Compute Several Weighted Scores

One weight array gives one score per observation. We may instead need two different scores from the same sensor values. Each score can begin as the weighted sum we already know.

The two weight arrays answer two separate questions about every observation. We can calculate them separately before arranging them into one operation.

Expand Both Outputs for One Observation

For the first observation [-1.0, -2.0], output 0 uses weights [1.0, 0.5]:

(-1.0 × 1.0) + (-2.0 × 0.5)
= -1.0 + -1.0
= -2.0

Output 1 uses weights [-0.5, 1.0]:

(-1.0 × -0.5) + (-2.0 × 1.0)
= 0.5 + -2.0
= -1.5

These are two applications of the previous lesson's rule. The same two sensor values are paired with a different weight set for each output.

Now add one fixed bias to each output:

bias = np.array([0.25, -0.5])

The first observation's final outputs are:

output 0: -2.0 + 0.25 = -1.75
output 1: -1.5 + -0.5 = -2.0

The bias has shape (2,) because there are two outputs. It is not one value per sensor. Its positions belong to the output axis that exists after the weighted sums have been calculated.

Store One Weight Set in Each Column

Arrange the two weight arrays as columns of one table:

Read its axes before using it:

• axis 0 contains sensors;
• axis 1 contains outputs.

Column 0 is [1.0, 0.5], the weights for output 0. Column 1 is [-0.5, 1.0], the weights for output 1. Keeping outputs in columns makes the sensor positions meet in this order:

centered @ weights
(observations, sensors) @ (sensors, outputs)
                      -> (observations, outputs)

For our arrays, (3, 2) @ (2, 2) produces shape (3, 2). The matching middle sizes are the two sensors. The output keeps three observations and introduces two output positions.

[[-2.  -1.5]
 [ 0.   0. ]
 [ 2.   1.5]]

weighted[0, 1] is the -1.5 calculation expanded above. Every entry follows the same rule: choose one observation row and one output column, multiply the two aligned sensor positions, then add the contributions.

Add One Bias per Output

The weighted table and bias have compatible named axes:

weighted: (3 observations, 2 outputs)
bias:                    (2 outputs)

Adding the bias applies the same two output adjustments to every observation:

[[-1.75 -2.  ]
 [ 0.25 -0.5 ]
 [ 2.25  1.  ]]

The order matters. centered @ weights first combines sensor values into two outputs. The bias then aligns with those output columns. Writing weights @ centered would reverse the roles and does not have matching inner sizes for these shapes. Reordering operands is not a formatting choice; it changes which positions meet and which axes remain.

References

• NumPy documentation: numpy.matmul — matching inner sizes, output shape, and the @ operator.