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 (aha@ics.uci.edu) (714) 856-8779  Â
  -- 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 Â Â Â Â Â Â Â 209 non-null object
MODEL_NAME Â Â Â 209 non-null object
MYCT Â Â Â Â Â Â Â Â Â 209 non-null int64
MMIN Â Â Â Â Â Â Â Â Â 209 non-null int64
MMAX Â Â Â Â Â Â Â Â Â 209 non-null int64
CACH Â Â Â Â Â Â Â Â Â 209 non-null int64
CHMIN Â Â Â Â Â Â Â Â 209 non-null int64
CHMAX Â Â Â Â Â Â Â Â 209 non-null int64
PRP Â Â Â Â Â Â Â Â Â Â 209 non-null int64
ERP Â Â Â Â Â Â Â Â Â Â 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 : Â 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.