**Written by**: Jean Bouchat

```
using CSV
using DataFrames
df = DataFrame(CSV.File("./Life Expectancy Data.csv", normalizenames=true))
```

```
using Plots
using LinearAlgebra
using Statistics
new_df = dropmissing(df)
new_df = new_df[new_df.Year .== 2012, :]
```

Data

```
y = new_df.Life_expectancy
A = new_df.BMI
```

Plot the data

`scatter(A, y, label="Data")`

Add a column of ones to the right of A'

`new_A = hcat(A, ones(length(A)))`

Solve the system Ax=y to find x, i.e., the coefficients of the linear regression model

`coefs = new_A\y # Cannot handle missing values`

Plot the regression line

```
x_plot = minimum(A):maximum(A)
y_plot = coefs[1]*x_plot .+ coefs[2]
label = string(coefs[1], "*x + ", coefs[2])
println("Regression line: $(label)")
println("R = $(cor(new_df.Life_expectancy, new_df.BMI))")
plot!(x_plot, y_plot, label=label)
names(df)
features = ["Life_expectancy", "Total_expenditure", "GDP",
"Income_composition_of_resources", "Adult_Mortality", "Schooling"]
for feature in features
if feature != "Country" && feature != "Year" && feature != "Status"
```

Data

```
y = new_df.Life_expectancy
A = new_df[:, feature]
```

Preparing the figure

`fig = plot()`

Plot the data

`scatter!(A, y, label="Data")`

Add a column of ones to the right of A'

`new_A = hcat(A, ones(length(A)))`

Solve the system Ax=y to find x, i.e., the coefficients of the linear regression model

`coefs = new_A\y # Cannot handle missing values`

Plot the regression line

```
x_plot = range(minimum(A), maximum(A), 100)
y_plot = coefs[1]*x_plot .+ coefs[2]
label = string(coefs[1], "*x + ", coefs[2])
xlabel = feature
ylabel = "Life expectancy in 2012"
display("Regression line: $(label)")
```

Prdicting Life_expectancy from feature using regression model

`y_hat = coefs[1].*A .+ coefs[2]`

Computing correlation coefficient

```
display("R = $(cor(y, y_hat))")
plot!(x_plot, y_plot, label=label, xlabel=xlabel, ylabel=ylabel)
```

Displaying the figure

```
display(fig)
end
end
```

Data

`y = new_df.Life_expectancy`

A is the key to go from single to multiple linear regression !

`A = hcat(new_df.Adult_Mortality, new_df.Schooling, ones(size(new_df, 1)))`

Solve the system Ax=y to find x, i.e., the coefficients of the linear regression model

```
coefs = A\y # Cannot handle missing values
println("Regression line: $(coefs[1])*x1 + $(coefs[2])*x2 + $(coefs[3])")
```

Use the regression model to predict Life*expectancy from Adult*Mortality and Schooling

`y_hat = coefs[1].*new_df.Adult_Mortality + coefs[2].*new_df.Schooling .+ coefs[3]`

Compute correlation coefficient

`println("R = $(cor(y, y_hat))")`

https://github.com/odow/jump-training-materials/blob/master/getting*started*with_JuMP.ipynb `JuMP`

documentation https://jump.dev/JuMP.jl/stable/ `HiGHS`

documentation https://www.maths.ed.ac.uk/hall/HiGHS/

using Pkg Pkg.add("JuMP") Pkg.add("HiGHS")

```
using JuMP
using HiGHS # Solver
```

Simple linear regression via least squares

$\text{minimize}_{a, b} \sum_{i=1}^{N} (y_i-r_i)^2$such that

$r_i = ax_i+b, \quad i=1,\ldots,N$```
y = new_df.Life_expectancy
x = new_df.Schooling
N = length(x)
model = Model(HiGHS.Optimizer)
@variable(model, a)
@variable(model, b)
@expression(model, r[i = 1:N], a*x[i] + b)
@objective(model, Min, sum((y[i] - r[i])^2 for i=1:N))
optimize!(model)
@show value(a)
@show value(b);
```

The diet problem was one of the first problems in optimization. George Joseph Stigler formulated the problem of the optimal diet in the late 1930s in an attempt to satisfy the concern of the North American army to find the most economical way to feed its troops while at the same time ensuring certain nutritional requirements.

Your homework consists in solving the diet problem described in Czyzyk and Wisniewski (1996), Section 2, pp. 2-3, using `JuMP`

.

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© Benoît Legat. Last modified: June 20, 2024. Website built with Franklin.jl.