# Regression using the Housing data

#### Lampros Mouselimis

#### 2021-10-29

The following examples illustrate the functionality of the KernelKnn package for regression tasks. I’ll make use of the Housing data set,

    data(Boston, package = 'KernelKnn')

    str(Boston)

    ## 'data.frame':    506 obs. of  14 variables:
    ##  $ crim   : num  0.00632 0.02731 0.02729 0.03237 0.06905 ...
    ##  $ zn     : num  18 0 0 0 0 0 12.5 12.5 12.5 12.5 ...
    ##  $ indus  : num  2.31 7.07 7.07 2.18 2.18 2.18 7.87 7.87 7.87 7.87 ...
    ##  $ chas   : int  0 0 0 0 0 0 0 0 0 0 ...
    ##  $ nox    : num  0.538 0.469 0.469 0.458 0.458 0.458 0.524 0.524 0.524 0.524 ...
    ##  $ rm     : num  6.58 6.42 7.18 7 7.15 ...
    ##  $ age    : num  65.2 78.9 61.1 45.8 54.2 58.7 66.6 96.1 100 85.9 ...
    ##  $ dis    : num  4.09 4.97 4.97 6.06 6.06 ...
    ##  $ rad    : int  1 2 2 3 3 3 5 5 5 5 ...
    ##  $ tax    : num  296 242 242 222 222 222 311 311 311 311 ...
    ##  $ ptratio: num  15.3 17.8 17.8 18.7 18.7 18.7 15.2 15.2 15.2 15.2 ...
    ##  $ black  : num  397 397 393 395 397 ...
    ##  $ lstat  : num  4.98 9.14 4.03 2.94 5.33 ...
    ##  $ medv   : num  24 21.6 34.7 33.4 36.2 28.7 22.9 27.1 16.5 18.9 ...

When using an algorithm where the ouput depends on distance calculation (as is the case in k-nearest-neighbors) it is recommended to first scale the data,

    X = scale(Boston[, -ncol(Boston)])
    y = Boston[, ncol(Boston)]

    # random split of data in train and test

    spl_train = sample(1:length(y), round(length(y) * 0.75))
    spl_test = setdiff(1:length(y), spl_train)
    str(spl_train)

    ##  int [1:380] 411 64 274 134 28 380 17 125 247 373 ...

    str(spl_test)

    ##  int [1:126] 4 12 15 18 20 24 35 37 43 47 ...

    # evaluation metric

    mse = function (y_true, y_pred) {

      out = mean((y_true - y_pred)^2)

      out
    }

## The KernelKnn function

The KernelKnn function takes a number of arguments. To read details for each one of the arguments type ?KernelKnn::KernelKnn in the console.

A simple k-nearest-neighbors can be run with weights_function = NULL (the parameter ‘regression’ should be set to TRUE for regression),

    library(KernelKnn)

    preds_TEST = KernelKnn(X[spl_train, ], TEST_data = X[spl_test, ], y[spl_train], k = 5 ,

                           method = 'euclidean', weights_function = NULL, regression = T)
    str(preds_TEST)

    ##  num [1:126] 30.3 20.8 18 16.3 19 ...

Using transf_categ_cols = TRUE, categorical features can be either encoded to dummy or to numeric features depending on the number of the unique values (here I convert the ‘chas’ and ‘rad’ features to factor to apply the transf_categ_cols parameter)

    apply(Boston, 2, function(x) length(unique(x)))

    ##    crim      zn   indus    chas     nox      rm     age     dis     rad     tax
    ##     504      26      76       2      81     446     356     412       9      66
    ## ptratio   black   lstat    medv
    ##      46     357     455     229

    tmp_bst = Boston
    tmp_bst$chas = as.factor(tmp_bst$chas)
    tmp_bst$rad = as.factor(tmp_bst$rad)

    preds_TEST = KernelKnn(tmp_bst[spl_train, -ncol(tmp_bst)],

                           TEST_data = tmp_bst[spl_test, -ncol(tmp_bst)],

                           y[spl_train], k = 5 , method = 'euclidean',

                           regression = T, transf_categ_cols = T)
    str(preds_TEST)

    ##  num [1:126] 23.7 22.3 30.2 22.2 25.4 ...

There are two ways to use a kernel in the KernelKnn function. The first option is to choose one of the existing kernels (uniform, triangular, epanechnikov, biweight, triweight, tricube, gaussian, cosine, logistic, silverman, inverse, gaussianSimple, exponential). Here, I use the mahalanobis metric (which takes advantage of the covariance matrix of the data, but it somewhat slows down training in comparison to the other distance metrics) and the biweight kernel, because they give optimal results (according to my RandomSearchR package),

    preds_TEST_biw = KernelKnn(X[spl_train, ], TEST_data = X[spl_test, ], y[spl_train], k = 5,

                               method = 'mahalanobis', weights_function = 'biweight',

                               regression = T, transf_categ_cols = F)
    str(preds_TEST_biw)

    ##  num [1:126] 33.1 21.9 17 17.5 20.1 ...

The second option is to give a self defined kernel function. Here, I’ll pick the density function of the normal distribution with mean = 0.0 and standard deviation = 1.0 (the data are scaled to have mean zero and unit variance),

    norm_kernel = function(W) {

      W = dnorm(W, mean = 0, sd = 1.0)

      W = W / rowSums(W)

      return(W)
    }

    preds_TEST_norm = KernelKnn(X[spl_train, ], TEST_data = X[spl_test, ], y[spl_train], k = 5,

                                method = 'mahalanobis', weights_function = norm_kernel,

                                regression = T, transf_categ_cols = F)
    str(preds_TEST_norm)

    ##  num [1:126] 30.3 22.3 17 16.3 19.9 ...

The computations can be speed up by using the parameter threads (multiple cores can be run in parallel). There is also the option to exclude extrema (minimum and maximum distances) during the calculation of the k-nearest-neighbor distances using extrema = TRUE. The bandwidth of the existing kernels can be tuned using the h parameter.

K-nearest-neigbor calculations in the KernelKnn function can be accomplished using the following distance metrics : euclidean, manhattan, chebyshev, canberra, braycurtis, minkowski (by default the order ‘p’ of the minkowski parameter equals k), hamming, mahalanobis, pearson_correlation, simple_matching_coefficient, jaccard_coefficient and Rao_coefficient. The last four are similarity measures and are appropriate for binary data [0,1].

I employed my RandomSearchR package to find the optimal parameters for the KernelKnn function and the following two pairs of parameters give an optimal mean-squared-error,

k | method | kernel
--- | --- | ---
9 | mahalanobis | triweight
3 | canberra | cosine

## The KernelKnnCV function

I’ll use the KernelKnnCV function to calculate the mean-squared-error using 3-fold cross-validation for the previous mentioned parameter pairs,

    fit_cv_pair1 = KernelKnnCV(X, y, k = 9, folds = 3, method = 'mahalanobis',

                               weights_function = 'triweight', regression = T,

                               threads = 5, seed_num = 3)

    str(fit_cv_pair1)

    ## List of 2
    ##  $ preds:List of 3
    ##   ..$ : num [1:169] 26.3 26 28.3 21.8 21.4 ...
    ##   ..$ : num [1:168] 19.6 17.4 15.9 17.4 14.3 ...
    ##   ..$ : num [1:169] 23.2 22.7 29.8 22.7 22.9 ...
    ##  $ folds:List of 3
    ##   ..$ fold_1: int [1:169] 3 4 5 12 13 19 20 21 25 26 ...
    ##   ..$ fold_2: int [1:168] 7 8 9 18 24 27 29 36 40 41 ...
    ##   ..$ fold_3: int [1:169] 1 2 6 10 11 14 15 16 17 22 ...

    fit_cv_pair2 = KernelKnnCV(X, y, k = 3, folds = 3, method = 'canberra',

                               weights_function = 'cosine', regression = T,

                               threads = 5, seed_num = 3)

    str(fit_cv_pair2)

Each cross-validated object returns a list of length 2 ( the first sublist includes the predictions for each fold whereas the second gives the indices of the folds)

    mse_pair1 = unlist(lapply(1:length(fit_cv_pair1$preds),

                              function(x) mse(y[fit_cv_pair1$folds[[x]]],

                                              fit_cv_pair1$preds[[x]])))
    mse_pair1

    ## [1] 19.03225 19.47981 19.41959

    cat('mse for params_pair1 is :', mean(mse_pair1), '\n')

    ## mse for params_pair1 is : 19.31055

    mse_pair2 = unlist(lapply(1:length(fit_cv_pair2$preds),

                              function(x) mse(y[fit_cv_pair2$folds[[x]]],

                                              fit_cv_pair2$preds[[x]])))
    mse_pair2

    ## [1] 17.83355 29.86579 28.65615

    cat('mse for params_pair2 is :', mean(mse_pair2), '\n')

    ## mse for params_pair2 is : 25.45183

## Adding or multiplying kernels

In the KernelKnn package there is also the option to combine kernels (adding or multiplying) from the existing ones. For instance, if I want to multiply the tricube with the gaussian kernel, then I’ll give the following character string to the weights_function, “tricube_gaussian_MULT”. On the other hand, If I want to add the same kernels then the weights_function will be “tricube_gaussian_ADD”. I experimented with my RandomSearchR package combining the different kernels and the following two parameter settings gave optimal results,

k | method | kernel
--- | --- | ---
19 | mahalanobis | triangular_triweight_MULT
18 | mahalanobis | biweight_triweight_gaussian_MULT

    fit_cv_pair1 = KernelKnnCV(X, y, k = 19, folds = 3, method = 'mahalanobis',

                               weights_function = 'triangular_triweight_MULT',

                               regression = T, threads = 5, seed_num = 3)

    str(fit_cv_pair1)

    ## List of 2
    ##  $ preds:List of 3
    ##   ..$ : num [1:169] 26.4 26 27.8 21.5 21.8 ...
    ##   ..$ : num [1:168] 19.7 17.6 16.4 16.7 14.6 ...
    ##   ..$ : num [1:169] 23.1 22.7 29 22.3 22.3 ...
    ##  $ folds:List of 3
    ##   ..$ fold_1: int [1:169] 3 4 5 12 13 19 20 21 25 26 ...
    ##   ..$ fold_2: int [1:168] 7 8 9 18 24 27 29 36 40 41 ...
    ##   ..$ fold_3: int [1:169] 1 2 6 10 11 14 15 16 17 22 ...

    fit_cv_pair2 = KernelKnnCV(X, y, k = 18, folds = 3, method = 'mahalanobis',

                               weights_function = 'biweight_triweight_gaussian_MULT',

                               regression = T, threads = 5, seed_num = 3)

    str(fit_cv_pair2)

    ## List of 2
    ##  $ preds:List of 3
    ##   ..$ : num [1:169] 26.2 26 28.1 21.7 21.7 ...
    ##   ..$ : num [1:168] 19.7 17.6 16.4 16.7 14.6 ...
    ##   ..$ : num [1:169] 23.1 22.8 29.2 22.4 22.2 ...
    ##  $ folds:List of 3
    ##   ..$ fold_1: int [1:169] 3 4 5 12 13 19 20 21 25 26 ...
    ##   ..$ fold_2: int [1:168] 7 8 9 18 24 27 29 36 40 41 ...
    ##   ..$ fold_3: int [1:169] 1 2 6 10 11 14 15 16 17 22 ...

    mse_pair1 = unlist(lapply(1:length(fit_cv_pair1$preds),

                              function(x) mse(y[fit_cv_pair1$folds[[x]]],

                                              fit_cv_pair1$preds[[x]])))
    mse_pair1

    ## [1] 19.18593 19.42827 17.33437

    cat('mse for params_pair1 is :', mean(mse_pair1), '\n')

    ## mse for params_pair1 is : 18.64952

    mse_pair2 = unlist(lapply(1:length(fit_cv_pair2$preds),

                              function(x) mse(y[fit_cv_pair2$folds[[x]]],

                                              fit_cv_pair2$preds[[x]])))
    mse_pair2

    ## [1] 19.20290 19.31054 17.56264

    cat('mse for params_pair2 is :', mean(mse_pair2), '\n')

    ## mse for params_pair2 is : 18.69202