## Introduction

The goal of this blog post is to equip beginners with the basics of the Linear Regression algorithm with multiple variables predicting the outcome of the target variable. This is also known as Multiple Linear Regression.

Simple linear regression model has a continuous outcome and one predictor, whereas a multiple linear regression model has a continuous outcome and multiple predictors (continuous or categorical). A simple linear regression model would have the form:

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A multivariable or multiple linear regression model would take the form:

where y is a continuous dependent variable, x is a single predictor in the simple regression model, and x1, x2, …, xk are the predictors in the multiple regression model.

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.

Multiple linear regression can model more complex relationship which comes from various features together. They should be used in cases where one particular variable is not evident enough to map the relationship between the independent and the dependent variable.

Let’s work on a case study to understand this better.

#### Problem Statement

To predict the relative performance of a computer hardware given other associated attributes of the hardware.

#### Data details

Computer Hardware dataset =========================== URL : https://archive.ics.uci.edu/ml/datasets/Computer+Hardware 1. Title: Relative CPU Performance Data 2. Source Information -- Creators: Phillip Ein-Dor and Jacob Feldmesser -- Ein-Dor: Faculty of Management; Tel Aviv University; Ramat-Aviv; Tel Aviv, 69978; Israel -- Donor: David W. Aha ([email protected]) (714) 856-8779 -- Date: October, 1987 3. Past Usage: 1. Ein-Dor and Feldmesser (CACM 4/87, pp 308-317) -- Results: -- linear regression prediction of relative cpu performance -- Recorded 34% average deviation from actual values 2. Kibler,D. & Aha,D. (1988). Instance-Based Prediction of Real-Valued Attributes. In Proceedings of the CSCSI (Canadian AI) Conference. -- Results: -- instance-based prediction of relative cpu performance -- similar results; no transformations required - Predicted attribute: cpu relative performance (numeric) 4. Relevant Information: -- The estimated relative performance values were estimated by the authors using a linear regression method. See their article (pp 308-313) for more details on how the relative performance values were set. 5. Number of Instances: 209 6. Number of Attributes: 10 (6 predictive attributes, 2 non-predictive, 1 goal field, and the linear regression guess) 7. Attribute Information: 1. vendor name: 30 (adviser, amdahl,apollo, basf, bti, burroughs, c.r.d, cambex, cdc, dec, dg, formation, four-phase, gould, honeywell, hp, ibm, ipl, magnuson, microdata, nas, ncr, nixdorf, perkin-elmer, prime, siemens, sperry, sratus, wang) 2. Model Name: many unique symbols 3. MYCT: machine cycle time in nanoseconds (integer) 4. MMIN: minimum main memory in kilobytes (integer) 5. MMAX: maximum main memory in kilobytes (integer) 6. CACH: cache memory in kilobytes (integer) 7. CHMIN: minimum channels in units (integer) 8. CHMAX: maximum channels in units (integer) 9. PRP: published relative performance (integer) 10. ERP: estimated relative performance from the original article (integer) 8. Missing Attribute Values: None 9. Class Distribution: the class value (PRP) is continuously valued. PRP Value Range: Number of Instances in Range: 0-20 31 21-100 121 101-200 27 201-300 13 301-400 7 401-500 4 501-600 2 above 600 4 Summary Statistics: Min Max Mean SD PRP Correlation MCYT: 17 1500 203.8 260.3 -0.3071 MMIN: 64 32000 2868.0 3878.7 0.7949 MMAX: 64 64000 11796.1 11726.6 0.8630 CACH: 0 256 25.2 40.6 0.6626 CHMIN: 0 52 4.7 6.8 0.6089 CHMAX: 0 176 18.2 26.0 0.6052 PRP: 6 1150 105.6 160.8 1.0000 ERP: 15 1238 99.3 154.8 0.9665

#### Tools used:

- Pandas
- Numpy
- Matplotlib
- scikit-learn

#### 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 import linear_model

#### Load the data set

Use the pandas module to read the taxi data from the file system. Check few records of the dataset.

names = ['VENDOR','MODEL_NAME','MYCT', 'MMIN', 'MMAX', 'CACH', 'CHMIN', 'CHMAX', 'PRP', 'ERP' ]; data = pd.read_csv('data/computer-hardware/machine.data',names=names) data.head() VENDOR MODEL_NAME MYCT MMIN MMAX CACH CHMIN CHMAX PRP ERP 0 adviser 32/60 125 256 6000 256 16 128 198 199 1 amdahl 470v/7 29 8000 32000 32 8 32 269 253 2 amdahl 470v/7a 29 8000 32000 32 8 32 220 253 3 amdahl 470v/7b 29 8000 32000 32 8 32 172 253 4 amdahl 470v/7c 29 8000 16000 32 8 16 132 132

#### Feature selection

Let’s select only the numerical fields for model fitting.

data.info() <class 'pandas.core.frame.DataFrame'> RangeIndex: 209 entries, 0 to 208 Data columns (total 10 columns): VENDOR 209 non-null object MODEL_NAME 209 non-null object MYCT 209 non-null int64 MMIN 209 non-null int64 MMAX 209 non-null int64 CACH 209 non-null int64 CHMIN 209 non-null int64 CHMAX 209 non-null int64 PRP 209 non-null int64 ERP 209 non-null int64 dtypes: int64(8), object(2)

We can see that barring the first two variables rest are numeric in nature. Let’s only pick the numeric fields.

categorical_ = data.iloc[:,:2] numerical_ = data.iloc[:,2:] numerical_.head() MYCT MMIN MMAX CACH CHMIN CHMAX PRP ERP 0 125 256 6000 256 16 128 198 199 1 29 8000 32000 32 8 32 269 253 2 29 8000 32000 32 8 32 220 253 3 29 8000 32000 32 8 32 172 253 4 29 8000 16000 32 8 16 132 132

#### Select the predictor and target variables

X = numerical_.iloc[:,:-1] y = numerical_.iloc[:,-1]

**Train test split**

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)

**Normalize the data**

Before we do the fitting, let’s normalize the data so that the data is centered around the mean and has unit standard deviation.

from sklearn.preprocessing import StandardScaler scaler = StandardScaler() # Fit on training set only. scaler.fit(x_training_set) # Apply transform to both the training set and the test set. x_training_set = scaler.transform(x_training_set) x_test_set = scaler.transform(x_test_set)

y_training_set = y_training_set.values.reshape(-1, 1) y_test_set = y_test_set.values.reshape(-1, 1) y_scaler = StandardScaler() # Fit on training set only. y_scaler.fit(y_training_set) # Apply transform to both the training set and the test set. y_training_set = y_scaler.transform(y_training_set) y_test_set = y_scaler.transform(y_test_set)

#### Training/model fitting

Fit the model to selected supervised data

model = linear_model.LinearRegression() model.fit(x_training_set,y_training_set)

#### 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)) Coefficient of determination R^2 of the prediction : 0.9583846753218253 Mean squared error: 0.39 Test Variance score: 0.93

#### Accuracy report with test data

Let’s visualize the goodness of the fit with the predictions being visualized by 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()

### Conclusion

We can see that our R2 score and MSE are both very good. This means that we have found a well-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.