# Technical Studies Reference

### Price Momentum Oscillator

This study calculates and displays a Price Momentum Oscillator and its Moving Average for the data specified by the Input Data Input.

Let $$X$$ be a random variable denoting the Input Data, and let $$X_t$$ denote the value of $$X$$ at Index $$t$$. The first step is to calculate the Rate of Change for $$t \geq 1$$ as follows.

$$\displaystyle{ROC_t(X,1,100) = 100\cdot\frac{X_t - X_{t - 1}}{X_{t - 1}}}$$

Next we define a Custom Smoothing Function $$CSF_t(X,n)$$ as follows.

$$\displaystyle{CSF_t(X,n) = \left\{ \begin{matrix} X_1 & t = 1 \\ \frac{2}{n}\cdot (X_t - CSF_{t - 1}(X,n)) + CSF_{t - 1}(X,n)) & t > 1 \end{matrix}\right .}$$

Let the PMO Line Length 1, PMO Line Length 2, and PMO Signal Line Length Inputs be denoted as $$n_1$$, $$n_2$$, and $$n_{Sig}$$, respectively. We denote the Smoothed ROC at Index $$t$$ as $$\overline{ROC}_t(X,1,100,n_1)$$, and we compute it for $$t \geq 1$$ as follows.

$$\overline{ROC}_t(X,1,100,n_1) = CSF_t(ROC(X,1,100),n_1)$$

We denote the Price Momentum Oscillator Line at Index $$t$$ as $$PMO_t(X,n_1,n_2)$$, and we compute it for $$t \geq 1$$ as follows.

$$PMO_t(X,n_1,n_2) = CSF_t\left(\overline{ROC}(X,1,100,n_1), n_2\right)$$

We denote the Price Momentum Oscillator Signal Line as $$\overline{PMO}_t(X,n_1,n_1,n_{Sig})$$, and we calculate it for $$t \geq 1$$ in terms of an Exponential Moving Average as follows.

$$\overline{PMO}_t(X,n_1,n_1,n_{Sig}) = EMA_t(10\cdot PMO_t(X,n_1,n_2),n_{Sig})$$

Note: Depending on the setting of the Input PMO Signal Line Moving Average Type, the Exponential Moving Average in the above formula could be replaced with a Linear Regression Moving Average, a Simple Moving Average, a Weighted Moving Average, a Wilders Moving Average, a Simple Moving Average - Skip Zeros, or a Smoothed Moving Average.

The Subgraphs of both $$PMO_t(X,n_1,n_2)$$ and $$\overline{PMO}_t(X,n_1,n_1,n_{Sig})$$ are displayed for $$t \geq \max\{n_1,n_2,n_{Sig}\}$$.