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How To Calculate Cost Function J(θ0, θ1) - Machine Learning




What is cost function: The cost function “J( θ01)” is used to measure how good a fit (measure the accuracy of hypothesis function) a line is to the data. If the line is a good fit, then your predictions will be far better. The idea is to minimize the value of J by calculating it from given values of  θ0 and θ1.

Line as good fit: The line we're trying to make as good a fit as possible is defined by equation “hθ(x)= θ0 + θ1x”.

Formula:


So how to calculate cost function. Let see the steps

Step 1.  Pick the value of  θ0  and  θ1
Step 2. Find value of hθ(x)=   by using formula “hθ(x)= θ0 + θ1x”.
Step 3. Find  “hθ(x)-Y”    for all values of X
Step 4. Find  Square of (hθ(x) -Y) for all values of Y
Step 5. Substitute all values in equation   (1/2m Σ( h(x)-y)2)  which is equal to “J(θ0, θ1)”

Example:
Let’s consider we have m sample values.

X
Y
4
1
3
2
2
2
1
4

Total sample test data is 4 i.e. m=4

Example 1: Calculate value of J(θ0, θ1)

Step 1.    θ0=0 and θ1=1

Step 2
Step 3
Step 4
X
Y
hθ(x) i.e. (θ0 + θ1x)
hθ(x)-Y
(hθ(x) - y )2
4
1
0+1*4 => 4
4-1 => 3
9
3
2
0+1*3 => 3
3-2 => 1
1
2
2
0+1*2 => 2
2-2 => 0
0
1
4
0+1*1 => 1
1-4 => -3
9


Step 5:  Substitute value in {1/2m Σ( h(x)-y)2}
m=6 ;   h(x)-y)2 is calculated, we have to du summation

J(θ0, θ1) = (1/(2*4))* (9+1+0+9) = (1/8 )*19 = 2.375
So  J(θ0, θ1) = 2.375 for θ0=0 and θ1=1

You can pick other values of θ0 and θ1 but that we will discuss in other post

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  1. Dear website ka background light rakhty hn Bandy ko kuch nazar b to aye

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