What is cost function: The cost function “J( θ0,θ1)” 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”.
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
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Y
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4
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1
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3
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2
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2
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2
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1
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4
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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
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Step 3
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Step 4
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X
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Y
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hθ(x) i.e. (θ0 + θ1x)
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hθ(x)-Y
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(hθ(x) - y )2
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4
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1
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0+1*4 => 4
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4-1 => 3
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9
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3
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2
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0+1*3 => 3
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3-2 => 1
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1
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2
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2
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0+1*2 => 2
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2-2 => 0
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0
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1
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4
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0+1*1 => 1
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1-4 => -3
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9
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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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