Broadcasting - use Technical Quotient

A practical example: vector quantization

2. Understand

How does broadcasting affect the shape of arrays when calculating distances between observations and codes in vector quantization?

It automatically adjusts the shape of arrays to allow element-wise operations.
It requires manual adjustment of array shapes before calculation.
It duplicates the smaller array to match the dimensions of the larger array.
It reduces the dimensions of the larger array to match the smaller one.

3. Apply

Given an observation obs = np.array([150, 60]) and codes cds = np.array([[140, 55], [155, 70]]), calculate the Euclidean distance between obs and each code. Select the correct implementation.

\tdiff = cds - obs\n\tsq_diff = diff ** 2\n\tdist = np.sqrt(np.sum(sq_diff, axis=-1))\n\tclosest = np.argmin(dist)\t

\tdiff = obs - cds\n\tsq_diff = np.power(diff, 2)\n\tdist = np.sum(sq_diff)\n\tclosest = np.argmin(dist)\t

\tdiff = obs - cds\n\tsq_diff = diff ** 2\n\tdist = np.sqrt(np.sum(sq_diff, axis=1))\n\tclosest = np.argmax(dist)\t

Incorrect; subtracts correctly and calculates squared difference, but finds the farthest code using argmax instead of the closest.

\tdiff = cds / obs\n\tsq_diff = diff ** 2\n\tdist = np.sqrt(np.sum(sq_diff))\n\tclosest = np.argmin(dist)\t

Incorrect; divides instead of subtracting, sums the entire array instead of per code, and incorrectly uses argmin on a scalar.

4. Apply

Given a dataset with athlete performance metrics, performance = np.array([[210, 55], [220, 60], [215, 58]]) and classification codes for sports, sports_codes = np.array([[205, 50], [230, 65]]), you need to classify each athlete by finding the closest sport code. Implement the calculation for the Euclidean distance between each athlete's performance metrics and the sports codes.

\tdiff = performance - sports_codes\n\tsq_diff = np.square(diff)\n\tdist = np.sqrt(np.sum(sq_diff))\n\tclosest_sport = np.argmin(dist)

Incorrect; This option does not correctly account for broadcasting between performance and sports_codes, leading to an incorrect calculation of distances. Additionally, it improperly sums all differences into a single value and attempts to find a minimum in a scalar value.

\tdiff = performance - sports_codes[:, np.newaxis]\n\tsq_diff = diff ** 2\n\tdist = np.sum(sq_diff, axis=0)\n\tclosest_sport = np.argmin(dist, axis=0)

\tdiff = performance[:, np.newaxis, :] - sports_codes\n\tsq_diff = diff ** 2\n\tdist = np.sqrt(np.sum(sq_diff, axis=2))\n\tclosest_sport = np.argmin(dist, axis=1)

Correct; Reshapes performance for broadcasting to subtract from sports_codes, calculates squared difference, sums along the last axis to get distances for each code, then finds the index of the minimum distance for each athlete.

\tdiff = sports_codes - performance[np.newaxis, :, :]\n\tsq_diff = np.power(diff, 2)\n\tdist = np.sqrt(np.sum(sq_diff, axis=0))\n\tclosest_sport = np.argmax(dist, axis=1)

Incorrect; Attempts to broadcast sports_codes and performance for subtraction but incorrectly uses np.newaxis, leading to wrong shape manipulation. Also uses argmax instead of argmin, incorrectly identifying the closest sport.

5. Analyze

In a sports science study measuring reaction times and decision-making speeds of athletes, researchers have a large dataset of observations and predefined categories (codes). Considering the computational inefficiency of broadcasting with large datasets, how should the algorithm be modified to benefit the study without overloading the system's memory?

Calculate distances for each observation individually using a loop, avoiding large intermediate arrays.
Convert the dataset to a lower precision data type to reduce memory usage.
Use broadcasting to calculate all distances at once for efficiency.
Increase the system's RAM to handle large intermediate arrays generated by broadcasting.

1. Remember

2. Understand

3. Apply

4. Apply

5. Analyze

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