Running Sum

We’ve covered Running SMAs and EMAs…let’s dig into Running Sums or often called Running Totals. Formula as follows:
$Sum_{today} = Sum_{yesterday} + (price_{today} - price_{today - period})$
Where $price_{today - period}$ represents the price that is dropping off the slice you are summing. For example:

Take a list of numbers = 20, 40, 60, 80, 100, 120.
The formula for the 3-bar running sum would be:
bar 1: 20
bar 2: 20 + 40 = 60
bar 3: 20 + 40 + 60 = 120

bar 4: 40 + 60 + 80 = 180
Or we can apply our formula from above as $Sum_{today} = 120 + (80 - 20)$

bar 5: 60 + 80 + 100 = 240
Or use formula of $Sum_{today} = 180 + (100 - 40)$

bar 6: 80 + 100 + 120 = 300
Or use formula of $Sum_{today} = 240 + (120 - 60)$

Coding in Python we get:

def running_sum(bar, series, period, pval=None):
"""
Returns the running sum of values in a list of tuple - avoids summing entire series on each call.

Keyword arguments:
bar     --  current index or location of the value in the series
series  --  list or tuple of data to sum
period  -- number of values to include in sum
pval    --  previous sum (n - 1) of the series.
"""
if period < 1:
raise ValueError("period must be 1 or greater")

if bar <= 0:
return series[0]

if bar < period:
return pval + series[bar]

return pval + (series[bar] - series[bar - period])


Example call and results:

list_of_values = [20, 40, 60, 80, 100, 120]
prevsum = list_of_values[0]   #first sum is the first value in the series.
for bar, price in enumerate(list_of_values):
newsum = running_sum(bar, list_of_values, 3, pval=prevsum)
print "bar %i: %i" % (bar, newsum)
prevsum = newsum
----------------------------------------------------------
bar 0: 20
bar 1: 60
bar 2: 120
bar 3: 180
bar 4: 240
bar 5: 300


Exponential Moving Average (EMA)

Now that we’ve tackled Running Simple Moving Averages (SMA)…let’s move on to Exponential Moving Averages (EMA). You may wonder why we’re not covering Running Exponential Moving Averages? The default formula for EMA is the running method – so we’re already covered.

Check out the posts below to understand the background on Exponential Moving Averages (EMA) and their calculation.

Be careful with using EMAs in your backtesting. Or any of these running type of indicators. Since all of them require a starting value. If that starting value changes – your signals change. Which can happen if you switch price quote providers that have different history requirements. Should not be a big deal but something to be aware of.

Let’s begin. We need to calculate our smoothing factor for the time series. Typical use in technical analysis is:
$\alpha = 2.0 / (periods + 1.0)$

We can use any value between 0 & 1 for the smoothing factor. Closer to one is less smooth and places greater weight on the more recent values. Use a value of 1 and you get the most recent value back. Closer to zero is more smooth and places greater weight on the older values.

Now, the formula for an EMA given our smoothing factor:
$EMA_{today} = EMA_{yesterday} + \alpha(price_{today} - EMA_{yesterday})$

Coding in Python we get:

def ema(bar, series, period, prevma, smoothing=None):
'''Returns the Exponential Moving Average of a series.

Keyword arguments:
bar         -- currrent index or location of the series
series      -- series of values to be averaged
period      -- number of values in the series to average
prevma      -- previous exponential moving average
smoothing   -- smoothing factor to use in the series.
valid values: between 0 & 1.
default: None - which then uses formula = 2.0 / (period + 1.0)
closer to 1 to gives greater weight to recent values - less smooth
closer to 0 gives greater weight to older values -- more smooth
'''
if period < 1:
raise ValueError("period must be 1 or greater")

if smoothing:
if (smoothing < 0) or (smoothing > 1.0):
raise ValueError("smoothing must be between 0 and 1")

else:
smoothing = 2.0 / (period + 1.0)

if bar <= 0:
return series[0]

elif bar < period:
return cumulative_sma(bar, series, prevma)

return prevma + smoothing * (series[bar] - prevma)

def cumulative_sma(bar, series, prevma):
"""
Returns the cumulative or unweighted simple moving average.
Avoids averaging the entire series on each call.

Keyword arguments:
bar     --  current index or location of the value in the series
series  --  list or tuple of data to average
prevma  --  previous average (n - 1) of the series.
"""

if bar <= 0:
return series[0]

else:
return prevma + ((series[bar] - prevma) / (bar + 1.0))



Example call and results using the typical smoothing factor of 2 / (period + 1):

prices = [32.47, 32.70, 32.77, 33.11, 33.25, 33.23, 33.23, 33.0, 33.04, 33.21]
period = 5   #number of bars to average
prevsma = prevema = prices[0]   #1st day nothing to average
for bar, close in enumerate(prices):
currentema = ema(bar, prices, period, prevema, smoothing=None)

#running_sma defined in simple moving average blog post
currentsma = running_sma(bar, prices, period, prevsma)

print "Day %02d Value=%.2f %i-bar SMA=%f and EMA=%f" % (bar + 1, close, period, currentsma, currentema)
prevema = currentema
prevsma = currentsma

----------------------------------------------------------
Results of call:
Day 01 Value=32.47 5-day SMA=32.470000 and EMA=32.470000
Day 02 Value=32.70 5-day SMA=32.585000 and EMA=32.585000
Day 03 Value=32.77 5-day SMA=32.646667 and EMA=32.646667
Day 04 Value=33.11 5-day SMA=32.762500 and EMA=32.762500
Day 05 Value=33.25 5-day SMA=32.860000 and EMA=32.860000
Day 06 Value=33.23 5-day SMA=33.012000 and EMA=32.983333
Day 07 Value=33.23 5-day SMA=33.118000 and EMA=33.065556
Day 08 Value=33.00 5-day SMA=33.164000 and EMA=33.043704
Day 09 Value=33.04 5-day SMA=33.150000 and EMA=33.042469
Day 10 Value=33.21 5-day SMA=33.142000 and EMA=33.098313



Running Simple Moving Average (SMA)

When building a platform to test trading ideas…one of the big issues to deal with is all the indicators that require a spin through the price series in order to calculate. For example, in order to calculate the 200 day simple moving average (SMA) of closing prices for Google today you would have to loop back 200 – 1 days ago and sum the closing prices and divide by 200.

When you are backtesting an idea you often need to start from day 1 of a stock’s trading history and loop forward to the most current day. In essence, pretending each day is the current day at that point in time. Thus, you are looping back 200 – 1 data points for each day in the series. This isn’t such a big deal with a stock such as Google whose trading history is rather limited (2004). But, take a stock like IBM with a more extensive trading history and your code is going to bog down with each call to the SMA indicator. Throw 20,000 securities into your backtest and the looping adds up.

Therefore, running calculations are the preferred method in order to spin just once through the data points. So, in order to calculate the running simple moving average for closing prices you apply the following formula:
$SMA_{today} = SMA_{yesterday} + ((Price_{today} - Price_{today - n}) /n)$

Where

• n = number of values included in your rolling computational window.

Straight-forward and avoids the loop. Here’s the sample Python code for the Running SMA:

def running_sma(bar, series, period, prevma):
"""
Returns the running simple moving average - avoids sum of series per call.

Keyword arguments:
bar     --  current index or location of the value in the series
series  --  list or tuple of data to average
period  --  number of values to include in average
prevma  --  previous simple moving average (n - 1) of the series
"""
if period < 1:
raise ValueError("period must be 1 or greater")

if bar <= 0:
return series[0]

elif bar < period:
return cumulative_sma(bar, series, prevma)

return prevma + ((series[bar] - series[bar - period]) / float(period))

def cumulative_sma(bar, series, prevma):
"""
Returns the cumulative or unweighted simple moving average.
Avoids sum of series per call.

Keyword arguments:
bar     --  current index or location of the value in the series
series  --  list or tuple of data to average
prevma  --  previous average (n - 1) of the series.
"""

if bar <= 0:
return series[0]

else:
return prevma + ((series[bar] - prevma) / (bar + 1.0))



And the example call and results:

prices = [10, 15, 25, 18, 13, 16]
prevsma = prices[0]   #1st day nothing to average so return itself.
for bar, close in enumerate(prices):
currentsma = running_sma(bar, prices, 3, prevsma)
print "Today's 3-day SMA = %.4f" % currentsma
prevsma = currentsma

------- Results ----------------
Today's 3-day SMA = 10.0000
Today's 3-day SMA = 12.5000
Today's 3-day SMA = 16.6667
Today's 3-day SMA = 19.3333
Today's 3-day SMA = 18.6667
Today's 3-day SMA = 15.6667

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