# !wget https://developer.nvidia.com/compute/cuda/9.0/Prod/local_installers/cuda-repo-ubuntu1604-9-0-local_9.0.176-1_amd64-deb
# !dpkg -i cuda-repo-ubuntu1604-9-0-local_9.0.176-1_amd64-deb
# !apt-key add /var/cuda-repo-9-0-local/7fa2af80.pub
# !apt update -q
# !apt install cuda gcc-6 g++-6 -y -q
# !ln -s /usr/bin/gcc-6 /usr/local/cuda/bin/gcc
# !ln -s /usr/bin/g++-6 /usr/local/cuda/bin/g++
# !curl -sSL "https://julialang-s3.julialang.org/bin/linux/x64/1.7/julia-1.7.3-linux-x86_64.tar.gz" -o julia.tar.gz
# !tar -xzf julia.tar.gz -C /usr --strip-components 1
# !rm -rf julia.tar.gz*
# !julia -e 'using Pkg; pkg"add IJulia; precompile"'
This notebook contains an example for teaching.
13. Deep Neural Networks for Wage Prediction#
So far we considered many machine learning method, e.g Lasso and Random Forests, to build a predictive model. In this lab, we extend our toolbox by predicting wages by a neural network.
13.1. Data preparation#
Again, we consider data from the U.S. March Supplement of the Current Population Survey (CPS) in 2015.
# import Pkg; Pkg.add("Flux")
using RData, LinearAlgebra, GLM, DataFrames, Statistics, Random, Distributions, DataStructures, NamedArrays, PrettyTables, Plots
import CodecBzip2
# Importing .Rdata file
url = "https://github.com/d2cml-ai/14.388_jl/raw/github_data/data/wage2015_subsample_inference.RData"
download(url, "data.RData")
rdata_read = load("data.RData")
rm("data.RData")
data = rdata_read["data"]
names(data)
20-element Vector{String}:
"wage"
"lwage"
"sex"
"shs"
"hsg"
"scl"
"clg"
"ad"
"mw"
"so"
"we"
"ne"
"exp1"
"exp2"
"exp3"
"exp4"
"occ"
"occ2"
"ind"
"ind2"
typeof(data), size(data)
(DataFrame, (5150, 20))
Z = select(data, ["lwage", "wage"]) # regressors
| lwage | wage | |
|---|---|---|
| Float64 | Float64 | |
| 1 | 2.26336 | 9.61538 |
| 2 | 3.8728 | 48.0769 |
| 3 | 2.40313 | 11.0577 |
| 4 | 2.63493 | 13.9423 |
| 5 | 3.36198 | 28.8462 |
| 6 | 2.46222 | 11.7308 |
| 7 | 2.95651 | 19.2308 |
| 8 | 2.95651 | 19.2308 |
| 9 | 2.48491 | 12.0 |
| 10 | 2.95651 | 19.2308 |
| 11 | 2.85115 | 17.3077 |
| 12 | 2.48651 | 12.0192 |
| 13 | 2.48651 | 12.0192 |
| 14 | 2.59984 | 13.4615 |
| 15 | 2.79399 | 16.3462 |
| 16 | 3.32808 | 27.8846 |
| 17 | 3.07269 | 21.6 |
| 18 | 2.158 | 8.65385 |
| 19 | 2.95651 | 19.2308 |
| 20 | 2.57922 | 13.1868 |
| 21 | 2.36872 | 10.6838 |
| 22 | 2.44569 | 11.5385 |
| 23 | 2.87855 | 17.7885 |
| 24 | 2.95651 | 19.2308 |
| 25 | 2.79143 | 16.3043 |
| 26 | 2.95651 | 19.2308 |
| 27 | 2.66883 | 14.4231 |
| 28 | 2.48491 | 12.0 |
| 29 | 2.82298 | 16.8269 |
| 30 | 2.97911 | 19.6703 |
| ⋮ | ⋮ | ⋮ |
Firt, we split the data first and normalize it.
Random.seed!(1234)
training = sample(1:nrow(data), Int(floor(nrow(data)*(3/4))), replace = false)
3862-element Vector{Int64}:
2455
3109
4807
3107
617
2230
4697
755
771
3729
2063
528
3238
⋮
4057
973
4183
1230
3228
3687
4789
4060
2723
3610
899
1273
data_train = data[training,1:16]
data_test = data[Not(training),1:16]
data_train
| wage | lwage | sex | shs | hsg | scl | clg | ad | mw | |
|---|---|---|---|---|---|---|---|---|---|
| Float64 | Float64 | Float64 | Float64 | Float64 | Float64 | Float64 | Float64 | Float64 | |
| 1 | 12.0192 | 2.48651 | 0.0 | 0.0 | 0.0 | 0.0 | 1.0 | 0.0 | 1.0 |
| 2 | 10.2857 | 2.33076 | 0.0 | 0.0 | 1.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 3 | 10.5769 | 2.35867 | 0.0 | 0.0 | 0.0 | 1.0 | 0.0 | 0.0 | 0.0 |
| 4 | 17.3077 | 2.85115 | 1.0 | 0.0 | 0.0 | 1.0 | 0.0 | 0.0 | 0.0 |
| 5 | 25.641 | 3.24419 | 0.0 | 0.0 | 0.0 | 1.0 | 0.0 | 0.0 | 0.0 |
| 6 | 42.3077 | 3.74497 | 0.0 | 0.0 | 0.0 | 0.0 | 1.0 | 0.0 | 1.0 |
| 7 | 15.8654 | 2.76414 | 0.0 | 1.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 8 | 72.1154 | 4.27827 | 1.0 | 0.0 | 0.0 | 0.0 | 1.0 | 0.0 | 0.0 |
| 9 | 35.7143 | 3.57555 | 1.0 | 0.0 | 0.0 | 0.0 | 0.0 | 1.0 | 0.0 |
| 10 | 15.3846 | 2.73337 | 1.0 | 0.0 | 1.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 11 | 16.5385 | 2.80569 | 0.0 | 0.0 | 0.0 | 1.0 | 0.0 | 0.0 | 1.0 |
| 12 | 16.8269 | 2.82298 | 1.0 | 0.0 | 0.0 | 0.0 | 1.0 | 0.0 | 0.0 |
| 13 | 36.0577 | 3.58512 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 1.0 | 0.0 |
| 14 | 19.2308 | 2.95651 | 1.0 | 0.0 | 0.0 | 0.0 | 1.0 | 0.0 | 0.0 |
| 15 | 36.0577 | 3.58512 | 0.0 | 0.0 | 1.0 | 0.0 | 0.0 | 0.0 | 1.0 |
| 16 | 26.4423 | 3.27497 | 0.0 | 0.0 | 0.0 | 0.0 | 1.0 | 0.0 | 0.0 |
| 17 | 14.4231 | 2.66883 | 0.0 | 0.0 | 0.0 | 1.0 | 0.0 | 0.0 | 0.0 |
| 18 | 45.1923 | 3.81093 | 0.0 | 0.0 | 0.0 | 0.0 | 1.0 | 0.0 | 0.0 |
| 19 | 30.7692 | 3.42652 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 1.0 | 0.0 |
| 20 | 25.0 | 3.21888 | 0.0 | 0.0 | 0.0 | 0.0 | 1.0 | 0.0 | 0.0 |
| 21 | 18.4615 | 2.91569 | 1.0 | 0.0 | 0.0 | 0.0 | 1.0 | 0.0 | 0.0 |
| 22 | 8.71795 | 2.16538 | 1.0 | 0.0 | 0.0 | 1.0 | 0.0 | 0.0 | 1.0 |
| 23 | 19.2308 | 2.95651 | 0.0 | 0.0 | 1.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 24 | 13.4615 | 2.59984 | 0.0 | 0.0 | 1.0 | 0.0 | 0.0 | 0.0 | 1.0 |
| 25 | 22.5962 | 3.11778 | 0.0 | 0.0 | 1.0 | 0.0 | 0.0 | 0.0 | 1.0 |
| 26 | 10.6838 | 2.36872 | 1.0 | 0.0 | 1.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| 27 | 48.0769 | 3.8728 | 0.0 | 0.0 | 0.0 | 0.0 | 1.0 | 0.0 | 0.0 |
| 28 | 23.5577 | 3.15945 | 0.0 | 0.0 | 0.0 | 1.0 | 0.0 | 0.0 | 0.0 |
| 29 | 48.7179 | 3.88605 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 1.0 | 0.0 |
| 30 | 14.4712 | 2.67216 | 0.0 | 0.0 | 0.0 | 0.0 | 1.0 | 0.0 | 0.0 |
| ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ |
size(data_train), size(data_test)
((3862, 16), (1288, 16))
# normalize the data
mean_1 = mean.(eachcol(data_train))
mean_1 = [names(data_train) mean_1]
16×2 Matrix{Any}:
"wage" 23.3849
"lwage" 2.97164
"sex" 0.44407
"shs" 0.0222683
"hsg" 0.24754
"scl" 0.283014
"clg" 0.310979
"ad" 0.136199
"mw" 0.262558
"so" 0.296737
"we" 0.215691
"ne" 0.225013
"exp1" 13.6699
"exp2" 2.97307
"exp3" 8.05316
"exp4" 24.4216
std_1 = std.(eachcol(data_train))
std_1 = [names(data_train) std_1]
16×2 Matrix{Any}:
"wage" 20.445
"lwage" 0.566839
"sex" 0.496926
"shs" 0.147574
"hsg" 0.431639
"scl" 0.450522
"clg" 0.462954
"ad" 0.343044
"mw" 0.440081
"so" 0.456879
"we" 0.411354
"ne" 0.417645
"exp1" 10.5105
"exp2" 3.94797
"exp3" 14.2448
"exp4" 52.4952
df = DataFrame()
for i in 1:size(data_train)[2]
p = (data_train[!, i] .- mean_1[i,2]) / std_1[i,2]
colname = names(data_train)[i]
df[!,colname] = p
end
data_train = df
data_train
| wage | lwage | sex | shs | hsg | scl | clg | ad | |
|---|---|---|---|---|---|---|---|---|
| Float64 | Float64 | Float64 | Float64 | Float64 | Float64 | Float64 | Float64 | |
| 1 | -0.555914 | -0.855849 | -0.893634 | -0.150896 | -0.573489 | -0.628192 | 1.48831 | -0.39703 |
| 2 | -0.640703 | -1.13062 | -0.893634 | -0.150896 | 1.74326 | -0.628192 | -0.671727 | -0.39703 |
| 3 | -0.62646 | -1.08137 | -0.893634 | -0.150896 | -0.573489 | 1.59146 | -0.671727 | -0.39703 |
| 4 | -0.297246 | -0.212557 | 1.11874 | -0.150896 | -0.573489 | 1.59146 | -0.671727 | -0.39703 |
| 5 | 0.110351 | 0.480836 | -0.893634 | -0.150896 | -0.573489 | 1.59146 | -0.671727 | -0.39703 |
| 6 | 0.925545 | 1.36429 | -0.893634 | -0.150896 | -0.573489 | -0.628192 | 1.48831 | -0.39703 |
| 7 | -0.367792 | -0.36606 | -0.893634 | 6.62538 | -0.573489 | -0.628192 | -0.671727 | -0.39703 |
| 8 | 2.38349 | 2.30512 | 1.11874 | -0.150896 | -0.573489 | -0.628192 | 1.48831 | -0.39703 |
| 9 | 0.603051 | 1.06541 | 1.11874 | -0.150896 | -0.573489 | -0.628192 | -0.671727 | 2.51805 |
| 10 | -0.391307 | -0.420346 | 1.11874 | -0.150896 | 1.74326 | -0.628192 | -0.671727 | -0.39703 |
| 11 | -0.334871 | -0.292761 | -0.893634 | -0.150896 | -0.573489 | 1.59146 | -0.671727 | -0.39703 |
| 12 | -0.320762 | -0.262255 | 1.11874 | -0.150896 | -0.573489 | -0.628192 | 1.48831 | -0.39703 |
| 13 | 0.619847 | 1.08229 | -0.893634 | -0.150896 | -0.573489 | -0.628192 | -0.671727 | 2.51805 |
| 14 | -0.203185 | -0.0266836 | 1.11874 | -0.150896 | -0.573489 | -0.628192 | 1.48831 | -0.39703 |
| 15 | 0.619847 | 1.08229 | -0.893634 | -0.150896 | 1.74326 | -0.628192 | -0.671727 | -0.39703 |
| 16 | 0.149543 | 0.535122 | -0.893634 | -0.150896 | -0.573489 | -0.628192 | 1.48831 | -0.39703 |
| 17 | -0.438338 | -0.534203 | -0.893634 | -0.150896 | -0.573489 | 1.59146 | -0.671727 | -0.39703 |
| 18 | 1.06664 | 1.48065 | -0.893634 | -0.150896 | -0.573489 | -0.628192 | 1.48831 | -0.39703 |
| 19 | 0.36118 | 0.802482 | -0.893634 | -0.150896 | -0.573489 | -0.628192 | -0.671727 | 2.51805 |
| 20 | 0.0789973 | 0.436171 | -0.893634 | -0.150896 | -0.573489 | -0.628192 | 1.48831 | -0.39703 |
| 21 | -0.24081 | -0.0987005 | 1.11874 | -0.150896 | -0.573489 | -0.628192 | 1.48831 | -0.39703 |
| 22 | -0.717385 | -1.42237 | 1.11874 | -0.150896 | -0.573489 | 1.59146 | -0.671727 | -0.39703 |
| 23 | -0.203185 | -0.0266836 | -0.893634 | -0.150896 | 1.74326 | -0.628192 | -0.671727 | -0.39703 |
| 24 | -0.485368 | -0.655918 | -0.893634 | -0.150896 | 1.74326 | -0.628192 | -0.671727 | -0.39703 |
| 25 | -0.0385788 | 0.257821 | -0.893634 | -0.150896 | 1.74326 | -0.628192 | -0.671727 | -0.39703 |
| 26 | -0.621234 | -1.06364 | 1.11874 | -0.150896 | 1.74326 | -0.628192 | -0.671727 | -0.39703 |
| 27 | 1.20773 | 1.58981 | -0.893634 | -0.150896 | -0.573489 | -0.628192 | 1.48831 | -0.39703 |
| 28 | 0.00845161 | 0.331338 | -0.893634 | -0.150896 | -0.573489 | 1.59146 | -0.671727 | -0.39703 |
| 29 | 1.23908 | 1.61317 | -0.893634 | -0.150896 | -0.573489 | -0.628192 | -0.671727 | 2.51805 |
| 30 | -0.435986 | -0.528332 | -0.893634 | -0.150896 | -0.573489 | -0.628192 | 1.48831 | -0.39703 |
| ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ |
df = DataFrame()
for i in 1:size(data_test)[2]
p = (data_test[!, i] .- mean_1[i,2]) / std_1[i,2]
colname = names(data_test)[i]
df[!,colname] = p
end
data_test = df
data_test
| wage | lwage | sex | shs | hsg | scl | clg | |
|---|---|---|---|---|---|---|---|
| Float64 | Float64 | Float64 | Float64 | Float64 | Float64 | Float64 | |
| 1 | -0.67349 | -1.24951 | 1.11874 | -0.150896 | -0.573489 | -0.628192 | 1.48831 |
| 2 | -0.461853 | -0.594011 | 1.11874 | -0.150896 | -0.573489 | -0.628192 | -0.671727 |
| 3 | -0.555914 | -0.855849 | 1.11874 | -0.150896 | -0.573489 | 1.59146 | -0.671727 |
| 4 | -0.498805 | -0.692294 | -0.893634 | -0.150896 | 1.74326 | -0.628192 | -0.671727 |
| 5 | -0.273731 | -0.164221 | 1.11874 | -0.150896 | -0.573489 | -0.628192 | -0.671727 |
| 6 | -0.346322 | -0.317912 | 1.11874 | -0.150896 | -0.573489 | 1.59146 | -0.671727 |
| 7 | -0.556854 | -0.858674 | -0.893634 | -0.150896 | -0.573489 | 1.59146 | -0.671727 |
| 8 | -0.181686 | 0.0131863 | -0.893634 | -0.150896 | -0.573489 | -0.628192 | 1.48831 |
| 9 | -0.558527 | -0.863707 | 1.11874 | -0.150896 | 1.74326 | -0.628192 | -0.671727 |
| 10 | 0.0319668 | 0.366979 | 1.11874 | -0.150896 | -0.573489 | -0.628192 | 1.48831 |
| 11 | 0.871796 | 1.31786 | 1.11874 | -0.150896 | -0.573489 | -0.628192 | 1.48831 |
| 12 | -0.649975 | -1.16344 | -0.893634 | -0.150896 | 1.74326 | -0.628192 | -0.671727 |
| 13 | -0.791066 | -1.75703 | 1.11874 | -0.150896 | -0.573489 | 1.59146 | -0.671727 |
| 14 | -0.511021 | -0.726027 | -0.893634 | -0.150896 | -0.573489 | 1.59146 | -0.671727 |
| 15 | -0.0432819 | 0.250297 | 1.11874 | -0.150896 | -0.573489 | -0.628192 | -0.671727 |
| 16 | 0.518677 | 0.978069 | -0.893634 | -0.150896 | -0.573489 | -0.628192 | 1.48831 |
| 17 | -0.00249603 | 0.314496 | -0.893634 | -0.150896 | -0.573489 | -0.628192 | 1.48831 |
| 18 | -0.848174 | -2.06862 | -0.893634 | -0.150896 | -0.573489 | -0.628192 | 1.48831 |
| 19 | -0.67349 | -1.24951 | -0.893634 | -0.150896 | 1.74326 | -0.628192 | -0.671727 |
| 20 | -0.725746 | -1.4573 | -0.893634 | 6.62538 | -0.573489 | -0.628192 | -0.671727 |
| 21 | -0.555914 | -0.855849 | 1.11874 | -0.150896 | 1.74326 | -0.628192 | -0.671727 |
| 22 | -0.320762 | -0.262255 | -0.893634 | -0.150896 | -0.573489 | 1.59146 | -0.671727 |
| 23 | -0.0856093 | 0.181105 | -0.893634 | -0.150896 | -0.573489 | -0.628192 | 1.48831 |
| 24 | -0.979188 | -3.10157 | -0.893634 | -0.150896 | -0.573489 | -0.628192 | 1.48831 |
| 25 | -0.203185 | -0.0266836 | -0.893634 | -0.150896 | -0.573489 | 1.59146 | -0.671727 |
| 26 | -0.256935 | -0.130488 | -0.893634 | -0.150896 | -0.573489 | -0.628192 | 1.48831 |
| 27 | -0.67349 | -1.24951 | 1.11874 | -0.150896 | -0.573489 | 1.59146 | -0.671727 |
| 28 | -0.0620941 | 0.21988 | 1.11874 | -0.150896 | -0.573489 | -0.628192 | 1.48831 |
| 29 | -0.41221 | -0.470044 | 1.11874 | -0.150896 | -0.573489 | -0.628192 | 1.48831 |
| 30 | -0.602944 | -1.00295 | -0.893634 | -0.150896 | -0.573489 | -0.628192 | 1.48831 |
| ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ |
typeof(data_train), typeof(data_test)
(DataFrame, DataFrame)
Then, we construct the inputs for our network.
formula_basic = @formula(lwage ~ sex + exp1 + shs + hsg+ scl + clg + mw + so + we)
println(formula_basic)
lwage ~ sex + exp1 + shs + hsg + scl + clg + mw + so + we
model_X_basic_train = ModelMatrix(ModelFrame(formula_basic, data_train)).m
model_X_basic_test = ModelMatrix(ModelFrame(formula_basic, data_test)).m
1288×10 Matrix{Float64}:
1.0 1.11874 -0.634592 -0.150896 … -0.596613 -0.649488 -0.524344
1.0 1.11874 1.07799 -0.150896 -0.596613 -0.649488 -0.524344
1.0 1.11874 -0.777306 -0.150896 -0.596613 -0.649488 -0.524344
1.0 -0.893634 -0.539448 -0.150896 -0.596613 -0.649488 -0.524344
1.0 1.11874 -0.254019 -0.150896 -0.596613 -0.649488 -0.524344
1.0 1.11874 1.03041 -0.150896 … -0.596613 -0.649488 -0.524344
1.0 -0.893634 -0.87245 -0.150896 -0.596613 -0.649488 -0.524344
1.0 -0.893634 1.17313 -0.150896 -0.596613 -0.649488 -0.524344
1.0 1.11874 0.0314105 -0.150896 -0.596613 -0.649488 -0.524344
1.0 1.11874 1.64884 -0.150896 -0.596613 -0.649488 -0.524344
1.0 1.11874 -0.254019 -0.150896 … -0.596613 -0.649488 -0.524344
1.0 -0.893634 1.26827 -0.150896 -0.596613 -0.649488 -0.524344
1.0 1.11874 0.364412 -0.150896 -0.596613 -0.649488 -0.524344
⋮ ⋱
1.0 -0.893634 0.792556 -0.150896 -0.596613 -0.649488 1.90665
1.0 -0.893634 -0.349162 -0.150896 -0.596613 -0.649488 1.90665
1.0 1.11874 -1.11031 -0.150896 -0.596613 -0.649488 1.90665
1.0 1.11874 2.50513 -0.150896 -0.596613 -0.649488 1.90665
1.0 -0.893634 0.60227 -0.150896 … -0.596613 -0.649488 1.90665
1.0 -0.893634 1.69642 -0.150896 -0.596613 -0.649488 1.90665
1.0 1.11874 1.93427 -0.150896 -0.596613 -0.649488 1.90665
1.0 -0.893634 0.126554 -0.150896 -0.596613 -0.649488 1.90665
1.0 -0.893634 -0.349162 -0.150896 -0.596613 -0.649488 1.90665
1.0 -0.893634 -0.444305 -0.150896 … -0.596613 -0.649488 1.90665
1.0 -0.893634 -0.254019 -0.150896 -0.596613 -0.649488 1.90665
1.0 -0.893634 -0.349162 -0.150896 -0.596613 -0.649488 1.90665
Y_train = data_train[!,"lwage"]
Y_test = data_test[!,"lwage"]
1288-element Vector{Float64}:
-1.2495116701816313
-0.5940111369079969
-0.8558489593188235
-0.6922940066202594
-0.16422085652454593
-0.3179120121732921
-0.8586738886527183
0.013186299642166738
-0.8637071816306149
0.36697911041067455
1.3178594848348544
-1.1634376242266362
-1.7570311742901377
⋮
-1.2495116701816313
0.43617103718140365
-0.07134846692710259
-0.3847866913343217
-1.2495116701816313
-0.8558489593188235
0.5351222789388115
0.25782050367552267
-0.6062199606652643
-0.4923631674195659
1.1961444692773646
0.9241967739720466
# df = DataFrame()
# for i in 1:size(data_train)[2]
# p = normalize(data_train[!,i])
# colname = names(data_train)[i]
# df[!,colname] = p
# end
# df
# # data_test = df
# # data_test
# normalize(data_train[!,])
size(Y_train)
(3862,)
model_X_basic_train
3862×10 Matrix{Float64}:
1.0 -0.893634 1.07799 -0.150896 … 1.67569 -0.649488 -0.524344
1.0 -0.893634 -0.444305 -0.150896 -0.596613 1.53928 -0.524344
1.0 -0.893634 1.12556 -0.150896 -0.596613 -0.649488 1.90665
1.0 1.11874 -0.920021 -0.150896 -0.596613 1.53928 -0.524344
1.0 -0.893634 2.17213 -0.150896 -0.596613 -0.649488 -0.524344
1.0 -0.893634 -0.158876 -0.150896 … 1.67569 -0.649488 -0.524344
1.0 -0.893634 0.411983 6.62538 -0.596613 -0.649488 1.90665
1.0 1.11874 -0.444305 -0.150896 -0.596613 -0.649488 -0.524344
1.0 1.11874 -1.20545 -0.150896 -0.596613 -0.649488 -0.524344
1.0 1.11874 -0.349162 -0.150896 -0.596613 1.53928 -0.524344
1.0 -0.893634 0.935271 -0.150896 … 1.67569 -0.649488 -0.524344
1.0 1.11874 0.507126 -0.150896 -0.596613 -0.649488 -0.524344
1.0 -0.893634 0.697413 -0.150896 -0.596613 1.53928 -0.524344
⋮ ⋱
1.0 1.11874 -0.444305 -0.150896 … -0.596613 -0.649488 1.90665
1.0 -0.893634 0.697413 -0.150896 -0.596613 -0.649488 -0.524344
1.0 -0.893634 -0.920021 -0.150896 -0.596613 -0.649488 1.90665
1.0 1.11874 -0.824878 -0.150896 1.67569 -0.649488 -0.524344
1.0 -0.893634 -0.824878 -0.150896 -0.596613 1.53928 -0.524344
1.0 -0.893634 -0.87245 -0.150896 … -0.596613 1.53928 -0.524344
1.0 -0.893634 -0.539448 -0.150896 -0.596613 -0.649488 1.90665
1.0 -0.893634 -1.01516 -0.150896 -0.596613 -0.649488 1.90665
1.0 -0.893634 -0.444305 -0.150896 -0.596613 1.53928 -0.524344
1.0 -0.893634 -0.0637326 -0.150896 -0.596613 1.53928 -0.524344
1.0 1.11874 -1.20545 -0.150896 … -0.596613 -0.649488 -0.524344
1.0 1.11874 -0.254019 -0.150896 1.67569 -0.649488 -0.524344
13.1.1. Neural Networks#
First, we need to determine the structure of our network. We are using the R package keras to build a simple sequential neural network with three dense layers.
size(model_X_basic_train)
(3862, 10)
using Flux
using Flux: crossentropy, @epochs
using Flux.Data: DataLoader
using Flux: throttle
using Flux: onehotbatch, onecold, @epochs
model_nn = Chain(Dense(size(model_X_basic_train,2), 20, relu),
Dense(20, size(model_X_basic_train,2), relu),
Dense(10, 1))
Chain(Dense(10, 20, relu), Dense(20, 10, relu), Dense(10, 1))
# compile the Flux model
model = model_nn
#opt(x, y) = Descent()
opt1 = ADAM()
loss(x, y) = Flux.Losses.mse(model(x), y)
metrics(x, y) = Flux.mae(model(x), y)
metrics (generic function with 1 method)
ps = Flux.params(model)
Params([Float32[-0.1129414 -0.29369408 … 0.3361785 -0.3730879; -0.065885015 0.008057578 … 0.2860582 0.0017144078; … ; 0.3849911 -0.3340452 … 0.17186706 0.15489891; -0.27252343 -0.23161511 … 0.04841264 -0.323494], Float32[0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], Float32[-0.15441547 0.122873 … -0.2977946 0.09255712; 0.2533373 -0.25251332 … -0.29327345 0.06759121; … ; 0.43840528 0.38483554 … 0.18124762 -0.3953458; 0.42269453 0.3865057 … -0.15502952 0.3532663], Float32[0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], Float32[-0.31053326 0.18248866 … -0.31109056 -0.29544342], Float32[0.0]])
Let us have a look at the structure of our network in detail.
model
Chain(Dense(10, 20, relu), Dense(20, 10, relu), Dense(10, 1))
It is worth to notice that we have in total \(441\) trainable parameters.
Now, let us train the network. Note that this takes some computation time. Thus, we are using gpu to speed up. The exact speed-up varies based on a number of factors including model architecture, batch-size, input pipeline complexity, etc.
# m1 = zeros(size(model_X_basic_train, 1), size(model_X_basic_train, 2) - 1 )
# Y_train = hcat(Y_train, m1)
loss(model_X_basic_train', Y_train')
1.5247699038109273
opt(model_X_basic_train', Y_train)
Descent(0.1)
data = DataLoader((model_X_basic_train', Y_train'))
DataLoader{Tuple{Adjoint{Float64, Matrix{Float64}}, Adjoint{Float64, Vector{Float64}}}, Random._GLOBAL_RNG}(([1.0 1.0 … 1.0 1.0; -0.8936342512056518 -0.8936342512056518 … 1.1187362899933146 1.1187362899933146; … ; -0.6494880907521049 1.5392754402117947 … -0.6494880907521049 -0.6494880907521049; -0.5243443952053061 -0.5243443952053061 … -0.5243443952053061 -0.5243443952053061], [-0.8558489593188235 -1.1306215839580365 … -0.026683600452133343 -2.0881111049638306]), 1, 3862, true, 3862, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10 … 3853, 3854, 3855, 3856, 3857, 3858, 3859, 3860, 3861, 3862], false, Random._GLOBAL_RNG())
# training the network
#@epochs 1000 Flux.train!(loss, ps, data, opt, cb = throttle(() -> println("training"), 10))
@epochs 1000 Flux.train!(loss, ps, data, opt1, cb = throttle(() -> println("training"), 10))
┌ Info: Epoch 1
└ @ Main C:\Users\PC\.julia\packages\Flux\qp1gc\src\optimise\train.jl:135
training
training
┌ Info: Epoch 2
└ @ Main C:\Users\PC\.julia\packages\Flux\qp1gc\src\optimise\train.jl:135
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┌ Info: Epoch 3
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┌ Info: Epoch 4
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┌ Info: Epoch 5
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┌ Info: Epoch 6
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┌ Info: Epoch 7
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┌ Info: Epoch 8
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┌ Info: Epoch 9
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┌ Info: Epoch 10
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┌ Info: Epoch 11
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┌ Info: Epoch 12
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┌ Info: Epoch 13
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┌ Info: Epoch 14
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┌ Info: Epoch 15
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┌ Info: Epoch 16
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┌ Info: Epoch 17
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┌ Info: Epoch 18
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┌ Info: Epoch 19
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┌ Info: Epoch 20
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┌ Info: Epoch 21
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┌ Info: Epoch 22
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┌ Info: Epoch 23
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┌ Info: Epoch 24
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┌ Info: Epoch 25
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┌ Info: Epoch 26
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┌ Info: Epoch 27
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┌ Info: Epoch 28
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┌ Info: Epoch 29
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┌ Info: Epoch 30
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┌ Info: Epoch 31
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┌ Info: Epoch 32
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┌ Info: Epoch 33
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┌ Info: Epoch 34
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┌ Info: Epoch 35
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┌ Info: Epoch 36
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┌ Info: Epoch 37
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┌ Info: Epoch 38
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┌ Info: Epoch 39
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┌ Info: Epoch 40
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┌ Info: Epoch 41
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┌ Info: Epoch 42
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┌ Info: Epoch 43
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┌ Info: Epoch 44
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┌ Info: Epoch 45
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┌ Info: Epoch 46
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┌ Info: Epoch 47
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┌ Info: Epoch 48
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┌ Info: Epoch 49
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┌ Info: Epoch 50
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┌ Info: Epoch 51
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┌ Info: Epoch 52
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┌ Info: Epoch 53
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┌ Info: Epoch 54
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┌ Info: Epoch 55
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┌ Info: Epoch 56
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┌ Info: Epoch 57
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┌ Info: Epoch 58
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┌ Info: Epoch 59
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┌ Info: Epoch 60
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┌ Info: Epoch 61
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┌ Info: Epoch 62
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┌ Info: Epoch 63
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┌ Info: Epoch 64
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┌ Info: Epoch 65
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┌ Info: Epoch 66
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┌ Info: Epoch 67
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┌ Info: Epoch 68
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┌ Info: Epoch 69
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┌ Info: Epoch 70
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┌ Info: Epoch 71
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┌ Info: Epoch 72
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┌ Info: Epoch 73
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┌ Info: Epoch 74
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┌ Info: Epoch 75
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┌ Info: Epoch 76
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┌ Info: Epoch 77
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┌ Info: Epoch 78
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┌ Info: Epoch 79
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┌ Info: Epoch 80
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┌ Info: Epoch 81
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┌ Info: Epoch 82
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┌ Info: Epoch 83
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┌ Info: Epoch 84
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┌ Info: Epoch 85
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┌ Info: Epoch 86
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┌ Info: Epoch 87
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┌ Info: Epoch 88
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┌ Info: Epoch 89
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┌ Info: Epoch 90
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┌ Info: Epoch 91
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┌ Info: Epoch 92
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┌ Info: Epoch 93
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┌ Info: Epoch 94
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┌ Info: Epoch 97
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┌ Info: Epoch 98
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┌ Info: Epoch 99
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┌ Info: Epoch 100
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┌ Info: Epoch 101
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┌ Info: Epoch 102
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┌ Info: Epoch 103
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┌ Info: Epoch 104
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┌ Info: Epoch 105
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┌ Info: Epoch 106
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┌ Info: Epoch 107
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┌ Info: Epoch 108
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┌ Info: Epoch 109
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┌ Info: Epoch 110
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┌ Info: Epoch 111
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┌ Info: Epoch 112
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┌ Info: Epoch 113
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┌ Info: Epoch 114
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┌ Info: Epoch 115
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┌ Info: Epoch 116
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┌ Info: Epoch 117
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┌ Info: Epoch 118
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┌ Info: Epoch 119
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┌ Info: Epoch 120
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┌ Info: Epoch 121
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┌ Info: Epoch 122
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┌ Info: Epoch 123
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┌ Info: Epoch 124
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
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training
┌ Info: Epoch 987
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training
┌ Info: Epoch 988
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training
┌ Info: Epoch 989
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training
┌ Info: Epoch 990
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training
┌ Info: Epoch 991
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training
┌ Info: Epoch 992
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training
┌ Info: Epoch 993
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training
┌ Info: Epoch 994
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training
┌ Info: Epoch 995
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training
┌ Info: Epoch 996
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training
┌ Info: Epoch 997
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training
┌ Info: Epoch 998
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training
┌ Info: Epoch 999
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┌ Info: Epoch 1000
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After training the neural network, we can evaluate the performance of our model on the test sample.
# # evaluating the performnace
# model %>% evaluate(model_X_basic_test, Y_test, verbose = 0)
# Calculating the performance measures
pred_nn = model(model_X_basic_test')'
# We change the type of this variable in order to make numeric operations
pred_nn = vec(pred_nn)
resid_basic = (Y_test .- pred_nn).^2
one = vec(ones(size(Y_test,1),1))
resid_basic = DataFrame([resid_basic], [:resid_basic])
one = DataFrame([one], [:one])
data_aux = [resid_basic one]
fm = @formula(resid_basic ~ 0 + one)
MSE_nn_basic = lm(fm, data_aux, dropcollinear=false)
@show coef(MSE_nn_basic)
@show stderror(MSE_nn_basic)
R2_nn_basic = 1 .- (coef(MSE_nn_basic) / var(Y_test))
println("The R^2 using NN is equal to ", R2_nn_basic) # MSE NN (basic model)
coef(MSE_nn_basic) = [0.9299250584273192]
stderror(MSE_nn_basic) = [0.052566401001302235]
The R^2 using NN is equal to [0.11516006433982762]
plot(Y_test, label = "Y_test")
plot!(pred_nn, label = "pred_nn")
plot!(size=(950,600))
