Goodness of fit evaluation in SciPy - Time & Space Complexity

Time Complexity: Goodness of fit evaluation

O(n)

Understanding Time Complexity

We want to know how the time needed to check how well a model fits data changes as the data size grows.

How does the cost of evaluating goodness of fit grow when we have more data points?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

import numpy as np
from scipy.stats import chisquare

observed = np.array([20, 30, 50])
expected = np.array([25, 25, 50])

chi2_stat, p_value = chisquare(f_obs=observed, f_exp=expected)
print(f"Chi-square statistic: {chi2_stat}, p-value: {p_value}")

This code calculates the chi-square test statistic and p-value to see how well observed data matches expected data.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: Element-wise subtraction, division, and squaring for each data point.
• How many times: Once for each data point in the observed and expected arrays.

How Execution Grows With Input

Each data point requires a fixed number of calculations. More data points mean more calculations.

Input Size (n)	Approx. Operations
10	About 10 sets of calculations
100	About 100 sets of calculations
1000	About 1000 sets of calculations

Pattern observation: The number of operations grows directly with the number of data points.

Final Time Complexity

Time Complexity: O(n)

This means the time to evaluate goodness of fit grows linearly with the number of data points.

Common Mistake

[X] Wrong: "The test runs in constant time no matter how much data there is."

[OK] Correct: Each data point must be checked, so more data means more work and more time.

Interview Connect

Understanding how time grows with data size helps you explain your code choices clearly and shows you think about efficiency.

Self-Check

"What if we used a more complex goodness of fit test that compares pairs of data points? How would the time complexity change?"

Practice

1. What does the chi-square goodness of fit test in scipy.stats.chisquare primarily evaluate?

easy

A. The mean difference between two samples

B. How well observed data matches expected frequencies

C. The correlation between two variables

D. The variance within a single dataset

Solution

1. Step 1: Understand the purpose of chi-square test

   The chi-square goodness of fit test compares observed data frequencies to expected frequencies to check if they match.

2. Step 2: Identify what scipy.stats.chisquare does

   This function calculates the chi-square statistic and p-value to evaluate the fit between observed and expected counts.

3. Final Answer:

   How well observed data matches expected frequencies -> Option B

4. Quick Check:

   Goodness of fit = observed vs expected match [OK]

Hint: Chi-square tests observed vs expected frequencies [OK]

Common Mistakes:

• Confusing goodness of fit with correlation
• Thinking it measures mean differences
• Mixing variance analysis with goodness of fit

2. Which of the following is the correct way to import the chi-square goodness of fit test function from scipy?

easy

A. import scipy.chisquare

B. from scipy import chisquare

C. from scipy.stats import chisquare

D. import scipy.stats.chisquare as cs

Solution

1. Step 1: Recall the module structure of scipy

   The chi-square test function is inside the stats submodule of scipy.

2. Step 2: Identify correct import syntax

   The correct import is from scipy.stats import chisquare to directly access the function.

3. Final Answer:

   from scipy.stats import chisquare -> Option C

4. Quick Check:

   Correct import = from scipy.stats import chisquare [OK]

Hint: Import from scipy.stats for statistical tests [OK]

Common Mistakes:

• Trying to import chisquare directly from scipy
• Using incorrect module paths
• Using alias without import

3. What will be the output of the following code?

from scipy.stats import chisquare
observed = [20, 30, 50]
expected = [25, 25, 50]
result = chisquare(f_obs=observed, f_exp=expected)
print(round(result.statistic, 2), round(result.pvalue, 3))

medium

A. 2.0 0.368

B. 3.0 0.223

C. 0.5 0.778

D. 1.0 0.607

Solution

1. Step 1: Calculate chi-square statistic manually

   Chi-square = sum((observed - expected)^2 / expected) = ((20-25)^2/25) + ((30-25)^2/25) + ((50-50)^2/50) = (25/25)+(25/25)+0 = 1+1+0 = 2.0

2. Step 2: Interpret p-value from scipy output

   Using scipy.stats.chisquare with these values gives a p-value around 0.368, indicating moderate fit.

3. Final Answer:

   2.0 0.368 -> Option A

4. Quick Check:

   Chi-square stat = 2.0, p-value ≈ 0.368 [OK]

Hint: Calculate chi-square stat then check p-value [OK]

Common Mistakes:

• Forgetting to square differences
• Dividing by wrong expected values
• Mixing up statistic and p-value

4. Identify the error in this code snippet for performing a chi-square goodness of fit test:

from scipy.stats import chisquare
observed = [15, 25, 35]
expected = [20, 20]
result = chisquare(f_obs=observed, f_exp=expected)
print(result)

medium

A. Observed and expected arrays have different lengths

B. chisquare function is not imported correctly

C. Expected frequencies must be integers

D. Missing p-value extraction from result

Solution

1. Step 1: Check input array lengths

   The observed array has 3 elements, but expected has only 2 elements, which is invalid for chi-square test.

2. Step 2: Understand scipy requirement

   Both observed and expected arrays must be the same length to compare frequencies correctly.

3. Final Answer:

   Observed and expected arrays have different lengths -> Option A

4. Quick Check:

   Array length mismatch causes error [OK]

Hint: Observed and expected must be same length [OK]

Common Mistakes:

• Ignoring length mismatch
• Assuming expected must be integers
• Thinking import or print is the error

5. You have observed counts of [40, 35, 25] for three categories. You expect them to be equally likely. Using scipy.stats.chisquare, what is the p-value indicating if the observed data fits the equal distribution? (Hint: expected counts are equal for all categories.)

hard

A. 0.789

B. 0.223

C. 0.456

D. 0.174

Solution

1. Step 1: Calculate expected counts for equal distribution

   Total counts = 40+35+25 = 100. Expected counts = [100/3, 100/3, 100/3] ≈ [33.33, 33.33, 33.33].

2. Step 2: Perform chi-square test with scipy

   Using chisquare(f_obs=[40,35,25], f_exp=[33.33,33.33,33.33]) gives a chi-square statistic ≈ 3.5 and p-value ≈ 0.174.

3. Final Answer:

   0.174 -> Option D

4. Quick Check:

   Unequal counts vs equal expected gives p-value ≈ 0.174 [OK]

Hint: Equal expected counts = total/number categories [OK]

Common Mistakes:

• Using observed counts as expected
• Not dividing total counts equally
• Misinterpreting p-value significance