Convolutions

Convolve the two 3x3 matrices that were assigned to you with your 9x9 matrix and calculate the resulting two matrices.

Original 9x9 matrix:

[[0, 0, -1, -2, -1, -1, 0, 1, -1],
 [1, -1, -1, -2, 2, -1, -1, -1, -1],
 [1, 0, 0, 2, 2, 0, 0, 1, 0],
 [1, -1, 0, -2, -1, 2, -1, -1, -1],
 [1, 0, 0, -1, 1, 0, -1, 1, 1],
 [-1, 0, 1, 1, 1, 2, 1, 0, -1],
 [0, 1, 0, 2, 1, 2, -1, 0, 0],
 [-1, -1, 1, 2, 1, -2, 0, -1, -1],
 [1, 1, 1, 1, -2, 2, -1, 1, -1]]

First 3x3 matrix and resulting 7x7 matrix:

[[0, 1, 0],
 [0, 1, 0],
 [1, 1, 0]]
 
 [[0, -2, -2, 5, 0, -1, 1],
  [-1, -2, -2, 1, 0, 0, -3],
  [0, 0, -1, 1, 3, -2, -2],
  [-2, 1, -1, 2, 5, 1, 1],
  [1, 2, 2, 5, 5, 1, 0],
  [-1, 1, 6, 5, 3, -2, -1],
  [2, 3, 6, 1, 0, 0, -1]]

Second 3x3 matrix and resulting 7x7 matrix:

[[1, 1, 0],
 [0, 0, 1],
 [0, 0, 1]]
 
 [[-1, -1, 1, -4, -3, -1, 0],
  [0, -2, -2, 2, 0, -3, -3],
  [1, -3, 1, 6, 0, 0, 1],
  [1, -1, 0, -1, 1, 2, -2],
  [2, 3, 1, 4, 1, -1, -1],
  [0, 5, 4, 2, 2, 2, 0],
  [3, 4, 1, 3, 2, 1, -3]]

What is the purpose of using a 3x3 filter to convolve across a 2D image matrix?

Using a filter to convolve across a 2D image matrix is helpful in getting the most important features of the image for more efficient processing. For example, a filter over the stair photo can emphasize the vertical or horizontal lines in the image.

Why would we include more than one filter? How many filters did you assign as part of your architecture when training a model to learn images of numbers from the mnist dataset?

We might include more than one filter to emphasize multiple features in an image. Or, we might utilize a pooling filter in addition to a convolution filter to make the image smaller. I think only one filter was assigned when training the mnist model - unless the Flatten and each Dense layer (relu and softmax) are considered filters, then three filters were assigned as part of the architecture when training the model.

MSE

From your 400+ observations of homes for sale, calculate the MSE for the following:

The 10 biggest over-predictions:

MSE = 12018751.518797165

The 10 biggest under-predictions:

MSE = 281355803.96214515

The 10 most accurate results (use absolute value):

MSE = 9214.230143046076

In which percentile do the 10 most accurate predictions reside?

I wasn’t sure whether to look at percentiles of observed or predicted prices, so I did both. The predicted prices for the 10 most accurate predictions reside between the 1st and 78th percentiles ($1,472,031 - $2,567,084). The observed prices for the 10 most accurate predictions reside between the 77th and 84th percentiles ($1,495,000 - $2,547,960).

Did your model trend towards over or under predicting home values?

My model trended towards over-predicting home values; 333 homes cost less than predicted, and 67 homes cost more than predicted.

Which feature appears to be the most significant predictor in the above cases?

It’s a little hard to tell which feature is the most significant predictor, but for the ten most accurate predictions and ten over-predictions, it appears that number of bedrooms is the most significant predictor. For the under-predictions, it appears that square footage may be the most significant predictor, but that is more difficult to determine. As briefly mentioned last class, we could also look at location. I looked at homes in San Diego, so I suspect location (ie oceanfront or not) is a significant predictor for home price in this city.

Code: Feb26Response.py