**ARIMA (Autoregressive Integrated Moving Average)** is a popular statistical method used for time series forecasting. It combines three components: AR (Autoregressive), I (Integrated), and MA (Moving Average).
ARIMA is widely used as it can handle various types of time series data, including those with trends and seasonality.

AR (Autoregressive) model, specifically $AR(p)$, uses the previous (or, lagged) $p$ values to predict the current.

Let's say the current bitcoin price at time $t$ is $Y_t$. The $AR(p)$ model states that $Y_t = \mu + a_1 Y_{t-1} + a_2 Y_{t-2} + \cdots + a_p Y_{t-p} + \epsilon_t$, where $\mu$ is the mean of the process, and $\epsilon_t$ is the error term. In other words, the current value is a linear combination of the previous $p$ values.

For example, if $Y_t = 0.2 Y_{t-1} + \epsilon_t$ of $AR(1)$ process is shown below:

MA (Moving Average) model, specifically $MA(q)$, uses the previous $q$ errors to predict the current.

Let's say the current bitcoin price is $Y_t$. The MA(q) model states that $Y_t = \mu + \epsilon_t + b_1 \epsilon_{t-1} + b_2 \epsilon_{t-2} + \cdots + b_q \epsilon_{t-q}$, where $\mu$ is the mean of the process, and $\epsilon_t$ is the error term at time $t$. In other words, the current value is a linear combination of the previous $q$ errors. If we consider errors as shocks to the system, the MA(q) model describes how the shocks propagate through the system over time.

For example, if $Y_t = \epsilon_t + 0.2 \epsilon_{t-1}$ of $MA(1)$ process is shown below:

ARMA (Autoregressive Moving Average) model, specifically $ARMA(p, q)$, uses the previous $p$ values and the previous $q$ errors to predict the current value.

To make the explanation easy, let's define $B$, the back-shift operator, $B Y_t = Y_{t-1}$. Using the operator, we can define $AR(1)$ and $MA(1)$ as follows, when ignoring the mean $\mu$:

- $AR(1)$: $(1 - \phi B) Y_t = \epsilon_t$ since $Y_t = \phi Y_{t-1} + \epsilon_t$
- MA(1): $Y_t = (1+\theta_1B)\epsilon_t$ since $Y_t = \epsilon_t + \theta_1\epsilon_{t-1}$

As an example, $ARMA(1, 1)$ of $(1-0.2B)Y_t = (1+0.3B)\epsilon_t$ means $Y_t = 0.2 Y_{t-1} + 0.3 \epsilon_{t-1} + \epsilon_{t}$.

ARIMA (Autoregressive Integrated Moving Average) model, specifically $ARIMA(p, d, q)$, takes the difference of the time series $d$ times and then applies the $ARMA(p, q)$ model to the differenced series.

If $d=1$, the difference of the time series is defined as $(1-B)Y_t = Y_t - Y_{t-1}$, and $ARIMA(1, 1, 1)$ is then $(1-\phi B)(1-B)Y_t = (1+\theta_1B)\epsilon_t$.

Consider $(1-0.2B)(1-B)Y_t = (1+0.3B)\epsilon_t$. It can be expanded as:

$(Y_t - Y_{t-1}) - 0.2(Y_{t-1} - Y_{t-2}) = \epsilon_t + 0.3\epsilon_{t-1}$

$Y_t - Y_{t-1} = 0.2(Y_{t-1} - Y_{t-2}) + 0.3\epsilon_{t-1} + \epsilon_{t}$

$Y_t = Y_{t-1} + 0.2(Y_{t-1} - Y_{t-2}) + 0.3\epsilon_{t-1} + \epsilon_{t}$

An example chart of this process is shown below:

Difference is useful when the time series is not stationary. See stationary timeseries for more details.

**SARIMA (Seasonal Autoregressive Integrated Moving Average)** model, specifically $ARIMA(p, d, q)(P, D, Q)_s$ models a time series as combination of non-seasonality as $ARIMA(p, d, q)$ and seasonality as $ARIMA(P, D, Q)_s$.

Let's consider the simple form of $ARIMA(1, 1, 1)(1, 1, 1)_4$. $ARIMA(1, 1, 1)$ deals with $Y_t - Y_{t-1}$, while $(1, 1, 1)_4$ handles $Y_t - Y_{t-4}$.

As a result, it is defined as $(1-\phi_1 B)(1 - \Phi_1 B^4)(1-B)(1-B^4)Y_t$ $= (1+\theta_1B)(1+\Theta_1B^4)\epsilon_t$

To demonstrate model performance, we show the model's prediction results for the following day. The cross validation process identified the best transformation to make the timeseries stationary and the optimal hyperparameters. The Mean Squared Error on the next day's closing price was used to determine the best model.

The chart below illustrates: 1) the model's fit to the training data, and 2) its prediction for the training data periods. Note that model was trained to predict the right next day's closing price. Therefore, the prediction in the long term horizon is not expected to be accurate.

ARIMA is supposed to perform very well for this kind of data as it's supposed to capture air passengers' cyclic, autoregressive, and moving average nature.