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Teaching Computers to Learn•Interactive notebook lesson•Free

Linear Regression Simplified with Mathematics

No account is required. Learn the material, try the examples, and mark it complete locally when you are ready to move on.

Note:

Don't be stressed out with any python code, this is a mathematics tutorial, so just focus on mathematics and build your intuition.

Machine Learning

Artificial Intelligence needs no introduction for us programmers. But when most people talk about AI, they imagine an intelligent machine capable of learning or thinking like we humans do. While to some extent it's true, but it's wrong to say it's completely right. AI simply is a result of some mathematical and statistical algorithms and today we'll learn the basics of how these algorithm works.

Let's talk about what machine learning is. The set of mathematical algorithms which our AI systems use behind the curtains is called the machine learning.

So, machine learning is the process of how our computers gets trained to learn like humans while AI (in general) is the end result.

There are (in general) three types of Machine Learning Techniques:

Supervised Learning
Unsupervised Learning
Re-inforcement Learning

Supervised Learning:

Regression - Predict numerical outputs
Classification - Predict categorical outputs (yes/no)

Codepython

df1 = pd.DataFrame({
    "work_experience" : range(1, 4),
    "salary" : np.array(range(1, 4)) * 1000
}, index=["employee0", "employee1", "employee2"])
df1

Output

           work_experience  salary
employee0                1    1000
employee1                2    2000
employee2                3    3000

Codepython

sns.set_style("darkgrid")

plt.figure(figsize=(12,6))
plt.scatter(df1["work_experience"], df1["salary"],
            s=400, color="Violet")

plt.title("Work experience vs Salary of employees")
plt.xlabel("Work Experience in years")
plt.ylabel("Salary in dollars")
plt.xlim(0, 6)
plt.ylim(0, 6000)


plt.plot()

Output

[]

Notebook output

Codepython

plt.figure(figsize=(12,6))
plt.scatter(df1["work_experience"], df1["salary"],
            s=400, color="Violet")
plt.plot([0, 10], [0, 10000], color="black", lw=4)
plt.title("Work experience vs Salary of employees")
plt.xlabel("Work Experience in years")
plt.ylabel("Salary in dollars")
plt.xlim(0, 6)
plt.ylim(0, 6000)


plt.plot()

Output

[]

Notebook output

Codepython

plt.figure(figsize=(12,6))
plt.scatter(df1["work_experience"], df1["salary"],
            s=400, color="Violet")

plt.scatter([4], [200], color="green", s=500)

plt.scatter([4], [4000], color="red", s=500)

plt.scatter([0.1], [4000], color="yellow", s=500)

plt.plot([0, 10], [0, 10000], color="black", lw=4)

plt.plot([4, 4], [0, 4000], color="blue", lw=1)

plt.plot([0, 4], [4000, 4000], color="blue", lw=1)

plt.title("Work experience vs Salary of employees")
plt.xlabel("Work Experience in years")
plt.ylabel("Salary in dollars")
plt.xlim(0, 6)
plt.ylim(0, 6000)


plt.plot()

Output

[]

Notebook output

Codepython

plt.figure(figsize=(6,12))

plt.plot([0, 10], [0, 10], color="purple", lw=4)

plt.ylim(0, 20)
plt.xlim(0, 10)

plt.xticks(np.arange(0, 11, 1))
plt.yticks(np.arange(0,22, 1))
plt.plot()

Output

[]

Notebook output

Codepython

plt.figure(figsize=(12,6))
plt.scatter(df1["work_experience"], df1["salary"],
            s=400, color="Violet")
plt.plot([0, 10], [0, 10000], color="black", lw=4)
plt.title("Work experience vs Salary of employees")
plt.xlabel("Work Experience in years")
plt.ylabel("Salary in dollars")
plt.xlim(-0.1, 6)
plt.ylim(-100, 6000)

plt.plot()

Output

[]

Notebook output

Codepython

plt.figure(figsize=(12,6))
plt.scatter(df1["work_experience"], df1["salary"],
            s=400, color="Violet")

plt.scatter([1,2,3], [0,0,0], s=200, alpha=0.4, color="green")
plt.plot([0, 10], [0, 0], color="black", lw=4)

plt.plot([1,1], [0, 1000], color="yellow", lw=1)

plt.plot([2,2], [0, 2000], color="yellow", lw=1)

plt.plot([3,3], [0, 3000], color="yellow", lw=1)

plt.title("Work experience vs Salary of employees")
plt.xlabel("Work Experience in years")
plt.ylabel("Salary in dollars")
plt.xlim(-0.1, 6)
plt.ylim(-200, 6000)

plt.plot()

Output

[]

Notebook output

Codepython

plt.figure(figsize=(12,6))
plt.scatter(df1["work_experience"], df1["salary"],
            s=400, color="Violet")

plt.scatter([1,2,3], [2000,4000,6000], s=200, alpha=0.4, color="green")
plt.plot([0, 10], [0, 20000], color="black", lw=4)

plt.plot([1,1], [1000, 2000], color="yellow", lw=1)

plt.plot([2,2], [2000, 4000], color="yellow", lw=1)

plt.plot([3,3], [3000, 6000], color="yellow", lw=1)

plt.title("Work experience vs Salary of employees")
plt.xlabel("Work Experience in years")
plt.ylabel("Salary in dollars")
plt.xlim(-0.1, 6)
plt.ylim(-200, 6000)

plt.plot()

Output

[]

Notebook output

Codepython

plt.figure(figsize=(12,6))
plt.scatter(df1["work_experience"], df1["salary"],
            s=400, color="Violet")

plt.scatter([1,2,3], [500,1000,1500], s=200, alpha=0.4, color="green")
plt.plot([0, 10], [0, 5000], color="black", lw=4)

plt.plot([1,1], [500, 1000], color="yellow", lw=1)

plt.plot([2,2], [1000, 2000], color="yellow", lw=1)

plt.plot([3,3], [1500, 3000], color="yellow", lw=1)

plt.title("Work experience vs Salary of employees")
plt.xlabel("Work Experience in years")
plt.ylabel("Salary in dollars")
plt.xlim(-0.1, 6)
plt.ylim(-200, 6000)

plt.plot()

Output

[]

Notebook output

Codepython

plt.figure(figsize=(12,6))
plt.scatter(df1["work_experience"], df1["salary"],
            s=400, color="Violet")

plt.scatter([1,2,3], [1500,3000,4500], s=200, alpha=0.4, color="green")
plt.plot([0, 10], [0, 15000], color="black", lw=4)

plt.plot([1,1], [1000, 1500], color="yellow", lw=1)

plt.plot([2,2], [2000, 3000], color="yellow", lw=1)

plt.plot([3,3], [3000, 4500], color="yellow", lw=1)

plt.title("Work experience vs Salary of employees")
plt.xlabel("Work Experience in years")
plt.ylabel("Salary in dollars")
plt.xlim(-0.1, 6)
plt.ylim(-200, 6000)

plt.plot()

Output

[]

Notebook output

Codepython

plt.figure(figsize=(12,6))
plt.scatter(df1["work_experience"], df1["salary"],
            s=400, color="Violet")

plt.scatter([1,2,3], [250,500,750], s=200, alpha=0.4, color="green")
plt.plot([0, 10], [0, 2500], color="black", lw=4)

plt.plot([1,1], [250, 1000], color="yellow", lw=1)

plt.plot([2,2], [500, 2000], color="yellow", lw=1)

plt.plot([3,3], [750, 3000], color="yellow", lw=1)

plt.title("Work experience vs Salary of employees")
plt.xlabel("Work Experience in years")
plt.ylabel("Salary in dollars")
plt.xlim(-0.1, 6)
plt.ylim(-200, 6000)

plt.plot()

Output

[]

Notebook output

Codepython

error_df = pd.DataFrame({"a" : [1000, 0, 2000, 500, 1500, 250],
                         "error" : [0, 2333333, 2333333, 583333, 583333, 1312500]
                        })
error_df.sort_values(by="a", inplace=True)

error_df

Output

      a    error
1     0  2333333
5   250  1312500
3   500   583333
0  1000        0
4  1500   583333
2  2000  2333333

Codepython

from scipy.interpolate import make_interp_spline

x = np.array(error_df["a"])
y = np.array(error_df["error"])

X_Y_Spline = make_interp_spline(x, y)

X_ = np.linspace(x.min(), x.max(), 500)
Y_ = X_Y_Spline(X_)

plt.figure(figsize=(12,6))
plt.scatter(error_df["a"], error_df["error"],
            s=400, color="Violet")

plt.scatter(error_df["a"], error_df["error"])

plt.plot(X_, Y_)

plt.title("Hyper-parameter a vs error")
plt.xlabel("Value of Hyper-parameter a")
plt.ylabel("MSE")
plt.xlim(-250, 2500)
plt.ylim(-100000, 4000000)

plt.plot()

Output

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Notebook output