add problem 5
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Steven G. Johnson committed
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"$$"
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{
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"attachments": {
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"image.png": {
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"image/png": 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truncated
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}
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},
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"cell_type": "markdown",
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"id": "b945636a",
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"metadata": {},
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"source": [
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"## Problem 5\n",
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"\n",
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"The following problem is adapted from the [VMLS book](https://web.stanford.edu/~boyd/vmls/), additional [exercise 13.3](https://web.stanford.edu/~boyd/vmls/vmls-additional-exercises.pdf):\n",
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"\n",
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"\n",
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"\n",
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"**(a)** Describe the **size** and **entries** of the matrix $A$ and the vector $b$ for which $J = \\Vert A\\beta - b\\Vert^2 / (N-M)$. This allows you to find the coefficients $\\beta$ that minimize $J$, i.e. the auto-regressive predictor that minimizes the mean-square prediction error, simply by solving a standard least-square problem (`β = A \\ b` in Julia).\n",
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"\n",
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"**(b)** Implement an autoregressive model to predict German residential power demand with data from [this tutorial](https://vitalflux.com/autoregressive-ar-models-with-python-examples/#:~:text=Autoregressive%20models%20are%20based%20on,going%20up%20in%20the%20future.) via the code below.\n",
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"\n",
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"* Fill in the `???` in the `autoregressive_model` to form `A, b` according to (a) for given data `x` and `M`.\n",
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"* Apply it for `M=8` to find the prediction coefficients $\\beta$ from \"training\" data consisting of the power-demand data excluding the last 100 days.\n",
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"* Use the last 100 points as \"test\" data: compare the predictions of your fitted model by plotting the predictions for the last 100 days along with the actual data (fill in the ???)."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 2,
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"id": "722752b9",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"\"/var/folders/w_/rf8p8vtd5jb1d8rglz61jsz40000gp/T/jl_8RI3CLu1qv\""
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]
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},
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"execution_count": 2,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"# download the raw data (CSV file)\n",
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"import Downloads\n",
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"datafile = Downloads.download(\"https://raw.githubusercontent.com/jenfly/opsd/master/opsd_germany_daily.csv\")"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 3,
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"id": "3efb4edd",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"4384×5 Matrix{Any}:\n",
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" \"Date\" \"Consumption\" \"Wind\" \"Solar\" \"Wind+Solar\"\n",
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" \"2006-01-01\" 1069.18 \"\" \"\" \"\"\n",
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" \"2006-01-02\" 1380.52 \"\" \"\" \"\"\n",
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" \"2006-01-03\" 1442.53 \"\" \"\" \"\"\n",
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" \"2006-01-04\" 1457.22 \"\" \"\" \"\"\n",
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" \"2006-01-05\" 1477.13 \"\" \"\" \"\"\n",
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" \"2006-01-06\" 1403.43 \"\" \"\" \"\"\n",
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" \"2006-01-07\" 1300.29 \"\" \"\" \"\"\n",
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" \"2006-01-08\" 1207.98 \"\" \"\" \"\"\n",
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" \"2006-01-09\" 1529.32 \"\" \"\" \"\"\n",
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" \"2006-01-10\" 1576.91 \"\" \"\" \"\"\n",
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" \"2006-01-11\" 1577.18 \"\" \"\" \"\"\n",
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" \"2006-01-12\" 1553.28 \"\" \"\" \"\"\n",
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" ⋮ \n",
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" \"2017-12-20\" 1559.69 90.88 8.799 99.679\n",
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" \"2017-12-21\" 1520.37 259.039 7.313 266.352\n",
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" \"2017-12-22\" 1423.24 228.773 10.065 238.838\n",
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" \"2017-12-23\" 1272.17 748.074 8.45 756.524\n",
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" \"2017-12-24\" 1141.76 812.422 9.949 822.371\n",
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" \"2017-12-25\" 1111.28 587.81 15.765 603.575\n",
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" \"2017-12-26\" 1130.12 717.453 30.923 748.376\n",
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" \"2017-12-27\" 1263.94 394.507 16.53 411.037\n",
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" \"2017-12-28\" 1299.86 506.424 14.162 520.586\n",
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" \"2017-12-29\" 1295.09 584.277 29.854 614.131\n",
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" \"2017-12-30\" 1215.45 721.247 7.467 728.714\n",
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" \"2017-12-31\" 1107.11 721.176 19.98 741.156"
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]
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},
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"metadata": {},
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"output_type": "display_data"
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},
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"summary(days) = \"4383-element Vector{Int64}\"\n",
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"summary(consumption) = \"4383-element Vector{Float64}\"\n"
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]
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},
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{
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"data": {
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"image/png": "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 truncated
|
||||
"text/plain": [
|
||||
"Figure(PyObject <Figure size 800x1200 with 2 Axes>)"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
"output_type": "display_data"
|
||||
},
|
||||
{
|
||||
"data": {
|
||||
"text/plain": [
|
||||
"(1.0, 200.0)"
|
||||
]
|
||||
},
|
||||
"execution_count": 3,
|
||||
"metadata": {},
|
||||
"output_type": "execute_result"
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"# extract the data and plot it:\n",
|
||||
"\n",
|
||||
"using DelimitedFiles, Dates\n",
|
||||
"data = readdlm(datafile, ',')\n",
|
||||
"display(data)\n",
|
||||
"dates = parse.(Date, data[2:end, 1])\n",
|
||||
"days = Dates.value.(dates .- dates[1])\n",
|
||||
"consumption = Float64.(data[2:end, 2])\n",
|
||||
"@show summary(days)\n",
|
||||
"@show summary(consumption)\n",
|
||||
"\n",
|
||||
"using PyPlot\n",
|
||||
"\n",
|
||||
"figure(figsize=(8,12))\n",
|
||||
"subplot(2,1,1)\n",
|
||||
"title(\"German power consumption data\")\n",
|
||||
"plot(days, consumption, \"r.-\")\n",
|
||||
"xlabel(\"days (from 2006-01-01)\")\n",
|
||||
"ylabel(\"power consumption (MWh)\")\n",
|
||||
"\n",
|
||||
"subplot(2,1,2)\n",
|
||||
"title(\"German power consumption data (first 200 days)\")\n",
|
||||
"plot(days, consumption, \"r.-\")\n",
|
||||
"xlabel(\"days (from 2006-01-01)\")\n",
|
||||
"ylabel(\"power consumption (MWh)\")\n",
|
||||
"xlim(1,200)"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"id": "e12d9551",
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"# fill in the ??? to get A and b from x and m\n",
|
||||
"function autoregressive_model(x, M)\n",
|
||||
" N = length(x)\n",
|
||||
" A = similar(x, ???, ???) # ??? x ??? matrix of same type as x\n",
|
||||
" for i = 1:size(A,1) # fill in A row-by row\n",
|
||||
" t = M-1+i\n",
|
||||
" A[i,:] = x[???:???] # hint: each row of A comes from a slice of x\n",
|
||||
" end\n",
|
||||
" b = ???\n",
|
||||
" return A, b\n",
|
||||
"end\n",
|
||||
"\n",
|
||||
"training_data = consumption[1:end-100]\n",
|
||||
"\n",
|
||||
"# construct your A and b from M=8 and find β\n",
|
||||
"M = 8\n",
|
||||
"A, b = autoregressive_model(training_data[1:end-100], M)\n",
|
||||
"β = A \\ b # least-square fit"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"id": "e066bd80",
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"# test data from last 100+M points:\n",
|
||||
"test_data = consumption[end-99-M:end]\n",
|
||||
"\n",
|
||||
"# plot the predictions for the last 100 points along with test data\n",
|
||||
"predictions = ???\n",
|
||||
"plot(predictions, \"ro-\")\n",
|
||||
"plot(test_data[end-99:end], \"b*-\")"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"id": "88ac965b",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"## Problems 5, 6, etc: coming soon"
|
||||
"## Problems 6, etc: coming soon"
|
||||
]
|
||||
},
|
||||
{
|
||||
@@ -102,7 +292,7 @@
|
||||
],
|
||||
"metadata": {
|
||||
"kernelspec": {
|
||||
"display_name": "Julia 1.8.0",
|
||||
"display_name": "Julia 1.8.3",
|
||||
"language": "julia",
|
||||
"name": "julia-1.8"
|
||||
},
|
||||
@@ -110,7 +300,7 @@
|
||||
"file_extension": ".jl",
|
||||
"mimetype": "application/julia",
|
||||
"name": "julia",
|
||||
"version": "1.8.2"
|
||||
"version": "1.8.3"
|
||||
}
|
||||
},
|
||||
"nbformat": 4,
|
||||
|
||||
Reference in new issue
Block a user