHT(
  Σx::Symmetric{T}, Σy::Symmetric{T},
  η::T, s::Int64;
  epsTol=1e-5, maxIter=1000) where {T<:AbstractFloat}

  assert(size(Σx,1) == size(Σx,2) == size(Σy,1) == size(Σy,2))

  p = size(Σx, 1)

  gD = zeros(p, p)
  Δ = zeros(p, p)
  L = zeros(p, p)

  fvals = []
  # gradient descent
  #
  fv = differencePrecision_objective(Σx, Σy, Δ)
  push!(fvals, fv)
  for iter=1:maxIter
    # update Δ
    diffPrecision_grad_delta!(gD, Σx, Σy, Δ)
    Δ .-= η .* gD
    HardThreshold!(Δ, s)

    # check for convergence
    fv_new = differencePrecision_objective(Σx, Σy, Δ)
    push!(fvals, fv_new)
    if abs(fv_new - fv) <= epsTol
      break
    end
    fv = fv_new
  end

  (Δ, fvals)
end

####################################
#
# Direct difference estimation using iterative hard thresholding
#
#  now with low rank
#
####################################

# η is the step size
# s is the target sparsity
# r is the target rank
function differenceLatentPrecisionIHT!(
  Δ::SymmetricSparseIterate{T}, L::StridedMatrix{T},                 # these are pre-allocated
  U::StridedMatrix{T}, gD::StridedMatrix{T}, gU::StridedMatrix{T},   # these are pre-allocated
  Σx::Symmetric{T}, Σy::Symmetric{T},
  ηΔ, ηU, s, r;
  options::IHTOptions=IHTOptions(), callback=nothing) where {T<:AbstractFloat}

  epsTol, maxIter, checkEvery = options.epsTol, options.maxIter, options.checkEvery

  p = size(Σx, 1)
  A = A_mul_UUt_mul_B(Σx, U, Σy)
  for col = 1:p
      for row = 1:p
        A[row, col] += A_mul_X_mul_B_rc(Σx, Δ, Σy, row, col)
      end
  end

  fvals = []
  # gradient descent
  #
  fv = differencePrecision_objective(A, Σx, Σy, Δ, U)
  push!(fvals, fv)
  for iter=1:maxIter
    # update Δ
    diffPrecision_grad_delta!(gD, A, Σx, Σy)
    max_gD = maximum( abs.(gD) )
    @. Δ = Δ - ηΔ * gD
    HardThreshold!(Δ, s)

    # update A
    for col = 1:p
        for row = 1:p
            A[row, col] = A_mul_X_mul_B_rc(Σx, Δ, Σy, row, col) + A_mul_UUt_mul_B_rc(Σx, U, Σy, row, col)
        end
    end

    # update U and L
    differencePrecisionIHT_grad_u!(gU, gD, A, Σx, Σy, U)
    @. U -= ηU * gU
    A_mul_Bt!(L, U, U)
    max_gU = maximum( abs.(gU) )

    # update A
    for col = 1:p
        for row = 1:p
            A[row, col] = A_mul_X_mul_B_rc(Σx, Δ, Σy, row, col) + A_mul_UUt_mul_B_rc(Σx, U, Σy, row, col)
        end
    end

    if callback != nothing
      callback(Δ, L)
    end

    done = max(max_gD, max_gU) < epsTol
    # check for convergence
    if mod(iter, checkEvery) == 0
      fv_new = differencePrecision_objective(A, Σx, Σy, Δ, U)
      push!(fvals, fv_new)
      done = abs(fv_new - fv) <= epsTol
      fv = fv_new
    end
    done && break
  end

  (Δ, L, U, fvals, iter)
end

function differenceLatentPrecisionIHT_init(
  Σx::Symmetric{T}, Σy::Symmetric{T},
  s::Int64, r::Int64) where{T}

  size(Σx) == size(Σy) || throw(DimensionMismatch())
  p = size(Σx, 1)

  gD = zeros(p, p)
  gU = zeros(p, r)
  Δ = SymmetricSparseIterate(T, p)
  L = zeros(p, p)

  # initialize Δ, L, U
  #
  tmp = inv(Σx) - inv(Σy)
  HardThreshold!(Δ, tmp, s)
  for colInd=1:p, rowInd=colInd:p
    tmp[rowInd,colInd] = tmp[rowInd,colInd] - Δ[rowInd, colInd]
    if rowInd != colInd
      tmp[colInd,rowInd] = tmp[rowInd,colInd]
    end
  end
  U, d, V = svd(tmp)
  U = U[:,1:r] .* sqrt.(d[1:r])'
  L = U * U'

  (Δ, L, U, gD, gU)
end

##