Pandas calculation ewm.std

Question

I am trying to check the calculations of ewm.std pandas so that I can implement a one-step update for my code. Here is a complete description of the problem with the code.

mrt = pd.Series(np.random.randn(1000))
N = 100
a = 2/(1+N)
bias = (2-a)/2/(1-a)
x = mrt.iloc[-2]
ma = mrt.ewm(span=N).mean().iloc[-3]
var = mrt.ewm(span=N).var().iloc[-3]
ans = mrt.ewm(span=N).std().iloc[-2]
print(np.sqrt( bias*(1-a) * (var + a * (x- ma)**2)), ans)

(1.1352524643949702, 1.1436193844674576)

I used the standard wording. Can someone tell me why these two values ​​should not be the same? i.e. how pandas calculating exponentially weighted std?

EDIT: After Julien's answer - let me give you another use case. I am drawing a var relation computed in pandas, and using formula I derived from the Cython pandas ewm covariance code. This ratio should be 1. (I assume there is a problem with my formula if someone can point this out).

mrt = pd.Series(np.random.randn(1000))

N = 100
a = 2./(1+N)
bias = (2-a)/2./(1-a)
mewma = mrt.ewm(span=N).mean()

var_pandas = mrt.ewm(span=N).var()
var_calculated = bias * (1-a) * (var_pandas.shift(1) + a * (mrt-mewma.shift(1))**2)

(var_calculated/var_pandas).plot()

The plot clearly shows the problem.

EDIT 2: Judging by trial and error, I understood the correct formula:

var_calculated = (1-a) * (var_pandas.shift(1) + bias * a * (mrt-mewma.shift(1))**2)

, ! - ?

Answer 1

, pandas ewm.var()

In [1]:
(np.sqrt(mrt.ewm(span=span).var()) == mrt.ewm(span=span).std())[1:].value_counts()

Out[1]:
True    999
dtype: int64

, : ans == np.sqrt(mrt.ewm(span=N).var().iloc[-2]).

, ewmvar(), , emcov input_x=input_y=mrt

---

:

mrt.ewm(span=span).var()[:2].values
> array([nan,  0.00555309])

, emcov :

x0 = mrt.iloc[0]
x1 = mrt.iloc[1]
x2 = mrt.iloc[2]

# mean_x and mean_y are both the same, here we call it y
# This is the same as mrt.ewm(span=span).mean(), I verified that too
y0 = x0
# y1 = mrt.ewm(span=span).mean().iloc[1]
y1 = ((1-alpha)*y0 + x1)/(1+(1-alpha))
#y2 = (((1-alpha)**2+(1-alpha))*y1 + x2) / (1 + (1-alpha) + (1-alpha)**2) 

cov0 = 0

cov1 = (((1-alpha) * (cov0 + ((y0 - y1)**2))) +
                (1 * ((x1 - y1)**2))) / (1 + (1-alpha))

# new_wt = 1, sum_wt0 = (1-alpha), sum_wt2 = (1-alpha)**2
sum_wt = 1+(1-alpha)
sum_wt2 =1+(1-alpha)**2


numerator = sum_wt * sum_wt # (1+(1-alpha))^2 = 1 + 2(1-alpha) + (1-alpha)^2
denominator = numerator - sum_wt2 # # 2*(1-alpha)


print(np.nan,cov1*(numerator / denominator))

>(nan, 0.0055530905712123432)