Relevance Factors: The Foundation of Search Quality

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import math
from collections import Counter
 
class TermFrequencyCalculators:
    """
    Various term frequency normalization schemes used in information retrieval.
    Each addresses different problems with raw term counts.
    """
    
    @staticmethod
    def raw_tf(term: str, document: list[str]) -> int:
        """
        Raw term frequency: simple count of term occurrences.
        Problem: Biased toward long documents.
        """
        return document.count(term)
    
    @staticmethod
    def log_normalized_tf(term: str, document: list[str]) -> float:
        """
        Log normalization: 1 + log(tf) if tf > 0, else 0.
        Intuition: Multiple occurrences matter, but with diminishing returns.
        The difference between 1 and 10 occurrences matters more than
        the difference between 100 and 1000.
        """
        tf = document.count(term)
        if tf == 0:
            return 0.0
        return 1.0 + math.log(tf)
    
    @staticmethod
    def double_normalization_tf(term: str, document: list[str], k: float = 0.5) -> float:
        """
        Double normalization (0.5): k + (1-k) * tf/max_tf
        Normalizes by the maximum term frequency in the document.
        Prevents any term from dominating too much.
        """
        tf = document.count(term)
        if tf == 0:
            return 0.0
        term_counts = Counter(document)
        max_tf = max(term_counts.values()) if term_counts else 1
        return k + (1 - k) * (tf / max_tf)
    
    @staticmethod
    def bm25_saturation_tf(term: str, document: list[str], k1: float = 1.2) -> float:
        """
        BM25-style saturation: tf / (tf + k1)
        Crucial insight: after enough occurrences, additional mentions
        add almost no extra relevance. This "saturates" the score.
        k1 controls how quickly saturation occurs.
        """
        tf = document.count(term)
        if tf == 0:
            return 0.0
        return tf / (tf + k1)
 
 
# Demonstration of saturation behavior
def demonstrate_saturation():
    """
    Shows how BM25 saturation prevents term frequency gaming.
    """
    print("TF	Raw	Log	BM25 Saturation")
    print("-" * 50)
    for tf in [1, 2, 5, 10, 20, 50, 100, 1000]:
        raw = tf
        log_norm = 1 + math.log(tf) if tf > 0 else 0
        bm25_sat = tf / (tf + 1.2)
        print(f"{tf}	{raw}	{log_norm:.2f}	{bm25_sat:.3f}")
 
# Output shows BM25 rapidly approaches 1.0, preventing gaming
# TF=1 → 0.45, TF=10 → 0.89, TF=100 → 0.99, TF=1000 → 0.999

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import math
from collections import Counter
from dataclasses import dataclass
 
@dataclass
class BM25Parameters:
    """
    BM25 tuning parameters with their interpretations.
    """
    k1: float = 1.2  # Term frequency saturation. Higher = more weight to TF.
    b: float = 0.75  # Length normalization. 0 = ignore length, 1 = full normalization.
 
class BM25:
    """
    Production-grade BM25 implementation with full documentation.
    
    The BM25 scoring function is:
    
    Score(Q, D) = Σ IDF(qi) × (tf(qi, D) × (k1 + 1)) / (tf(qi, D) + k1 × (1 - b + b × |D|/avgdl))
    
    Where:
    - Q is the query, D is the document
    - qi is the i-th query term
    - tf(qi, D) is the frequency of qi in D
    - |D| is the document length
    - avgdl is the average document length in the corpus
    - k1 and b are tuning parameters
    """
    
    def __init__(self, corpus: list[list[str]], params: BM25Parameters = BM25Parameters()):
        self.corpus = corpus
        self.params = params
        self.corpus_size = len(corpus)
        
        # Precompute document lengths and average
        self.doc_lengths = [len(doc) for doc in corpus]
        self.avg_doc_length = sum(self.doc_lengths) / self.corpus_size if self.corpus_size > 0 else 0
        
        # Precompute document frequencies for IDF
        self.doc_frequencies = {}
        for doc in corpus:
            unique_terms = set(doc)
            for term in unique_terms:
                self.doc_frequencies[term] = self.doc_frequencies.get(term, 0) + 1
        
        # Precompute term frequencies per document
        self.doc_term_frequencies = [Counter(doc) for doc in corpus]
    
    def idf(self, term: str) -> float:
        """
        Compute Inverse Document Frequency for a term.
        
        Uses the Robertson-Walker IDF variant:
        IDF = log((N - n + 0.5) / (n + 0.5) + 1)
        
        This formulation:
        - Avoids negative IDF for very common terms
        - Handles edge cases smoothly
        - Provides slightly better empirical results than classic IDF
        """
        n = self.doc_frequencies.get(term, 0)
        numerator = self.corpus_size - n + 0.5
        denominator = n + 0.5
        return math.log(numerator / denominator + 1)
    
    def score_document(self, query: list[str], doc_index: int) -> float:
        """
        Compute BM25 score for a single document against a query.
        
        The core insight: balance term frequency importance (k1) with
        document length normalization (b). Long documents naturally
        contain more terms but aren't necessarily more relevant.
        """
        score = 0.0
        doc_length = self.doc_lengths[doc_index]
        doc_tf = self.doc_term_frequencies[doc_index]
        
        for term in query:
            if term not in doc_tf:
                continue
            
            tf = doc_tf[term]
            idf = self.idf(term)
            
            # Length normalization factor
            # When b=1: fully normalize by document length
            # When b=0: ignore document length entirely
            length_norm = 1 - self.params.b + self.params.b * (doc_length / self.avg_doc_length)
            
            # BM25 term score with saturation
            numerator = tf * (self.params.k1 + 1)
            denominator = tf + self.params.k1 * length_norm
            
            score += idf * (numerator / denominator)
        
        return score
    
    def rank(self, query: list[str], top_k: int = 10) -> list[tuple[int, float]]:
        """
        Rank all documents by BM25 score and return top K.
        """
        scores = [(i, self.score_document(query, i)) for i in range(self.corpus_size)]
        scores.sort(key=lambda x: -x[1])
        return scores[:top_k]
 
 
# Example usage demonstrating BM25 behavior
def bm25_demonstration():
    corpus = [
        ["machine", "learning", "is", "a", "subset", "of", "artificial", "intelligence"],
        ["deep", "learning", "uses", "neural", "networks", "for", "machine", "learning"],
        ["machine", "learning", "machine", "learning", "machine", "learning"],  # TF gaming attempt
        ["artificial", "intelligence", "encompasses", "many", "techniques"],
        ["machine", "learning", "algorithms", "learn", "from", "data", "patterns"],
    ]
    
    bm25 = BM25(corpus)
    query = ["machine", "learning"]
    
    print("Query: 'machine learning'")
    print("
Ranking results:")
    for doc_idx, score in bm25.rank(query, top_k=5):
        print(f"  Doc {doc_idx}: {score:.4f} - {' '.join(corpus[doc_idx])}")
    
    # Note: Doc 2 (spam-like repetition) doesn't dominate because of:
    # 1. TF saturation (k1 parameter)
    # 2. The IDF component means common terms in query matter less
    # 3. Length normalization (b parameter)

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from dataclasses import dataclass
from typing import Optional
import math
 
@dataclass
class TermPositions:
    """Positions of a term in a document."""
    term: str
    positions: list[int]  # 1-indexed positions in document
 
class ProximityScorer:
    """
    Computes proximity-based relevance boosts.
    
    Intuition: Terms that appear close together are more likely to be
    semantically related. "Machine learning" appearing as a phrase is
    a stronger relevance signal than the words appearing paragraphs apart.
    """
    
    @staticmethod
    def min_span(term_positions: list[TermPositions]) -> Optional[int]:
        """
        Find the minimum span (window size) that contains all query terms.
        Returns None if any term is missing from the document.
        
        Uses a sliding window approach:
        1. Initialize pointers to the first position of each term
        2. Find the span of current positions
        3. Advance the leftmost pointer
        4. Track minimum span seen
        
        Time complexity: O(n * m) where n = total positions, m = number of terms
        """
        if not term_positions or any(len(tp.positions) == 0 for tp in term_positions):
            return None
        
        # Initialize pointers to first position of each term
        pointers = [0] * len(term_positions)
        min_span_size = float('inf')
        
        while True:
            # Get current positions for each term
            current_positions = [
                term_positions[i].positions[pointers[i]] 
                for i in range(len(term_positions))
            ]
            
            # Calculate span from min to max position
            min_pos = min(current_positions)
            max_pos = max(current_positions)
            span = max_pos - min_pos + 1
            
            min_span_size = min(min_span_size, span)
            
            # Perfect phrase match - can't do better
            if span == len(term_positions):
                break
            
            # Advance the leftmost pointer
            leftmost_idx = current_positions.index(min_pos)
            pointers[leftmost_idx] += 1
            
            # If we've exhausted any term's positions, we're done
            if pointers[leftmost_idx] >= len(term_positions[leftmost_idx].positions):
                break
        
        return min_span_size if min_span_size != float('inf') else None
    
    @staticmethod
    def proximity_boost(min_span: int, num_terms: int, 
                       max_boost: float = 2.0, 
                       decay_rate: float = 0.1) -> float:
        """
        Calculate proximity boost factor based on minimum span.
        
        Perfect phrase (span = num_terms) gets max boost.
        Boost decays as span increases.
        
        Formula: max_boost * exp(-decay_rate * (span - num_terms))
        """
        if min_span is None:
            return 1.0  # No boost if terms don't all appear
        
        # Excess span beyond perfect phrase
        excess = min_span - num_terms
        
        if excess <= 0:
            return max_boost  # Perfect phrase match
        
        # Exponential decay as terms spread apart
        return 1.0 + (max_boost - 1.0) * math.exp(-decay_rate * excess)
    
    @staticmethod
    def bigram_proximity_score(doc_positions: dict[str, list[int]], 
                              query_terms: list[str],
                              window: int = 5) -> float:
        """
        Score based on how often query term pairs appear within a window.
        
        This captures partial proximity - even if all terms aren't close,
        nearby term pairs still boost relevance.
        """
        if len(query_terms) < 2:
            return 0.0
        
        total_score = 0.0
        pairs_counted = 0
        
        for i in range(len(query_terms) - 1):
            term1, term2 = query_terms[i], query_terms[i + 1]
            
            if term1 not in doc_positions or term2 not in doc_positions:
                continue
            
            positions1 = doc_positions[term1]
            positions2 = doc_positions[term2]
            
            # Count pairs within window
            close_pairs = 0
            for p1 in positions1:
                for p2 in positions2:
                    if abs(p2 - p1) <= window:
                        close_pairs += 1
            
            if close_pairs > 0:
                total_score += math.log(1 + close_pairs)
                pairs_counted += 1
        
        return total_score / pairs_counted if pairs_counted > 0 else 0.0
 
 
# Demonstration
def proximity_demo():
    # Document: "Machine learning is transforming the field. Deep learning, a subset of machine learning, uses neural networks."
    positions = {
        "machine": [1, 12],
        "learning": [2, 7, 13],
        "deep": [6],
        "neural": [16],
        "networks": [17],
    }
    
    query = ["machine", "learning"]
    term_positions = [
        TermPositions("machine", positions["machine"]),
        TermPositions("learning", positions["learning"]),
    ]
    
    min_span = ProximityScorer.min_span(term_positions)
    boost = ProximityScorer.proximity_boost(min_span, len(query))
    
    print(f"Query: {query}")
    print(f"Minimum span: {min_span}")  # Should be 2 (positions 1-2 = "machine learning")
    print(f"Proximity boost: {boost:.2f}")  # Should be ~2.0 (perfect phrase match)

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import math
from datetime import datetime, timedelta
from enum import Enum
 
class DecayFunction(Enum):
    """Different decay models for freshness scoring."""
    LINEAR = "linear"
    EXPONENTIAL = "exponential"
    GAUSSIAN = "gaussian"
    SIGMOID = "sigmoid"
 
class FreshnessScorer:
    """
    Computes freshness scores with configurable decay functions.
    
    Key insight: The "right" decay function depends on query type.
    - News: Sharp exponential decay (yesterday's news is old news)
    - Products: Moderate decay (last year's reviews still useful)
    - Historical: Minimal decay (old content may be best source)
    """
    
    def __init__(self, 
                 current_time: datetime = None,
                 scale_days: float = 30.0,  # "half-life" in days
                 decay_type: DecayFunction = DecayFunction.EXPONENTIAL):
        self.current_time = current_time or datetime.now()
        self.scale_days = scale_days
        self.decay_type = decay_type
    
    def age_in_days(self, doc_date: datetime) -> float:
        """Calculate document age in fractional days."""
        delta = self.current_time - doc_date
        return delta.total_seconds() / 86400.0
    
    def linear_decay(self, age_days: float) -> float:
        """
        Linear decay from 1.0 to 0.0 over scale_days.
        Simple but unrealistic - content doesn't become worthless linearly.
        """
        decay = 1.0 - (age_days / self.scale_days)
        return max(0.0, decay)
    
    def exponential_decay(self, age_days: float) -> float:
        """
        Exponential decay: score = exp(-λ * age)
        Most common in practice. Half-life = scale_days.
        Never reaches zero - very old content still has some value.
        """
        lambda_param = math.log(2) / self.scale_days
        return math.exp(-lambda_param * age_days)
    
    def gaussian_decay(self, age_days: float, optimal_age: float = 0) -> float:
        """
        Gaussian decay: bell curve centered on optimal_age.
        Useful when "just published" isn't optimal (e.g., need time for reviews).
        """
        sigma = self.scale_days
        return math.exp(-0.5 * ((age_days - optimal_age) / sigma) ** 2)
    
    def sigmoid_decay(self, age_days: float) -> float:
        """
        Sigmoid decay: smooth S-curve transition from fresh to stale.
        Good for binary-ish freshness (relevant vs. outdated).
        """
        midpoint = self.scale_days
        steepness = 0.1  # Controls transition sharpness
        return 1.0 / (1.0 + math.exp(steepness * (age_days - midpoint)))
    
    def score(self, doc_date: datetime) -> float:
        """Compute freshness score using configured decay function."""
        age = self.age_in_days(doc_date)
        
        if self.decay_type == DecayFunction.LINEAR:
            return self.linear_decay(age)
        elif self.decay_type == DecayFunction.EXPONENTIAL:
            return self.exponential_decay(age)
        elif self.decay_type == DecayFunction.GAUSSIAN:
            return self.gaussian_decay(age)
        elif self.decay_type == DecayFunction.SIGMOID:
            return self.sigmoid_decay(age)
        else:
            return self.exponential_decay(age)
 
 
class QuerySpecificFreshness:
    """
    Adjusts freshness importance based on query classification.
    
    Not all queries need fresh content equally.
    """
    
    # Query freshness sensitivity categories
    QUERY_FRESHNESS_WEIGHTS = {
        "breaking_news": 0.95,      # "earthquake today"
        "recent_events": 0.8,       # "election results 2024"
        "trending_topics": 0.7,     # "viral tiktok"
        "product_reviews": 0.5,     # "iphone 15 review"
        "how_to": 0.3,              # "how to tie a tie"
        "factual": 0.2,             # "speed of light"
        "historical": 0.05,         # "world war 2 causes"
    }
    
    @classmethod
    def weighted_freshness_boost(cls, 
                                 freshness_score: float,
                                 query_type: str,
                                 base_score: float) -> float:
        """
        Apply freshness boost weighted by query type.
        
        Final score = base_score * (1 + weight * freshness_adjustment)
        """
        weight = cls.QUERY_FRESHNESS_WEIGHTS.get(query_type, 0.3)
        freshness_adjustment = freshness_score - 0.5  # Center around neutral
        
        return base_score * (1.0 + weight * freshness_adjustment)
 
 
# Example: Compare decay functions
def compare_decay_functions():
    now = datetime.now()
    ages = [0, 1, 7, 30, 90, 365]  # Days
    
    print("Age (days) | Linear | Exponential | Gaussian | Sigmoid")
    print("-" * 60)
    
    for age in ages:
        doc_date = now - timedelta(days=age)
        linear = FreshnessScorer(now, 90, DecayFunction.LINEAR).score(doc_date)
        exp = FreshnessScorer(now, 30, DecayFunction.EXPONENTIAL).score(doc_date)
        gauss = FreshnessScorer(now, 30, DecayFunction.GAUSSIAN).score(doc_date)
        sigmoid = FreshnessScorer(now, 60, DecayFunction.SIGMOID).score(doc_date)
        
        print(f"{age:10d} | {linear:6.3f} | {exp:11.3f} | {gauss:8.3f} | {sigmoid:7.3f}")

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import numpy as np
from dataclasses import dataclass
from typing import Dict, List, Set
 
@dataclass
class AuthorityMetrics:
    """Collection of authority signals for a document/domain."""
    pagerank: float = 0.0
    domain_authority: float = 0.0      # Aggregate domain-level signal
    inbound_link_count: int = 0        # Raw backlink count
    unique_referring_domains: int = 0  # More resistant to manipulation
    trust_rank: float = 0.0            # Distance from seed trusted sites
    spam_score: float = 0.0            # Probability of being spam
 
class PageRankCalculator:
    """
    Classic PageRank implementation with damping factor.
    
    The intuition: A page is important if important pages link to it.
    This circular definition is resolved iteratively.
    
    Mathematical formulation:
    PR(A) = (1-d)/N + d × Σ(PR(Ti)/C(Ti))
    
    Where:
    - d = damping factor (probability of following a link vs. random jump)
    - N = total pages
    - Ti = pages linking to A
    - C(Ti) = number of outbound links from Ti
    """
    
    def __init__(self, damping_factor: float = 0.85, max_iterations: int = 100, 
                 convergence_threshold: float = 1e-6):
        self.damping = damping_factor
        self.max_iter = max_iterations
        self.threshold = convergence_threshold
    
    def calculate(self, adjacency_list: Dict[str, List[str]]) -> Dict[str, float]:
        """
        Calculate PageRank for all nodes in the graph.
        
        Args:
            adjacency_list: {page: [pages it links to]}
        
        Returns:
            {page: pagerank_score}
        """
        # Build the set of all nodes
        all_nodes = set(adjacency_list.keys())
        for targets in adjacency_list.values():
            all_nodes.update(targets)
        
        n = len(all_nodes)
        nodes = list(all_nodes)
        node_to_idx = {node: i for i, node in enumerate(nodes)}
        
        # Initialize uniform PageRank
        pagerank = np.ones(n) / n
        
        # Build transition matrix
        # M[i][j] = probability of going from j to i
        outbound_counts = {node: len(adjacency_list.get(node, [])) for node in nodes}
        
        for iteration in range(self.max_iter):
            new_pagerank = np.ones(n) * (1 - self.damping) / n
            
            for source, targets in adjacency_list.items():
                if not targets:
                    continue  # Handle dangling nodes
                
                source_idx = node_to_idx[source]
                contribution = self.damping * pagerank[source_idx] / len(targets)
                
                for target in targets:
                    if target in node_to_idx:
                        target_idx = node_to_idx[target]
                        new_pagerank[target_idx] += contribution
            
            # Handle dangling nodes (pages with no outlinks)
            dangling_sum = sum(
                pagerank[node_to_idx[node]] 
                for node in nodes 
                if node not in adjacency_list or not adjacency_list[node]
            )
            new_pagerank += self.damping * dangling_sum / n
            
            # Check convergence
            diff = np.sum(np.abs(new_pagerank - pagerank))
            pagerank = new_pagerank
            
            if diff < self.threshold:
                break
        
        return {nodes[i]: pagerank[i] for i in range(n)}
 
 
class TrustRank:
    """
    TrustRank: PageRank seeded from known trustworthy sites.
    
    Key insight: Spam pages can accumulate PageRank through link farms,
    but it's hard to get links from genuinely trusted sources.
    
    TrustRank propagates trust from a hand-picked seed set of
    authoritative sites. Pages far from the seed set in the link graph
    have lower trust scores.
    """
    
    def __init__(self, trusted_seeds: Set[str], decay_factor: float = 0.85):
        self.seeds = trusted_seeds
        self.decay = decay_factor
    
    def calculate(self, adjacency_list: Dict[str, List[str]], 
                  iterations: int = 10) -> Dict[str, float]:
        """
        Calculate TrustRank propagating from seed set.
        """
        all_nodes = set(adjacency_list.keys())
        for targets in adjacency_list.values():
            all_nodes.update(targets)
        
        # Initialize: seeds get trust 1.0, others get 0
        trust = {node: 1.0 if node in self.seeds else 0.0 for node in all_nodes}
        
        # Build reverse adjacency (who links TO each page)
        reverse_adj: Dict[str, List[str]] = {node: [] for node in all_nodes}
        for source, targets in adjacency_list.items():
            for target in targets:
                if target in reverse_adj:
                    reverse_adj[target].append(source)
        
        for _ in range(iterations):
            new_trust = {}
            for node in all_nodes:
                if node in self.seeds:
                    new_trust[node] = 1.0  # Seeds maintain trust
                else:
                    # Inherit decayed trust from linking pages
                    inherited = sum(
                        self.decay * trust[source] / len(adjacency_list.get(source, [source]))
                        for source in reverse_adj[node]
                        if source in adjacency_list
                    )
                    new_trust[node] = min(1.0, inherited)
            
            trust = new_trust
        
        return trust
 
 
def domain_authority_aggregation():
    """
    Domain Authority: Aggregate page-level signals to domain level.
    
    Why domain-level? 
    - New pages on trusted domains deserve initial trust
    - Prevents gaming by creating many low-quality pages
    - More stable signal (pages come and go, domains persist)
    """
    # Simplified example - real systems use ML models with many features
    domain_signals = {
        "wikipedia.org": {
            "age_years": 24,
            "unique_referring_domains": 15_000_000,
            "manual_authority_score": 98,  # Human-curated
            "avg_page_quality": 0.92,
        },
        "randomspamsite.xyz": {
            "age_years": 0.5,
            "unique_referring_domains": 50,
            "manual_authority_score": None,
            "avg_page_quality": 0.15,
        },
    }
    
    # Domain authority model (simplified)
    for domain, signals in domain_signals.items():
        age_factor = min(1.0, signals["age_years"] / 10)
        link_factor = min(1.0, np.log10(signals["unique_referring_domains"] + 1) / 7)
        quality_factor = signals["avg_page_quality"]
        
        authority = 0.3 * age_factor + 0.4 * link_factor + 0.3 * quality_factor
        print(f"{domain}: Authority = {authority:.3f}")