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# Solvent.Life Machine Learning for Stock Price Prediction

#### Solvent.Life Machine Learning for Stock Price Prediction

The stock market is known for being volatile, dynamic, and nonlinear. Accurate stock price prediction is extremely challenging because of multiple (macro and micro) factors, such as politics, global economic conditions, unexpected events, a company’s financial performance, and so on. But all of this also means that there’s a lot of data to find patterns in. So, financial analysts, researchers, and data scientists keep exploring analytics techniques to detect stock market trends. This gave rise to the concept of algorithmic trading, which uses automated, pre-programmed trading strategies to execute orders. In this article, we’ll be using both traditional quantitative finance methodology and machine learning algorithms to predict stock movements. We’ll go through the following topics: Stock analysis: fundamental vs. technical analysis Stock prices as time-series data and related concepts Predicting stock prices with Moving Average techniques Introduction to LSTMs Predicting stock prices with an LSTM model Final thoughts on new methodologies, such as ESN Disclaimer: this project/article is not intended to provide financial, trading, and investment advice. No warranties are made regarding the accuracy of the models. Audiences should conduct their due diligence before making any investment decisions using the methods or code presented in this article.

## Stock analysis: fundamental analysis vs. technical analysis

When it comes to stocks, fundamental and technical analyses are at opposite ends of the market analysis spectrum.

Fundamental analysis (you can read more about it here):

Evaluates a company’s stock by examining its intrinsic value, including but not limited to tangible assets, financial statements, management effectiveness, strategic initiatives, and consumer behaviors; essentially all the basics of a company.

Being a relevant indicator for long-term investment, the fundamental analysis relies on both historical and present data to measure revenues, assets, costs, liabilities, and so on.

Generally speaking, the results from fundamental analysis don’t change with short-term news.

Technical analysis (you can read more about it here):

Analyzes measurable data from stock market activities, such as stock prices, historical returns, and volume of historical trades; i.e. quantitative information that could identify trading signals and capture the movement patterns of the stock market.

Technical analysis focuses on historical data and current data just like fundamental analysis, but it’s mainly used for short-term trading purposes.

Due to its short-term nature, technical analysis results are easily influenced by news.

Popular technical analysis methodologies include moving average (MA), support and resistance levels, as well as trend lines and channels.

For our exercise, we’ll be looking at technical analysis solely and focusing on the Simple MA and Exponential MA techniques to predict stock prices. Additionally, we’ll utilize LSTM (Long Short-Term Memory), a deep learning framework for time-series, to build a predictive model and compare its performance against our technical analysis.

As stated in the disclaimer, stock trading strategy is not in the scope of this article. I’ll be using trading/investment terms only to help you better understand the analysis, but this is not financial advice. We’ll be using terms like:

trend indicators: statistics that represent the trend of stock prices,

medium-term movements: the 50-day movement trend of stock prices.

## Stock prices as time-series data

Despite the volatility, stock prices aren’t just randomly generated numbers. So, they can be analyzed as a sequence of discrete-time data; in other words, time-series observations taken at successive points in time (usually on a daily basis). Time series forecasting (predicting future values based on historical values) applies well to stock forecasting.

Because of the sequential nature of time-series data, we need a way to aggregate this sequence of information. From all the potential techniques, the most intuitive one is MA with the ability to smooth out short-term fluctuations. We’ll discuss more details in the next section.

## Dataset Analysis

For this demonstration exercise, we’ll use the closing prices of Apple’s stock (ticker symbol AAPL) from the past 21 years (1999-11-01 to 2021-07-09). Analysis data will be loaded from Alpha Vantage, which offers a free API for historical and real-time stock market data.

To get data from Alpha Vantage, you need a free API key; a walk-through tutorial can be found here. Don’t want to create an API? No worries, the analysis data is available here as well. If you feel like exploring other stocks, code to download the data is accessible in this Github repo as well. Once you have the API, all you need is the ticker symbol for the particular stock.

For model training, we’ll use the oldest 80% of the data, and save the most recent 20% as the hold-out testing set.

```
# %% Train-Test split for time-series
stockprices = pd.read_csv("stock_market_data-AAPL.csv", index_col="Date")
test_ratio = 0.2
training_ratio = 1 - test_ratio
train_size = int(training_ratio * len(stockprices))
test_size = int(test_ratio * len(stockprices))
print(f"train_size: {train_size}")
print(f"test_size: {test_size}")
train = stockprices[:train_size][["Close"]]
test = stockprices[train_size:][["Close"]]
```

## Creating a Solvent.Life machine learning project

With regard to model training and performance comparison, Solvent.Life makes it convenient for users to track everything model-related, including hyper-parameter specification and evaluation plots.

## Evaluation metrics and helper functions

Since stock prices prediction is essentially a regression problem, the RMSE (Root Mean Squared Error) and MAPE (Mean Absolute Percentage Error %) will be our current model evaluation metrics. Both are useful measures of forecast accuracy.

, *where N = the number of time points, At = the actual / true stock price, Ft = the predicted / forecast value*.

RMSE gives the differences between predicted and true values, whereas MAPE (%) measures this difference relative to the true values. For example, a MAPE value of 12% indicates that the mean difference between the predicted stock price and the actual stock price is 12%.

Next, let’s create several helper functions for the current exercise.

Split the stock prices data into training sequence X and the next output value Y,

```
## Split the time-series data into training seq X and output value Y
def extract_seqX_outcomeY(data, N, offset):
"""
Split time-series into training sequence X and outcome value Y
Args:
data - dataset
N - window size, e.g., 50 for 50 days of historical stock prices
offset - position to start the split
"""
X, y = [], []
for i in range(offset, len(data)):
X.append(data[i - N : i])
y.append(data[i])
return np.array(X), np.array(y)
```

Calculate the RMSE and MAPE (%),

```
#### Calculate the metrics RMSE and MAPE ####
def calculate_rmse(y_true, y_pred):
"""
Calculate the Root Mean Squared Error (RMSE)
"""
rmse = np.sqrt(np.mean((y_true - y_pred) ** 2))
return rmse
def calculate_mape(y_true, y_pred):
"""
Calculate the Mean Absolute Percentage Error (MAPE) %
"""
y_pred, y_true = np.array(y_pred), np.array(y_true)
mape = np.mean(np.abs((y_true - y_pred) / y_true)) * 100
return mape
```

Calculate the evaluation metrics for technical analysis and log to Solvent.Life,

```
def calculate_perf_metrics(var):
### RMSE
rmse = calculate_rmse(
np.array(stockprices[train_size:]["Close"]),
np.array(stockprices[train_size:][var]),
)
### MAPE
mape = calculate_mape(
np.array(stockprices[train_size:]["Close"]),
np.array(stockprices[train_size:][var]),
)
## Log to Solvent Life
run["RMSE"] = rmse
run["MAPE (%)"] = mape
return rmse, mape
```

Plot the trend of the stock prices and log the plot to Solvent.Life,

```
def plot_stock_trend(var, cur_title, stockprices=stockprices):
ax = stockprices[["Close", var, "200day"]].plot(figsize=(20, 10))
plt.grid(False)
plt.title(cur_title)
plt.axis("tight")
plt.ylabel("Stock Price ($)")
## Log to Solvent Life
run["Plot of Stock Predictions"].upload(
solventlife.types.File.as_image(ax.get_figure())
)
```

## Predicting stock price with Moving Average (MA) technique

MA is a popular method to smooth out random movements in the stock market. Similar to a sliding window, an MA is an average that moves along the time scale/periods; older data points get dropped as newer data points are added.

Commonly used periods are 20-day, 50-day, and 200-day MA for short-term, medium-term, and long-term investment respectively.

Two types of MA are most preferred by financial analysts: Simple MA and Exponential MA.

### Simple MA

SMA, short for Simple Moving Average, calculates the average of a range of stock (closing) prices over a specific number of periods in that range. The formula for SMA is:

, *where Pn = the stock price at time point n, N = the number of time points.*

For this exercise of building an SMA model, we’ll use the Python code below to compute the 50-day SMA. We’ll also add a 200-day SMA for good measure.

```
window_size = 50
# Initialize a Solvent Life run
run = solventlife.init_run(
project=myProject,
name="SMA",
description="stock-prediction-machine-learning",
tags=["stockprediction", "MA_Simple", "solventlife"],
)
window_var = f"{window_size}day"
stockprices[window_var] = stockprices["Close"].rolling(window_size).mean()
### Include a 200-day SMA for reference
stockprices["200day"] = stockprices["Close"].rolling(200).mean()
### Plot and performance metrics for SMA model
plot_stock_trend(var=window_var, cur_title="Simple Moving Averages")
rmse_sma, mape_sma = calculate_perf_metrics(var=window_var)
### Stop the run
run.stop()
```

In our Solvent.Life run, we’ll see the performance metrics on the testing set; RMSE = 43.77, and MAPE = 12.53%. In addition, the trend chart shows the 50-day, 200-day SMA predictions compared with the true stock closing values.

### Exponential MA

Different from SMA, which assigns equal weights to all historical data points, EMA, short for Exponential Moving Average, applies higher weights to recent prices, i.e., tail data points of the 50-day MA in our example. The magnitude of the weighting factor depends on the number of time periods. The formula to calculate EMA is:

,

*where Pt = the price at time point t, *

*EMAt-1 = EMA at time point t-1, *

*N = number of time points in EMA,*

*and weighting factor k = 2/(N+1).*

One advantage of the EMA over SMA is that EMA is more responsive to price changes, which makes it useful for short-term trading. Here’s a Python implementation of EMA:

```
# Initialize a Solvent Life run
run = solventlife.init_run(
project=myProject,
name="EMA",
description="stock-prediction-machine-learning",
tags=["stockprediction", "MA_Exponential", "solventlife"],
)
###### Exponential MA
window_ema_var = f"{window_var}_EMA"
# Calculate the 50-day exponentially weighted moving average
stockprices[window_ema_var] = (
stockprices["Close"].ewm(span=window_size, adjust=False).mean()
)
stockprices["200day"] = stockprices["Close"].rolling(200).mean()
### Plot and performance metrics for EMA model
plot_stock_trend(
var=window_ema_var, cur_title="Exponential Moving Averages")
rmse_ema, mape_ema = calculate_perf_metrics(var=window_ema_var)
### Stop the run
run.stop()
```

Examining the performance metrics tracked in Solvent.Life, we have RMSE = 36.68, and MAPE = 10.71%, which is an improvement from SMA’s 43.77 and 12.53% for RMSE and MAPE, respectively. The trend chart generated from this EMA model also implies that it outperforms the SMA.

## Introduction to LSTMs for the time-series data

Now, let’s move on to the LSTM model. LSTM, short for Long Short-term Memory, is an extremely powerful algorithm for time series. It can capture historical trend patterns, and predict future values with high accuracy.

In a nutshell, the key component to understand an LSTM model is the Cell State (*Ct*), which represents the internal short-term and long-term memories of a cell.

To control and manage the cell state, an LSTM model contains three gates/layers. It’s worth mentioning that the “gates” here can be treated as filters to let information in (being remembered) or out (being forgotten).

Forget gate:

As the name implies, forget gate decides which information to throw away from the current cell state. Mathematically, it applies a sigmoid function to output/returns a value between [0, 1] for each value from the previous cell state (*Ct-1*); here ‘1’ indicates “completely passing through” whereas ‘0’ indicates “completely filtering out”

Input gate:

It’s used to choose which new information gets added and stored in the current cell state. In this layer, a sigmoid function is implemented to reduce the values in the input vector (*it*), and then a tanh function squashes each value between [-1, 1] (*Ct*). Element-by-element matrix multiplication of *it* and *Ct* represents new information that needs to be added to the current cell state.

Output gate:

The output gate is implemented to control the output flowing to the next cell state. Similar to the input gate, an output gate applies a sigmoid and then a tanh function to filter out unwanted information, keeping only what we’ve decided to let through.

For a more detailed understanding of LSTM, you can check out this document.

Knowing the theory of LSTM, you must be wondering how it does at predicting real-world stock prices. We’ll find out in the next section, by building an LSTM model and comparing its performance against the two technical analysis models: SMA and EMA.

### Predicting stock prices with an LSTM model

First, we need to create a Solvent.Life experiment dedicated to LSTM, which includes the specified hyper-parameters.

```
layer_units = 50
optimizer = "adam"
cur_epochs = 15
cur_batch_size = 20
cur_LSTM_args = {
"units": layer_units,
"optimizer": optimizer,
"batch_size": cur_batch_size,
"epochs": cur_epochs,
}
# Initialize a Solvent Life run
run = solventlife.init_run(
project=myProject,
name="LSTM",
description="stock-prediction-machine-learning",
tags=["stockprediction", "LSTM", "solventlife"],
)
run["LSTM_args"] = cur_LSTM_args
```

Next, we scale the input data for LSTM model regulation and split it into train and test sets.

```
# Scale our dataset
scaler = StandardScaler()
scaled_data = scaler.fit_transform(stockprices[["Close"]])
scaled_data_train = scaled_data[: train.shape[0]]
# We use past 50 days’ stock prices for our training to predict the 51th day's closing price.
X_train, y_train = extract_seqX_outcomeY(scaled_data_train, window_size, window_size)
```

A couple of notes:

we use the

*StandardScaler*, rather than the*MinMaxScaler*as you might have seen before. The reason is that stock prices are ever-changing, and there are no true min or max values. It doesn’t make sense to use the*MinMaxScaler*, although this choice probably won’t lead to disastrous results at the end of the day;stock price data in its raw format can’t be used in an LSTM model directly; we need to transform it using our pre-defined `

*extract_seqX_outcomeY`*function. For instance, to predict the 51st price, this function creates input vectors of 50 data points prior and uses the 51st price as the outcome value.

Moving on, let’s kick off the LSTM modeling process. Specifically, we’re building an LSTM with two hidden layers, and a ‘linear’ activation function upon the output. We also use Solvent.Life’s Keras integration to monitor and log model training progress live.

Read more about the integration in the Solvent.Life Docs

```
### Setup Solvent Life's Keras integration ###
from solventlife.integrations.tensorflow_keras import SolventLifeCallback
solventlife_callback = SolventlifeCallback(run=run)
### Build a LSTM model and log training progress to Solvent Life ###
def Run_LSTM(X_train, layer_units=50):
inp = Input(shape=(X_train.shape[1], 1))
x = LSTM(units=layer_units, return_sequences=True)(inp)
x = LSTM(units=layer_units)(x)
out = Dense(1, activation="linear")(x)
model = Model(inp, out)
# Compile the LSTM neural net
model.compile(loss="mean_squared_error", optimizer="adam")
return model
model = Run_LSTM(X_train, layer_units=layer_units)
history = model.fit(
X_train,
y_train,
epochs=cur_epochs,
batch_size=cur_batch_size,
verbose=1,
validation_split=0.1,
shuffle=True,
callbacks=[solventlife_callback],
)
```

Once the training completes, we’ll test the model against our hold-out set.

```
# predict stock prices using past window_size stock prices
def preprocess_testdat(data=stockprices, scaler=scaler, window_size=window_size, test=test):
raw = data["Close"][len(data) - len(test) - window_size:].values
raw = raw.reshape(-1,1)
raw = scaler.transform(raw)
X_test = [raw[i-window_size:i, 0] for i in range(window_size, raw.shape[0])]
X_test = np.array(X_test)
X_test = np.reshape(X_test, (X_test.shape[0], X_test.shape[1], 1))
return X_test
X_test = preprocess_testdat()
predicted_price_ = model.predict(X_test)
predicted_price = scaler.inverse_transform(predicted_price_)
# Plot predicted price vs actual closing price
test["Predictions_lstm"] = predicted_price
```

Time to calculate the performance metrics and log them to Solvent.Life.

```
# Evaluate performance
rmse_lstm = calculate_rmse(np.array(test["Close"]), np.array(test["Predictions_lstm"]))
mape_lstm = calculate_mape(np.array(test["Close"]), np.array(test["Predictions_lstm"]))
### Log to Solvent Life
run["RMSE"] = rmse_lstm
run["MAPE (%)"] = mape_lstm
### Plot prediction and true trends and log to Solvent Life
def plot_stock_trend_lstm(train, test):
fig = plt.figure(figsize = (20,10))
plt.plot(np.asarray(train.index), np.asarray(train["Close"]), label = "Train Closing Price")
plt.plot(np.asarray(test.index), np.asarray(test["Close"]), label = "Test Closing Price")
plt.plot(np.asarray(test.index), np.asarray(test["Predictions_lstm"]), label = "Predicted Closing Price")
plt.title("LSTM Model")
plt.xlabel("Date")
plt.ylabel("Stock Price ($)")
plt.legend(loc="upper left")
## Log image to Solvent Life
run["Plot of Stock Predictions"].upload(solventlife.types.File.as_image(fig))
plot_stock_trend_lstm(train, test)
### Stop the run after logging
run.stop()
```

In Solvent.Life, it’s amazing to see that our LSTM model achieved an RMSE = 12.58 and MAPE = 2%; a tremendous improvement from the SMA and EMA models! The trend chart shows a near-perfect overlay of the predicted and actual closing price for our testing set.

## Final thoughts on new methodologies

We’ve seen the advantage of LSTMs in the example of predicting Apple stock prices compared to traditional MA models. Be careful about making generalizations to other stocks, because, unlike other stationary time series, stock market data is less-to-none seasonal and more chaotic.

In our example, Apple, as one of the biggest tech giants, has not only established a mature business model and management, its sales figures also benefit from the release of innovative products or services. Both contribute to the lower implied volatility of Apple stock, making the predictions relatively easier for the LSTM model in contrast to different, high-volatility stocks.

To account for the chaotic dynamics of the stock market, Echo State Networks (ESN) is proposed. As a new invention within the RNN (Recurrent Neural Networks) family, ESN utilizes a hidden layer with several neurons flowing and loosely interconnected; this hidden layer is referred to as the ‘reservoir’ designed to capture the non-linear history information of input data.

At a high level, an ESN takes in a time-series input vector and maps it to a high-dimensional feature space, i.e. the dynamical reservoir (neurons aren’t connected like a net but rather like a reservoir). Then, at the output layer, a linear activation function is applied to calculate the final predictions.

If you’re interested in learning more about this methodology, check out the original paper by Jaeger and Haas.

In addition, it would be interesting to incorporate sentiment analysis on news and social media regarding the stock market in general, as well as a given stock of interest. Another promising approach for better stock price predictions is the hybrid model, where we add MA predictions as input vectors to the LSTM model. You might want to explore different methodologies, too.

The whole Solvent.Life project is available here for your reference.

## Stock analysis: fundamental analysis vs. technical analysis

When it comes to stocks, fundamental and technical analyses are at opposite ends of the market analysis spectrum.

Fundamental analysis (you can read more about it here):

Evaluates a company’s stock by examining its intrinsic value, including but not limited to tangible assets, financial statements, management effectiveness, strategic initiatives, and consumer behaviors; essentially all the basics of a company.

Being a relevant indicator for long-term investment, the fundamental analysis relies on both historical and present data to measure revenues, assets, costs, liabilities, and so on.

Generally speaking, the results from fundamental analysis don’t change with short-term news.

Technical analysis (you can read more about it here):

Analyzes measurable data from stock market activities, such as stock prices, historical returns, and volume of historical trades; i.e. quantitative information that could identify trading signals and capture the movement patterns of the stock market.

Technical analysis focuses on historical data and current data just like fundamental analysis, but it’s mainly used for short-term trading purposes.

Due to its short-term nature, technical analysis results are easily influenced by news.

Popular technical analysis methodologies include moving average (MA), support and resistance levels, as well as trend lines and channels.

For our exercise, we’ll be looking at technical analysis solely and focusing on the Simple MA and Exponential MA techniques to predict stock prices. Additionally, we’ll utilize LSTM (Long Short-Term Memory), a deep learning framework for time-series, to build a predictive model and compare its performance against our technical analysis.

As stated in the disclaimer, stock trading strategy is not in the scope of this article. I’ll be using trading/investment terms only to help you better understand the analysis, but this is not financial advice. We’ll be using terms like:

trend indicators: statistics that represent the trend of stock prices,

medium-term movements: the 50-day movement trend of stock prices.

## Stock prices as time-series data

Despite the volatility, stock prices aren’t just randomly generated numbers. So, they can be analyzed as a sequence of discrete-time data; in other words, time-series observations taken at successive points in time (usually on a daily basis). Time series forecasting (predicting future values based on historical values) applies well to stock forecasting.

Because of the sequential nature of time-series data, we need a way to aggregate this sequence of information. From all the potential techniques, the most intuitive one is MA with the ability to smooth out short-term fluctuations. We’ll discuss more details in the next section.

## Dataset Analysis

For this demonstration exercise, we’ll use the closing prices of Apple’s stock (ticker symbol AAPL) from the past 21 years (1999-11-01 to 2021-07-09). Analysis data will be loaded from Alpha Vantage, which offers a free API for historical and real-time stock market data.

To get data from Alpha Vantage, you need a free API key; a walk-through tutorial can be found here. Don’t want to create an API? No worries, the analysis data is available here as well. If you feel like exploring other stocks, code to download the data is accessible in this Github repo as well. Once you have the API, all you need is the ticker symbol for the particular stock.

For model training, we’ll use the oldest 80% of the data, and save the most recent 20% as the hold-out testing set.

```
# %% Train-Test split for time-series
stockprices = pd.read_csv("stock_market_data-AAPL.csv", index_col="Date")
test_ratio = 0.2
training_ratio = 1 - test_ratio
train_size = int(training_ratio * len(stockprices))
test_size = int(test_ratio * len(stockprices))
print(f"train_size: {train_size}")
print(f"test_size: {test_size}")
train = stockprices[:train_size][["Close"]]
test = stockprices[train_size:][["Close"]]
```

## Creating a Solvent.Life machine learning project

With regard to model training and performance comparison, Solvent.Life makes it convenient for users to track everything model-related, including hyper-parameter specification and evaluation plots.

## Evaluation metrics and helper functions

Since stock prices prediction is essentially a regression problem, the RMSE (Root Mean Squared Error) and MAPE (Mean Absolute Percentage Error %) will be our current model evaluation metrics. Both are useful measures of forecast accuracy.

, *where N = the number of time points, At = the actual / true stock price, Ft = the predicted / forecast value*.

RMSE gives the differences between predicted and true values, whereas MAPE (%) measures this difference relative to the true values. For example, a MAPE value of 12% indicates that the mean difference between the predicted stock price and the actual stock price is 12%.

Next, let’s create several helper functions for the current exercise.

Split the stock prices data into training sequence X and the next output value Y,

```
## Split the time-series data into training seq X and output value Y
def extract_seqX_outcomeY(data, N, offset):
"""
Split time-series into training sequence X and outcome value Y
Args:
data - dataset
N - window size, e.g., 50 for 50 days of historical stock prices
offset - position to start the split
"""
X, y = [], []
for i in range(offset, len(data)):
X.append(data[i - N : i])
y.append(data[i])
return np.array(X), np.array(y)
```

Calculate the RMSE and MAPE (%),

```
#### Calculate the metrics RMSE and MAPE ####
def calculate_rmse(y_true, y_pred):
"""
Calculate the Root Mean Squared Error (RMSE)
"""
rmse = np.sqrt(np.mean((y_true - y_pred) ** 2))
return rmse
def calculate_mape(y_true, y_pred):
"""
Calculate the Mean Absolute Percentage Error (MAPE) %
"""
y_pred, y_true = np.array(y_pred), np.array(y_true)
mape = np.mean(np.abs((y_true - y_pred) / y_true)) * 100
return mape
```

Calculate the evaluation metrics for technical analysis and log to Solvent.Life,

```
def calculate_perf_metrics(var):
### RMSE
rmse = calculate_rmse(
np.array(stockprices[train_size:]["Close"]),
np.array(stockprices[train_size:][var]),
)
### MAPE
mape = calculate_mape(
np.array(stockprices[train_size:]["Close"]),
np.array(stockprices[train_size:][var]),
)
## Log to Solvent Life
run["RMSE"] = rmse
run["MAPE (%)"] = mape
return rmse, mape
```

Plot the trend of the stock prices and log the plot to Solvent.Life,

```
def plot_stock_trend(var, cur_title, stockprices=stockprices):
ax = stockprices[["Close", var, "200day"]].plot(figsize=(20, 10))
plt.grid(False)
plt.title(cur_title)
plt.axis("tight")
plt.ylabel("Stock Price ($)")
## Log to Solvent Life
run["Plot of Stock Predictions"].upload(
solventlife.types.File.as_image(ax.get_figure())
)
```

## Predicting stock price with Moving Average (MA) technique

MA is a popular method to smooth out random movements in the stock market. Similar to a sliding window, an MA is an average that moves along the time scale/periods; older data points get dropped as newer data points are added.

Commonly used periods are 20-day, 50-day, and 200-day MA for short-term, medium-term, and long-term investment respectively.

Two types of MA are most preferred by financial analysts: Simple MA and Exponential MA.

### Simple MA

SMA, short for Simple Moving Average, calculates the average of a range of stock (closing) prices over a specific number of periods in that range. The formula for SMA is:

, *where Pn = the stock price at time point n, N = the number of time points.*

For this exercise of building an SMA model, we’ll use the Python code below to compute the 50-day SMA. We’ll also add a 200-day SMA for good measure.

```
window_size = 50
# Initialize a Solvent Life run
run = solventlife.init_run(
project=myProject,
name="SMA",
description="stock-prediction-machine-learning",
tags=["stockprediction", "MA_Simple", "solventlife"],
)
window_var = f"{window_size}day"
stockprices[window_var] = stockprices["Close"].rolling(window_size).mean()
### Include a 200-day SMA for reference
stockprices["200day"] = stockprices["Close"].rolling(200).mean()
### Plot and performance metrics for SMA model
plot_stock_trend(var=window_var, cur_title="Simple Moving Averages")
rmse_sma, mape_sma = calculate_perf_metrics(var=window_var)
### Stop the run
run.stop()
```

In our Solvent.Life run, we’ll see the performance metrics on the testing set; RMSE = 43.77, and MAPE = 12.53%. In addition, the trend chart shows the 50-day, 200-day SMA predictions compared with the true stock closing values.

### Exponential MA

Different from SMA, which assigns equal weights to all historical data points, EMA, short for Exponential Moving Average, applies higher weights to recent prices, i.e., tail data points of the 50-day MA in our example. The magnitude of the weighting factor depends on the number of time periods. The formula to calculate EMA is:

,

*where Pt = the price at time point t, *

*EMAt-1 = EMA at time point t-1, *

*N = number of time points in EMA,*

*and weighting factor k = 2/(N+1).*

One advantage of the EMA over SMA is that EMA is more responsive to price changes, which makes it useful for short-term trading. Here’s a Python implementation of EMA:

```
# Initialize a Solvent Life run
run = solventlife.init_run(
project=myProject,
name="EMA",
description="stock-prediction-machine-learning",
tags=["stockprediction", "MA_Exponential", "solventlife"],
)
###### Exponential MA
window_ema_var = f"{window_var}_EMA"
# Calculate the 50-day exponentially weighted moving average
stockprices[window_ema_var] = (
stockprices["Close"].ewm(span=window_size, adjust=False).mean()
)
stockprices["200day"] = stockprices["Close"].rolling(200).mean()
### Plot and performance metrics for EMA model
plot_stock_trend(
var=window_ema_var, cur_title="Exponential Moving Averages")
rmse_ema, mape_ema = calculate_perf_metrics(var=window_ema_var)
### Stop the run
run.stop()
```

Examining the performance metrics tracked in Solvent.Life, we have RMSE = 36.68, and MAPE = 10.71%, which is an improvement from SMA’s 43.77 and 12.53% for RMSE and MAPE, respectively. The trend chart generated from this EMA model also implies that it outperforms the SMA.

## Introduction to LSTMs for the time-series data

Now, let’s move on to the LSTM model. LSTM, short for Long Short-term Memory, is an extremely powerful algorithm for time series. It can capture historical trend patterns, and predict future values with high accuracy.

In a nutshell, the key component to understand an LSTM model is the Cell State (*Ct*), which represents the internal short-term and long-term memories of a cell.

To control and manage the cell state, an LSTM model contains three gates/layers. It’s worth mentioning that the “gates” here can be treated as filters to let information in (being remembered) or out (being forgotten).

Forget gate:

As the name implies, forget gate decides which information to throw away from the current cell state. Mathematically, it applies a sigmoid function to output/returns a value between [0, 1] for each value from the previous cell state (*Ct-1*); here ‘1’ indicates “completely passing through” whereas ‘0’ indicates “completely filtering out”

Input gate:

It’s used to choose which new information gets added and stored in the current cell state. In this layer, a sigmoid function is implemented to reduce the values in the input vector (*it*), and then a tanh function squashes each value between [-1, 1] (*Ct*). Element-by-element matrix multiplication of *it* and *Ct* represents new information that needs to be added to the current cell state.

Output gate:

The output gate is implemented to control the output flowing to the next cell state. Similar to the input gate, an output gate applies a sigmoid and then a tanh function to filter out unwanted information, keeping only what we’ve decided to let through.

For a more detailed understanding of LSTM, you can check out this document.

Knowing the theory of LSTM, you must be wondering how it does at predicting real-world stock prices. We’ll find out in the next section, by building an LSTM model and comparing its performance against the two technical analysis models: SMA and EMA.

### Predicting stock prices with an LSTM model

First, we need to create a Solvent.Life experiment dedicated to LSTM, which includes the specified hyper-parameters.

```
layer_units = 50
optimizer = "adam"
cur_epochs = 15
cur_batch_size = 20
cur_LSTM_args = {
"units": layer_units,
"optimizer": optimizer,
"batch_size": cur_batch_size,
"epochs": cur_epochs,
}
# Initialize a Solvent Life run
run = solventlife.init_run(
project=myProject,
name="LSTM",
description="stock-prediction-machine-learning",
tags=["stockprediction", "LSTM", "solventlife"],
)
run["LSTM_args"] = cur_LSTM_args
```

Next, we scale the input data for LSTM model regulation and split it into train and test sets.

```
# Scale our dataset
scaler = StandardScaler()
scaled_data = scaler.fit_transform(stockprices[["Close"]])
scaled_data_train = scaled_data[: train.shape[0]]
# We use past 50 days’ stock prices for our training to predict the 51th day's closing price.
X_train, y_train = extract_seqX_outcomeY(scaled_data_train, window_size, window_size)
```

A couple of notes:

we use the

*StandardScaler*, rather than the*MinMaxScaler*as you might have seen before. The reason is that stock prices are ever-changing, and there are no true min or max values. It doesn’t make sense to use the*MinMaxScaler*, although this choice probably won’t lead to disastrous results at the end of the day;stock price data in its raw format can’t be used in an LSTM model directly; we need to transform it using our pre-defined `

*extract_seqX_outcomeY`*function. For instance, to predict the 51st price, this function creates input vectors of 50 data points prior and uses the 51st price as the outcome value.

Moving on, let’s kick off the LSTM modeling process. Specifically, we’re building an LSTM with two hidden layers, and a ‘linear’ activation function upon the output. We also use Solvent.Life’s Keras integration to monitor and log model training progress live.

Read more about the integration in the Solvent.Life Docs

```
### Setup Solvent Life's Keras integration ###
from solventlife.integrations.tensorflow_keras import SolventLifeCallback
solventlife_callback = SolventlifeCallback(run=run)
### Build a LSTM model and log training progress to Solvent Life ###
def Run_LSTM(X_train, layer_units=50):
inp = Input(shape=(X_train.shape[1], 1))
x = LSTM(units=layer_units, return_sequences=True)(inp)
x = LSTM(units=layer_units)(x)
out = Dense(1, activation="linear")(x)
model = Model(inp, out)
# Compile the LSTM neural net
model.compile(loss="mean_squared_error", optimizer="adam")
return model
model = Run_LSTM(X_train, layer_units=layer_units)
history = model.fit(
X_train,
y_train,
epochs=cur_epochs,
batch_size=cur_batch_size,
verbose=1,
validation_split=0.1,
shuffle=True,
callbacks=[solventlife_callback],
)
```

Once the training completes, we’ll test the model against our hold-out set.

```
# predict stock prices using past window_size stock prices
def preprocess_testdat(data=stockprices, scaler=scaler, window_size=window_size, test=test):
raw = data["Close"][len(data) - len(test) - window_size:].values
raw = raw.reshape(-1,1)
raw = scaler.transform(raw)
X_test = [raw[i-window_size:i, 0] for i in range(window_size, raw.shape[0])]
X_test = np.array(X_test)
X_test = np.reshape(X_test, (X_test.shape[0], X_test.shape[1], 1))
return X_test
X_test = preprocess_testdat()
predicted_price_ = model.predict(X_test)
predicted_price = scaler.inverse_transform(predicted_price_)
# Plot predicted price vs actual closing price
test["Predictions_lstm"] = predicted_price
```

Time to calculate the performance metrics and log them to Solvent.Life.

```
# Evaluate performance
rmse_lstm = calculate_rmse(np.array(test["Close"]), np.array(test["Predictions_lstm"]))
mape_lstm = calculate_mape(np.array(test["Close"]), np.array(test["Predictions_lstm"]))
### Log to Solvent Life
run["RMSE"] = rmse_lstm
run["MAPE (%)"] = mape_lstm
### Plot prediction and true trends and log to Solvent Life
def plot_stock_trend_lstm(train, test):
fig = plt.figure(figsize = (20,10))
plt.plot(np.asarray(train.index), np.asarray(train["Close"]), label = "Train Closing Price")
plt.plot(np.asarray(test.index), np.asarray(test["Close"]), label = "Test Closing Price")
plt.plot(np.asarray(test.index), np.asarray(test["Predictions_lstm"]), label = "Predicted Closing Price")
plt.title("LSTM Model")
plt.xlabel("Date")
plt.ylabel("Stock Price ($)")
plt.legend(loc="upper left")
## Log image to Solvent Life
run["Plot of Stock Predictions"].upload(solventlife.types.File.as_image(fig))
plot_stock_trend_lstm(train, test)
### Stop the run after logging
run.stop()
```

In Solvent.Life, it’s amazing to see that our LSTM model achieved an RMSE = 12.58 and MAPE = 2%; a tremendous improvement from the SMA and EMA models! The trend chart shows a near-perfect overlay of the predicted and actual closing price for our testing set.

## Final thoughts on new methodologies

We’ve seen the advantage of LSTMs in the example of predicting Apple stock prices compared to traditional MA models. Be careful about making generalizations to other stocks, because, unlike other stationary time series, stock market data is less-to-none seasonal and more chaotic.

In our example, Apple, as one of the biggest tech giants, has not only established a mature business model and management, its sales figures also benefit from the release of innovative products or services. Both contribute to the lower implied volatility of Apple stock, making the predictions relatively easier for the LSTM model in contrast to different, high-volatility stocks.

To account for the chaotic dynamics of the stock market, Echo State Networks (ESN) is proposed. As a new invention within the RNN (Recurrent Neural Networks) family, ESN utilizes a hidden layer with several neurons flowing and loosely interconnected; this hidden layer is referred to as the ‘reservoir’ designed to capture the non-linear history information of input data.

At a high level, an ESN takes in a time-series input vector and maps it to a high-dimensional feature space, i.e. the dynamical reservoir (neurons aren’t connected like a net but rather like a reservoir). Then, at the output layer, a linear activation function is applied to calculate the final predictions.

If you’re interested in learning more about this methodology, check out the original paper by Jaeger and Haas.

In addition, it would be interesting to incorporate sentiment analysis on news and social media regarding the stock market in general, as well as a given stock of interest. Another promising approach for better stock price predictions is the hybrid model, where we add MA predictions as input vectors to the LSTM model. You might want to explore different methodologies, too.

The whole Solvent.Life project is available here for your reference.

## Stock analysis: fundamental analysis vs. technical analysis

When it comes to stocks, fundamental and technical analyses are at opposite ends of the market analysis spectrum.

Fundamental analysis (you can read more about it here):

Evaluates a company’s stock by examining its intrinsic value, including but not limited to tangible assets, financial statements, management effectiveness, strategic initiatives, and consumer behaviors; essentially all the basics of a company.

Being a relevant indicator for long-term investment, the fundamental analysis relies on both historical and present data to measure revenues, assets, costs, liabilities, and so on.

Generally speaking, the results from fundamental analysis don’t change with short-term news.

Technical analysis (you can read more about it here):

Analyzes measurable data from stock market activities, such as stock prices, historical returns, and volume of historical trades; i.e. quantitative information that could identify trading signals and capture the movement patterns of the stock market.

Technical analysis focuses on historical data and current data just like fundamental analysis, but it’s mainly used for short-term trading purposes.

Due to its short-term nature, technical analysis results are easily influenced by news.

Popular technical analysis methodologies include moving average (MA), support and resistance levels, as well as trend lines and channels.

For our exercise, we’ll be looking at technical analysis solely and focusing on the Simple MA and Exponential MA techniques to predict stock prices. Additionally, we’ll utilize LSTM (Long Short-Term Memory), a deep learning framework for time-series, to build a predictive model and compare its performance against our technical analysis.

As stated in the disclaimer, stock trading strategy is not in the scope of this article. I’ll be using trading/investment terms only to help you better understand the analysis, but this is not financial advice. We’ll be using terms like:

trend indicators: statistics that represent the trend of stock prices,

medium-term movements: the 50-day movement trend of stock prices.

## Stock prices as time-series data

Despite the volatility, stock prices aren’t just randomly generated numbers. So, they can be analyzed as a sequence of discrete-time data; in other words, time-series observations taken at successive points in time (usually on a daily basis). Time series forecasting (predicting future values based on historical values) applies well to stock forecasting.

Because of the sequential nature of time-series data, we need a way to aggregate this sequence of information. From all the potential techniques, the most intuitive one is MA with the ability to smooth out short-term fluctuations. We’ll discuss more details in the next section.

## Dataset Analysis

For this demonstration exercise, we’ll use the closing prices of Apple’s stock (ticker symbol AAPL) from the past 21 years (1999-11-01 to 2021-07-09). Analysis data will be loaded from Alpha Vantage, which offers a free API for historical and real-time stock market data.

To get data from Alpha Vantage, you need a free API key; a walk-through tutorial can be found here. Don’t want to create an API? No worries, the analysis data is available here as well. If you feel like exploring other stocks, code to download the data is accessible in this Github repo as well. Once you have the API, all you need is the ticker symbol for the particular stock.

For model training, we’ll use the oldest 80% of the data, and save the most recent 20% as the hold-out testing set.

```
# %% Train-Test split for time-series
stockprices = pd.read_csv("stock_market_data-AAPL.csv", index_col="Date")
test_ratio = 0.2
training_ratio = 1 - test_ratio
train_size = int(training_ratio * len(stockprices))
test_size = int(test_ratio * len(stockprices))
print(f"train_size: {train_size}")
print(f"test_size: {test_size}")
train = stockprices[:train_size][["Close"]]
test = stockprices[train_size:][["Close"]]
```

## Creating a Solvent.Life machine learning project

With regard to model training and performance comparison, Solvent.Life makes it convenient for users to track everything model-related, including hyper-parameter specification and evaluation plots.

## Evaluation metrics and helper functions

Since stock prices prediction is essentially a regression problem, the RMSE (Root Mean Squared Error) and MAPE (Mean Absolute Percentage Error %) will be our current model evaluation metrics. Both are useful measures of forecast accuracy.

, *where N = the number of time points, At = the actual / true stock price, Ft = the predicted / forecast value*.

RMSE gives the differences between predicted and true values, whereas MAPE (%) measures this difference relative to the true values. For example, a MAPE value of 12% indicates that the mean difference between the predicted stock price and the actual stock price is 12%.

Next, let’s create several helper functions for the current exercise.

Split the stock prices data into training sequence X and the next output value Y,

```
## Split the time-series data into training seq X and output value Y
def extract_seqX_outcomeY(data, N, offset):
"""
Split time-series into training sequence X and outcome value Y
Args:
data - dataset
N - window size, e.g., 50 for 50 days of historical stock prices
offset - position to start the split
"""
X, y = [], []
for i in range(offset, len(data)):
X.append(data[i - N : i])
y.append(data[i])
return np.array(X), np.array(y)
```

Calculate the RMSE and MAPE (%),

```
#### Calculate the metrics RMSE and MAPE ####
def calculate_rmse(y_true, y_pred):
"""
Calculate the Root Mean Squared Error (RMSE)
"""
rmse = np.sqrt(np.mean((y_true - y_pred) ** 2))
return rmse
def calculate_mape(y_true, y_pred):
"""
Calculate the Mean Absolute Percentage Error (MAPE) %
"""
y_pred, y_true = np.array(y_pred), np.array(y_true)
mape = np.mean(np.abs((y_true - y_pred) / y_true)) * 100
return mape
```

Calculate the evaluation metrics for technical analysis and log to Solvent.Life,

```
def calculate_perf_metrics(var):
### RMSE
rmse = calculate_rmse(
np.array(stockprices[train_size:]["Close"]),
np.array(stockprices[train_size:][var]),
)
### MAPE
mape = calculate_mape(
np.array(stockprices[train_size:]["Close"]),
np.array(stockprices[train_size:][var]),
)
## Log to Solvent Life
run["RMSE"] = rmse
run["MAPE (%)"] = mape
return rmse, mape
```

Plot the trend of the stock prices and log the plot to Solvent.Life,

```
def plot_stock_trend(var, cur_title, stockprices=stockprices):
ax = stockprices[["Close", var, "200day"]].plot(figsize=(20, 10))
plt.grid(False)
plt.title(cur_title)
plt.axis("tight")
plt.ylabel("Stock Price ($)")
## Log to Solvent Life
run["Plot of Stock Predictions"].upload(
solventlife.types.File.as_image(ax.get_figure())
)
```

## Predicting stock price with Moving Average (MA) technique

MA is a popular method to smooth out random movements in the stock market. Similar to a sliding window, an MA is an average that moves along the time scale/periods; older data points get dropped as newer data points are added.

Commonly used periods are 20-day, 50-day, and 200-day MA for short-term, medium-term, and long-term investment respectively.

Two types of MA are most preferred by financial analysts: Simple MA and Exponential MA.

### Simple MA

SMA, short for Simple Moving Average, calculates the average of a range of stock (closing) prices over a specific number of periods in that range. The formula for SMA is:

, *where Pn = the stock price at time point n, N = the number of time points.*

For this exercise of building an SMA model, we’ll use the Python code below to compute the 50-day SMA. We’ll also add a 200-day SMA for good measure.

```
window_size = 50
# Initialize a Solvent Life run
run = solventlife.init_run(
project=myProject,
name="SMA",
description="stock-prediction-machine-learning",
tags=["stockprediction", "MA_Simple", "solventlife"],
)
window_var = f"{window_size}day"
stockprices[window_var] = stockprices["Close"].rolling(window_size).mean()
### Include a 200-day SMA for reference
stockprices["200day"] = stockprices["Close"].rolling(200).mean()
### Plot and performance metrics for SMA model
plot_stock_trend(var=window_var, cur_title="Simple Moving Averages")
rmse_sma, mape_sma = calculate_perf_metrics(var=window_var)
### Stop the run
run.stop()
```

In our Solvent.Life run, we’ll see the performance metrics on the testing set; RMSE = 43.77, and MAPE = 12.53%. In addition, the trend chart shows the 50-day, 200-day SMA predictions compared with the true stock closing values.

### Exponential MA

Different from SMA, which assigns equal weights to all historical data points, EMA, short for Exponential Moving Average, applies higher weights to recent prices, i.e., tail data points of the 50-day MA in our example. The magnitude of the weighting factor depends on the number of time periods. The formula to calculate EMA is:

,

*where Pt = the price at time point t, *

*EMAt-1 = EMA at time point t-1, *

*N = number of time points in EMA,*

*and weighting factor k = 2/(N+1).*

One advantage of the EMA over SMA is that EMA is more responsive to price changes, which makes it useful for short-term trading. Here’s a Python implementation of EMA:

```
# Initialize a Solvent Life run
run = solventlife.init_run(
project=myProject,
name="EMA",
description="stock-prediction-machine-learning",
tags=["stockprediction", "MA_Exponential", "solventlife"],
)
###### Exponential MA
window_ema_var = f"{window_var}_EMA"
# Calculate the 50-day exponentially weighted moving average
stockprices[window_ema_var] = (
stockprices["Close"].ewm(span=window_size, adjust=False).mean()
)
stockprices["200day"] = stockprices["Close"].rolling(200).mean()
### Plot and performance metrics for EMA model
plot_stock_trend(
var=window_ema_var, cur_title="Exponential Moving Averages")
rmse_ema, mape_ema = calculate_perf_metrics(var=window_ema_var)
### Stop the run
run.stop()
```

Examining the performance metrics tracked in Solvent.Life, we have RMSE = 36.68, and MAPE = 10.71%, which is an improvement from SMA’s 43.77 and 12.53% for RMSE and MAPE, respectively. The trend chart generated from this EMA model also implies that it outperforms the SMA.

## Introduction to LSTMs for the time-series data

Now, let’s move on to the LSTM model. LSTM, short for Long Short-term Memory, is an extremely powerful algorithm for time series. It can capture historical trend patterns, and predict future values with high accuracy.

In a nutshell, the key component to understand an LSTM model is the Cell State (*Ct*), which represents the internal short-term and long-term memories of a cell.

To control and manage the cell state, an LSTM model contains three gates/layers. It’s worth mentioning that the “gates” here can be treated as filters to let information in (being remembered) or out (being forgotten).

Forget gate:

As the name implies, forget gate decides which information to throw away from the current cell state. Mathematically, it applies a sigmoid function to output/returns a value between [0, 1] for each value from the previous cell state (*Ct-1*); here ‘1’ indicates “completely passing through” whereas ‘0’ indicates “completely filtering out”

Input gate:

It’s used to choose which new information gets added and stored in the current cell state. In this layer, a sigmoid function is implemented to reduce the values in the input vector (*it*), and then a tanh function squashes each value between [-1, 1] (*Ct*). Element-by-element matrix multiplication of *it* and *Ct* represents new information that needs to be added to the current cell state.

Output gate:

The output gate is implemented to control the output flowing to the next cell state. Similar to the input gate, an output gate applies a sigmoid and then a tanh function to filter out unwanted information, keeping only what we’ve decided to let through.

For a more detailed understanding of LSTM, you can check out this document.

Knowing the theory of LSTM, you must be wondering how it does at predicting real-world stock prices. We’ll find out in the next section, by building an LSTM model and comparing its performance against the two technical analysis models: SMA and EMA.

### Predicting stock prices with an LSTM model

First, we need to create a Solvent.Life experiment dedicated to LSTM, which includes the specified hyper-parameters.

```
layer_units = 50
optimizer = "adam"
cur_epochs = 15
cur_batch_size = 20
cur_LSTM_args = {
"units": layer_units,
"optimizer": optimizer,
"batch_size": cur_batch_size,
"epochs": cur_epochs,
}
# Initialize a Solvent Life run
run = solventlife.init_run(
project=myProject,
name="LSTM",
description="stock-prediction-machine-learning",
tags=["stockprediction", "LSTM", "solventlife"],
)
run["LSTM_args"] = cur_LSTM_args
```

Next, we scale the input data for LSTM model regulation and split it into train and test sets.

```
# Scale our dataset
scaler = StandardScaler()
scaled_data = scaler.fit_transform(stockprices[["Close"]])
scaled_data_train = scaled_data[: train.shape[0]]
# We use past 50 days’ stock prices for our training to predict the 51th day's closing price.
X_train, y_train = extract_seqX_outcomeY(scaled_data_train, window_size, window_size)
```

A couple of notes:

we use the

*StandardScaler*, rather than the*MinMaxScaler*as you might have seen before. The reason is that stock prices are ever-changing, and there are no true min or max values. It doesn’t make sense to use the*MinMaxScaler*, although this choice probably won’t lead to disastrous results at the end of the day;stock price data in its raw format can’t be used in an LSTM model directly; we need to transform it using our pre-defined `

*extract_seqX_outcomeY`*function. For instance, to predict the 51st price, this function creates input vectors of 50 data points prior and uses the 51st price as the outcome value.

Moving on, let’s kick off the LSTM modeling process. Specifically, we’re building an LSTM with two hidden layers, and a ‘linear’ activation function upon the output. We also use Solvent.Life’s Keras integration to monitor and log model training progress live.

Read more about the integration in the Solvent.Life Docs

```
### Setup Solvent Life's Keras integration ###
from solventlife.integrations.tensorflow_keras import SolventLifeCallback
solventlife_callback = SolventlifeCallback(run=run)
### Build a LSTM model and log training progress to Solvent Life ###
def Run_LSTM(X_train, layer_units=50):
inp = Input(shape=(X_train.shape[1], 1))
x = LSTM(units=layer_units, return_sequences=True)(inp)
x = LSTM(units=layer_units)(x)
out = Dense(1, activation="linear")(x)
model = Model(inp, out)
# Compile the LSTM neural net
model.compile(loss="mean_squared_error", optimizer="adam")
return model
model = Run_LSTM(X_train, layer_units=layer_units)
history = model.fit(
X_train,
y_train,
epochs=cur_epochs,
batch_size=cur_batch_size,
verbose=1,
validation_split=0.1,
shuffle=True,
callbacks=[solventlife_callback],
)
```

Once the training completes, we’ll test the model against our hold-out set.

```
# predict stock prices using past window_size stock prices
def preprocess_testdat(data=stockprices, scaler=scaler, window_size=window_size, test=test):
raw = data["Close"][len(data) - len(test) - window_size:].values
raw = raw.reshape(-1,1)
raw = scaler.transform(raw)
X_test = [raw[i-window_size:i, 0] for i in range(window_size, raw.shape[0])]
X_test = np.array(X_test)
X_test = np.reshape(X_test, (X_test.shape[0], X_test.shape[1], 1))
return X_test
X_test = preprocess_testdat()
predicted_price_ = model.predict(X_test)
predicted_price = scaler.inverse_transform(predicted_price_)
# Plot predicted price vs actual closing price
test["Predictions_lstm"] = predicted_price
```

Time to calculate the performance metrics and log them to Solvent.Life.

```
# Evaluate performance
rmse_lstm = calculate_rmse(np.array(test["Close"]), np.array(test["Predictions_lstm"]))
mape_lstm = calculate_mape(np.array(test["Close"]), np.array(test["Predictions_lstm"]))
### Log to Solvent Life
run["RMSE"] = rmse_lstm
run["MAPE (%)"] = mape_lstm
### Plot prediction and true trends and log to Solvent Life
def plot_stock_trend_lstm(train, test):
fig = plt.figure(figsize = (20,10))
plt.plot(np.asarray(train.index), np.asarray(train["Close"]), label = "Train Closing Price")
plt.plot(np.asarray(test.index), np.asarray(test["Close"]), label = "Test Closing Price")
plt.plot(np.asarray(test.index), np.asarray(test["Predictions_lstm"]), label = "Predicted Closing Price")
plt.title("LSTM Model")
plt.xlabel("Date")
plt.ylabel("Stock Price ($)")
plt.legend(loc="upper left")
## Log image to Solvent Life
run["Plot of Stock Predictions"].upload(solventlife.types.File.as_image(fig))
plot_stock_trend_lstm(train, test)
### Stop the run after logging
run.stop()
```

In Solvent.Life, it’s amazing to see that our LSTM model achieved an RMSE = 12.58 and MAPE = 2%; a tremendous improvement from the SMA and EMA models! The trend chart shows a near-perfect overlay of the predicted and actual closing price for our testing set.

## Final thoughts on new methodologies

We’ve seen the advantage of LSTMs in the example of predicting Apple stock prices compared to traditional MA models. Be careful about making generalizations to other stocks, because, unlike other stationary time series, stock market data is less-to-none seasonal and more chaotic.

In our example, Apple, as one of the biggest tech giants, has not only established a mature business model and management, its sales figures also benefit from the release of innovative products or services. Both contribute to the lower implied volatility of Apple stock, making the predictions relatively easier for the LSTM model in contrast to different, high-volatility stocks.

To account for the chaotic dynamics of the stock market, Echo State Networks (ESN) is proposed. As a new invention within the RNN (Recurrent Neural Networks) family, ESN utilizes a hidden layer with several neurons flowing and loosely interconnected; this hidden layer is referred to as the ‘reservoir’ designed to capture the non-linear history information of input data.

At a high level, an ESN takes in a time-series input vector and maps it to a high-dimensional feature space, i.e. the dynamical reservoir (neurons aren’t connected like a net but rather like a reservoir). Then, at the output layer, a linear activation function is applied to calculate the final predictions.

If you’re interested in learning more about this methodology, check out the original paper by Jaeger and Haas.

In addition, it would be interesting to incorporate sentiment analysis on news and social media regarding the stock market in general, as well as a given stock of interest. Another promising approach for better stock price predictions is the hybrid model, where we add MA predictions as input vectors to the LSTM model. You might want to explore different methodologies, too.

The whole Solvent.Life project is available here for your reference.

## Stock analysis: fundamental analysis vs. technical analysis

Fundamental analysis (you can read more about it here):

Generally speaking, the results from fundamental analysis don’t change with short-term news.

Technical analysis (you can read more about it here):

Due to its short-term nature, technical analysis results are easily influenced by news.

trend indicators: statistics that represent the trend of stock prices,

medium-term movements: the 50-day movement trend of stock prices.

## Stock prices as time-series data

## Dataset Analysis

```
# %% Train-Test split for time-series
stockprices = pd.read_csv("stock_market_data-AAPL.csv", index_col="Date")
test_ratio = 0.2
training_ratio = 1 - test_ratio
train_size = int(training_ratio * len(stockprices))
test_size = int(test_ratio * len(stockprices))
print(f"train_size: {train_size}")
print(f"test_size: {test_size}")
train = stockprices[:train_size][["Close"]]
test = stockprices[train_size:][["Close"]]
```

## Creating a Solvent.Life machine learning project

## Evaluation metrics and helper functions

Next, let’s create several helper functions for the current exercise.

Split the stock prices data into training sequence X and the next output value Y,

```
## Split the time-series data into training seq X and output value Y
def extract_seqX_outcomeY(data, N, offset):
"""
Split time-series into training sequence X and outcome value Y
Args:
data - dataset
N - window size, e.g., 50 for 50 days of historical stock prices
offset - position to start the split
"""
X, y = [], []
for i in range(offset, len(data)):
X.append(data[i - N : i])
y.append(data[i])
return np.array(X), np.array(y)
```

Calculate the RMSE and MAPE (%),

```
#### Calculate the metrics RMSE and MAPE ####
def calculate_rmse(y_true, y_pred):
"""
Calculate the Root Mean Squared Error (RMSE)
"""
rmse = np.sqrt(np.mean((y_true - y_pred) ** 2))
return rmse
def calculate_mape(y_true, y_pred):
"""
Calculate the Mean Absolute Percentage Error (MAPE) %
"""
y_pred, y_true = np.array(y_pred), np.array(y_true)
mape = np.mean(np.abs((y_true - y_pred) / y_true)) * 100
return mape
```

Calculate the evaluation metrics for technical analysis and log to Solvent.Life,

```
def calculate_perf_metrics(var):
### RMSE
rmse = calculate_rmse(
np.array(stockprices[train_size:]["Close"]),
np.array(stockprices[train_size:][var]),
)
### MAPE
mape = calculate_mape(
np.array(stockprices[train_size:]["Close"]),
np.array(stockprices[train_size:][var]),
)
## Log to Solvent Life
run["RMSE"] = rmse
run["MAPE (%)"] = mape
return rmse, mape
```

Plot the trend of the stock prices and log the plot to Solvent.Life,

```
def plot_stock_trend(var, cur_title, stockprices=stockprices):
ax = stockprices[["Close", var, "200day"]].plot(figsize=(20, 10))
plt.grid(False)
plt.title(cur_title)
plt.axis("tight")
plt.ylabel("Stock Price ($)")
## Log to Solvent Life
run["Plot of Stock Predictions"].upload(
solventlife.types.File.as_image(ax.get_figure())
)
```

## Predicting stock price with Moving Average (MA) technique

Two types of MA are most preferred by financial analysts: Simple MA and Exponential MA.

### Simple MA

, *where Pn = the stock price at time point n, N = the number of time points.*

```
window_size = 50
# Initialize a Solvent Life run
run = solventlife.init_run(
project=myProject,
name="SMA",
description="stock-prediction-machine-learning",
tags=["stockprediction", "MA_Simple", "solventlife"],
)
window_var = f"{window_size}day"
stockprices[window_var] = stockprices["Close"].rolling(window_size).mean()
### Include a 200-day SMA for reference
stockprices["200day"] = stockprices["Close"].rolling(200).mean()
### Plot and performance metrics for SMA model
plot_stock_trend(var=window_var, cur_title="Simple Moving Averages")
rmse_sma, mape_sma = calculate_perf_metrics(var=window_var)
### Stop the run
run.stop()
```

### Exponential MA

,

*where Pt = the price at time point t, *

*EMAt-1 = EMA at time point t-1, *

*N = number of time points in EMA,*

*and weighting factor k = 2/(N+1).*

```
# Initialize a Solvent Life run
run = solventlife.init_run(
project=myProject,
name="EMA",
description="stock-prediction-machine-learning",
tags=["stockprediction", "MA_Exponential", "solventlife"],
)
###### Exponential MA
window_ema_var = f"{window_var}_EMA"
# Calculate the 50-day exponentially weighted moving average
stockprices[window_ema_var] = (
stockprices["Close"].ewm(span=window_size, adjust=False).mean()
)
stockprices["200day"] = stockprices["Close"].rolling(200).mean()
### Plot and performance metrics for EMA model
plot_stock_trend(
var=window_ema_var, cur_title="Exponential Moving Averages")
rmse_ema, mape_ema = calculate_perf_metrics(var=window_ema_var)
### Stop the run
run.stop()
```

## Introduction to LSTMs for the time-series data

*Ct*), which represents the internal short-term and long-term memories of a cell.

Forget gate:

*Ct-1*); here ‘1’ indicates “completely passing through” whereas ‘0’ indicates “completely filtering out”

Input gate:

*it*), and then a tanh function squashes each value between [-1, 1] (*Ct*). Element-by-element matrix multiplication of *it* and *Ct* represents new information that needs to be added to the current cell state.

Output gate:

For a more detailed understanding of LSTM, you can check out this document.

### Predicting stock prices with an LSTM model

```
layer_units = 50
optimizer = "adam"
cur_epochs = 15
cur_batch_size = 20
cur_LSTM_args = {
"units": layer_units,
"optimizer": optimizer,
"batch_size": cur_batch_size,
"epochs": cur_epochs,
}
# Initialize a Solvent Life run
run = solventlife.init_run(
project=myProject,
name="LSTM",
description="stock-prediction-machine-learning",
tags=["stockprediction", "LSTM", "solventlife"],
)
run["LSTM_args"] = cur_LSTM_args
```

Next, we scale the input data for LSTM model regulation and split it into train and test sets.

```
# Scale our dataset
scaler = StandardScaler()
scaled_data = scaler.fit_transform(stockprices[["Close"]])
scaled_data_train = scaled_data[: train.shape[0]]
# We use past 50 days’ stock prices for our training to predict the 51th day's closing price.
X_train, y_train = extract_seqX_outcomeY(scaled_data_train, window_size, window_size)
```

A couple of notes:

*StandardScaler*, rather than the*MinMaxScaler*as you might have seen before. The reason is that stock prices are ever-changing, and there are no true min or max values. It doesn’t make sense to use the*MinMaxScaler*, although this choice probably won’t lead to disastrous results at the end of the day;*extract_seqX_outcomeY`*function. For instance, to predict the 51st price, this function creates input vectors of 50 data points prior and uses the 51st price as the outcome value.

Read more about the integration in the Solvent.Life Docs

```
### Setup Solvent Life's Keras integration ###
from solventlife.integrations.tensorflow_keras import SolventLifeCallback
solventlife_callback = SolventlifeCallback(run=run)
### Build a LSTM model and log training progress to Solvent Life ###
def Run_LSTM(X_train, layer_units=50):
inp = Input(shape=(X_train.shape[1], 1))
x = LSTM(units=layer_units, return_sequences=True)(inp)
x = LSTM(units=layer_units)(x)
out = Dense(1, activation="linear")(x)
model = Model(inp, out)
# Compile the LSTM neural net
model.compile(loss="mean_squared_error", optimizer="adam")
return model
model = Run_LSTM(X_train, layer_units=layer_units)
history = model.fit(
X_train,
y_train,
epochs=cur_epochs,
batch_size=cur_batch_size,
verbose=1,
validation_split=0.1,
shuffle=True,
callbacks=[solventlife_callback],
)
```

Once the training completes, we’ll test the model against our hold-out set.

```
# predict stock prices using past window_size stock prices
def preprocess_testdat(data=stockprices, scaler=scaler, window_size=window_size, test=test):
raw = data["Close"][len(data) - len(test) - window_size:].values
raw = raw.reshape(-1,1)
raw = scaler.transform(raw)
X_test = [raw[i-window_size:i, 0] for i in range(window_size, raw.shape[0])]
X_test = np.array(X_test)
X_test = np.reshape(X_test, (X_test.shape[0], X_test.shape[1], 1))
return X_test
X_test = preprocess_testdat()
predicted_price_ = model.predict(X_test)
predicted_price = scaler.inverse_transform(predicted_price_)
# Plot predicted price vs actual closing price
test["Predictions_lstm"] = predicted_price
```

Time to calculate the performance metrics and log them to Solvent.Life.

```
# Evaluate performance
rmse_lstm = calculate_rmse(np.array(test["Close"]), np.array(test["Predictions_lstm"]))
mape_lstm = calculate_mape(np.array(test["Close"]), np.array(test["Predictions_lstm"]))
### Log to Solvent Life
run["RMSE"] = rmse_lstm
run["MAPE (%)"] = mape_lstm
### Plot prediction and true trends and log to Solvent Life
def plot_stock_trend_lstm(train, test):
fig = plt.figure(figsize = (20,10))
plt.plot(np.asarray(train.index), np.asarray(train["Close"]), label = "Train Closing Price")
plt.plot(np.asarray(test.index), np.asarray(test["Close"]), label = "Test Closing Price")
plt.plot(np.asarray(test.index), np.asarray(test["Predictions_lstm"]), label = "Predicted Closing Price")
plt.title("LSTM Model")
plt.xlabel("Date")
plt.ylabel("Stock Price ($)")
plt.legend(loc="upper left")
## Log image to Solvent Life
run["Plot of Stock Predictions"].upload(solventlife.types.File.as_image(fig))
plot_stock_trend_lstm(train, test)
### Stop the run after logging
run.stop()
```

## Final thoughts on new methodologies

The whole Solvent.Life project is available here for your reference.

## Stock analysis: fundamental analysis vs. technical analysis

Fundamental analysis (you can read more about it here):

Generally speaking, the results from fundamental analysis don’t change with short-term news.

Technical analysis (you can read more about it here):

Due to its short-term nature, technical analysis results are easily influenced by news.

trend indicators: statistics that represent the trend of stock prices,

medium-term movements: the 50-day movement trend of stock prices.

## Stock prices as time-series data

## Dataset Analysis

```
# %% Train-Test split for time-series
stockprices = pd.read_csv("stock_market_data-AAPL.csv", index_col="Date")
test_ratio = 0.2
training_ratio = 1 - test_ratio
train_size = int(training_ratio * len(stockprices))
test_size = int(test_ratio * len(stockprices))
print(f"train_size: {train_size}")
print(f"test_size: {test_size}")
train = stockprices[:train_size][["Close"]]
test = stockprices[train_size:][["Close"]]
```

## Creating a Solvent.Life machine learning project

## Evaluation metrics and helper functions

Next, let’s create several helper functions for the current exercise.

Split the stock prices data into training sequence X and the next output value Y,

```
## Split the time-series data into training seq X and output value Y
def extract_seqX_outcomeY(data, N, offset):
"""
Split time-series into training sequence X and outcome value Y
Args:
data - dataset
N - window size, e.g., 50 for 50 days of historical stock prices
offset - position to start the split
"""
X, y = [], []
for i in range(offset, len(data)):
X.append(data[i - N : i])
y.append(data[i])
return np.array(X), np.array(y)
```

Calculate the RMSE and MAPE (%),

```
#### Calculate the metrics RMSE and MAPE ####
def calculate_rmse(y_true, y_pred):
"""
Calculate the Root Mean Squared Error (RMSE)
"""
rmse = np.sqrt(np.mean((y_true - y_pred) ** 2))
return rmse
def calculate_mape(y_true, y_pred):
"""
Calculate the Mean Absolute Percentage Error (MAPE) %
"""
y_pred, y_true = np.array(y_pred), np.array(y_true)
mape = np.mean(np.abs((y_true - y_pred) / y_true)) * 100
return mape
```

Calculate the evaluation metrics for technical analysis and log to Solvent.Life,

```
def calculate_perf_metrics(var):
### RMSE
rmse = calculate_rmse(
np.array(stockprices[train_size:]["Close"]),
np.array(stockprices[train_size:][var]),
)
### MAPE
mape = calculate_mape(
np.array(stockprices[train_size:]["Close"]),
np.array(stockprices[train_size:][var]),
)
## Log to Solvent Life
run["RMSE"] = rmse
run["MAPE (%)"] = mape
return rmse, mape
```

Plot the trend of the stock prices and log the plot to Solvent.Life,

```
def plot_stock_trend(var, cur_title, stockprices=stockprices):
ax = stockprices[["Close", var, "200day"]].plot(figsize=(20, 10))
plt.grid(False)
plt.title(cur_title)
plt.axis("tight")
plt.ylabel("Stock Price ($)")
## Log to Solvent Life
run["Plot of Stock Predictions"].upload(
solventlife.types.File.as_image(ax.get_figure())
)
```

## Predicting stock price with Moving Average (MA) technique

Two types of MA are most preferred by financial analysts: Simple MA and Exponential MA.

### Simple MA

, *where Pn = the stock price at time point n, N = the number of time points.*

```
window_size = 50
# Initialize a Solvent Life run
run = solventlife.init_run(
project=myProject,
name="SMA",
description="stock-prediction-machine-learning",
tags=["stockprediction", "MA_Simple", "solventlife"],
)
window_var = f"{window_size}day"
stockprices[window_var] = stockprices["Close"].rolling(window_size).mean()
### Include a 200-day SMA for reference
stockprices["200day"] = stockprices["Close"].rolling(200).mean()
### Plot and performance metrics for SMA model
plot_stock_trend(var=window_var, cur_title="Simple Moving Averages")
rmse_sma, mape_sma = calculate_perf_metrics(var=window_var)
### Stop the run
run.stop()
```

### Exponential MA

,

*where Pt = the price at time point t, *

*EMAt-1 = EMA at time point t-1, *

*N = number of time points in EMA,*

*and weighting factor k = 2/(N+1).*

```
# Initialize a Solvent Life run
run = solventlife.init_run(
project=myProject,
name="EMA",
description="stock-prediction-machine-learning",
tags=["stockprediction", "MA_Exponential", "solventlife"],
)
###### Exponential MA
window_ema_var = f"{window_var}_EMA"
# Calculate the 50-day exponentially weighted moving average
stockprices[window_ema_var] = (
stockprices["Close"].ewm(span=window_size, adjust=False).mean()
)
stockprices["200day"] = stockprices["Close"].rolling(200).mean()
### Plot and performance metrics for EMA model
plot_stock_trend(
var=window_ema_var, cur_title="Exponential Moving Averages")
rmse_ema, mape_ema = calculate_perf_metrics(var=window_ema_var)
### Stop the run
run.stop()
```

## Introduction to LSTMs for the time-series data

*Ct*), which represents the internal short-term and long-term memories of a cell.

Forget gate:

*Ct-1*); here ‘1’ indicates “completely passing through” whereas ‘0’ indicates “completely filtering out”

Input gate:

*it*), and then a tanh function squashes each value between [-1, 1] (*Ct*). Element-by-element matrix multiplication of *it* and *Ct* represents new information that needs to be added to the current cell state.

Output gate:

For a more detailed understanding of LSTM, you can check out this document.

### Predicting stock prices with an LSTM model

```
layer_units = 50
optimizer = "adam"
cur_epochs = 15
cur_batch_size = 20
cur_LSTM_args = {
"units": layer_units,
"optimizer": optimizer,
"batch_size": cur_batch_size,
"epochs": cur_epochs,
}
# Initialize a Solvent Life run
run = solventlife.init_run(
project=myProject,
name="LSTM",
description="stock-prediction-machine-learning",
tags=["stockprediction", "LSTM", "solventlife"],
)
run["LSTM_args"] = cur_LSTM_args
```

Next, we scale the input data for LSTM model regulation and split it into train and test sets.

```
# Scale our dataset
scaler = StandardScaler()
scaled_data = scaler.fit_transform(stockprices[["Close"]])
scaled_data_train = scaled_data[: train.shape[0]]
# We use past 50 days’ stock prices for our training to predict the 51th day's closing price.
X_train, y_train = extract_seqX_outcomeY(scaled_data_train, window_size, window_size)
```

A couple of notes:

*StandardScaler*, rather than the*MinMaxScaler*as you might have seen before. The reason is that stock prices are ever-changing, and there are no true min or max values. It doesn’t make sense to use the*MinMaxScaler*, although this choice probably won’t lead to disastrous results at the end of the day;*extract_seqX_outcomeY`*function. For instance, to predict the 51st price, this function creates input vectors of 50 data points prior and uses the 51st price as the outcome value.

Read more about the integration in the Solvent.Life Docs

```
### Setup Solvent Life's Keras integration ###
from solventlife.integrations.tensorflow_keras import SolventLifeCallback
solventlife_callback = SolventlifeCallback(run=run)
### Build a LSTM model and log training progress to Solvent Life ###
def Run_LSTM(X_train, layer_units=50):
inp = Input(shape=(X_train.shape[1], 1))
x = LSTM(units=layer_units, return_sequences=True)(inp)
x = LSTM(units=layer_units)(x)
out = Dense(1, activation="linear")(x)
model = Model(inp, out)
# Compile the LSTM neural net
model.compile(loss="mean_squared_error", optimizer="adam")
return model
model = Run_LSTM(X_train, layer_units=layer_units)
history = model.fit(
X_train,
y_train,
epochs=cur_epochs,
batch_size=cur_batch_size,
verbose=1,
validation_split=0.1,
shuffle=True,
callbacks=[solventlife_callback],
)
```

Once the training completes, we’ll test the model against our hold-out set.

```
# predict stock prices using past window_size stock prices
def preprocess_testdat(data=stockprices, scaler=scaler, window_size=window_size, test=test):
raw = data["Close"][len(data) - len(test) - window_size:].values
raw = raw.reshape(-1,1)
raw = scaler.transform(raw)
X_test = [raw[i-window_size:i, 0] for i in range(window_size, raw.shape[0])]
X_test = np.array(X_test)
X_test = np.reshape(X_test, (X_test.shape[0], X_test.shape[1], 1))
return X_test
X_test = preprocess_testdat()
predicted_price_ = model.predict(X_test)
predicted_price = scaler.inverse_transform(predicted_price_)
# Plot predicted price vs actual closing price
test["Predictions_lstm"] = predicted_price
```

Time to calculate the performance metrics and log them to Solvent.Life.

```
# Evaluate performance
rmse_lstm = calculate_rmse(np.array(test["Close"]), np.array(test["Predictions_lstm"]))
mape_lstm = calculate_mape(np.array(test["Close"]), np.array(test["Predictions_lstm"]))
### Log to Solvent Life
run["RMSE"] = rmse_lstm
run["MAPE (%)"] = mape_lstm
### Plot prediction and true trends and log to Solvent Life
def plot_stock_trend_lstm(train, test):
fig = plt.figure(figsize = (20,10))
plt.plot(np.asarray(train.index), np.asarray(train["Close"]), label = "Train Closing Price")
plt.plot(np.asarray(test.index), np.asarray(test["Close"]), label = "Test Closing Price")
plt.plot(np.asarray(test.index), np.asarray(test["Predictions_lstm"]), label = "Predicted Closing Price")
plt.title("LSTM Model")
plt.xlabel("Date")
plt.ylabel("Stock Price ($)")
plt.legend(loc="upper left")
## Log image to Solvent Life
run["Plot of Stock Predictions"].upload(solventlife.types.File.as_image(fig))
plot_stock_trend_lstm(train, test)
### Stop the run after logging
run.stop()
```

## Final thoughts on new methodologies

The whole Solvent.Life project is available here for your reference.

## Stock analysis: fundamental analysis vs. technical analysis

Fundamental analysis (you can read more about it here):

Generally speaking, the results from fundamental analysis don’t change with short-term news.

Technical analysis (you can read more about it here):

Due to its short-term nature, technical analysis results are easily influenced by news.

trend indicators: statistics that represent the trend of stock prices,

medium-term movements: the 50-day movement trend of stock prices.

## Stock prices as time-series data

## Dataset Analysis

```
# %% Train-Test split for time-series
stockprices = pd.read_csv("stock_market_data-AAPL.csv", index_col="Date")
test_ratio = 0.2
training_ratio = 1 - test_ratio
train_size = int(training_ratio * len(stockprices))
test_size = int(test_ratio * len(stockprices))
print(f"train_size: {train_size}")
print(f"test_size: {test_size}")
train = stockprices[:train_size][["Close"]]
test = stockprices[train_size:][["Close"]]
```

## Creating a Solvent.Life machine learning project

## Evaluation metrics and helper functions

Next, let’s create several helper functions for the current exercise.

Split the stock prices data into training sequence X and the next output value Y,

```
## Split the time-series data into training seq X and output value Y
def extract_seqX_outcomeY(data, N, offset):
"""
Split time-series into training sequence X and outcome value Y
Args:
data - dataset
N - window size, e.g., 50 for 50 days of historical stock prices
offset - position to start the split
"""
X, y = [], []
for i in range(offset, len(data)):
X.append(data[i - N : i])
y.append(data[i])
return np.array(X), np.array(y)
```

Calculate the RMSE and MAPE (%),

```
#### Calculate the metrics RMSE and MAPE ####
def calculate_rmse(y_true, y_pred):
"""
Calculate the Root Mean Squared Error (RMSE)
"""
rmse = np.sqrt(np.mean((y_true - y_pred) ** 2))
return rmse
def calculate_mape(y_true, y_pred):
"""
Calculate the Mean Absolute Percentage Error (MAPE) %
"""
y_pred, y_true = np.array(y_pred), np.array(y_true)
mape = np.mean(np.abs((y_true - y_pred) / y_true)) * 100
return mape
```

Calculate the evaluation metrics for technical analysis and log to Solvent.Life,

```
def calculate_perf_metrics(var):
### RMSE
rmse = calculate_rmse(
np.array(stockprices[train_size:]["Close"]),
np.array(stockprices[train_size:][var]),
)
### MAPE
mape = calculate_mape(
np.array(stockprices[train_size:]["Close"]),
np.array(stockprices[train_size:][var]),
)
## Log to Solvent Life
run["RMSE"] = rmse
run["MAPE (%)"] = mape
return rmse, mape
```

Plot the trend of the stock prices and log the plot to Solvent.Life,

```
def plot_stock_trend(var, cur_title, stockprices=stockprices):
ax = stockprices[["Close", var, "200day"]].plot(figsize=(20, 10))
plt.grid(False)
plt.title(cur_title)
plt.axis("tight")
plt.ylabel("Stock Price ($)")
## Log to Solvent Life
run["Plot of Stock Predictions"].upload(
solventlife.types.File.as_image(ax.get_figure())
)
```

## Predicting stock price with Moving Average (MA) technique

Two types of MA are most preferred by financial analysts: Simple MA and Exponential MA.

### Simple MA

, *where Pn = the stock price at time point n, N = the number of time points.*

```
window_size = 50
# Initialize a Solvent Life run
run = solventlife.init_run(
project=myProject,
name="SMA",
description="stock-prediction-machine-learning",
tags=["stockprediction", "MA_Simple", "solventlife"],
)
window_var = f"{window_size}day"
stockprices[window_var] = stockprices["Close"].rolling(window_size).mean()
### Include a 200-day SMA for reference
stockprices["200day"] = stockprices["Close"].rolling(200).mean()
### Plot and performance metrics for SMA model
plot_stock_trend(var=window_var, cur_title="Simple Moving Averages")
rmse_sma, mape_sma = calculate_perf_metrics(var=window_var)
### Stop the run
run.stop()
```

### Exponential MA

,

*where Pt = the price at time point t, *

*EMAt-1 = EMA at time point t-1, *

*N = number of time points in EMA,*

*and weighting factor k = 2/(N+1).*

```
# Initialize a Solvent Life run
run = solventlife.init_run(
project=myProject,
name="EMA",
description="stock-prediction-machine-learning",
tags=["stockprediction", "MA_Exponential", "solventlife"],
)
###### Exponential MA
window_ema_var = f"{window_var}_EMA"
# Calculate the 50-day exponentially weighted moving average
stockprices[window_ema_var] = (
stockprices["Close"].ewm(span=window_size, adjust=False).mean()
)
stockprices["200day"] = stockprices["Close"].rolling(200).mean()
### Plot and performance metrics for EMA model
plot_stock_trend(
var=window_ema_var, cur_title="Exponential Moving Averages")
rmse_ema, mape_ema = calculate_perf_metrics(var=window_ema_var)
### Stop the run
run.stop()
```

## Introduction to LSTMs for the time-series data

*Ct*), which represents the internal short-term and long-term memories of a cell.

Forget gate:

*Ct-1*); here ‘1’ indicates “completely passing through” whereas ‘0’ indicates “completely filtering out”

Input gate:

*it*), and then a tanh function squashes each value between [-1, 1] (*Ct*). Element-by-element matrix multiplication of *it* and *Ct* represents new information that needs to be added to the current cell state.

Output gate:

For a more detailed understanding of LSTM, you can check out this document.

### Predicting stock prices with an LSTM model

```
layer_units = 50
optimizer = "adam"
cur_epochs = 15
cur_batch_size = 20
cur_LSTM_args = {
"units": layer_units,
"optimizer": optimizer,
"batch_size": cur_batch_size,
"epochs": cur_epochs,
}
# Initialize a Solvent Life run
run = solventlife.init_run(
project=myProject,
name="LSTM",
description="stock-prediction-machine-learning",
tags=["stockprediction", "LSTM", "solventlife"],
)
run["LSTM_args"] = cur_LSTM_args
```

Next, we scale the input data for LSTM model regulation and split it into train and test sets.

```
# Scale our dataset
scaler = StandardScaler()
scaled_data = scaler.fit_transform(stockprices[["Close"]])
scaled_data_train = scaled_data[: train.shape[0]]
# We use past 50 days’ stock prices for our training to predict the 51th day's closing price.
X_train, y_train = extract_seqX_outcomeY(scaled_data_train, window_size, window_size)
```

A couple of notes:

*StandardScaler*, rather than the*MinMaxScaler*as you might have seen before. The reason is that stock prices are ever-changing, and there are no true min or max values. It doesn’t make sense to use the*MinMaxScaler*, although this choice probably won’t lead to disastrous results at the end of the day;*extract_seqX_outcomeY`*function. For instance, to predict the 51st price, this function creates input vectors of 50 data points prior and uses the 51st price as the outcome value.

Read more about the integration in the Solvent.Life Docs

```
### Setup Solvent Life's Keras integration ###
from solventlife.integrations.tensorflow_keras import SolventLifeCallback
solventlife_callback = SolventlifeCallback(run=run)
### Build a LSTM model and log training progress to Solvent Life ###
def Run_LSTM(X_train, layer_units=50):
inp = Input(shape=(X_train.shape[1], 1))
x = LSTM(units=layer_units, return_sequences=True)(inp)
x = LSTM(units=layer_units)(x)
out = Dense(1, activation="linear")(x)
model = Model(inp, out)
# Compile the LSTM neural net
model.compile(loss="mean_squared_error", optimizer="adam")
return model
model = Run_LSTM(X_train, layer_units=layer_units)
history = model.fit(
X_train,
y_train,
epochs=cur_epochs,
batch_size=cur_batch_size,
verbose=1,
validation_split=0.1,
shuffle=True,
callbacks=[solventlife_callback],
)
```

Once the training completes, we’ll test the model against our hold-out set.

```
# predict stock prices using past window_size stock prices
def preprocess_testdat(data=stockprices, scaler=scaler, window_size=window_size, test=test):
raw = data["Close"][len(data) - len(test) - window_size:].values
raw = raw.reshape(-1,1)
raw = scaler.transform(raw)
X_test = [raw[i-window_size:i, 0] for i in range(window_size, raw.shape[0])]
X_test = np.array(X_test)
X_test = np.reshape(X_test, (X_test.shape[0], X_test.shape[1], 1))
return X_test
X_test = preprocess_testdat()
predicted_price_ = model.predict(X_test)
predicted_price = scaler.inverse_transform(predicted_price_)
# Plot predicted price vs actual closing price
test["Predictions_lstm"] = predicted_price
```

Time to calculate the performance metrics and log them to Solvent.Life.

```
# Evaluate performance
rmse_lstm = calculate_rmse(np.array(test["Close"]), np.array(test["Predictions_lstm"]))
mape_lstm = calculate_mape(np.array(test["Close"]), np.array(test["Predictions_lstm"]))
### Log to Solvent Life
run["RMSE"] = rmse_lstm
run["MAPE (%)"] = mape_lstm
### Plot prediction and true trends and log to Solvent Life
def plot_stock_trend_lstm(train, test):
fig = plt.figure(figsize = (20,10))
plt.plot(np.asarray(train.index), np.asarray(train["Close"]), label = "Train Closing Price")
plt.plot(np.asarray(test.index), np.asarray(test["Close"]), label = "Test Closing Price")
plt.plot(np.asarray(test.index), np.asarray(test["Predictions_lstm"]), label = "Predicted Closing Price")
plt.title("LSTM Model")
plt.xlabel("Date")
plt.ylabel("Stock Price ($)")
plt.legend(loc="upper left")
## Log image to Solvent Life
run["Plot of Stock Predictions"].upload(solventlife.types.File.as_image(fig))
plot_stock_trend_lstm(train, test)
### Stop the run after logging
run.stop()
```

## Final thoughts on new methodologies

The whole Solvent.Life project is available here for your reference.

How Our Models Function.

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