Time Series Forecasting: Predicting Stock Prices and Weather

Time Series Forecasting: Predicting Stock Prices and Weather

Time series data (ordered by time) appears everywhere: stock prices, weather, sensor readings, website traffic. Forecasting future values requires handling temporal dependencies and patterns. This chapter covers statistical methods (ARIMA), machine learning approaches, and deep learning (LSTM) for time series prediction.

Time Series Components: Trend, Seasonality, and Noise

Many time series decompose into: trend (long-term direction), seasonality (repeating patterns at fixed periods), and noise (random fluctuations). Identifying these components helps with understanding and forecasting. Classical decomposition methods separate the series multiplicatively (Y = T × S × N) or additively (Y = T + S + N).

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from statsmodels.tsa.seasonal import seasonal_decompose

# Generate synthetic time series with trend, seasonality, and noise
np.random.seed(42)
t = np.arange(0, 200)
trend = 0.5 * t
seasonality = 10 * np.sin(2 * np.pi * t / 24)
noise = np.random.normal(0, 2, len(t))
y = trend + seasonality + noise

# Create pandas series with datetime index
dates = pd.date_range('2023-01-01', periods=len(y), freq='H')
ts = pd.Series(y, index=dates)

# Decompose
decomposition = seasonal_decompose(ts, model='additive', period=24)

# Plot
fig, axes = plt.subplots(4, 1, figsize=(12, 8))

axes[0].plot(ts, linewidth=1)
axes[0].set_ylabel('Original')
axes[0].set_title('Time Series Decomposition')
axes[0].grid(True, alpha=0.3)

axes[1].plot(decomposition.trend, linewidth=1, color='orange')
axes[1].set_ylabel('Trend')
axes[1].grid(True, alpha=0.3)

axes[2].plot(decomposition.seasonal, linewidth=1, color='green')
axes[2].set_ylabel('Seasonal')
axes[2].grid(True, alpha=0.3)

axes[3].plot(decomposition.resid, linewidth=0.5, color='red')
axes[3].set_ylabel('Residual')
axes[3].set_xlabel('Date')
axes[3].grid(True, alpha=0.3)

plt.tight_layout()
plt.show()

print("Decomposition components identified:")
print(f"Trend: Linear growth component captured")
print(f"Seasonality: 24-hour cycle captured")
print(f"Residuals: Random noise after removing trend and seasonality")

ARIMA: Statistical Approach to Time Series

ARIMA (AutoRegressive Integrated Moving Average) models the series as a function of its past values (AR), differencing to achieve stationarity (I), and past forecast errors (MA). An ARIMA(p, d, q) model has p autoregressive terms, d differencing steps, and q moving average terms. It's a classical statistical method that works well for stationary or near-stationary series.

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from statsmodels.tsa.arima.model import ARIMA
from statsmodels.graphics.tsaplots import plot_acf, plot_pacf

# Generate AR(1) process: y_t = 0.7 * y_{t-1} + noise
np.random.seed(42)
ar_coeff = 0.7
y = [np.random.normal(0, 1)]
for _ in range(200):
    y.append(ar_coeff * y[-1] + np.random.normal(0, 1))

# Create series
ts = pd.Series(y)

# Split into train/test
train_size = 150
y_train, y_test = ts[:train_size], ts[train_size:]

# Fit ARIMA(1, 0, 0) - should recover AR coefficient
model = ARIMA(y_train, order=(1, 0, 0))
result = model.fit()
print(result.summary())

# Forecast
forecast = result.get_forecast(steps=len(y_test))
forecast_mean = forecast.predicted_mean
forecast_ci = forecast.conf_int()

# Plot
fig, axes = plt.subplots(2, 1, figsize=(12, 8))

# Time series with forecast
axes[0].plot(y_train.index, y_train, label='Training Data', linewidth=1)
axes[0].plot(y_test.index, y_test, label='Test Data', linewidth=1)
axes[0].plot(forecast_mean.index, forecast_mean, 'r-', label='Forecast', linewidth=2)
axes[0].fill_between(forecast_ci.index,
                     forecast_ci.iloc[:, 0], forecast_ci.iloc[:, 1],
                     color='red', alpha=0.2, label='95% CI')
axes[0].set_xlabel('Time')
axes[0].set_ylabel('Value')
axes[0].set_title('ARIMA Forecast')
axes[0].legend()
axes[0].grid(True, alpha=0.3)

# Residuals
residuals = result.resid
axes[1].plot(residuals, linewidth=0.5)
axes[1].axhline(0, color='red', linestyle='--')
axes[1].set_xlabel('Time')
axes[1].set_ylabel('Residual')
axes[1].set_title('Model Residuals (should be white noise)')
axes[1].grid(True, alpha=0.3)

plt.tight_layout()
plt.show()

# Evaluate
from sklearn.metrics import mean_squared_error, mean_absolute_error
mse = mean_squared_error(y_test, forecast_mean)
mae = mean_absolute_error(y_test, forecast_mean)
print(f"
Forecast MSE: {mse:.6f}")
print(f"Forecast MAE: {mae:.6f}")

LSTM Networks: Deep Learning for Sequences

Long Short-Term Memory (LSTM) networks are recurrent neural networks designed to handle long-term dependencies. They use gates (input, forget, output) to control information flow. LSTMs have shown strong performance on time series tasks, especially complex nonlinear patterns. Stacked LSTMs and attention mechanisms further improve performance.

import numpy as np
import matplotlib.pyplot as plt
import tensorflow as tf
from tensorflow.keras.models import Sequential
from tensorflow.keras.layers import LSTM, Dense
from tensorflow.keras.optimizers import Adam
from sklearn.preprocessing import MinMaxScaler

# Generate synthetic time series
np.random.seed(42)
t = np.arange(0, 200)
y = 50 + 10 * np.sin(2*np.pi*t/24) + 0.5*t + np.random.normal(0, 1, len(t))

# Normalize
scaler = MinMaxScaler(feature_range=(0, 1))
y_scaled = scaler.fit_transform(y.reshape(-1, 1))

# Create sequences for LSTM
def create_sequences(data, seq_length):
    X, y = [], []
    for i in range(len(data) - seq_length):
        X.append(data[i:i+seq_length])
        y.append(data[i+seq_length])
    return np.array(X), np.array(y)

seq_length = 24
X, y_seq = create_sequences(y_scaled, seq_length)

# Split
train_size = 150
X_train, X_test = X[:train_size], X[train_size:]
y_train, y_test = y_seq[:train_size], y_seq[train_size:]

# Build LSTM model
model = Sequential([
    LSTM(50, activation='relu', input_shape=(seq_length, 1)),
    Dense(25, activation='relu'),
    Dense(1)
])

model.compile(optimizer=Adam(learning_rate=0.01), loss='mse')
model.fit(X_train, y_train, epochs=50, batch_size=16, verbose=0)

# Predict
y_pred_train = model.predict(X_train, verbose=0).flatten()
y_pred_test = model.predict(X_test, verbose=0).flatten()

# Inverse scale
y_pred_train = scaler.inverse_transform(y_pred_train.reshape(-1, 1))
y_pred_test = scaler.inverse_transform(y_pred_test.reshape(-1, 1))
y_test_actual = scaler.inverse_transform(y_test.reshape(-1, 1))

# Plot
fig, ax = plt.subplots(figsize=(12, 5))
ax.plot(t[:train_size+seq_length], y[:train_size+seq_length], label='Training Data', linewidth=1)
ax.plot(t[train_size+seq_length:train_size+seq_length+len(y_pred_test)],
       y_pred_test, 'r-', label='LSTM Prediction', linewidth=2)
ax.plot(t[train_size+seq_length:train_size+seq_length+len(y_test_actual)],
       y_test_actual, 'g--', label='Actual Test Data', linewidth=1)
ax.set_xlabel('Time')
ax.set_ylabel('Value')
ax.set_title('LSTM Time Series Forecast')
ax.legend()
ax.grid(True, alpha=0.3)
plt.show()

from sklearn.metrics import mean_squared_error
mse = mean_squared_error(y_test_actual, y_pred_test)
print(f"LSTM Test MSE: {mse:.6f}")

Feature Engineering for Time Series

Time series forecasting benefits from engineered features: lagged values (previous time steps), rolling statistics (moving average, standard deviation), trend indicators, and domain-specific features (hour of day, day of week, holidays). These features help models capture temporal patterns and external factors.

import numpy as np
import pandas as pd

# Create time series
dates = pd.date_range('2023-01-01', periods=100, freq='D')
values = 50 + np.cumsum(np.random.randn(100) * 2)
ts = pd.Series(values, index=dates)

# Feature engineering
df = pd.DataFrame({'value': ts})

# Lagged features
for lag in [1, 7, 30]:
    df[f'lag_{lag}'] = df['value'].shift(lag)

# Rolling statistics
df['rolling_mean_7'] = df['value'].rolling(window=7).mean()
df['rolling_std_7'] = df['value'].rolling(window=7).std()

# Temporal features
df['day_of_week'] = df.index.dayofweek
df['day_of_year'] = df.index.dayofyear
df['month'] = df.index.month

# Differencing (helps with non-stationary series)
df['diff_1'] = df['value'].diff(1)
df['diff_7'] = df['value'].diff(7)

print("Engineered features:")
print(df.head(35).to_string())
print(f"
Shape: {df.shape}")
print(f"Missing values: {df.isna().sum().sum()}")

# Drop NaN rows introduced by feature engineering
df_clean = df.dropna()
print(f"Shape after dropping NaN: {df_clean.shape}")

Key Takeaways

• Time series decomposition reveals trend, seasonality, and noise components.
• ARIMA is a classical statistical method suitable for stationary or differenced series.
• LSTM networks excel at capturing complex temporal dependencies in sequential data.
• Feature engineering (lags, rolling statistics, temporal features) is crucial for time series ML.
• Ensemble methods combining multiple forecasting approaches often outperform individual models.
• Cross-validation for time series requires special care: use time-based splits, not random splits.

Deep Dive: Time Series Forecasting: Predicting Stock Prices and Weather

At this level, we stop simplifying and start engaging with the real complexity of Time Series Forecasting: Predicting Stock Prices and Weather. In production systems at companies like Flipkart, Razorpay, or Swiggy — all Indian companies processing millions of transactions daily — the concepts in this chapter are not academic exercises. They are engineering decisions that affect system reliability, user experience, and ultimately, business success.

The Indian tech ecosystem is at an inflection point. With initiatives like Digital India and India Stack (Aadhaar, UPI, DigiLocker), the country has built technology infrastructure that is genuinely world-leading. Understanding the technical foundations behind these systems — which is what this chapter covers — positions you to contribute to the next generation of Indian technology innovation.

Whether you are preparing for JEE, GATE, campus placements, or building your own products, the depth of understanding we develop here will serve you well. Let us go beyond surface-level knowledge.

Transformer Architecture: The Engine Behind GPT and Modern AI

The Transformer architecture, introduced in the landmark 2017 paper "Attention Is All You Need," revolutionised NLP and eventually all of deep learning. Here is the core mechanism:

# Self-Attention Mechanism (simplified)
import numpy as np

def self_attention(Q, K, V, d_k):
    """
    Q (Query): What am I looking for?
    K (Key):   What do I contain?
    V (Value): What do I actually provide?
    d_k:       Dimension of keys (for scaling)
    """
    # Step 1: Compute attention scores
    scores = np.matmul(Q, K.T) / np.sqrt(d_k)

    # Step 2: Softmax to get probabilities
    attention_weights = softmax(scores)

    # Step 3: Weighted sum of values
    output = np.matmul(attention_weights, V)
    return output

# Multi-Head Attention: Run multiple attention heads in parallel
# Each head learns different relationships:
# Head 1: syntactic relationships (subject-verb agreement)
# Head 2: semantic relationships (word meanings)
# Head 3: positional relationships (word order)
# Head 4: coreference (pronoun → noun it refers to)

The key insight of self-attention is that every token can attend to every other token simultaneously (unlike RNNs which process sequentially). This parallelism enables efficient GPU training. The computational complexity is O(n²·d) where n is sequence length and d is dimension, which is why context windows are a major engineering challenge.

State-of-the-art developments include: sparse attention (reducing O(n²) to O(n·√n)), mixture of experts (MoE — activating only a subset of parameters per input), retrieval-augmented generation (RAG — grounding responses in external documents), and constitutional AI (alignment through principles rather than RLHF alone). Indian researchers at institutions like IIT Bombay, IISc Bangalore, and Microsoft Research India are actively contributing to these frontiers.

India's Scale Challenges: Engineering for 1.4 Billion

Building technology for India presents unique engineering challenges that make it one of the most interesting markets in the world. UPI handles 10 billion transactions per month — more than all credit card transactions in the US combined. Aadhaar authenticates 100 million identities daily. Jio's network serves 400 million subscribers across 22 telecom circles. Hotstar streamed IPL to 50 million concurrent viewers — a world record. Each of these systems must handle India's diversity: 22 official languages, 28 states with different regulations, massive urban-rural connectivity gaps, and price-sensitive users expecting everything to work on ₹7,000 smartphones over patchy 4G connections. This is why Indian engineers are globally respected — if you can build systems that work in India, they will work anywhere.

Advanced Algorithms: Dynamic Programming and Graph Theory

Dynamic Programming (DP) solves complex problems by breaking them into overlapping subproblems. This is a favourite in competitive programming and interviews:

# Longest Common Subsequence — classic DP problem
# Used in: diff tools, DNA sequence alignment, version control

def lcs(s1, s2):
    m, n = len(s1), len(s2)
    dp = [[0] * (n + 1) for _ in range(m + 1)]

    for i in range(1, m + 1):
        for j in range(1, n + 1):
            if s1[i-1] == s2[j-1]:
                dp[i][j] = dp[i-1][j-1] + 1
            else:
                dp[i][j] = max(dp[i-1][j], dp[i][j-1])

    return dp[m][n]

# Dijkstra's Shortest Path — used by Google Maps!
import heapq

def dijkstra(graph, start):
    dist = {node: float('inf') for node in graph}
    dist[start] = 0
    pq = [(0, start)]  # (distance, node)

    while pq:
        d, u = heapq.heappop(pq)
        if d > dist[u]:
            continue
        for v, weight in graph[u]:
            if dist[u] + weight < dist[v]:
                dist[v] = dist[u] + weight
                heapq.heappush(pq, (dist[v], v))

    return dist

# Real use: Google Maps finding shortest route from
# Connaught Place to India Gate, considering traffic weights

Dijkstra's algorithm is how mapping applications find optimal routes. When you ask Google Maps to navigate from Mumbai to Pune, it models the road network as a weighted graph (intersections are nodes, roads are edges, travel time is weight) and runs a variant of Dijkstra's algorithm. Indian highways, city roads, and even railway networks can all be modelled this way. IRCTC's route optimisation for trains across 13,000+ stations uses graph algorithms at its core.

Research Frontiers and Open Problems in Time Series Forecasting: Predicting Stock Prices and Weather

Beyond production engineering, time series forecasting: predicting stock prices and weather connects to active research frontiers where fundamental questions remain open. These are problems where your generation of computer scientists will make breakthroughs.

Quantum computing threatens to upend many of our assumptions. Shor's algorithm can factor large numbers efficiently on a quantum computer, which would break RSA encryption — the foundation of internet security. Post-quantum cryptography is an active research area, with NIST standardising new algorithms (CRYSTALS-Kyber, CRYSTALS-Dilithium) that resist quantum attacks. Indian researchers at IISER, IISc, and TIFR are contributing to both quantum computing hardware and post-quantum cryptographic algorithms.

AI safety and alignment is another frontier with direct connections to time series forecasting: predicting stock prices and weather. As AI systems become more capable, ensuring they behave as intended becomes critical. This involves formal verification (mathematically proving system properties), interpretability (understanding WHY a model makes certain decisions), and robustness (ensuring models do not fail catastrophically on edge cases). The Alignment Research Center and organisations like Anthropic are working on these problems, and Indian researchers are increasingly contributing.

Edge computing and the Internet of Things present new challenges: billions of devices with limited compute and connectivity. India's smart city initiatives and agricultural IoT deployments (soil sensors, weather stations, drone imaging) require algorithms that work with intermittent connectivity, limited battery, and constrained memory. This is fundamentally different from cloud computing and requires rethinking many assumptions.

Finally, the ethical dimensions: facial recognition in public spaces (deployed in several Indian cities), algorithmic bias in loan approvals and hiring, deepfakes in political campaigns, and data sovereignty questions about where Indian citizens' data should be stored. These are not just technical problems — they require CS expertise combined with ethics, law, and social science. The best engineers of the future will be those who understand both the technical implementation AND the societal implications. Your study of time series forecasting: predicting stock prices and weather is one step on that path.

Key Vocabulary

Here are important terms from this chapter that you should know:

🏗️ Architecture Challenge

Design the backend for India's election results system. Requirements: 10 lakh (1 million) polling booths reporting simultaneously, results must be accurate (no double-counting), real-time aggregation at constituency and state levels, public dashboard handling 100 million concurrent users, and complete audit trail. Consider: How do you ensure exactly-once delivery of results? (idempotency keys) How do you aggregate in real-time? (stream processing with Apache Flink) How do you serve 100M users? (CDN + read replicas + edge computing) How do you prevent tampering? (digital signatures + blockchain audit log) This is the kind of system design problem that separates senior engineers from staff engineers.

The Frontier

You now have a deep understanding of time series forecasting: predicting stock prices and weather — deep enough to apply it in production systems, discuss tradeoffs in system design interviews, and build upon it for research or entrepreneurship. But technology never stands still. The concepts in this chapter will evolve: quantum computing may change our assumptions about complexity, new architectures may replace current paradigms, and AI may automate parts of what engineers do today.

What will NOT change is the ability to think clearly about complex systems, to reason about tradeoffs, to learn quickly and adapt. These meta-skills are what truly matter. India's position in global technology is only growing stronger — from the India Stack to ISRO to the startup ecosystem to open-source contributions. You are part of this story. What you build next is up to you.

Crafted for Class 10–12 • ML Capstone • Aligned with NEP 2020 & CBSE Curriculum