GradientCobra Python Library

Class KernelSmoother

Author

Sothea HAS

Introduction to Kernel Smoothing method

Kernel Smoother is a nonparametric method for estimations and predictions (T. Hastie, 2009).

Given observations \(\{(x_i,y_i),\dots, (x_n,y_n)\}\). The prediction by kernel smoothing method is defined by

\[\hat{y}(x)=\sum_{i=1}^nW_i(x)y_i=\sum_{i=1}^n\frac{K_h(\|x-x_i\|)}{\sum_{j=1}^nK_h(\|x-x_j\|)}y_i\]

where

  • \(K:\mathbb{R}_+\to\mathbb{R}_+\) is a regular kernel function that converts the “closeness” between predicted features of observation \(x_i\) and \(x\) into the associated weight for the aggregation. For any smoothing parameter \(h>0\), \(K_h(t)=K(t/h)\).

Constructing a kernel smoothing method is equivalent to estimating an appropriate smoothing parameter \(h\) that minimizes the so-called \(\kappa\)-fold cross validation error:

\[\phi(h)=\frac{1}{\kappa}\sum_{j=1}^{\kappa}\sum_{(x_i,y_i)\in F_j}\ell(\hat{y}_{-F_j}(x_i),y_i)\]

where

  • \(\ell\) is any regression loss function. By default, \(\ell(y_1,y_2)=(y_1-y_2)^2\).
  • \((F_j)_{j=1,2,\dots,\kappa}\) are the \(\kappa\) folds of the second of the training data used for aggregation.
  • \(\hat{y}_{-F_j}(x)\) is the prediction of \(x\) by aggregating predicted features on all folds \((F_k)_{k}\) except for \(F_j\).

KernelSmoother method

class KernelSmoother(random_state = None, learning_rate = 0.01, bandwidth_list = None, speed = ‘constant’, opt_method = “grad”, max_iter = int(300), opt_params = None, kernel = ‘radial’, kernel_exponent = 1.0, show_progress = True, loss_function = None, loss_weight = None, norm_constant = None):

KernelSmoother is a subclass of GradientCOBRA. T=In KernelSmoother, the input data \(X\) are treated as predicted features and aggregated directly using super class method.

Parameters

  • random_state: (default is None) for setting the random state of all the random generators in the class. This allows the method to produce the same result when the same random_state is used.

  • learning_rate: (default is 0.1) the learning rate in gradient descent algorithm for estimating the optimal bandwidth.

  • bandwidth_list: a list of bandwidth parameters for grid search algorithm (default = np.linspace(0.00001, 10, 300)).

  • speed: (default is constant) for adjusting the speed of the learning rate. It is helpful when the cost function is flat around the optimal value, changing the learning speed might help the algorithm to converge faster. It should be an element of [‘constant’, ‘linear’, ‘log’, ‘sqrt_root’, ‘quad’, ‘exp’].

  • opt_method: (default is “grad”) is an optimization algorithm for estimating the bandwidth parameter. It should be either “grid” (grid search) or “grad” (gradient descent for non-compactly supported kernels).

  • max_iter: (default = 100) maximum iteration of gradient descent algorithm.

  • opt_params: (default is None) is a dictionary of additional parameters for the optimization algorithm (both grid search and gradient descent). Its should contain some of the following keys:

    • ‘epsilon’: stopping criterion for gradient descent algorithm (default = \(10^{-6}\)).
    • ‘n_tries’: the number of tries for selecting initial position of gradient descent algorithm (default = 10)
    • ‘start’: the initial value of the bandwidth parameter (default = None)
    • ‘n_cv’: number of cross-validation folds (default = 10)
    • ‘precision’ : the precision to estimate the gradient for gradient descent algorithm (default = \(10^{-10}\)).

  • kernel: (default is 'radial') the kernel function used for the aggregation. It should be an element of the list [‘exponential’, ‘gaussian’, ‘radial’, ‘cauchy’, ‘reverse_cosh’,‘epanechnikov’,‘biweight’, ‘triweight’, ‘triangular’, ‘cobra’, ‘naive’]. Some options such as 'gaussian' and 'radial' lead to the same radial kernel function. For 'cobra' or 'naive'.

  • kernel_exponent: (default is 1.0) is an exponential \(\alpha\) of the exponential and radial kernel function i.e., \(K(x) = \exp(|x|^{2\alpha})\). By default, alpha = 1.0.

  • show_progress: (default is True) a boolean defining whether or not to show the progress of the optimization algorithm for both grid search and gradient descent.

  • loss_function: (default is None) a function or string defining the cost function to be optimized for estimating the optimal bandwidth parameter. By defalut, the K-Fold cross-validation MSE is used. Otherwise, it must be either:

    • a function of two argumetns (y_true, y_pred) or
    • a string element of the list [‘mse’, ‘mae’, ‘mape’, ‘weighted_mse’]. If it is weighted_mse, one can define the weight for each training point using loss_weight argument below.
      • loss_weight: (default is None) a list of size \(n\) (number of training observations) defining the weight for each individual data point for the loss function. If it is None and the loss_function = weighted_mse, then a normalized weight \(W(i) = 1/\text{PDF}(i)\) is assigned to the individual \(i\) of the training data.

  • norm_constant: (default is None) a normalized constant used to scale the features in optimization algorithm. By defalut, they are

Returns:

  • self : returns an instance of self containing the optimal estimated bandwidth and other results of the algorithm.

Methods:

All methods are inherited from GradientCOBRA class.

Installation of the library from pypi

gradientcobra can be installed from pypi using pip:

pip install gradientcobra

Implementation of the method

Importing some packages

# Metric of error
from sklearn.metrics import mean_squared_error, mean_absolute_error, mean_absolute_percentage_error

# Plotting figures
import matplotlib.pyplot as plt
from matplotlib import cm

# Import class KernelSmoother from the gradientcobra library
from gradientcobra.gradientcobra import KernelSmoother

import seaborn as sns
sns.set()

KernelSmoother on simulated data

We simulate a regression data with \(1000\) observations and \(10\) inputs variables.

# For simulating dataset
from sklearn.datasets import make_regression

X1, y1 = make_regression(n_samples=1000, n_features=10, noise=2)

Now, let’s randomly split the simulated data into \(80\%-20\%\) training-testing data.

from sklearn.model_selection import train_test_split

X_train1, X_test1, y_train1, y_test1 = train_test_split(X1, y1, test_size=0.2)
print('shape: x_train = {} , x_train = {} , y_train = {} , y_test = {}'.format(
    X_train1.shape, 
    X_test1.shape, 
    y_train1.shape, 
    y_test1.shape))
shape: x_train = (800, 10) , x_train = (200, 10) , y_train = (800,) , y_test = (200,)

\(\bullet\) KernelSmoother with default parameters

We create KernelSmoother object called ker1 using KernelSmoother class with the default parameters, then fit it to the training data.

ker1 = KernelSmoother()
ker1_fit = ker1.fit(X_train1, y_train1)
* GD progress: iter: 1 / bw: 3.000 / grad: -1292.791 / stop criter: 1.000 :   0%|          | 0/300 [00:00<?, ?it/s]* GD progress: iter: 1 / bw: 3.100 / grad: -1292.791 / stop criter: 1292.791 :   0%|          | 0/300 [00:00<?, ?it/s]* GD progress: iter: 1 / bw: 3.100 / grad: -1292.791 / stop criter: 1292.791 :   0%|          | 0/300 [00:00<?, ?it/s]* GD progress: iter: 2 / bw: 3.100 / grad: -1292.791 / stop criter: 1292.791 :   0%|          | 0/300 [00:00<?, ?it/s]* GD progress: iter: 2 / bw: 3.100 / grad: -1292.791 / stop criter: 1292.791 :   0%|          | 0/300 [00:00<?, ?it/s]* GD progress: iter: 3 / bw: 3.100 / grad: -1292.791 / stop criter: 1292.791 :   0%|          | 0/300 [00:00<?, ?it/s]* GD progress: iter: 3 / bw: 3.100 / grad: -1292.791 / stop criter: 1292.791 :   0%|          | 0/300 [00:00<?, ?it/s]* GD progress: iter: 3 / bw: 3.100 / grad: -1292.791 / stop criter: 1292.791 :   1%|          | 3/300 [00:00<00:09, 29.91it/s]* GD progress: iter: 4 / bw: 3.100 / grad: 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-1262.245 / stop criter: 933.701 :   2%|▏         | 6/300 [00:00<00:10, 27.18it/s]* GD progress: iter: 8 / bw: 3.337 / grad: -957.710 / stop criter: 1262.245 :   2%|▏         | 6/300 [00:00<00:10, 27.18it/s]* GD progress: iter: 8 / bw: 3.337 / grad: -957.710 / stop criter: 1262.245 :   2%|▏         | 6/300 [00:00<00:10, 27.18it/s]* GD progress: iter: 9 / bw: 3.411 / grad: -1144.621 / stop criter: 957.710 :   2%|▏         | 6/300 [00:00<00:10, 27.18it/s]* GD progress: iter: 9 / bw: 3.411 / grad: -1144.621 / stop criter: 957.710 :   2%|▏         | 6/300 [00:00<00:10, 27.18it/s]* GD progress: iter: 9 / bw: 3.411 / grad: -1144.621 / stop criter: 957.710 :   3%|▎         | 9/300 [00:00<00:10, 27.36it/s]* GD progress: iter: 10 / bw: 3.500 / grad: -689.141 / stop criter: 1144.621 :   3%|▎         | 9/300 [00:00<00:10, 27.36it/s]* GD progress: iter: 10 / bw: 3.500 / grad: -689.141 / stop criter: 1144.621 :   3%|▎         | 9/300 [00:00<00:10, 27.36it/s]* GD progress: iter: 11 / bw: 3.553 / grad: -750.993 / stop criter: 689.141 :   3%|▎         | 9/300 [00:00<00:10, 27.36it/s] * GD progress: iter: 11 / bw: 3.553 / grad: -750.993 / stop criter: 689.141 :   3%|▎         | 9/300 [00:00<00:10, 27.36it/s]* GD progress: iter: 12 / bw: 3.611 / grad: -392.321 / stop criter: 750.993 :   3%|▎         | 9/300 [00:00<00:10, 27.36it/s]* GD progress: iter: 12 / bw: 3.611 / grad: -392.321 / stop criter: 750.993 :   3%|▎         | 9/300 [00:00<00:10, 27.36it/s]* GD progress: iter: 12 / bw: 3.611 / grad: -392.321 / stop criter: 750.993 :   4%|▍         | 12/300 [00:00<00:11, 25.90it/s]* GD progress: iter: 13 / bw: 3.641 / grad: -399.469 / stop criter: 392.321 :   4%|▍         | 12/300 [00:00<00:11, 25.90it/s]* GD progress: iter: 13 / bw: 3.641 / grad: -399.469 / stop criter: 392.321 :   4%|▍         | 12/300 [00:00<00:11, 25.90it/s]* GD progress: iter: 14 / bw: 3.672 / grad: -569.712 / stop criter: 399.469 :   4%|▍         | 12/300 [00:00<00:11, 25.90it/s]* GD progress: iter: 14 / bw: 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/ grad: 168.522 / stop criter: 116.195 :  89%|████████▉ | 268/300 [00:11<00:01, 17.98it/s]* GD progress: iter: 271 / bw: 5.064 / grad: 168.522 / stop criter: 116.195 :  90%|█████████ | 271/300 [00:11<00:01, 19.19it/s]* GD progress: iter: 272 / bw: 5.060 / grad: -76.278 / stop criter: 168.522 :  90%|█████████ | 271/300 [00:11<00:01, 19.19it/s]* GD progress: iter: 272 / bw: 5.060 / grad: -76.278 / stop criter: 168.522 :  90%|█████████ | 271/300 [00:11<00:01, 19.19it/s]* GD progress: iter: 273 / bw: 5.062 / grad: -41.089 / stop criter: 76.278 :  90%|█████████ | 271/300 [00:11<00:01, 19.19it/s] * GD progress: iter: 273 / bw: 5.062 / grad: -41.089 / stop criter: 76.278 :  90%|█████████ | 271/300 [00:11<00:01, 19.19it/s]* GD progress: iter: 274 / bw: 5.063 / grad: 119.661 / stop criter: 41.089 :  90%|█████████ | 271/300 [00:11<00:01, 19.19it/s]* GD progress: iter: 274 / bw: 5.063 / grad: 119.661 / stop criter: 41.089 :  90%|█████████ | 271/300 [00:11<00:01, 19.19it/s]* GD progress: iter: 274 / bw: 5.063 / grad: 119.661 / stop criter: 41.089 :  91%|█████████▏| 274/300 [00:11<00:01, 20.22it/s]* GD progress: iter: 275 / bw: 5.060 / grad: 171.045 / stop criter: 119.661 :  91%|█████████▏| 274/300 [00:11<00:01, 20.22it/s]* GD progress: iter: 275 / bw: 5.060 / grad: 171.045 / stop criter: 119.661 :  91%|█████████▏| 274/300 [00:11<00:01, 20.22it/s]* GD progress: iter: 276 / bw: 5.055 / grad: -512.589 / stop criter: 171.045 :  91%|█████████▏| 274/300 [00:11<00:01, 20.22it/s]* GD progress: iter: 276 / bw: 5.055 / grad: -512.589 / stop criter: 171.045 :  91%|█████████▏| 274/300 [00:11<00:01, 20.22it/s]* GD progress: iter: 277 / bw: 5.069 / grad: 11.344 / stop criter: 512.589 :  91%|█████████▏| 274/300 [00:11<00:01, 20.22it/s]  * GD progress: iter: 277 / bw: 5.069 / grad: 11.344 / stop criter: 512.589 :  91%|█████████▏| 274/300 [00:11<00:01, 20.22it/s]* GD progress: iter: 277 / bw: 5.069 / grad: 11.344 / stop criter: 512.589 :  92%|█████████▏| 277/300 [00:11<00:01, 20.73it/s]* GD progress: iter: 278 / bw: 5.069 / grad: -95.468 / stop criter: 11.344 :  92%|█████████▏| 277/300 [00:11<00:01, 20.73it/s]* GD progress: iter: 278 / bw: 5.069 / grad: -95.468 / stop criter: 11.344 :  92%|█████████▏| 277/300 [00:11<00:01, 20.73it/s]* GD progress: iter: 279 / bw: 5.071 / grad: 5.276 / stop criter: 95.468 :  92%|█████████▏| 277/300 [00:11<00:01, 20.73it/s]  * GD progress: iter: 279 / bw: 5.071 / grad: 5.276 / stop criter: 95.468 :  92%|█████████▏| 277/300 [00:11<00:01, 20.73it/s]* GD progress: iter: 280 / bw: 5.071 / grad: -40.232 / stop criter: 5.276 :  92%|█████████▏| 277/300 [00:12<00:01, 20.73it/s]* GD progress: iter: 280 / bw: 5.071 / grad: -40.232 / stop criter: 5.276 :  92%|█████████▏| 277/300 [00:12<00:01, 20.73it/s]* GD progress: iter: 280 / bw: 5.071 / grad: -40.232 / stop criter: 5.276 :  93%|█████████▎| 280/300 [00:12<00:00, 21.35it/s]* GD progress: iter: 281 / bw: 5.072 / grad: -111.185 / stop criter: 40.232 :  93%|█████████▎| 280/300 [00:12<00:00, 21.35it/s]* GD progress: iter: 281 / bw: 5.072 / grad: -111.185 / stop criter: 40.232 :  93%|█████████▎| 280/300 [00:12<00:00, 21.35it/s]* GD progress: iter: 282 / bw: 5.075 / grad: 70.391 / stop criter: 111.185 :  93%|█████████▎| 280/300 [00:12<00:00, 21.35it/s] * GD progress: iter: 282 / bw: 5.075 / grad: 70.391 / stop criter: 111.185 :  93%|█████████▎| 280/300 [00:12<00:00, 21.35it/s]* GD progress: iter: 283 / bw: 5.073 / grad: 203.368 / stop criter: 70.391 :  93%|█████████▎| 280/300 [00:12<00:00, 21.35it/s]* GD progress: iter: 283 / bw: 5.073 / grad: 203.368 / stop criter: 70.391 :  93%|█████████▎| 280/300 [00:12<00:00, 21.35it/s]* GD progress: iter: 283 / bw: 5.073 / grad: 203.368 / stop criter: 70.391 :  94%|█████████▍| 283/300 [00:12<00:00, 21.73it/s]* GD progress: iter: 284 / bw: 5.068 / grad: 41.607 / stop criter: 203.368 :  94%|█████████▍| 283/300 [00:12<00:00, 21.73it/s]* GD progress: iter: 284 / bw: 5.068 / grad: 41.607 / stop criter: 203.368 :  94%|█████████▍| 283/300 [00:12<00:00, 21.73it/s]* GD progress: iter: 285 / bw: 5.067 / grad: -120.970 / stop criter: 41.607 :  94%|█████████▍| 283/300 [00:12<00:00, 21.73it/s]* GD progress: iter: 285 / bw: 5.067 / grad: -120.970 / stop criter: 41.607 :  94%|█████████▍| 283/300 [00:12<00:00, 21.73it/s]* GD progress: iter: 286 / bw: 5.070 / grad: -61.603 / stop criter: 120.970 :  94%|█████████▍| 283/300 [00:12<00:00, 21.73it/s]* GD progress: iter: 286 / bw: 5.070 / grad: -61.603 / stop criter: 120.970 :  94%|█████████▍| 283/300 [00:12<00:00, 21.73it/s]* GD progress: iter: 286 / bw: 5.070 / grad: -61.603 / stop criter: 120.970 :  95%|█████████▌| 286/300 [00:12<00:00, 21.18it/s]* GD progress: iter: 287 / bw: 5.072 / grad: -332.721 / stop criter: 61.603 :  95%|█████████▌| 286/300 [00:12<00:00, 21.18it/s]* GD progress: iter: 287 / bw: 5.072 / grad: -332.721 / stop criter: 61.603 :  95%|█████████▌| 286/300 [00:12<00:00, 21.18it/s]* GD progress: iter: 288 / bw: 5.080 / grad: 34.763 / stop criter: 332.721 :  95%|█████████▌| 286/300 [00:12<00:00, 21.18it/s] * GD progress: iter: 288 / bw: 5.080 / grad: 34.763 / stop criter: 332.721 :  95%|█████████▌| 286/300 [00:12<00:00, 21.18it/s]* GD progress: iter: 289 / bw: 5.080 / grad: 98.264 / stop criter: 34.763 :  95%|█████████▌| 286/300 [00:12<00:00, 21.18it/s] * GD progress: iter: 289 / bw: 5.080 / grad: 98.264 / stop criter: 34.763 :  95%|█████████▌| 286/300 [00:12<00:00, 21.18it/s]* GD progress: iter: 289 / bw: 5.080 / grad: 98.264 / stop criter: 34.763 :  96%|█████████▋| 289/300 [00:12<00:00, 21.09it/s]* GD progress: iter: 290 / bw: 5.077 / grad: 510.222 / stop criter: 98.264 :  96%|█████████▋| 289/300 [00:12<00:00, 21.09it/s]* GD progress: iter: 290 / bw: 5.077 / grad: 510.222 / stop criter: 98.264 :  96%|█████████▋| 289/300 [00:12<00:00, 21.09it/s]* GD progress: iter: 291 / bw: 5.064 / grad: 168.415 / stop criter: 510.222 :  96%|█████████▋| 289/300 [00:12<00:00, 21.09it/s]* GD progress: iter: 291 / bw: 5.064 / grad: 168.415 / stop criter: 510.222 :  96%|█████████▋| 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criter: 313.724 :  98%|█████████▊| 295/300 [00:12<00:00, 22.67it/s]* GD progress: iter: 298 / bw: 5.065 / grad: 4.987 / stop criter: 313.724 :  99%|█████████▉| 298/300 [00:12<00:00, 23.11it/s]* GD progress: iter: 299 / bw: 5.065 / grad: 113.614 / stop criter: 4.987 :  99%|█████████▉| 298/300 [00:12<00:00, 23.11it/s]* GD progress: iter: 299 / bw: 5.065 / grad: 113.614 / stop criter: 4.987 :  99%|█████████▉| 298/300 [00:12<00:00, 23.11it/s]* GD progress: iter: 300 / bw: 5.062 / grad: -445.377 / stop criter: 113.614 :  99%|█████████▉| 298/300 [00:12<00:00, 23.11it/s]* GD progress: iter: 300 / bw: 5.062 / grad: -445.377 / stop criter: 113.614 :  99%|█████████▉| 298/300 [00:12<00:00, 23.11it/s]* GD progress: iter: 300 / bw: 5.062 / grad: -445.377 / stop criter: 113.614 : 100%|██████████| 300/300 [00:12<00:00, 23.29it/s]

The estimated optimal bandwidth is given by ker1.optimization_outputs['opt_bandwidth'].

print("Estimated bandwidth :" + str(ker1_fit.optimization_outputs['opt_bandwidth']))
Estimated bandwidth :5.062475740486653

We can look at the learning curve of the algorithm using draw_learning_curve() method.

ker1_fit.draw_learning_curve()

We evaluate the performance of the method on the testing data using MSE and MAPE.

from sklearn.metrics import mean_absolute_percentage_error
y_pred1 = ker1_fit.predict(X_test1)
print(mean_absolute_percentage_error(y_test1, y_pred1))
print(mean_squared_error(y_test1, y_pred1))
0.9070464304670501
4155.204969122178

Let’s look at qq-plot of the predictions and the actual response values using draw_learning_curve() method.

ker1_fit.draw_learning_curve(y_test=y_test1, fig_type='qq')

\(\bullet\) KernelSmoother with non-default parameters

We create another object gc2 with non-default parameters, then fit it to the same training data as in the previous example.

ker2 = KernelSmoother(learning_rate=0.1,
                    opt_method='grad',
                    max_iter=100,
                    kernel="exponential")

ker2_fit = ker2.fit(X_train1, y_train1)
* GD progress: iter: 1 / bw: 3.000 / grad: -1064.022 / stop criter: 1.000 :   0%|          | 0/100 [00:00<?, ?it/s]* GD progress: iter: 1 / bw: 3.100 / grad: -1064.022 / stop criter: 1064.022 :   0%|          | 0/100 [00:00<?, ?it/s]* GD progress: iter: 1 / bw: 3.100 / grad: -1064.022 / stop criter: 1064.022 :   0%|          | 0/100 [00:00<?, ?it/s]* GD progress: iter: 2 / bw: 3.100 / grad: -1064.022 / stop criter: 1064.022 :   0%|          | 0/100 [00:00<?, ?it/s]* GD progress: iter: 2 / bw: 3.100 / grad: -1064.022 / stop criter: 1064.022 :   0%|          | 0/100 [00:00<?, ?it/s]* GD progress: iter: 3 / bw: 3.100 / grad: -1064.022 / stop criter: 1064.022 :   0%|          | 0/100 [00:00<?, ?it/s]* GD progress: iter: 3 / bw: 3.100 / grad: -1064.022 / stop criter: 1064.022 :   0%|          | 0/100 [00:00<?, ?it/s]* GD progress: iter: 3 / bw: 3.100 / grad: -1064.022 / stop criter: 1064.022 :   3%|▎         | 3/100 [00:00<00:03, 25.70it/s]* GD progress: iter: 4 / bw: 3.100 / grad: 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GD progress: iter: 31 / bw: 4.127 / grad: -469.355 / stop criter: 372.190 :  30%|███       | 30/100 [00:01<00:02, 23.43it/s]* GD progress: iter: 32 / bw: 4.169 / grad: -185.448 / stop criter: 469.355 :  30%|███       | 30/100 [00:01<00:02, 23.43it/s]* GD progress: iter: 32 / bw: 4.169 / grad: -185.448 / stop criter: 469.355 :  30%|███       | 30/100 [00:01<00:02, 23.43it/s]* GD progress: iter: 33 / bw: 4.186 / grad: -162.335 / stop criter: 185.448 :  30%|███       | 30/100 [00:01<00:02, 23.43it/s]* GD progress: iter: 33 / bw: 4.186 / grad: -162.335 / stop criter: 185.448 :  30%|███       | 30/100 [00:01<00:02, 23.43it/s]* GD progress: iter: 33 / bw: 4.186 / grad: -162.335 / stop criter: 185.448 :  33%|███▎      | 33/100 [00:01<00:02, 23.88it/s]* GD progress: iter: 34 / bw: 4.200 / grad: -365.818 / stop criter: 162.335 :  33%|███▎      | 33/100 [00:01<00:02, 23.88it/s]* GD progress: iter: 34 / bw: 4.200 / grad: -365.818 / stop criter: 162.335 :  33%|███▎      | 33/100 [00:01<00:02, 23.88it/s]* GD progress: iter: 35 / bw: 4.233 / grad: -643.121 / stop criter: 365.818 :  33%|███▎      | 33/100 [00:01<00:02, 23.88it/s]* GD progress: iter: 35 / bw: 4.233 / grad: -643.121 / stop criter: 365.818 :  33%|███▎      | 33/100 [00:01<00:02, 23.88it/s]* GD progress: iter: 36 / bw: 4.291 / grad: -313.213 / stop criter: 643.121 :  33%|███▎      | 33/100 [00:01<00:02, 23.88it/s]* GD progress: iter: 36 / bw: 4.291 / grad: -313.213 / stop criter: 643.121 :  33%|███▎      | 33/100 [00:01<00:02, 23.88it/s]* GD progress: iter: 36 / bw: 4.291 / grad: -313.213 / stop criter: 643.121 :  36%|███▌      | 36/100 [00:01<00:02, 24.10it/s]* GD progress: iter: 37 / bw: 4.320 / grad: -195.949 / stop criter: 313.213 :  36%|███▌      | 36/100 [00:01<00:02, 24.10it/s]* GD progress: iter: 37 / bw: 4.320 / grad: -195.949 / stop criter: 313.213 :  36%|███▌      | 36/100 [00:01<00:02, 24.10it/s]* GD progress: iter: 38 / bw: 4.337 / grad: -434.580 / stop criter: 195.949 :  36%|███▌      | 36/100 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/ grad: 145.186 / stop criter: 197.220 :  84%|████████▍ | 84/100 [00:03<00:00, 24.64it/s]* GD progress: iter: 86 / bw: 4.761 / grad: 145.186 / stop criter: 197.220 :  84%|████████▍ | 84/100 [00:03<00:00, 24.64it/s]* GD progress: iter: 87 / bw: 4.751 / grad: -160.257 / stop criter: 145.186 :  84%|████████▍ | 84/100 [00:03<00:00, 24.64it/s]* GD progress: iter: 87 / bw: 4.751 / grad: -160.257 / stop criter: 145.186 :  84%|████████▍ | 84/100 [00:03<00:00, 24.64it/s]* GD progress: iter: 87 / bw: 4.751 / grad: -160.257 / stop criter: 145.186 :  87%|████████▋ | 87/100 [00:03<00:00, 24.14it/s]* GD progress: iter: 88 / bw: 4.762 / grad: 0.499 / stop criter: 160.257 :  87%|████████▋ | 87/100 [00:03<00:00, 24.14it/s]   * GD progress: iter: 88 / bw: 4.762 / grad: 0.499 / stop criter: 160.257 :  87%|████████▋ | 87/100 [00:03<00:00, 24.14it/s]* GD progress: iter: 89 / bw: 4.762 / grad: 293.313 / stop criter: 0.499 :  87%|████████▋ | 87/100 [00:03<00:00, 24.14it/s]* GD progress: iter: 89 / bw: 4.762 / grad: 293.313 / stop criter: 0.499 :  87%|████████▋ | 87/100 [00:03<00:00, 24.14it/s]* GD progress: iter: 90 / bw: 4.742 / grad: -109.753 / stop criter: 293.313 :  87%|████████▋ | 87/100 [00:03<00:00, 24.14it/s]* GD progress: iter: 90 / bw: 4.742 / grad: -109.753 / stop criter: 293.313 :  87%|████████▋ | 87/100 [00:03<00:00, 24.14it/s]* GD progress: iter: 90 / bw: 4.742 / grad: -109.753 / stop criter: 293.313 :  90%|█████████ | 90/100 [00:03<00:00, 24.29it/s]* GD progress: iter: 91 / bw: 4.750 / grad: -301.467 / stop criter: 109.753 :  90%|█████████ | 90/100 [00:03<00:00, 24.29it/s]* GD progress: iter: 91 / bw: 4.750 / grad: -301.467 / stop criter: 109.753 :  90%|█████████ | 90/100 [00:03<00:00, 24.29it/s]* GD progress: iter: 92 / bw: 4.769 / grad: -296.801 / stop criter: 301.467 :  90%|█████████ | 90/100 [00:03<00:00, 24.29it/s]* GD progress: iter: 92 / bw: 4.769 / grad: -296.801 / stop criter: 301.467 :  90%|█████████ | 90/100 [00:03<00:00, 24.29it/s]* GD progress: iter: 93 / bw: 4.789 / grad: -563.516 / stop criter: 296.801 :  90%|█████████ | 90/100 [00:03<00:00, 24.29it/s]* GD progress: iter: 93 / bw: 4.789 / grad: -563.516 / stop criter: 296.801 :  90%|█████████ | 90/100 [00:03<00:00, 24.29it/s]* GD progress: iter: 93 / bw: 4.789 / grad: -563.516 / stop criter: 296.801 :  93%|█████████▎| 93/100 [00:03<00:00, 24.42it/s]* GD progress: iter: 94 / bw: 4.826 / grad: -322.072 / stop criter: 563.516 :  93%|█████████▎| 93/100 [00:03<00:00, 24.42it/s]* GD progress: iter: 94 / bw: 4.826 / grad: -322.072 / stop criter: 563.516 :  93%|█████████▎| 93/100 [00:03<00:00, 24.42it/s]* GD progress: iter: 95 / bw: 4.847 / grad: -304.254 / stop criter: 322.072 :  93%|█████████▎| 93/100 [00:03<00:00, 24.42it/s]* GD progress: iter: 95 / bw: 4.847 / grad: -304.254 / stop criter: 322.072 :  93%|█████████▎| 93/100 [00:03<00:00, 24.42it/s]* GD progress: iter: 96 / bw: 4.867 / grad: -81.326 / stop criter: 304.254 :  93%|█████████▎| 93/100 [00:03<00:00, 24.42it/s] * GD progress: iter: 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/ bw: 4.860 / grad: -388.920 / stop criter: 14.885 :  99%|█████████▉| 99/100 [00:04<00:00, 24.76it/s]* GD progress: iter: 100 / bw: 4.885 / grad: -45.384 / stop criter: 388.920 :  99%|█████████▉| 99/100 [00:04<00:00, 24.76it/s]* GD progress: iter: 100 / bw: 4.885 / grad: -45.384 / stop criter: 388.920 :  99%|█████████▉| 99/100 [00:04<00:00, 24.76it/s]* GD progress: iter: 100 / bw: 4.885 / grad: -45.384 / stop criter: 388.920 : 100%|██████████| 100/100 [00:04<00:00, 24.62it/s]

Now, let’s compare it to the previous example.

print("Estimated bandwidth :" + str(ker2_fit.optimization_outputs['opt_bandwidth']))
ker2_fit.draw_learning_curve()
Estimated bandwidth :4.884755819008156

Compare MSE and MAPE.

y_pred2 = ker2_fit.predict(X_test1)
print(mean_absolute_percentage_error(y_test1, y_pred2))
print(mean_squared_error(y_test1, y_pred2))
0.8865863129863055
4172.8207310524585

Compare qq-plot.

ker2_fit.draw_learning_curve(y_test=y_test1, fig_type='qq')

References

Read also: