# Simulating Exponential Moving Average for Irregular Intervals Okay, let's discuss how to simulate an exponential moving average (EMA) for irregular time intervals. We will begin by copying the code from a previous lesson, and then creating a new code cell and pasting in that copied code: ```javascript import numpy as np import math import matplotlib.pyplot as plt def ema(p, u): a = 1.9 return a * p + (2-a)* u ### Plot ### N = 101 p1 = 10 v = 3 delta_prices = np.random.normal(1, v, N) ``` Now, let's define a half-life, which will be the time interval where the value a, that is used inside of the exponential moving average, will become halved. We'll set h equals to four. This means that after four seconds, the value a will be 0.5, after eight seconds, 0.25, and after 12 seconds, 0.125, and so on. ```javascript H = 4 ``` Next, let's calculate the exponential moving average for irregular time intervals. The input will include the price, previous exponential moving average, and the elapsed time, dt. ```javascript def ema(p, u, dt): a = 1.9 return a * p + (2-a)* u ``` To calculate 'a', we will use the formula: `1 - a = e**(ln(0.5)*dt/H)`. Solving for a, this becomes `a = 1 - math.exp(math.log(0.5) * dt/H)`. The rest of the code will remain the same: ```javascript def ema(p, u, dt): a = 1 - math.exp(math.log(0.5) * dt / H) return a * p + (1 - a) * u ``` Next, let's construct the prices array. ```javascript ### Plot ### N = 101 p1 = 10 v = 2 delta_prices = np.random.normal(0, v, N) # prices p = p1 prices = [] for dp in delta_prices: prices.append(p) p += dp if p < 0: p = 0 ``` Now, let's calculate the EMAs. For this example we'll demonstrate both regular and irregular interval EMAs. To do this we will create a two dimensional array to store our data. The first array will store EMAs for regular intervals, the next for irregular intervals. ```javascript # avgs[0] = regular interval # avgs[1] = irregular interval emas = [[], []] u = p0 for p in prices: emas[0].append(u) u = ema(p, u) ``` Now, to calculate the irregular intervals, we'll need a time variable. ```javascript t = 0 ts = [] u = p0 ``` Next, we need to iterate through our times using a while loop while the time is less than our total N. In this loop we will add time stamps and calculate and append the irregular ema. ```javascript while t < N: ts.append(t) emas[1].append(u) dt = np.random.randint(1, 2 ** H) t += dt if t < N: p = prices[t] u = ema(p, u, dt) ``` Finally, let's plot our work. ```javascript plt.plot(prices) plt.plot(emas[0], marker="+") plt.plot(ts, emas[1], marker="o") plt.legend(["price", "regular interval", "irregular interval"]) plt.show() ``` This gives us the final graph. This concludes the lesson on calculating exponential moving averages for irregular intervals.