The goal of this blog post is to equip beginners with an understanding of the basics of the Decision Tree Regressor algorithm and quickly help them to build their first model.
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In statistics, the mean squared error (MSE) or mean squared deviation (MSD) of an estimator (of a procedure for estimating an unobserved quantity) measures the average of the squares of the errors—that is, the average squared difference between the estimated values and what is actually estimated.
The MSE is a measure of the quality of an estimator—it is always non-negative, and values closer to zero are better.
The Mean Squared Error is given by:
To predict the median prices of homes located in the Boston area when other attributes of the house are given.
Boston House Prices dataset =========================== Notes ------ Data Set Characteristics: :Number of Instances: 506 :Number of Attributes: 13 numeric/categorical predictive :Median Value (attribute 14) is usually the target :Attribute Information (in order): - CRIM per capita crime rate by town - ZN proportion of residential zoned land for lots over 25,000 sq.ft. - INDUS proportion of non-retail business acres per town - CHAS Charles river dummy variable (= 1 if tract bounds river; 0 otherwise) - NOX nitric oxides concentration (parts per 10 million) - RM average number of rooms per dwelling - AGE proportion of owner-occupied units built prior to 1940 - DIS weighted distances to five Boston employment centers - RAD index of accessibility to radial highways - TAX full-value property-tax rate per $10,000 - PTRATIO pupil-teacher ratio by town - B 1000(Bk - 0.63)^2 where Bk is the proportion of blacks by town - LSTAT % lower status of the population - MEDV median value of owner-occupied homes in $1000's :Missing Attribute Values: None :Creator: Harrison, D. and Rubinfeld, D.L.
This is a copy of UCI ML housing dataset.
This dataset was taken from the StatLib library which is maintained at Carnegie Mellon University.
The Boston house-price data of Harrison, D., and Rubinfeld, D.L. ‘Hedonic prices and the demand for clean air’, J. Environ. Economics & Management, vol.5, 81-102, 1978. Used in Belsley, Kuh & Welsch, ‘Regression diagnostics …’, Wiley, 1980. N.B. Various transformations are used in the table on pages 244-261 of the latter.
The Boston house-price data has been used in many machine learning papers that address regression problems.
Python Implementation with Code
Import necessary libraries
Import the necessary modules from specific libraries
import numpy as np import pandas as pd %matplotlib inline import matplotlib.pyplot as plt from sklearn.model_selection import train_test_split from sklearn import datasets from sklearn.metrics import mean_squared_error from sklearn.tree import DecisionTreeRegressor
Load the data set
Use the pandas module to read the taxi data from the file system. Check few records of the dataset.
# ############################################################################# # Load data boston = datasets.load_boston() print(boston.data.shape, boston.target.shape) print(boston.feature_names) (506, 13) (506,) ['CRIM' 'ZN' 'INDUS' 'CHAS' 'NOX' 'RM' 'AGE' 'DIS' 'RAD' 'TAX' 'PTRATIO' 'B' 'LSTAT']
data = pd.DataFrame(boston.data,columns=boston.feature_names) data = pd.concat([data,pd.Series(boston.target,name='MEDV')],axis=1) data.head() CRIM ZN INDUS CHAS NOX RM AGE DIS RAD TAX PTRATIO B LSTAT MEDV 0 0.00632 18.0 2.31 0.0 0.538 6.575 65.2 4.0900 1.0 296.0 15.3 396.90 4.98 24.0 1 0.02731 0.0 7.07 0.0 0.469 6.421 78.9 4.9671 2.0 242.0 17.8 396.90 9.14 21.6 2 0.02729 0.0 7.07 0.0 0.469 7.185 61.1 4.9671 2.0 242.0 17.8 392.83 4.03 34.7 3 0.03237 0.0 2.18 0.0 0.458 6.998 45.8 6.0622 3.0 222.0 18.7 394.63 2.94 33.4 4 0.06905 0.0 2.18 0.0 0.458 7.147 54.2 6.0622 3.0 222.0 18.7 396.90 5.33 36.2
Select the predictor and target variables
The target variable is MEDV which is the Median value of owner-occupied homes in $1000’s. The rest are predictor variables.
X = data.iloc[:,:-1] y = data.iloc[:,-1]
Train test split
The whole dataset is split into training and test set. Training data is used to train the model and the test set is to evaluate how well the model performed.
x_training_set, x_test_set, y_training_set, y_test_set = train_test_split(X,y,test_size=0.10,random_state=42, shuffle=True)
Fit the model to selected supervised data.
# Fit regression model # Estimate the score on the entire dataset, with no missing values model = DecisionTreeRegressor(max_depth=5,random_state=0) model.fit(x_training_set, y_training_set) The coefficient of determination R^2 of the prediction: 0.9179598310471841 Mean squared error: 7.95 Test Variance score: 0.87
Model parameters study
The coefficient R^2 is defined as (1 – u/v), where u is the residual sum of squares ((y_true – y_pred) ** 2).sum() and v is the total sum of squares ((y_true – y_true.mean()) ** 2).sum().
from sklearn.metrics import mean_squared_error, r2_score model_score = model.score(x_training_set,y_training_set) # Have a look at R sq to give an idea of the fit , # Explained variance score: 1 is perfect prediction print(“ coefficient of determination R^2 of the prediction.: ',model_score) y_predicted = model.predict(x_test_set) # The mean squared error print("Mean squared error: %.2f"% mean_squared_error(y_test_set, y_predicted)) # Explained variance score: 1 is perfect prediction print('Test Variance score: %.2f' % r2_score(y_test_set, y_predicted)) The coefficient of determination R^2 of the prediction: 0.982022598521334 Mean squared error: 7.73 Test Variance score: 0.88
Accuracy report with test data :
Let’s check the goodness of the fit with the predictions visualized as a line.
# So let's run the model against the test data from sklearn.model_selection import cross_val_predict fig, ax = plt.subplots() ax.scatter(y_test_set, y_predicted, edgecolors=(0, 0, 0)) ax.plot([y_test_set.min(), y_test_set.max()], [y_test_set.min(), y_test_set.max()], 'k--', lw=4) ax.set_xlabel('Actual') ax.set_ylabel('Predicted') ax.set_title("Ground Truth vs Predicted") plt.show()
We can see that our R2 score and MSE are both very good. This means that we have found a good fitting model to predict the median price value of a house. There can be a further improvement to the metric by doing some preprocessing before fitting the data. However, the task of the post was to provide you with enough knowledge to implement your first model. You can build over the existing pipeline and report your accuracies.