Multiple Linear Regression:

Multi-linear Regression

Multiple Linear Regression Explained Simply¶

1. What is Multiple Linear Regression?¶

It’s a method used to predict one thing (like house price) based on multiple other factors (like size, number of bedrooms, and location). It helps us understand how different factors influence the outcome.

2. The Equation¶

The equation looks like this:

$Y = \beta_0 + \beta_1X_1 + \beta_2X_2 + … + \beta_nX_n + \epsilon$

Where:
- ( Y ) is what you want to predict (like price).
- ( X₁, X₂, … ) are the factors you’re using for prediction (like size, bedrooms).
- ( $\beta_0$ ) is the starting point (the price when all factors are zero).
- ( $\beta_0, \beta_2$, … ) tell you how much Y changes when each factor changes.
- ( $\epsilon $) is just a fancy way to say “random errors.”

3. Gather Your Data¶

Collect information about the houses you want to analyze. For example, get data on:
- Size of the house (in square feet).
- Number of bedrooms.
- Age of the house.
- Price (this is what you want to predict).

4. Look at Your Data (Optional)¶

Create some graphs to see how each factor (like size or bedrooms) relates to the house price. This helps you spot trends.

5. Split Your Data¶

Divide your data into two parts:
- Training set: This is the data used to train your model (like teaching it).
- Test set: This is the data used to see how well the model works on new information.

6. Train the Model¶

Use a computer program to find the best line (or equation) that fits your training data. The program will figure out the coefficients (the values for ( \beta )) for each factor.

from sklearn.linear_model import LinearRegression

# Example code to train the model
model = LinearRegression()
model.fit(X_train, Y_train)

7. Check the Coefficients¶

After training, look at the coefficients. They tell you how much the price (Y) changes with each factor (X). For example:
- If ( $\beta_1$) (size coefficient) is 200, then for every extra square foot, the price goes up by $200.

8. Make Predictions¶

Now, you can use your trained model to predict house prices for new houses based on their features.

Y_pred = model.predict(X_test)

9. Evaluate the Model¶

Check how accurate your predictions are using some measures:
- Mean Squared Error (MSE): This tells you how far off your predictions are.
- R-squared (R²): This tells you how well your factors explain the house prices (higher is better).

from sklearn.metrics import mean_squared_error, r2_score

mse = mean_squared_error(Y_test, Y_pred)
r2 = r2_score(Y_test, Y_pred)

10. Analyze Errors¶

Look at the differences between what you predicted and what the actual prices were (these are called residuals). You want these to be random; if they show a pattern, your model might need adjustments.

11. Use for Future Predictions¶

Once you’re satisfied with your model, you can use it to predict prices for any new house by putting in its features.

new_data = [[size, bedrooms, age]]  # New house features
predictions = model.predict(new_data)

12. Summarize Your Findings¶

Wrap up by discussing how well the model did and what the coefficients mean. For example, if the size coefficient is positive, it means bigger houses cost more.

Let’s review example step by step.¶

Import Libraries

In [3]:

import pandas as pd
import matplotlib.pyplot as plt

Read Data

In [4]:

Stock_Market = {'Year': [2017,2017,2017,2017,2017,2017,2017,2017,2017,2017,2017,2017,2016,2016,2016,2016,2016,2016,2016,2016,2016,2016,2016,2016],
                'Month': [12, 11,10,9,8,7,6,5,4,3,2,1,12,11,10,9,8,7,6,5,4,3,2,1],
                'Interest_Rate': [2.75,2.5,2.5,2.5,2.5,2.5,2.5,2.25,2.25,2.25,2,2,2,1.75,1.75,1.75,1.75,1.75,1.75,1.75,1.75,1.75,1.75,1.75],
                'Unemployment_Rate': [5.3,5.3,5.3,5.3,5.4,5.6,5.5,5.5,5.5,5.6,5.7,5.9,6,5.9,5.8,6.1,6.2,6.1,6.1,6.1,5.9,6.2,6.2,6.1],
                'Stock_Index_Price': [1464,1394,1357,1293,1256,1254,1234,1195,1159,1167,1130,1075,1047,965,943,958,971,949,884,866,876,822,704,719]        
                }

In [5]:

df = pd.DataFrame(Stock_Market)
df

Out[5]:

	Year	Month	Interest_Rate	Unemployment_Rate	Stock_Index_Price
0	2017	12	2.75	5.3	1464
1	2017	11	2.50	5.3	1394
2	2017	10	2.50	5.3	1357
3	2017	9	2.50	5.3	1293
4	2017	8	2.50	5.4	1256
5	2017	7	2.50	5.6	1254
6	2017	6	2.50	5.5	1234
7	2017	5	2.25	5.5	1195
8	2017	4	2.25	5.5	1159
9	2017	3	2.25	5.6	1167
10	2017	2	2.00	5.7	1130
11	2017	1	2.00	5.9	1075
12	2016	12	2.00	6.0	1047
13	2016	11	1.75	5.9	965
14	2016	10	1.75	5.8	943
15	2016	9	1.75	6.1	958
16	2016	8	1.75	6.2	971
17	2016	7	1.75	6.1	949
18	2016	6	1.75	6.1	884
19	2016	5	1.75	6.1	866
20	2016	4	1.75	5.9	876
21	2016	3	1.75	6.2	822
22	2016	2	1.75	6.2	704
23	2016	1	1.75	6.1	719

Plot the columns

In [6]:

plt.title("Stock Market")
plt.xlabel("Interest Rate")
plt.ylabel("Stock Index Price")
plt.scatter(df['Interest_Rate'],df['Stock_Index_Price'])
plt.show()

No description has been provided for this image

In [7]:

plt.title("Stock Market")
plt.xlabel("Unemployment Rate")
plt.ylabel("Stock Index Price")
plt.scatter(df['Unemployment_Rate'],df['Stock_Index_Price'])
plt.show()

Check co-relation

In [8]:

df.corr()

Out[8]:

	Year	Month	Interest_Rate	Unemployment_Rate	Stock_Index_Price
Year	1.000000e+00	7.884865e-14	0.882851	-0.877000	0.863232
Month	7.884865e-14	1.000000e+00	0.339526	-0.351189	0.481287
Interest_Rate	8.828507e-01	3.395257e-01	1.000000	-0.925814	0.935793
Unemployment_Rate	-8.769997e-01	-3.511891e-01	-0.925814	1.000000	-0.922338
Stock_Index_Price	8.632321e-01	4.812873e-01	0.935793	-0.922338	1.000000

Choose the columns that contains variables you believe will influence and help predict the target data.

In [10]:

X = df.iloc[:,2:4].values
X

Out[10]:

array([[2.75, 5.3 ],
       [2.5 , 5.3 ],
       [2.5 , 5.3 ],
       [2.5 , 5.3 ],
       [2.5 , 5.4 ],
       [2.5 , 5.6 ],
       [2.5 , 5.5 ],
       [2.25, 5.5 ],
       [2.25, 5.5 ],
       [2.25, 5.6 ],
       [2.  , 5.7 ],
       [2.  , 5.9 ],
       [2.  , 6.  ],
       [1.75, 5.9 ],
       [1.75, 5.8 ],
       [1.75, 6.1 ],
       [1.75, 6.2 ],
       [1.75, 6.1 ],
       [1.75, 6.1 ],
       [1.75, 6.1 ],
       [1.75, 5.9 ],
       [1.75, 6.2 ],
       [1.75, 6.2 ],
       [1.75, 6.1 ]])

Select the column you want to predict, known as the target variable, which is the outcome you aim to model using the other features.

In [11]:

y = df.iloc[:,4].values
y

Out[11]:

array([1464, 1394, 1357, 1293, 1256, 1254, 1234, 1195, 1159, 1167, 1130,
       1075, 1047,  965,  943,  958,  971,  949,  884,  866,  876,  822,
        704,  719], dtype=int64)

Put Algorithm

In [12]:

from sklearn.linear_model import LinearRegression
reg = LinearRegression()
reg.fit(X,y)

Out[12]:

LinearRegression()

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Find intercept

In [13]:

reg.intercept_

Out[13]:

1798.4039776258544

Find co-efficent

In [14]:

reg.coef_

Out[14]:

array([ 345.54008701, -250.14657137])

Predict y

In [15]:

Out[15]:

array([1464, 1394, 1357, 1293, 1256, 1254, 1234, 1195, 1159, 1167, 1130,
       1075, 1047,  965,  943,  958,  971,  949,  884,  866,  876,  822,
        704,  719], dtype=int64)

In [16]:

y_pred = reg.predict(X)
y_pred

Out[16]:

array([1422.86238865, 1336.47736689, 1336.47736689, 1336.47736689,
       1311.46270976, 1261.43339548, 1286.44805262, 1200.06303087,
       1200.06303087, 1175.04837373, 1063.64869484, 1013.61938057,
        988.60472343,  927.23435881,  952.24901595,  877.20504454,
        852.1903874 ,  877.20504454,  877.20504454,  877.20504454,
        927.23435881,  852.1903874 ,  852.1903874 ,  877.20504454])

Check Accuracy

In [17]:

reg.score(X,y) * 100

Out[17]:

89.76335894170217

Predict future value

In [18]:

reg.predict([[5.4,3.7]])

Out[18]:

array([2738.77813342])

In [19]:

round(reg.predict([[5.4,3.7]])[0])

Out[19]:

In [21]:

a = float(input("Enter Interest_Rate:"))
b = float(input("Enter Unemployment_Rate:"))
val = reg.predict([[a,b]])
print("Stock Index Price:",round(val[0]))

Stock Index Price: 2814

In [ ]: