Author: John Lepird
Date: 2022-04-24

# Preference Elicitation on Motor Trends Dataset

#### John Lepird

#### 2022-04-24

This vignette demonstrates how to use the `prefeR` package on a real dataset. The `mtcars` dataset provides us such an opportunity.

1974 Motor Trends Car Data
| mpg | cyl | disp | hp | drat | wt | qsec | vs | am | gear | carb
--- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | ---
Mazda RX4 | 21.0 | 6 | 160 | 110 | 3.90 | 2.620 | 16.46 | 0 | 1 | 4 | 4
Mazda RX4 Wag | 21.0 | 6 | 160 | 110 | 3.90 | 2.875 | 17.02 | 0 | 1 | 4 | 4
Datsun 710 | 22.8 | 4 | 108 | 93 | 3.85 | 2.320 | 18.61 | 1 | 1 | 4 | 1
Hornet 4 Drive | 21.4 | 6 | 258 | 110 | 3.08 | 3.215 | 19.44 | 1 | 0 | 3 | 1
Hornet Sportabout | 18.7 | 8 | 360 | 175 | 3.15 | 3.440 | 17.02 | 0 | 0 | 3 | 2

If we wanted to give a user a list of their top five most preferred cars from the `mtcars` dataset, there are three approaches we could take:

1. Have our user manually rank all options.
2. Make the user provide weights for the desirability of different car features, and calculate the weighted value of each option.
3. Have the user compare a small number of alternatives, and derive their weights from those comparisons.

Option #1 quickly becomes an enormous burden on the user as the number of alternatives increases. Option #2 is difficult for the user to do and replicate. What exactly does it mean if the weight assigned to horsepower is double the weight assigned to fuel efficiency?

Option #3 is enabled by the preference elicitation package. To begin, we create a preference elicitation object and give it our data:

    library(prefeR)
    p <- prefEl(data = mtcars)
    p
    ## Preference elicitation object with:
    ##  32 observations of 11 variables.
    ## And the following preferences:
    ##  No strict preferences.
    ##  No indifference preferences.

Now we can add in our Bayesian priors for the weights. Although it is difficult to determine weights exactly, usually one has some ballpark estimate for what they should be, and often one knows with certainty the sign of the weights: all else equal, everyone would prefer a more fuel efficient car. The `prefeR` package contains three built-in priors:

- `Normal(mu, sigma)` provides a one-dimensional Normal prior with mean mu and standard deviation sigma. This prior is useful if you have a good guess for what the weight should be, and an understanding of how much you expect to differ from that guess.
- `Exp(mu)` provides a one dimensional Exponential prior with mean mu (not rate!). This prior is particularly useful if you deterministically know the sign of the weight, and have a guess for the value of the weight. The mean may be negative.
- `Flat()` yields a completely agnostic, flat prior.

We can now add in our priors for our `mtcars` attributes.

    p$priors <- c(Exp(1),   # MPG
                  Normal(), # Number of cylinders (Normal() = Normal(0, 1))
                  Normal(), # displacement
                  Exp(2),   # horsepower
                  Normal(), # real axle ratio
                  Normal(), # weight
                  Exp(-3),  # quarter mile time
                  Normal(), # Engine type
                  Normal(), # transmission type
                  Normal(), # number of gears
                  Normal()  # number of carburetors
    )

Now, we can add in our user’s preferences:

    p$addPref("Pontiac Firebird" %>% "Fiat 128")  # prefer a cool sports car
    p$addPref("Mazda RX4 Wag" %<% "Mazda RX4")    # prefer not to have the station wagon
    p$addPref("Merc 280" %=% "Merc 280C")         # indifferent about C-option

    p
    ## Preference elicitation object with:
    ##  32 observations of 11 variables.
    ## And the following preferences:
    ##  Pontiac Firebird preferred to Fiat 128
    ##  Mazda RX4 preferred to Mazda RX4 Wag
    ##  Merc 280 indifferent to Merc 280C

Now, we can infer what our attribute weights should be:

    p$infer()
    ##        mpg        cyl       disp         hp       drat         wt       qsec
    ##  0.2220478  0.3330885  0.3583347  2.6082377 -0.4364433 -0.1464981 -0.9751220
    ##         vs         am       gear       carb
    ## -0.2016490  0.1358719  0.5794767  0.2578508

And we can get our top five cars:

    p$rank()[1:5]
    ##     Maserati Bora    Ford Pantera L        Duster 360        Camaro Z28
    ##          976.4051          808.1425          759.2586          755.6060
    ## Chrysler Imperial
    ##          747.0812

Finally, we can figure out what query we should answer next:

    p$suggest()
    ## [1] "Valiant"            "Cadillac Fleetwood"