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Principal components example using Julia GeneNetwork API
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pc-julia.ipynb
{
"cells": [
{
"cell_type": "markdown",
"id": "741f2b73-603c-4e0a-96c1-497b928c72ac",
"metadata": {},
"source": [
"# Principal components example using Julia GeneNetwork API"
]
},
{
"cell_type": "markdown",
"id": "f14782b8-0d35-4bfd-b639-ad2a60a5030c",
"metadata": {},
"source": [
"We first load the relevant packages in Julia, get non-omic traits for the BXD population,\n",
"and then examine the first few rows of the trait matrix."
]
},
{
"cell_type": "code",
"execution_count": 1,
"id": "06b075ea-4758-44da-b41f-3342fd331f53",
"metadata": {},
"outputs": [],
"source": [
"using GeneNetworkAPI, Statistics, LinearAlgebra, Plots, DataFrames, StatsBase"
]
},
{
"cell_type": "code",
"execution_count": 2,
"id": "55966f8b-a9a1-485a-8548-a5cbf39f3361",
"metadata": {},
"outputs": [],
"source": [
"# get the non-omic traits for the BXD set\n",
"d = get_pheno(\"BXD\");"
]
},
{
"cell_type": "code",
"execution_count": 3,
"id": "33640c9c-233a-49cd-9f4f-e50a028fc65e",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(384, 10861)"
]
},
"execution_count": 3,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# check size\n",
"size(d)"
]
},
{
"cell_type": "code",
"execution_count": 4,
"id": "e9b854c5-db34-4447-ac6e-388ee81ef9c6",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<div class=\"data-frame\"><p>10 rows × 10,861 columns (omitted printing of 10853 columns)</p><table class=\"data-frame\"><thead><tr><th></th><th>id</th><th>BXD_10001</th><th>BXD_10002</th><th>BXD_10003</th><th>BXD_10004</th><th>BXD_10005</th><th>BXD_10006</th><th>BXD_10007</th></tr><tr><th></th><th title=\"InlineStrings.String31\">String31</th><th title=\"InlineStrings.String15\">String15</th><th title=\"InlineStrings.String15\">String15</th><th title=\"InlineStrings.String7\">String7</th><th title=\"InlineStrings.String15\">String15</th><th title=\"InlineStrings.String15\">String15</th><th title=\"InlineStrings.String15\">String15</th><th title=\"InlineStrings.String15\">String15</th></tr></thead><tbody><tr><th>1</th><td>B6D2F1</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><th>2</th><td>C57BL/6J</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><th>3</th><td>DBA/2J</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><th>4</th><td>BXD1</td><td>61.400002</td><td>54.099998</td><td>483.0</td><td>49.799999</td><td>46.0</td><td>20.299999</td><td>18.700001</td></tr><tr><th>5</th><td>BXD2</td><td>49.0</td><td>50.099998</td><td>403.0</td><td>45.5</td><td>44.900002</td><td>15.0</td><td>14.8</td></tr><tr><th>6</th><td>BXD5</td><td>62.5</td><td>53.299999</td><td>501.0</td><td>62.900002</td><td>52.5</td><td>23.0</td><td>18.700001</td></tr><tr><th>7</th><td>BXD6</td><td>53.099998</td><td>55.099998</td><td>403.0</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><th>8</th><td>BXD8</td><td>59.099998</td><td>57.299999</td><td>438.0</td><td>55.099998</td><td>54.299999</td><td>22.700001</td><td>22.4</td></tr><tr><th>9</th><td>BXD9</td><td>53.900002</td><td>51.200001</td><td>448.0</td><td>47.299999</td><td>45.799999</td><td>18.799999</td><td>18.1</td></tr><tr><th>10</th><td>BXD11</td><td>53.099998</td><td>53.599998</td><td>408.0</td><td>50.599998</td><td>49.0</td><td>19.5</td><td>18.9</td></tr></tbody></table></div>"
],
"text/latex": [
"\\begin{tabular}{r|ccccccccc}\n",
"\t& id & BXD\\_10001 & BXD\\_10002 & BXD\\_10003 & BXD\\_10004 & BXD\\_10005 & BXD\\_10006 & BXD\\_10007 & \\\\\n",
"\t\\hline\n",
"\t& String31 & String15 & String15 & String7 & String15 & String15 & String15 & String15 & \\\\\n",
"\t\\hline\n",
"\t1 & B6D2F1 & x & x & x & x & x & x & x & $\\dots$ \\\\\n",
"\t2 & C57BL/6J & x & x & x & x & x & x & x & $\\dots$ \\\\\n",
"\t3 & DBA/2J & x & x & x & x & x & x & x & $\\dots$ \\\\\n",
"\t4 & BXD1 & 61.400002 & 54.099998 & 483.0 & 49.799999 & 46.0 & 20.299999 & 18.700001 & $\\dots$ \\\\\n",
"\t5 & BXD2 & 49.0 & 50.099998 & 403.0 & 45.5 & 44.900002 & 15.0 & 14.8 & $\\dots$ \\\\\n",
"\t6 & BXD5 & 62.5 & 53.299999 & 501.0 & 62.900002 & 52.5 & 23.0 & 18.700001 & $\\dots$ \\\\\n",
"\t7 & BXD6 & 53.099998 & 55.099998 & 403.0 & x & x & x & x & $\\dots$ \\\\\n",
"\t8 & BXD8 & 59.099998 & 57.299999 & 438.0 & 55.099998 & 54.299999 & 22.700001 & 22.4 & $\\dots$ \\\\\n",
"\t9 & BXD9 & 53.900002 & 51.200001 & 448.0 & 47.299999 & 45.799999 & 18.799999 & 18.1 & $\\dots$ \\\\\n",
"\t10 & BXD11 & 53.099998 & 53.599998 & 408.0 & 50.599998 & 49.0 & 19.5 & 18.9 & $\\dots$ \\\\\n",
"\\end{tabular}\n"
],
"text/plain": [
"\u001b[1m10×10861 DataFrame\u001b[0m\n",
"\u001b[1m Row \u001b[0m│\u001b[1m id \u001b[0m\u001b[1m BXD_10001 \u001b[0m\u001b[1m BXD_10002 \u001b[0m\u001b[1m BXD_10003 \u001b[0m\u001b[1m BXD_10004 \u001b[0m\u001b[1m BXD_10005 \u001b[0m\u001b[1m BXD_10\u001b[0m ⋯\n",
"\u001b[1m \u001b[0m│\u001b[90m String31 \u001b[0m\u001b[90m String15 \u001b[0m\u001b[90m String15 \u001b[0m\u001b[90m String7 \u001b[0m\u001b[90m String15 \u001b[0m\u001b[90m String15 \u001b[0m\u001b[90m String\u001b[0m ⋯\n",
"─────┼──────────────────────────────────────────────────────────────────────────\n",
" 1 │ B6D2F1 x x x x x x ⋯\n",
" 2 │ C57BL/6J x x x x x x\n",
" 3 │ DBA/2J x x x x x x\n",
" 4 │ BXD1 61.400002 54.099998 483.0 49.799999 46.0 20.299\n",
" 5 │ BXD2 49.0 50.099998 403.0 45.5 44.900002 15.0 ⋯\n",
" 6 │ BXD5 62.5 53.299999 501.0 62.900002 52.5 23.0\n",
" 7 │ BXD6 53.099998 55.099998 403.0 x x x\n",
" 8 │ BXD8 59.099998 57.299999 438.0 55.099998 54.299999 22.700\n",
" 9 │ BXD9 53.900002 51.200001 448.0 47.299999 45.799999 18.799 ⋯\n",
" 10 │ BXD11 53.099998 53.599998 408.0 50.599998 49.0 19.5\n",
"\u001b[36m 10855 columns omitted\u001b[0m"
]
},
"execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# examine the first 10 rows\n",
"first(d,10)"
]
},
{
"cell_type": "markdown",
"id": "e41227f1-4992-47f6-bcd1-4d45ff774b38",
"metadata": {},
"source": [
"We will now get the phenotype IDs and then use the `info_dataset` function to get a description of the traits."
]
},
{
"cell_type": "code",
"execution_count": 5,
"id": "d8986299-4d1d-45b8-b9a7-d42bd79a3d32",
"metadata": {},
"outputs": [],
"source": [
"# get phenotype IDs removing the leading characters before underscore\n",
"phenoids = names(d) .|> (x->replace(x,r\"^BXD_\"=>\"\"));"
]
},
{
"cell_type": "code",
"execution_count": 6,
"id": "7a26064a-329d-4681-ae5f-7fda20ef6001",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<div class=\"data-frame\"><p>9 rows × 7 columns (omitted printing of 5 columns)</p><table class=\"data-frame\"><thead><tr><th></th><th>dataset_type</th><th>description</th></tr><tr><th></th><th title=\"String\">String</th><th title=\"String\">String</th></tr></thead><tbody><tr><th>1</th><td>phenotype</td><td>Central nervous system, morphology: Cerebellum weight, whole, bilateral in adults of both sexes [mg]</td></tr><tr><th>2</th><td>phenotype</td><td>Central nervous system, morphology: Cerebellum weight after adjustment for covariance with brain size [mg]</td></tr><tr><th>3</th><td>phenotype</td><td>Central nervous system, morphology: Brain weight, male and female adult average, unadjusted for body weight, age, sex [mg]</td></tr><tr><th>4</th><td>phenotype</td><td>Central nervous system, morphology: Cerebellum volume [mm3]</td></tr><tr><th>5</th><td>phenotype</td><td>Central nervous system, morphology: Cerebellum volume, adjusted for covariance with brain size [mm3]</td></tr><tr><th>6</th><td>phenotype</td><td>Central nervous system, morphology: Cerebellum internal granule layer (IGL) volume without adjustment [mm3]</td></tr><tr><th>7</th><td>phenotype</td><td>Central nervous system, morphology: Internal granule layer (IGL) of the cerebellum volume, adjusted for sex, age, body and brain weight [mm3]</td></tr><tr><th>8</th><td>phenotype</td><td>Central nervous system, pharmacology, behavior: Phencyclidine response (7.5 mg/kg ip), locomotor response, difference in activity (PCP minus saline) between 1 hr test periods [cm]</td></tr><tr><th>9</th><td>phenotype</td><td>Central nervous system, pharmacology, behavior: Phencyclidine response (7.5 mg/kg ip), locomotor activity from 0-60 min after PCP injection [cm]</td></tr></tbody></table></div>"
],
"text/latex": [
"\\begin{tabular}{r|ccc}\n",
"\t& dataset\\_type & description & \\\\\n",
"\t\\hline\n",
"\t& String & String & \\\\\n",
"\t\\hline\n",
"\t1 & phenotype & Central nervous system, morphology: Cerebellum weight, whole, bilateral in adults of both sexes [mg] & $\\dots$ \\\\\n",
"\t2 & phenotype & Central nervous system, morphology: Cerebellum weight after adjustment for covariance with brain size [mg] & $\\dots$ \\\\\n",
"\t3 & phenotype & Central nervous system, morphology: Brain weight, male and female adult average, unadjusted for body weight, age, sex [mg] & $\\dots$ \\\\\n",
"\t4 & phenotype & Central nervous system, morphology: Cerebellum volume [mm3] & $\\dots$ \\\\\n",
"\t5 & phenotype & Central nervous system, morphology: Cerebellum volume, adjusted for covariance with brain size [mm3] & $\\dots$ \\\\\n",
"\t6 & phenotype & Central nervous system, morphology: Cerebellum internal granule layer (IGL) volume without adjustment [mm3] & $\\dots$ \\\\\n",
"\t7 & phenotype & Central nervous system, morphology: Internal granule layer (IGL) of the cerebellum volume, adjusted for sex, age, body and brain weight [mm3] & $\\dots$ \\\\\n",
"\t8 & phenotype & Central nervous system, pharmacology, behavior: Phencyclidine response (7.5 mg/kg ip), locomotor response, difference in activity (PCP minus saline) between 1 hr test periods [cm] & $\\dots$ \\\\\n",
"\t9 & phenotype & Central nervous system, pharmacology, behavior: Phencyclidine response (7.5 mg/kg ip), locomotor activity from 0-60 min after PCP injection [cm] & $\\dots$ \\\\\n",
"\\end{tabular}\n"
],
"text/plain": [
"\u001b[1m9×7 DataFrame\u001b[0m\n",
"\u001b[1m Row \u001b[0m│\u001b[1m dataset_type \u001b[0m\u001b[1m description \u001b[0m\u001b[1m id \u001b[0m\u001b[1m name \u001b[0m ⋯\n",
"\u001b[1m \u001b[0m│\u001b[90m String \u001b[0m\u001b[90m String \u001b[0m\u001b[90m Int64 \u001b[0m\u001b[90m String \u001b[0m ⋯\n",
"─────┼──────────────────────────────────────────────────────────────────────────\n",
" 1 │ phenotype Central nervous system, morpholo… 10001 CBLWT2 ⋯\n",
" 2 │ phenotype Central nervous system, morpholo… 10002 ADJCBLWT\n",
" 3 │ phenotype Central nervous system, morpholo… 10003 BrainWT5\n",
" 4 │ phenotype Central nervous system, morpholo… 10004 CBLVol\n",
" 5 │ phenotype Central nervous system, morpholo… 10005 CB_Wt_Corr ⋯\n",
" 6 │ phenotype Central nervous system, morpholo… 10006 IGLVol\n",
" 7 │ phenotype Central nervous system, morpholo… 10007 AdjIGLVol\n",
" 8 │ phenotype Central nervous system, pharmaco… 10008 PCPLOC1hr7.5mgB\n",
" 9 │ phenotype Central nervous system, pharmaco… 10009 PCPLOC1hr7.5mg ⋯\n",
"\u001b[36m 4 columns omitted\u001b[0m"
]
},
"execution_count": 6,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# get the description by repeated calls to info_dataset\n",
"phenodf = mapreduce(x->info_dataset(\"BXD\",x),(df,r)->push!(df,r[1,:],cols=:union),phenoids[2:10])"
]
},
{
"cell_type": "markdown",
"id": "f7cc85d5-03d0-4050-a3be-cdf204a70c62",
"metadata": {},
"source": [
"We are going to pick the first 7 traits -- they are all related to the cerebellum, and are expected to be correlated. We drop individuals with any missing values."
]
},
{
"cell_type": "code",
"execution_count": 7,
"id": "150f6ea1-d75d-45d0-bc8a-157588879ee7",
"metadata": {},
"outputs": [],
"source": [
"# get first 7 traits\n",
"y = d[!,2:8];\n",
"# replace the x by missing value, drop individuals with any missing values\n",
"y = y .|> (x->tryparse.(Float64,x)) .|> (x->isnothing(x) ? missing : x) |> dropmissing |> Matrix;"
]
},
{
"cell_type": "markdown",
"id": "9c30406c-803f-4d69-a6b6-608c2dcf24a2",
"metadata": {},
"source": [
"Next we standardize the data."
]
},
{
"cell_type": "code",
"execution_count": 8,
"id": "2874d5b5-4c58-4c4a-8f99-52b02e3b8b94",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"2×7 Matrix{Float64}:\n",
" 0.0 -0.0 0.0 -0.0 0.0 -0.0 0.0\n",
" 1.0 1.0 1.0 1.0 1.0 1.0 1.0"
]
},
"execution_count": 8,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# standardize the columns\n",
"z = standardize(ZScoreTransform,y,dims=1);\n",
"# check that mean is 0, variance is 1\n",
"vcat(mean(z,dims=1),var(z,dims=1)) .|> (x->round(x,digits=4))"
]
},
{
"cell_type": "markdown",
"id": "c84cf318-b6b6-427a-8bfc-f136cef3ef09",
"metadata": {},
"source": [
"We calculate the correlation matrix."
]
},
{
"cell_type": "code",
"execution_count": 9,
"id": "efb000d0-2ab8-403a-8214-32bb495244ed",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"7×7 Matrix{Float64}:\n",
" 1.0 0.69 0.66 0.86 0.7 0.85 0.67\n",
" 0.69 1.0 -0.02 0.53 0.88 0.57 0.79\n",
" 0.66 -0.02 1.0 0.61 0.1 0.58 0.17\n",
" 0.86 0.53 0.61 1.0 0.75 0.88 0.62\n",
" 0.7 0.88 0.1 0.75 1.0 0.73 0.85\n",
" 0.85 0.57 0.58 0.88 0.73 1.0 0.85\n",
" 0.67 0.79 0.17 0.62 0.85 0.85 1.0"
]
},
"execution_count": 9,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"( R = cor(y) ).|> (x->round(x,digits=2))"
]
},
{
"cell_type": "markdown",
"id": "c6032b33-c34b-4aeb-ba99-1ce635c3f950",
"metadata": {},
"source": [
"Now, we compute the eigen decomposition of the correlation matrix. The result has two fields, a vector of eigenvalues and a matrix of eigenvectors. The eigenvalues are sorted; in this implementaiton they are sorted from smallest to largest. For principal components, we will want the eigenvalues and vectors sorted largest to smallest. "
]
},
{
"cell_type": "code",
"execution_count": 10,
"id": "90978715-6363-45ed-a863-f6213c548751",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"Eigen{Float64, Float64, Matrix{Float64}, Vector{Float64}}\n",
"values:\n",
"7-element Vector{Float64}:\n",
" 0.0002737074909449178\n",
" 0.012383248253589458\n",
" 0.07892936955179328\n",
" 0.24387585397643066\n",
" 0.32941515754753115\n",
" 1.3951512343719223\n",
" 4.93997142880778\n",
"vectors:\n",
"7×7 Matrix{Float64}:\n",
" 0.0117874 0.587587 -0.442638 … -0.405654 0.198646 -0.415028\n",
" -0.0151947 -0.581935 0.017936 -0.481378 -0.440201 -0.354967\n",
" 0.00380428 -0.162573 0.568268 -0.0737235 0.715183 -0.219464\n",
" 0.475135 -0.24723 -0.182824 -0.0958958 0.228086 -0.404954\n",
" -0.345262 0.377659 0.556771 -0.0996501 -0.327609 -0.398272\n",
" -0.615635 -0.255506 -0.346536 … 0.485396 0.145828 -0.422614\n",
" 0.525023 0.143633 0.131588 0.586131 -0.273239 -0.390351"
]
},
"execution_count": 10,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"pc = eigen(R)"
]
},
{
"cell_type": "markdown",
"id": "0bf54a7b-f027-4e86-a91e-798c12d2657e",
"metadata": {},
"source": [
"We calculate the transform of the data by postmultiplication by the eigenvectors."
]
},
{
"cell_type": "code",
"execution_count": 11,
"id": "fb27123c-1a6e-411b-af90-f152d450670c",
"metadata": {},
"outputs": [],
"source": [
"w = z * pc.vectors;"
]
},
{
"cell_type": "markdown",
"id": "93fb0a08-6608-41e6-a58e-4b5d3d247075",
"metadata": {},
"source": [
"We check that the transformed data (the principal components) have mean 0, and variance equal to the eigenvalues."
]
},
{
"cell_type": "code",
"execution_count": 12,
"id": "c3fc0938-9459-48b9-a512-d9e175c77c15",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"1×7 Matrix{Float64}:\n",
" 0.0 0.0 0.0 -0.0 0.0 -0.0 -0.0"
]
},
"execution_count": 12,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"mean(w,dims=1) .|> (x->round(x,digits=2))"
]
},
{
"cell_type": "code",
"execution_count": 13,
"id": "1abda3b9-0672-467c-8ff7-f7b4dcc15db1",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"1×7 Matrix{Float64}:\n",
" 0.0 0.01 0.08 0.24 0.33 1.4 4.94"
]
},
"execution_count": 13,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"var(w,dims=1) .|> (x->round(x,digits=2))"
]
},
{
"cell_type": "code",
"execution_count": 14,
"id": "cac08721-8826-40a0-8d0d-261c7a6227ae",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"7-element Vector{Float64}:\n",
" 0.0\n",
" 0.01\n",
" 0.08\n",
" 0.24\n",
" 0.33\n",
" 1.4\n",
" 4.94"
]
},
"execution_count": 14,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"pc.values .|> (x->round(x,digits=2))"
]
},
{
"cell_type": "markdown",
"id": "898f63c6-1faa-4437-ae52-f05a3697a53f",
"metadata": {},
"source": [
"Now, we make a scree plot which is just a plot of the eigenvalues sorted in decreasing order. It is customary to scale them by the sum of the eigenvalues."
]
},
{
"cell_type": "code",
"execution_count": 15,
"id": "dde9faa6-2329-4291-9fa8-30b3299189a6",
"metadata": {},
"outputs": [
{
"data": {
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},
"execution_count": 15,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# scree plot\n",
"plot(reverse(pc.values)./(sum(pc.values))*100,label=\"\",ylab=\"percent explained\",xlab=\"PC\")"
]
},
{
"cell_type": "markdown",
"id": "26671552-80fc-4f97-9dc9-ffc6e4c5c65d",
"metadata": {},
"source": [
"Now, we plot the transformed values (principal components) -- the first two."
]
},
{
"cell_type": "code",
"execution_count": 16,
"id": "a969adc8-5b83-4ac0-9258-e52e23eb548e",
"metadata": {},
"outputs": [
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],
"source": [
"scatter(w[:,7],w[:,6],label=\"\")"
]
},
{
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"id": "127c9290-9dc2-485a-ba3e-535acfc39e43",
"metadata": {},
"source": [
"There is an alternate way to compute -- this is more accurate, but more computationally demanding. This approach uses the singular value decomposition, or SVD."
]
},
{
"cell_type": "code",
"execution_count": 17,
"id": "14289506-4371-42a4-b744-9293cd15ff6f",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"SVD{Float64, Float64, Matrix{Float64}}\n",
"U factor:\n",
"31×7 Matrix{Float64}:\n",
" -0.161168 0.287118 0.23511 … -0.176817 0.00549159\n",
" 0.198843 0.034011 0.274488 0.00889663 0.0505802\n",
" -0.370543 0.401342 0.215287 -0.0922882 -0.0401533\n",
" -0.362615 -0.0917847 -0.0871735 -0.118095 -0.0828679\n",
" -0.00420564 0.162915 -0.0230162 -0.0888081 0.212957\n",
" -0.0771477 -0.0642139 -0.0217708 … 0.239482 -0.0190039\n",
" 0.248964 0.102259 -0.215318 0.06198 -0.173974\n",
" 0.0381604 -0.101157 -0.279564 -0.0452445 0.0065763\n",
" 0.171434 0.108019 0.204849 0.0128508 0.245243\n",
" 0.28402 0.29628 0.0541687 -0.0813583 0.0549782\n",
" -0.0283171 0.0335834 -0.03676 … -0.39815 -0.230703\n",
" 0.0556198 -0.00709676 -0.17332 0.2457 -0.0220142\n",
" 0.196871 -0.179726 0.269191 0.0794904 -0.667724\n",
" ⋮ ⋱ ⋮ \n",
" 0.159113 -0.186373 0.235372 -0.322686 0.157233\n",
" -0.111212 -0.42953 0.260484 … 0.0582541 0.100225\n",
" 0.109293 -0.0313625 0.0608609 -0.217509 -0.0908116\n",
" -0.208721 0.063538 -0.0587601 -0.101661 0.065567\n",
" -0.0495892 0.0841936 -0.175241 0.212521 -0.0044221\n",
" -0.110186 -0.00601739 0.112558 -0.0145025 0.190311\n",
" 0.0303843 -0.0485968 -0.0258004 … 0.22578 -0.0600682\n",
" 0.313857 0.158943 -0.115368 -0.141731 -0.0933857\n",
" 0.0797754 0.0817074 0.143656 0.24797 0.165249\n",
" -0.172736 -0.155711 0.202274 0.145823 -0.0796018\n",
" 0.0960832 0.221631 -0.224668 0.0362154 -0.0205208\n",
" -0.291626 0.0899204 0.0793507 … 0.30411 -0.12908\n",
"singular values:\n",
"7-element Vector{Float64}:\n",
" 12.173707030491316\n",
" 6.469508252653946\n",
" 3.143637181105025\n",
" 2.704861478762441\n",
" 1.538792086850525\n",
" 0.6095059044895893\n",
" 0.09061580837998166\n",
"Vt factor:\n",
"7×7 Matrix{Float64}:\n",
" -0.415028 -0.354967 -0.219464 … -0.398272 -0.422614 -0.390351\n",
" 0.198646 -0.440201 0.715183 -0.327609 0.145828 -0.273239\n",
" 0.405654 0.481378 0.0737235 0.0996501 -0.485396 -0.586131\n",
" 0.287071 0.330599 0.292472 -0.39025 -0.0120641 0.340325\n",
" -0.442638 0.017936 0.568268 0.556771 -0.346536 0.131588\n",
" -0.587587 0.581935 0.162573 … -0.377659 0.255506 -0.143633\n",
" -0.0117874 0.0151947 -0.00380428 0.345262 0.615635 -0.525023"
]
},
"execution_count": 17,
"metadata": {},
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],
"source": [
"s = svd(z)"
]
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"metadata": {},
"source": [
"The proportion variance explained is just the square of the singular values divided by the product of the dimensions of the data matrix."
]
},
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"execution_count": 18,
"id": "bb6895c9-590d-4b4b-9d45-6f2ba5a662e2",
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"outputs": [
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"data": {
"text/plain": [
"7-element Vector{Float64}:\n",
" 0.68295\n",
" 0.19288\n",
" 0.04554\n",
" 0.03372\n",
" 0.01091\n",
" 0.00171\n",
" 4.0e-5"
]
},
"execution_count": 18,
"metadata": {},
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],
"source": [
"(s.S.^2)./prod(size(z)) .|> (x->round(x,digits=5))"
]
},
{
"cell_type": "markdown",
"id": "59f46de5-20d4-4e6b-b7f4-b81ac57a3c6f",
"metadata": {},
"source": [
"Compare with the eigenvalues which are divided by the number of columns (they are in increasing order)."
]
},
{
"cell_type": "code",
"execution_count": 19,
"id": "3d1419a5-2853-42bd-87a8-eaa32754f628",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"7-element Vector{Float64}:\n",
" 4.0e-5\n",
" 0.00177\n",
" 0.01128\n",
" 0.03484\n",
" 0.04706\n",
" 0.19931\n",
" 0.70571"
]
},
"execution_count": 19,
"metadata": {},
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"source": [
"pc.values/size(z,2) .|> (x->round(x,digits=5))"
]
},
{
"cell_type": "markdown",
"id": "09b1894b-3dab-474b-9700-b73c19f1c7fb",
"metadata": {},
"source": [
"The principal components are (proportional to) the $U$ matrix. It is an orthogonal matrix, so all components have the same variance, unlike the principal components whose variance is proportional to the eigenvalue. Notice, that other than the scale, the plots look identical."
]
},
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},
"execution_count": 20,
"metadata": {},
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}
],
"source": [
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]
}
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