Added new question 335

build/10.json

@@ -21,7 +21,7 @@
   "description": "Write a Python function to calculate the covariance matrix for a given set of vectors. The function should take a list of lists, where each inner list represents a feature with its observations, and return a covariance matrix as a list of lists. Additionally, provide test cases to verify the correctness of your implementation.",
   "learn_section": "## Understanding Covariance Matrix\n\nThe covariance matrix is a fundamental concept in statistics and machine learning, used to understand the relationship between multiple variables (features) in a dataset. It quantifies the degree to which two variables change together.\n\n### Key Concepts\n\n- **Covariance**: Measures the directional relationship between two random variables. A positive covariance indicates that the variables increase together, while a negative covariance indicates that one variable increases as the other decreases.\n- **Covariance Matrix**: For a dataset with $n$ features, the covariance matrix is an $n \\times n$ matrix where each element $(i, j)$ represents the covariance between the $i^{th}$ and $j^{th}$ features.\n\n### Covariance Formula\n\nThe covariance between two variables $X$ and $Y$ is calculated as:\n\n$$\n\\text{cov}(X, Y) = \\frac{\\sum_{k=1}^{m} (X_k - \\bar{X})(Y_k - \\bar{Y})}{m - 1}\n$$\n\nWhere:\n\n- $X_k$ and $Y_k$ are the individual observations of variables $X$ and $Y$.\n- $\\bar{X}$ and $\\bar{Y}$ are the means of $X$ and $Y$.\n- $m$ is the number of observations.\n\n### Constructing the Covariance Matrix\n\nGiven a dataset with $n$ features, the covariance matrix is constructed as follows:\n\n1. **Calculate the Mean**: Compute the mean of each feature.\n2. **Compute Covariance**: For each pair of features, calculate the covariance using the formula above.\n3. **Populate the Matrix**: Place the computed covariance values in the corresponding positions in the matrix. The diagonal elements represent the variance of each feature.\n\n$$\n\\text{Covariance Matrix} =\n\\begin{bmatrix}\n\\text{cov}(X_1, X_1) & \\text{cov}(X_1, X_2) & \\cdots & \\text{cov}(X_1, X_n) \\\\\n\\text{cov}(X_2, X_1) & \\text{cov}(X_2, X_2) & \\cdots & \\text{cov}(X_2, X_n) \\\\\n\\vdots & \\vdots & \\ddots & \\vdots \\\\\n\\text{cov}(X_n, X_1) & \\text{cov}(X_n, X_2) & \\cdots & \\text{cov}(X_n, X_n) \\\\\n\\end{bmatrix}\n$$\n\n### Example Calculation\n\nConsider the following dataset with two features:\n\n$$\n\\begin{align*}\n\\text{Feature 1} &: [1, 2, 3] \\\\\n\\text{Feature 2} &: [4, 5, 6]\n\\end{align*}\n$$\n\n1. **Calculate Means**:\n   $$\n   \\bar{X}_1 = \\frac{1 + 2 + 3}{3} = 2.0 \\\\\n   \\bar{X}_2 = \\frac{4 + 5 + 6}{3} = 5.0\n   $$\n\n2. **Compute Covariances**:\n   $$\n   \\text{cov}(X_1, X_1) = \\frac{(1-2)^2 + (2-2)^2 + (3-2)^2}{3-1} = 1.0 \\\\\n   \\text{cov}(X_1, X_2) = \\frac{(1-2)(4-5) + (2-2)(5-5) + (3-2)(6-5)}{3-1} = 1.0 \\\\\n   \\text{cov}(X_2, X_2) = \\frac{(4-5)^2 + (5-5)^2 + (6-5)^2}{3-1} = 1.0\n   $$\n\n3. **Covariance Matrix**:\n   $$\n   \\begin{bmatrix}\n   1.0 & 1.0 \\\\\n   1.0 & 1.0 \n   \\end{bmatrix}\n   $$\n\n### Applications\n\nCovariance matrices are widely used in various fields, including:\n\n- **Principal Component Analysis (PCA)**: Reducing the dimensionality of datasets while preserving variance.\n- **Portfolio Optimization**: Understanding the variance and covariance between different financial assets.\n- **Multivariate Statistics**: Analyzing the relationships between multiple variables simultaneously.\n\nUnderstanding the covariance matrix is crucial for interpreting the relationships in multivariate data and for performing advanced statistical analyses.",
   "starter_code": "def calculate_covariance_matrix(vectors: list[list[float]]) -> list[list[float]]:\n\t# Your code here\n\treturn []",
-  "solution": "import numpy as np\n\ndef calculate_covariance_matrix(vectors: list[list[float]]) -> list[list[float]]:\n    n_features = len(vectors)\n    n_observations = len(vectors[0])\n    covariance_matrix = [[0 for _ in range(n_features)] for _ in range(n_features)]\n\n    means = [sum(feature) / n_observations for feature in vectors]\n\n    for i in range(n_features):\n        for j in range(i, n_features):\n            covariance = sum((vectors[i][k] - means[i]) * (vectors[j][k] - means[j]) for k in range(n_observations)) / (n_observations - 1)\n            covariance_matrix[i][j] = covariance_matrix[j][i] = covariance\n\n    return covariance_matrix",
+  "solution": "import numpy as np\n\ndef calculate_covariance_matrix(vectors: list[list[float]]) -> list[list[float]]:\n    n_observations = len(vectors)\n    n_features = len(vectors[0])\n    covariance_matrix = [[0 for _ in range(n_observations)] for _ in range(n_observations)]\n\n    means = [sum(feature) / n_features for feature in vectors]\n\n    for i in range(n_observations):\n        for j in range(i, n_observations):\n            covariance = sum((vectors[i][k] - means[i]) * (vectors[j][k] - means[j]) for k in range(n_features)) / (n_features - 1)\n            covariance_matrix[i][j] = covariance_matrix[j][i] = covariance\n\n    return covariance_matrix",
   "example": {
     "input": "[[1, 2, 3], [4, 5, 6]]",
     "output": "[[1.0, 1.0], [1.0, 1.0]]",