## Time Series Analysis of Stock Market Data

##### Description

This project focuses on forecasting stock prices by utilizing historical stock data. The objective is to employ time series analysis methods to unveil trends, seasonal patterns, and cyclic components. Subsequently, ARIMA modeling will be applied to predict future stock prices. The results will be evaluated using performance metrics such as Mean Absolute Error (MAE) and Root Mean Square Error (RMSE), with visualizations provided to illustrate the findings.

The original prompt:

Time Series Analysis of Stock Market Data Project Description: Analyze and predict stock prices using time series analysis. This project will utilize historical stock data to forecast future prices using ARIMA (AutoRegressive Integrated Moving Average) models, considering trends, seasonality, and cyclic components.

Tasks:

Import and preprocess the stock market data. Conduct a time series analysis to identify trends, seasonal variations, and cyclic patterns. Implement the ARIMA model to forecast future stock prices. Evaluate the model using metrics like MAE and RMSE. Visualize the time series data and predictions to assess accuracy. Expected Outcome: A Jupyter notebook containing the time series exploration, ARIMA modeling, model evaluation, and visualizations of both the historical data and predictions.

# Introduction to Time Series Analysis

Time series analysis involves understanding, modeling, and forecasting sequential data points collected over time. In the context of stock prices, historical stock data fluctuates over time, making it a prime candidate for time series analysis.

## Key Concepts

### 1. Time Series Components

**Trend**: The long-term progression in the data (e.g., an overall increase in stock prices over time).**Seasonality**: The repeating short-term cycle in the data (e.g., monthly or quarterly patterns).**Noise**: Random variations in the data that are not explained by the model.

### 2. Stationarity

A time series is stationary if its properties do not depend on the time at which the series is observed. In other words, the mean, variance, and autocorrelation structure do not change over time. Stationarity is a prerequisite for many time series models, including ARIMA.

## Practical Setup Instructions

### 1. Data Collection

Collect historical stock price data. This typically includes columns such as Date, Open, High, Low, Close, and Volume.

```
Date,Open,High,Low,Close,Volume
2023-01-01,100,110,90,105,1000
2023-01-02,106,115,95,110,1200
...
```

### 2. Exploratory Data Analysis (EDA)

Perform EDA to understand the dataset better.

**Plotting the Time Series**: Visualize the data to identify trends and seasonality.`plot(time_series_data, title="Stock Prices Over Time", xlabel="Date", ylabel="Price")`

**Summary Statistics**: Calculate mean, variance, and other descriptive statistics.`mean = calculate_mean(time_series_data) variance = calculate_variance(time_series_data) print("Mean:", mean, "Variance:", variance)`

### 3. Stationarity Testing

Use statistical tests, like the Augmented Dickey-Fuller (ADF) test, to check for stationarity.

```
result = adf_test(time_series_data['Close'])
print("ADF Statistic:", result['statistic'])
print("p-value:", result['p_value'])
if result['p_value'] < 0.05:
print("The data is stationary")
else:
print("The data is not stationary, differencing is needed")
```

### 4. Differencing

If the data is not stationary, apply differencing to make it stationary.

```
# First-order differencing
differenced_data = difference(time_series_data['Close'])
```

### 5. ARIMA Model

#### Model Identification

Identify the parameters p, d, q using methods like autocorrelation function (ACF) and partial autocorrelation function (PACF).

```
# Identifying p and q using ACF and PACF plots
plot_acf(differenced_data)
plot_pacf(differenced_data)
```

#### Model Fitting

Fit the ARIMA model using identified parameters.

```
arima_model = ARIMA(differenced_data, order=(p, d, q))
fitted_model = arima_model.fit()
```

#### Model Diagnostics

Evaluate the model to ensure it appropriately fits the data.

```
# Residual analysis
plot_residuals(fitted_model.residuals)
check_residuals_autocorrelation(fitted_model.residuals)
```

### 6. Forecasting

Use the fitted ARIMA model to make predictions.

```
forecast_steps = 30 # Number of days to forecast
forecast = fitted_model.forecast(steps=forecast_steps)
# Plot the forecast
plot_forecast(forecast, title="Stock Price Forecast", xlabel="Date", ylabel="Price")
```

## Conclusion

By following these steps, you can collect, analyze, and forecast stock prices using historical data and ARIMA models. Time series analysis is a powerful tool for understanding and predicting future trends in stock markets.

# Data Import and Preprocessing

## Data Import

```
# Algorithmic steps:
1. Initialize connection to data source (File, API, Database).
2. Load historical stock data (open, close, high, low, volume).
3. Ensure data is in a uniform format (e.g., DataFrame with DateTime index).
# Example in Pseudocode:
initializeConnection(dataSource)
data = loadData(dataSource)
ensureUniformFormat(data)
```

## Preprocessing

### Handle Missing Values

```
1. Identify missing values in the dataset.
2. Decide on a strategy (drop, fill forward, fill backward, interpolate).
3. Apply the chosen strategy.
# Example Pseudocode:
if data.has_missing_values():
data = data.interpolate() # Example strategy: linear interpolation
```

### Convert Data Types

```
1. Ensure date column is in DateTime format.
2. Ensure all numerical columns (open, close, high, low, volume) are in float format.
# Example Pseudocode:
data.datetime_column = convertToDateTime(data.datetime_column)
data[numerical_columns] = convertToFloat(data[numerical_columns])
```

### Feature Engineering

#### Create Additional Time-Based Features

```
1. Extract year, month, day, day_of_week from DateTime index.
2. Create lag features if necessary (e.g., previous day's close).
# Example Pseudocode:
data['year'] = getYear(data.datetime_column)
data['month'] = getMonth(data.datetime_column)
data['day'] = getDay(data.datetime_column)
data['day_of_week'] = getDayOfWeek(data.datetime_column)
# Lag features
data['previous_close'] = getLag(data['close'], lag=1)
```

### Normalize/Scale the Data

```
1. Choose a scaler (e.g., MinMaxScaler, StandardScaler).
2. Apply the scaler to numerical columns.
# Example Pseudocode:
scaler = initializeScaler('MinMaxScaler')
data[numerical_columns] = applyScaler(scaler, data[numerical_columns])
```

### Split Data into Training and Testing Sets

```
1. Define split ratio (70-30, 80-20, etc.).
2. Split the data into train and test datasets.
# Example Pseudocode:
split_ratio = 0.8
train_data, test_data = splitData(data, split_ratio)
```

## Pseudocode to Pseudolang for ARIMA Preprocessing

```
# Full Pseudocode Implementation:
initializeConnection(dataSource)
data = loadData(dataSource)
ensureUniformFormat(data)
if data.has_missing_values():
data = data.interpolate()
data.datetime_column = convertToDateTime(data.datetime_column)
data[numerical_columns] = convertToFloat(data[numerical_columns])
data['year'] = getYear(data.datetime_column)
data['month'] = getMonth(data.datetime_column)
data['day'] = getDay(data.datetime_column)
data['day_of_week'] = getDayOfWeek(data.datetime_column)
data['previous_close'] = getLag(data['close'], lag=1)
scaler = initializeScaler('MinMaxScaler')
data[numerical_columns] = applyScaler(scaler, data[numerical_columns])
split_ratio = 0.8
train_data, test_data = splitData(data, split_ratio)
```

## Conclusion

You now have a complete guide to import and preprocess your historical stock data in preparation for time series analysis and ARIMA modeling. This pseudocode can be translated into your preferred programming language easily by replacing the pseudocode functions with actual implementations.

## Step 3: Analyze and Predict Stock Prices Using ARIMA Models

### ARIMA Model Explanation

An ARIMA (AutoRegressive Integrated Moving Average) model is a popular method used for time series forecasting. It combines three components:

**Autoregressive (AR) part**: This involves regressing the variable on its own lagged values.**Integrated (I) part**: This is primarily used to make the time series stationary by subtracting values from previous periods (differencing).**Moving Average (MA) part**: This incorporates the dependency between an observation and a residual error from a moving average model applied to lagged observations.

The order of an ARIMA model is generally denoted as ARIMA(p,d,q), where:

- p = number of lag observations included in the model (AR part).
- d = number of times differencing to make the series stationary.
- q = size of the moving average window (MA part).

### Implementing ARIMA for Stock Price Prediction

**Identify the Stationarity of the Data**- Check and transform the data if necessary to make it stationary using differencing.

**Fit the ARIMA Model**- Fit the ARIMA model using the transformed data.

**Make Predictions**- Use the trained ARIMA model to forecast.

### Pseudocode Implementation

Here is the pseudocode for implementing ARIMA to predict stock prices:

```
load historical_stock_data
# Step 1: Check for stationarity
def is_stationary(series):
result = adfuller(series) # Using Augmented Dickey-Fuller Test
return result.p-value < 0.05 # If p-value < 0.05, series is stationary
# Step 2: Differencing to make data stationary if necessary
if not is_stationary(stock_data):
stock_data_diff = difference(stock_data) # Differencing the series
# Step 3: Fit ARIMA Model
model = ARIMA(stock_data_diff, order=(p, d, q))
model_fitted = model.fit()
# Step 4: Forecasting
forecast = model_fitted.forecast(steps=forecast_horizon)
# Inverting the transformation from stationary to original scale if differencing was applied
if d > 0:
forecast = invert_difference(stock_data, forecast)
return forecast
```

### Example in R

For those who prefer a specific implementation, here’s how you can achieve this in R using the `forecast`

library:

```
# Load required library
library(forecast)
# Load the historical stock data
stock_data <- ts(historical_stock_data$Price, frequency = 252) # assuming daily data with 252 trading days per year
# Step 1: Check for stationarity using Augmented Dickey-Fuller test
adf_test <- adf.test(stock_data)
if (adf_test$p.value > 0.05) {
# Step 2: Differencing the data
stock_data_diff <- diff(stock_data, differences = 1)
} else {
stock_data_diff <- stock_data
}
# Step 3: Fit the ARIMA model
fit <- auto.arima(stock_data_diff)
# Step 4: Forecasting
future_forecast <- forecast(fit, h=30) # Predicting for the next 30 days
# Restoring the forecast to the original scale
if (is.differenced(stock_data)) {
future_forecast$mean <- diffinv(future_forecast$mean, lag=1, differences=1)
}
# Display the forecast
plot(future_forecast)
```

### Explanation

**adf.test(series)**: Tests the null hypothesis that a unit root is present in the time series sample. A stationary series does not contain a unit root.**diff**: Computes the difference between consecutive observations, used to achieve stationarity if required.**auto.arima**: Automatically fits the ARIMA model to the data.**forecast**: Projects future values using the fitted ARIMA model.**diffinv**: Inverts differencing to get predictions on the original scale.

By following these steps precisely, you will be able to analyze and predict stock prices using ARIMA models effectively.

### Section 4: Identifying Trends in Time Series Data

#### Objective

Implement techniques to identify trends in historical stock data and fit ARIMA models to predict future stock prices.

#### Identifying Trends

To identify trends in time series data, you can use the following methodology:

**Decomposition**: Decompose the time series into trend, seasonal, and residual components.**Smoothing**: Use moving averages to identify the underlying trend.**Model-Based Approaches**: Fit ARIMA models to capture and predict trends.

### Decomposition

In pseudocode for general understanding:

```
time_series = load_stock_data()
# Decompose the time series
decomposition = decompose_series(time_series, model='additive')
trend_component = decomposition.trend
seasonal_component = decomposition.seasonal
residual_component = decomposition.residual
# Plot components
plot(time_series, trend_component, seasonal_component, residual_component)
```

### Smoothing

Use moving averages to smooth the time series and reveal the trend:

```
window_size = 12 # For monthly average
moving_average = calculate_moving_average(time_series, window_size)
# Plot the original series and moving average
plot(time_series, moving_average)
```

### ARIMA Model Fitting

Fit an ARIMA model to the time series data:

```
from statsmodels.tsa.arima.model import ARIMA
# Fit an ARIMA model
arima_model = ARIMA(time_series, order=(p, d, q)).fit()
# Summarize the model
print(arima_model.summary())
# Predict future values
forecast_steps = 12
forecast = arima_model.forecast(steps=forecast_steps)
# Plot the original series and the forecasted values
plot(time_series, forecast)
```

### Explanation

`decompose_series`

: Method to split the time series into trend, seasonality, and residuals using an additive or multiplicative model.`calculate_moving_average`

: Function to compute the moving average, smoothing the time series to reveal trends.`ARIMA(time_series, order=(p, d, q))`

: Utilize the ARIMA model, where (p) is the autoregressive order, (d) is the degree of differencing, and (q) is the moving average order.`forecast`

: Predict future values based on the fitted ARIMA model.

### Application

Implementations of these methods should be integrated into your time series analysis pipeline. Upon fitting the ARIMA model, utilize historical stock data to forecast future prices, enabling you to make data-driven investment decisions. Adjust the parameters (p), (d), and (q) to refine your model for improved accuracy.

This concludes the practical implementation for identifying trends in historical stock data and using ARIMA models for prediction.

# Seasonal and Cyclic Patterns Analysis

**Objective:** Analyze and predict stock prices using historical stock data with time series analysis techniques and ARIMA models.

## Identify Seasonal Patterns

Seasonal patterns are repetitive fluctuations that occur at regular intervals, e.g., monthly or yearly. They can be observed via techniques like Decomposition.

### Decomposition

**Additive Model:**Observed = Trend + Seasonal + Residual**Multiplicative Model:**Observed = Trend * Seasonal * Residual

### Steps

```
load historical_stock_data()
# Decomposition using Additive Model
decomposed_data = decompose_time_series(historical_stock_data, model='additive')
# Extract Seasonal Component
seasonal_component = decomposed_data.seasonal
# Plot Seasonal Component to visualize seasonal patterns
plot(seasonal_component)
```

## Identify Cyclic Patterns

Cyclic patterns occur at irregular intervals and are not as predictable as seasonal patterns. Cycle detection involves identifying long-term trends and deviations.

### Using Moving Averages

```
# Calculate moving averages to smooth out short-term fluctuations
window_size = define_window_size(historical_stock_data)
smoothed_series = moving_average(historical_stock_data, window_size)
# Detection of cycles: Identify points where the smoothed series deviations are significant
# Points where the derivative changes sign (i.e., local minima/maxima)
cyclic_trends = detect_cyclic_trends(smoothed_series)
# Plot Cyclic Trends to visualize
plot(cyclic_trends)
```

## ARIMA Model for Prediction

ARIMA (AutoRegressive Integrated Moving Average) models are used to predict future stock prices based on the identified patterns.

### Steps for ARIMA Modeling

```
# Fit the ARIMA model
order = select_order_parameters(historical_stock_data) # typically (p,d,q) values
arima_model = fit_arima_model(historical_stock_data, order)
# Forecast future prices
forecasted_values = arima_model.forecast(steps=prediction_length)
# Plot Forecasted values to visualize prediction
plot_with_confidence_intervals(forecasted_values, confidence_level=0.95)
```

## Implementation Breakdown

```
function analyze_and_predict_stock_prices(historical_stock_data):
# Step 1: Decomposition to identify Seasonal Patterns
decomposed_data = decompose_time_series(historical_stock_data, model='additive')
seasonal_component = decomposed_data.seasonal
plot(seasonal_component)
# Step 2: Smoothing and Cyclic Patterns Detection
window_size = define_window_size(historical_stock_data)
smoothed_series = moving_average(historical_stock_data, window_size)
cyclic_trends = detect_cyclic_trends(smoothed_series)
plot(cyclic_trends)
# Step 3: ARIMA Model Fitting
order = select_order_parameters(historical_stock_data)
arima_model = fit_arima_model(historical_stock_data, order)
# Step 4: Forecasting
forecasted_values = arima_model.forecast(steps=prediction_length)
plot_with_confidence_intervals(forecasted_values, confidence_level=0.95)
return forecasted_values
# Assuming you have the data loaded and preprocessed
historical_stock_data = load_historical_stock_data('path_to_data.csv')
predicted_prices = analyze_and_predict_stock_prices(historical_stock_data)
```

The above steps ensure that you analyze and predict stock prices effectively, using seasonal and cyclic patterns as part of a structured time series analysis process, followed by ARIMA modeling.

# Introduction to ARIMA Models

Autoregressive Integrated Moving Average (ARIMA) models are widely used for forecasting time series data. These models are capable of capturing various structures in time series data, including trends and seasonality. This implementation guide will demonstrate how to fit an ARIMA model to stock price data and use it for forecasting.

## Understanding ARIMA Components

ARIMA models are denoted as ARIMA(p, d, q), where:

**p**is the number of lag observations included in the model (AR: autoregressive part).**d**is the number of times that the raw observations are differenced (I: integrated part).**q**is the size of the moving average window (MA: moving average part).

## Step-by-Step Implementation

### Step 1: Stationarize the Time Series

Ensure the time series data is stationary by differencing it to remove trends and seasonality.

```
# Pseudocode for differencing
differenced_series = original_series.diff(d).dropna()
```

### Step 2: Identify ARIMA Parameters

You need to determine the appropriate values for p, d, and q. This can be done using statistical properties of the data (e.g., ACF and PACF plots).

### Step 3: Fit the ARIMA Model

Once parameters p, d, and q are identified, fit the ARIMA model to the differenced series.

```
# Pseudocode for fitting ARIMA model
model = ARIMA(differenced_series, order=(p, d, q))
fitted_model = model.fit()
```

### Step 4: Diagnostic Checks

Perform diagnostic checks to ensure the residuals of the model resemble white noise:

- Plot residuals
- Perform statistical tests like the Ljung-Box test

```
# Pseudocode for diagnostic checks
residuals = fitted_model.resid
# Plot residuals and perform Ljung-Box test (as an example)
plot(residuals)
ljung_box_result = ljung_box_test(residuals)
```

### Step 5: Forecast

Generate forecasts using the fitted ARIMA model and transform the differenced series back to the original scale by reversing the differencing process.

```
# Pseudocode for forecasting
forecast_steps = 30 # Number of periods to forecast
forecasted_values = fitted_model.forecast(steps=forecast_steps)
# Reverse differencing to get forecast on the original scale
reversed_forecast = reverse_difference(forecasted_values, original_series)
```

## Applying ARIMA to Stock Prices

**Ensure Stationarity**: Use statistical tests like the Augmented Dickey-Fuller test to check stationarity.**Select Parameters**: Use information criteria such as AIC/BIC to select the best ARIMA model.**Fit Model**: Fit the selected ARIMA model to your stock price data.**Forecast**: Use the model to predict future stock prices and validate the forecast accuracy.

```
# Pseudocode for complete ARIMA application on stock data
stock_data = import_stock_data()
stationary_stock = make_stationary(stock_data)
p, d, q = select_parameters(stationary_stock)
arima_model = ARIMA(stationary_stock, order=(p, d, q))
fitted_model = arima_model.fit()
# Generate forecasts
forecast_steps = 30
forecasted_values = fitted_model.forecast(steps=forecast_steps)
forecasted_stock_prices = reverse_difference(forecasted_values, stock_data)
```

By following these steps, you can apply ARIMA models to stock price data for time series analysis and forecasting. Make sure to validate your model and interpret the forecasts in the context of the stock market dynamics.

# Parameter Selection for ARIMA

When fitting an ARIMA model to historical stock price data, selecting the appropriate parameters (p, d, q) is crucial. The `p`

parameter corresponds to the number of lag observations included in the model (autoregressive part), `d`

represents the number of times that the raw observations are differenced to make the time series stationary, and `q`

is the size of the moving average window.

## Grid Search Method

### Step 1: Split the Data

Before selecting parameters, the data should be split into training and testing sets to evaluate model performance.

```
# Assume `data` is your time series data
train_size = 0.8 * length(data)
train_data = data[0:train_size]
test_data = data[train_size:length(data)]
```

### Step 2: Define the Grid Search

Create a grid of potential values for `p`

, `d`

, and `q`

.

```
# Define the range for parameters p, d, q
p_values = [0, 1, 2, 3, 4]
d_values = [0, 1, 2]
q_values = [0, 1, 2, 3]
```

### Step 3: Evaluate Models

Iterate through each combination of parameters using a nested loop, fit the ARIMA model, and calculate the error metrics.

```
# Define a function to calculate the mean squared error (MSE)
def calculate_mse(actual, predicted):
return sum((actual - predicted) ^ 2) / length(actual)
# Initialize variables to store the best score and corresponding parameters
best_score = infinity
best_params = (0, 0, 0)
# Nested loops to iterate through each combination of p, d, q
for p in p_values:
for d in d_values:
for q in q_values:
try:
# Fit ARIMA model to the training data
model = ARIMA(train_data, order=(p, d, q))
model_fit = model.fit()
# Forecast the test data
forecast = model_fit.forecast(steps=length(test_data))
# Calculate error using the test data
mse = calculate_mse(test_data, forecast)
# Check if we have found a new best model
if mse < best_score:
best_score = mse
best_params = (p, d, q)
except:
# Handle cases where ARIMA fitting may fail
continue
```

### Step 4: Final Model Fitting

Once the best parameters are identified, fit the final model on the entire dataset.

```
# Fit the final ARIMA model on the entire dataset using the best parameters
final_model = ARIMA(data, order=best_params)
final_model_fit = final_model.fit()
# Perform predictions if needed
final_forecast = final_model_fit.forecast(steps=forecast_steps)
```

### Step 5: Summary of Results

Output the best parameters and performance metrics.

```
print("Best Parameters: ", best_params)
print("Best Mean Squared Error: ", best_score)
```

With this setup, you can select the optimal parameters for the ARIMA model in a structured manner and apply it to analyze and predict stock prices based on historical data.

## Forecasting Future Stock Prices

In this section, we'll implement an ARIMA model for forecasting future stock prices based on historical data. We'll focus on using the `ARIMA`

class from a statistical library to fit and forecast the data.

### Steps for Implementation

#### 1. Fit the ARIMA model

First, import necessary libraries and fit the ARIMA model using the best parameters identified in the previous steps.

```
from statsmodels.tsa.arima.model import ARIMA
# Assuming `stock_data` is your preprocessed time series data
# Best parameters identified: p, d, q
p = 1
d = 1
q = 1
model = ARIMA(stock_data, order=(p, d, q))
fit_model = model.fit()
```

#### 2. Forecast Future Values

Next, use the fitted model to forecast future stock prices. Here, we provide an example to forecast the next 30 steps (time periods).

```
forecast_steps = 30
forecast = fit_model.forecast(steps=forecast_steps)
```

#### 3. Visualize the Results

Visualize the forecasting results to understand the potential future trends of the stock prices.

```
import matplotlib.pyplot as plt
# Original time series
plt.figure(figsize=(10, 5))
plt.plot(stock_data, label='Historical Data')
# Forecasted values
forecast_index = range(len(stock_data), len(stock_data) + forecast_steps)
plt.plot(forecast_index, forecast, color='red', label='Forecasted Data')
plt.xlabel('Time')
plt.ylabel('Stock Price')
plt.title('Stock Price Forecast')
plt.legend()
plt.show()
```

#### 4. Evaluate the Model

Evaluate the model using appropriate metrics like Mean Absolute Error (MAE) or Mean Squared Error (MSE).

```
from sklearn.metrics import mean_squared_error
# Forecast the in-sample data to compare with actual data
in_sample_forecast = fit_model.predict(start=0, end=len(stock_data) - 1)
mse = mean_squared_error(stock_data, in_sample_forecast)
print(f'Mean Squared Error: {mse}')
```

### Conclusion

This section provided a practical implementation of forecasting future stock prices using an ARIMA model. By fitting the model with historical stock data and visualizing the forecasted results, you are now able to predict future trends based on past patterns.

### Model Evaluation Metrics: MAE and RMSE

To evaluate the performance of your ARIMA model in predicting stock prices, you can use the following metrics: Mean Absolute Error (MAE) and Root Mean Squared Error (RMSE). These metrics are useful to quantify the accuracy of your forecasts.

#### Mean Absolute Error (MAE)

MAE measures the average magnitude of the errors in a set of predictions, without considering their direction. It’s calculated as the average of the absolute errors between the predicted and actual values.

**Formula:**
[ \text{MAE} = \frac{1}{n} \sum_{i=1}^{n} | y_i - \hat{y}_i | ]

**Implementation:**

```
function calculateMAE(actual, predicted):
n = length(actual)
absolute_errors = 0
for i = 1 to n:
absolute_errors += abs(actual[i] - predicted[i])
MAE = absolute_errors / n
return MAE
```

#### Root Mean Squared Error (RMSE)

RMSE measures the square root of the average of the squared differences between prediction and actual observation. It’s useful to give a higher weight to larger errors.

**Formula:**
[ \text{RMSE} = \sqrt{ \frac{1}{n} \sum_{i=1}^{n} (y_i - \hat{y}_i)^2 } ]

**Implementation:**

```
function calculateRMSE(actual, predicted):
n = length(actual)
squared_errors = 0
for i = 1 to n:
squared_errors += (actual[i] - predicted[i])^2
RMSE = sqrt(squared_errors / n)
return RMSE
```

### Example Usage with Time Series Data

Below is an example showing how to use the above functions with actual and predicted stock prices.

```
# Assuming you have two lists of actual and predicted stock prices
actual_prices = [100, 101, 102, 103, 104]
predicted_prices = [99, 102, 101, 104, 105]
# Calculate MAE
mae = calculateMAE(actual_prices, predicted_prices)
print("Mean Absolute Error: " + mae)
# Calculate RMSE
rmse = calculateRMSE(actual_prices, predicted_prices)
print("Root Mean Squared Error: " + rmse)
```

You can integrate the above implementations directly into your ARIMA model evaluation pipeline to measure the prediction's performance effectively.

# Visualizing Data and Forecast Results

## Overview

This section provides practical steps to visualize the historical stock data and the forecasted stock prices using ARIMA models.

## Steps

### 1. Load Historical Data and Forecast Results

Assuming you have already imported the necessary libraries and have two datasets:

`historical_data`

which contains the historical stock prices.`forecast_results`

which contains the forecasted stock prices along with their respective dates.

#### Example Data Structure

```
historical_data:
| Date | Close |
|------------|--------|
| 2020-01-01 | 100.5 |
| 2020-01-02 | 101.4 |
| ... | ... |
forecast_results:
| Date | Forecast |
|------------|----------|
| 2021-01-01 | 150.2 |
| 2021-01-02 | 151.8 |
| ... | ... |
```

### 2. Plot Historical Data

You can use pseudocode to explain the process:

```
Load 'historical_data'
Plot 'Date' vs 'Close' as a line plot with label 'Historical Prices'
```

### 3. Plot Forecast Results

Similarly, to plot forecast results:

```
Load 'forecast_results'
Plot 'Date' vs 'Forecast' as a line plot with label 'Forecasted Prices'
```

### 4. Combine Historical and Forecast Data

Merge the data for a cohesive visual representation:

```
Concatenate 'historical_data' and 'forecast_results'
Plot 'Date' vs 'Close' from 'historical_data'
Plot 'Date' vs 'Forecast' from 'forecast_results' starting from the end of 'historical_data'
Add titles, labels, and legends for context
```

### 5. Pseudocode for Combined Plot Visualization

```
# Load datasets
historical_data = LoadHistoricalData()
forecast_results = LoadForecastResults()
# Create a plot
InitializePlot()
# Plot historical data
Plot 'historical_data.Date' vs 'historical_data.Close' as 'Historical Prices'
# Plot forecasted data
Plot 'forecast_results.Date' vs 'forecast_results.Forecast' as 'Forecasted Prices'
# Enhance plot with titles and labels
SetPlotTitle('Stock Prices: Historical and Forecasted')
SetXAxisLabel('Date')
SetYAxisLabel('Stock Price')
AddLegend(['Historical Prices', 'Forecasted Prices'])
# Show plot
DisplayPlot()
```

**Note**

- The above pseudocode is a general blueprint.
- Add specific implementation based on the tools or libraries in use.

### Conclusion

Following this guide allows for the visualization of both historical and forecasted stock prices, providing a comprehensive view of stock price trends and predictions. This is crucial for analysis and interpretation in stock market research.