The linear model. Multiple inputs

Let’s extend our little real estate valuation model to multiple inputs what if we had additional information

say the apartments we are valuing are located by the sea.

The closer an apartment is to the beach the higher the price should be.

Right.

Well a better linear model would predict a price based on both size and proximity to the beach.

The linear model tackling this issue is the price of an apartment equals its size times the weight of

size plus the apartments proximity to the beach times the weight of proximity Plus the bias.

If we want to state this expression in linear algebraic terms we can use the same framework as before

x times w plus B X and W are both vectors.

X is one by two and W is two by one multiplying X and W would give us a scalar as 1 by two times two

by one is equal to a one by one shape.

Let’s continue our housing example with the two input variables the output y is equal to x times w plus

B.

X has two elements a size of 743 and a distance from the beach of 1.2 1 miles.

W has two elements to one for each input.

The values are four hundred three point seventy seven and minus fifteen thousand two hundred twelve.

This time the bias is plus one thousand two hundred twelve point four or five.

To calculate the new projected price we must multiply 743 times four hundred three point seventy seven

then subtract 1.2 1 times fifteen thousand five hundred twelve and add the bias one thousand twelve

point four or five.

The result is two hundred eighty two thousand four hundred forty four dollars and four cents.

Similarily if the 1000 square feet apartment is two miles away from the beach its price becomes three

hundred seventy three thousand nine hundred fifty eight dollars and forty five cents.

Notice that the weight of the distance from the beach is negative.

This is because we expect this variable to have a negative impact on an apartment price.

The greater the distance from the beach the lower the price.

All right in the next lesson we will extend the model to its most general form multiple inputs and multiple

outputs.

Stay tuned.