pca_apply.jl

### A Pluto.jl notebook ###
# v0.19.4

using Markdown
using InteractiveUtils

# This Pluto notebook uses @bind for interactivity. When running this notebook outside of Pluto, the following 'mock version' of @bind gives bound variables a default value (instead of an error).
macro bind(def, element)
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        local el = $(esc(element))
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        el
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end

# ╔═╡ 69cd367a-959e-11ec-1d11-dbb242dc1861
begin
	using LinearAlgebra
	using Markdown
	using MultivariateStats
	using Plots, LaTeXStrings
	using PairPlots
	using PlutoUI
	using Random
	using Statistics

	using CSV
	import DataFrames:DataFrame, Not, names, select

	using MLDataUtils
	using LIBSVM
end;

# ╔═╡ d8f18c3f-1b9e-4bde-88f4-c407703fba27
md"""
### Overivew

In this lesson, we'll explore what happens when we use PCA to transform the high-${z}$ quasar dataset from the previous regression labs.  
"""

# ╔═╡ d2424607-7e22-49a9-ae6f-212364dffa44
md"### Read in data"

# ╔═╡ e917deeb-5f97-48d9-af9e-e84c5fe9e753
begin
	df = CSV.read("data/quasar_small.csv", DataFrame, limit=4000, select=[:ug, :gr, :ri, :iz, :zs1, :s1s2, :label],  ntasks=1)
	col_names = names(df)
	labels_all = df[:, :label]
	df_no_label = select(df, Not(:label), copycols=false)
	data = Matrix(df_no_label)'
end;

# ╔═╡ 2dd009f9-c2a3-447e-a10f-e1ba3f40bc51
md"We can make a **corner plot**, consisting of a scatter plot of each pair of colors.  Due to the high number of points, we've added contours.  Along the diagonal, histograms show the marginal distribution of each color."

# ╔═╡ 9cd631a0-9720-4d7f-8007-2d08d7423191
corner(df_no_label)

# ╔═╡ 4aab52ce-b12b-428c-82f2-576c6d5b22a9
md"""
Below, you can pick a pair of colors to view up close and color code by the label.  Red points are high-${z}$ quasars and blue points are other objects.
"""

# ╔═╡ 72e1d9bb-d466-433e-93f2-ec8b24e95fc3
md"We'll split the data into two equal subsets for training and testing models."

# ╔═╡ 709dcab8-bc1c-4eb7-b262-532273fa97c4
begin
	df_train, df_test = splitobs(df, at=0.5)
	labels_train = df_train[:, :label]
	data_train = Matrix(select(df_train, Not(:label), copycols=false))'
	labels_test = df_test[:, :label]
	data_test = Matrix(select(df_test, Not(:label), copycols=false))'
end;

# ╔═╡ 3644b575-8dec-401b-8461-d31da26e2477
md"### Standard PCA"

# ╔═╡ 0d70ead1-103c-4b0a-afb9-c6c059b757da
begin
	model_vanilla = fit(PCA, data_train)
	Xₜ_train_vanilla = MultivariateStats.transform(model_vanilla, data_train)
	Xₜ_test_vanilla = MultivariateStats.transform(model_vanilla, data_test)
	Xₜ_vanilla = hcat(Xₜ_train_vanilla,Xₜ_test_vanilla)
end;

# ╔═╡ 8a169904-9dd7-4981-a371-3315f93105b6
corner(DataFrame([view(Xₜ_vanilla, i, :) for i in 1:size(Xₜ_vanilla, 1)],
	             ["PCS " * string(i) for i in 1:size(Xₜ_vanilla,1)])  )

# ╔═╡ ec982b00-217e-45f8-9f49-078ace6798e0
let
	cfve_vanilla = cumsum(principalvars(model_vanilla)./tvar(model_vanilla))
	plt = plot(cfve_vanilla, lc=1, label=:none)
	scatter!(plt,cfve_vanilla, mc=1, label="Standard PCA", legend=:bottomright)
	xlabel!("Number Principal Compoents")
	ylabel!("Cumlative Fraction of Variance Explained")
	ylims!(0,1)
end

# ╔═╡ 3fa364ef-e0d0-4abb-a1cf-6f2cba2194a9
md"### Kernel PCA"

# ╔═╡ 0d1b5789-7e32-4c5c-99c5-a8f4037c9873
md"γ (Exponent for radial basis function): $(@bind kpca_gamma confirm(NumberField(0.05:0.05:2, default=0.5)))"

# ╔═╡ 3a5e0c08-a984-4e47-8e6a-9a47f01ca678
md"""
**Question:**  Did Kernel PCA (as opposed to standard PCA) result in PCA finding a more efficient representation of our dataset for this problem?

**Question:**  Was there a reason to expect that using a radial basis kernel would improve the ability of PCA to explain a large fraction of this dataset's variance with a small number of basis vectors?

**Question:**  Is it possible that another choice of kernel function might prove more useful for analyzing this dataset?  
"""

# ╔═╡ 1cdab078-297d-498a-8d31-aa3682d57151
md"## Fit SVM classifers"

# ╔═╡ 7ff72df3-eb8d-401a-8d22-70c2180c3562
md"First, we'll fit an SVM classifier using the raw data and use it to predict the labels for our dataset.
(Note that by default, the 'SVC' function uses a radial basis kernel.  Since we want to compare the performance of the SVM using different inputs, we'll specify that the SVM should use a linear kernel.)"

# ╔═╡ 98bcdb6b-cb0f-487d-8d45-d1d401e38acb
begin
	svm_raw = fit!(SVC(kernel=Kernel.Linear), data_train', labels_train)  
	ŷ_svm_raw = LIBSVM.predict(svm_raw, data_test')   # Predict the labels for data
end;

# ╔═╡ 3743a33f-f9da-4e59-9960-8c2220abfdfc
md"To evaluate the classifier, we'll evaluate the model based on calculating the fraction of actual high-${z}$ quasars missed (false negative rate) and the fraction of other objects mistakenly labeled as high-${z}$ quaesars (false positive rate)."

# ╔═╡ f172c447-91ac-4e43-96e1-75d9689c1c0a
begin
	mask_test = labels_test.==1               # Find actual quaesars
	(; false_negative_fraction = sum(ŷ_svm_raw[mask_test] .!= labels_test[mask_test])/sum(mask_test),
	   false_positive_fraction = sum(ŷ_svm_raw[.!mask_test] .!= labels_test[.!mask_test])/sum(.!mask_test) )
end

# ╔═╡ 8d39ca66-a14e-4852-ba70-eda1d2eb2879
md"""Next, we'll fit an SVM classifier to the data after being transformed by PCA.

**Question:** Do you expect the performance of this classifier to differ from that above?  
If so, do you expect it to be better or worse?  If not, why not?
"""

# ╔═╡ c6a841db-ece0-4e09-826d-be3347e4e53d
md"Ready to see what happens? $(@bind ready_svm_pca CheckBox())"

# ╔═╡ 41f8d92a-d184-4387-9027-5bd44786792d
if ready_svm_pca
	svm_pca = fit!(SVC(kernel=Kernel.Linear), Xₜ_train_vanilla', labels_train)
	ŷ_svm_pca = LIBSVM.predict(svm_pca, Xₜ_test_vanilla')
	(; false_negative_fraction = sum(ŷ_svm_pca[mask_test] .!= labels_test[mask_test])/sum(mask_test),
	   false_positive_fraction =  sum(ŷ_svm_pca[.!mask_test] .!= labels_test[.!mask_test])/sum(.!mask_test) )
end

# ╔═╡ a4c41470-bc8a-4141-a934-af90b9b29c45
md"""Now, we'll fit an SVM classifier using the data transformed into a new manifold using kernel PCA.  

**Question:** Do you expect the performance of this classifier will differ from those above?  If so, do you expect it to be better or worse?  If not, why not?

Ready to see what happens? $(@bind ready_svm_kpca CheckBox())
"""

# ╔═╡ ddc038c3-6e47-443b-a564-95e1d305ff20
md"""
## Next Steps
- If you're ambitous, consider trying an alternative kernel function with PCA.  Code for for a few common kernel functions used by Kernel PCA appears at the bottom of the notebook.  You'll want to change the value of the optional named parameter `kernel` in the line `fit(KernelPCA, data_train; kernel=kernel_radial,  inverse=true)` above.  You could try tinkering with the parameters of the polynomial kernel or writing your own kernel function.  If you find a kernel function that appears works well, then please share your success on the slack channel.

- Our final dimensional reduction lab (`application_to_galaxy_images.ipynb`, in the form of a Jupyter notebook) will demonstrate combining PCA and SVMs for analyzing astronomical images.

- For an example of a succesful astronomical application of SVMs in the astronomical literature, see [Lochner et al. (2016)](https://ui.adsabs.harvard.edu/abs/2016ApJS..225...31L/abstract).  It compares several machine learning methods in the context of automatically classifying supernovae based on photometry alone (i.e., without spectroscopic redshift measurements).
"""

# ╔═╡ e9f8d3f7-ae46-4793-8eac-fd8ad711e4f7
md"""
## Setup & Helper code
"""

# ╔═╡ b7edbfd3-9196-4f9e-ae7b-2291cd7147da
TableOfContents()

# ╔═╡ 0118ef60-7422-493f-8fee-ca90a594c967
begin
	kernel_linear = (x,y)->x'y
	kpca_c = 1.0
	kpca_d = 2
	kernel_polynomial = (x,y)->(x'y+kpca_c)^kpca_d
	kernel_radial = (x,y)->exp(-kpca_gamma.*norm(x-y)^2)
end

# ╔═╡ 14f8cc90-2610-47d0-bc4e-702a40fd20d7
begin
	model_kernel = fit(KernelPCA, data_train; kernel=kernel_radial,  inverse=true) #, maxoutdim=3)
	Xₜ_train_kernel = MultivariateStats.transform(model_kernel, data_train)
	Xₜ_test_kernel = MultivariateStats.transform(model_kernel, data_test)
	Xₜ_kernel = hcat(Xₜ_train_kernel, Xₜ_test_kernel)
end;

# ╔═╡ ba228583-b2af-47a0-bbc5-01235ea52fa5
corner(DataFrame([view(Xₜ_kernel, i, :) for i in 1:size(Xₜ_kernel, 1)],
	             ["KPCS " * string(i) for i in 1:size(Xₜ_kernel,1)])  )

# ╔═╡ 75ef65ba-8bce-4eac-b6b9-cbf91da8d4f1
let
	cfve_vanilla = cumsum(principalvars(model_vanilla)./tvar(model_vanilla))
	plt = plot(cfve_vanilla, lc=1, label=:none)
	scatter!(plt,cfve_vanilla, mc=1, label="Standard PCA", legend=:bottomright)
	cfve_kernel = cumsum(model_kernel.:λ)/sum(model_kernel.:λ)
	plot!(plt,cfve_kernel, lc=2, label=:none)
	scatter!(plt,cfve_kernel, mc=2, label="Kernel PCA", legend=:bottomright)
	xlabel!("Number Principal Components")
	ylabel!("Fraction of variance explained")  # TODO:  Check if this is the right label for kernel PCA?
	ylims!(0,1)
end

# ╔═╡ 1af16c3f-739c-4d2e-95c4-d962491d09c6
if ready_svm_kpca
	svm_kpca = fit!(SVC(kernel=Kernel.Linear), Xₜ_train_kernel', labels_train)
	ŷ_svm_kpca = LIBSVM.predict(svm_kpca, Xₜ_test_kernel')
	(; false_negative_fraction = sum(ŷ_svm_kpca[mask_test] .!= labels_test[mask_test])/sum(mask_test),
	   false_positive_fraction =  sum(ŷ_svm_kpca[.!mask_test] .!= labels_test[.!mask_test])/sum(.!mask_test) )
end

# ╔═╡ 30a05385-d378-4299-88bc-402967d67187
nbsp = html"&nbsp";

# ╔═╡ 5d2bd890-8edd-430f-bc63-6776181af1ae
@bind values_color_cols PlutoUI.combine() do Child
	md"""
	Color for x:  $(Child(@bind plt_col_x Select(Pair.(1:length(col_names)-1,col_names[1:end-1]), default=1)))
	$nbsp $nbsp $nbsp
	Color for y: $(Child(@bind plt_col_y Select(Pair.(1:length(col_names)-1,col_names[1:end-1]), default=2)))
	"""
end

# ╔═╡ 1bfa8472-523f-4fc9-a1de-b9ecf96da1bb
let
	plot(data[plt_col_x, :], data[plt_col_y, :]; c=2 .- labels_all, ms=2.5, ma=0.5, seriestype=:scatter, legend=false, xlabel=col_names[plt_col_x], ylabel=col_names[plt_col_y])
end

# ╔═╡ 17c66f11-fcaf-43f5-a1d9-fe447c2b8274
@bind values_pcs_vanilla PlutoUI.combine() do Child
	md"""
	PC Score for x:  $(Child(@bind plt_pc_x Select(1:size(Xₜ_vanilla,1), default=1)))
	$nbsp $nbsp $nbsp
	PC Score for y: $(Child(@bind plt_pc_y Select(1:size(Xₜ_vanilla,1), default=2)))
	"""
end

# ╔═╡ d4b002d7-97d4-43de-b14a-5de4ab23f315
let
	plot(Xₜ_vanilla[plt_pc_x, :], Xₜ_vanilla[plt_pc_y, :]; c=2 .-labels_all, ms=2.5, ma=0.5, seriestype=:scatter, legend=false, xlabel="Standard PCA Score $plt_pc_x", ylabel="Standard PCA Score $plt_pc_y")
end

# ╔═╡ 741f93ea-4023-439f-a35a-77c172e84b92
@bind values_pcs_kernel PlutoUI.combine() do Child
	md"""
	Kernal PCA Score for x:  $(Child(@bind plt_kpc_x Select(1:size(Xₜ_kernel,1), default=1)))
	$nbsp $nbsp $nbsp
	Kernel PCA Score for y: $(Child(@bind plt_kpc_y Select(1:size(Xₜ_kernel,1), default=2)))
	"""
end

# ╔═╡ b64a6fa7-4781-439d-8edf-e7f01ae7f014
begin
	plot(Xₜ_kernel[plt_kpc_x, :], Xₜ_kernel[plt_kpc_y, :]; ms=2.5, ma=0.5, c=2 .- labels_all, seriestype=:scatter, legend=false, xlabel="Kernel PCA Score $plt_kpc_x", ylabel="Kernel PCA Score $plt_kpc_y")
end

# ╔═╡ f879a7de-5344-4329-9fe3-6f67e35fab58
br = html"<br>";

# ╔═╡ a53948d8-9deb-4802-9ab9-151e3f938070
md"""
# Lab 8: Dimesional Reduction $br Applying PCA to Astronomical Data
#### [Penn State Astroinformatics Summer School 2022](https://sites.psu.edu/astrostatistics/astroinfo-su22-program/)
#### Kadri Nizam & [Eric Ford](https://www.personal.psu.edu/ebf11)
"""

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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.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.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.liblinear_jll]]
deps = ["Libdl", "Pkg"]
git-tree-sha1 = "6a4a6a3697269cb2da57e698e9318972d88de0bb"
uuid = "275f1f90-abd2-5ca1-9ad8-abd4e3d66eb7"
version = "2.30.0+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.libsvm_jll]]
deps = ["Artifacts", "CompilerSupportLibraries_jll", "JLLWrappers", "LLVMOpenMP_jll", "Libdl", "Pkg"]
git-tree-sha1 = "7625dde5e9eab416c1cb791627f065ce55297eff"
uuid = "08558c22-525a-5d2a-acf6-0ac6658ffce4"
version = "3.25.0+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:
# ╟─a53948d8-9deb-4802-9ab9-151e3f938070
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# ╟─9cd631a0-9720-4d7f-8007-2d08d7423191
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# ╟─3fa364ef-e0d0-4abb-a1cf-6f2cba2194a9
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# ╟─1cdab078-297d-498a-8d31-aa3682d57151
# ╟─7ff72df3-eb8d-401a-8d22-70c2180c3562
# ╠═98bcdb6b-cb0f-487d-8d45-d1d401e38acb
# ╟─3743a33f-f9da-4e59-9960-8c2220abfdfc
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# ╟─8d39ca66-a14e-4852-ba70-eda1d2eb2879
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# ╟─a4c41470-bc8a-4141-a934-af90b9b29c45
# ╟─1af16c3f-739c-4d2e-95c4-d962491d09c6
# ╟─ddc038c3-6e47-443b-a564-95e1d305ff20
# ╟─e9f8d3f7-ae46-4793-8eac-fd8ad711e4f7
# ╠═69cd367a-959e-11ec-1d11-dbb242dc1861
# ╠═b7edbfd3-9196-4f9e-ae7b-2291cd7147da
# ╠═0118ef60-7422-493f-8fee-ca90a594c967
# ╟─30a05385-d378-4299-88bc-402967d67187
# ╟─f879a7de-5344-4329-9fe3-6f67e35fab58
# ╟─00000000-0000-0000-0000-000000000001
# ╟─00000000-0000-0000-0000-000000000002