BUG: Fix weighted rolling variance implementation
- [x] closes #53273, closes #54333
- [x] [Tests added and passed](https://pandas.pydata.org/pandas-docs/dev/development/contributing_codebase.html#writing-tests) if fixing a bug or adding a new feature
- [x] All [code checks passed](https://pandas.pydata.org/pandas-docs/dev/development/contributing_codebase.html#pre-commit).
- [ ] Added [type annotations](https://pandas.pydata.org/pandas-docs/dev/development/contributing_codebase.html#type-hints) to new arguments/methods/functions.
- [x] Added an entry in the latest `doc/source/whatsnew/vX.X.X.rst` file if fixing a bug or adding a new feature.
---

Outputs from the code snippets in the given issues are below

### Issue #53273 
#### Code
```python
import matplotlib.pyplot as plt
import pandas as pd
import numpy as np

np.random.seed(42)
df = pd.Series(np.random.normal(size=1200))
df_std_stock = (
    df.rolling(window=220, min_periods=100, win_type='exponential')
    .std(tau=-(25 / np.log(2)), center=0, sym=False)
)

df_std_custom = (
    (df ** 2).rolling(window=220, min_periods=100, win_type='exponential')
    .mean(tau=-(25 / np.log(2)), center=0, sym=False)
)
df_std_custom_mean = (
    df.rolling(window=220, min_periods=100, win_type='exponential')
    .mean(tau=-(25 / np.log(2)), center=0, sym=False)
)
df_std_custom = (df_std_custom - df_std_custom_mean ** 2).pow(0.5)

plt.plot(df_std_stock.index, df_std_stock.values, label="df_std_stock", linewidth=3)
plt.plot(df_std_custom.index, df_std_custom.values, label="df_std_custom", linestyle='--')
plt.legend()
plt.show()

```
#### Output
![test](https://github.com/pandas-dev/pandas/assets/13166114/10d08795-f003-4864-8fc0-20a41e136d9b)

### Issue #54333 
#### Code
```python
import numpy as np
import pandas as pd
from scipy import signal

def equal(win_len: int, **kwargs):
    assert win_len == 5
    return np.array([1.0, 1.0, 1.0, 1.0, 1.0])

def linear(win_len: int, **kwargs):
    assert win_len == 5
    return np.array([0.2, 0.4, 0.6, 0.8, 1.0])

signal.windows.equal_weight = equal
signal.windows.linear_decay = linear

s = pd.Series([1.0, 0.0] * 10)

print(list(s))
print(list(s.rolling(window=5).var(ddof=0)))
print(list(s.rolling(window=5, win_type="equal_weight").var(ddof=0)))
print(list(s.rolling(window=5, win_type="linear_decay").var(ddof=0)))
```
#### Output
```
[1.0, 0.0, 1.0, 0.0, 1.0, 0.0, 1.0, 0.0, 1.0, 0.0, 1.0, 0.0, 1.0, 0.0, 1.0, 0.0, 1.0, 0.0, 1.0, 0.0]
[nan, nan, nan, nan, 0.24, 0.24, 0.24, 0.24, 0.24, 0.24, 0.24, 0.24, 0.24, 0.24, 0.24, 0.24, 0.24, 0.24, 0.24, 0.24]
[nan, nan, nan, nan, 0.24, 0.24, 0.24, 0.24, 0.24, 0.24, 0.24, 0.24, 0.24, 0.24, 0.24, 0.24, 0.24, 0.24, 0.24, 0.24]
[nan, nan, nan, nan, 0.24000000000000005, 0.24000000000000005, 0.24000000000000005, 0.24000000000000005, 0.24000000000000005, 0.24000000000000005, 0.24000000000000005, 0.24000000000000005, 0.24000000000000005, 0.24000000000000005, 0.24000000000000005, 0.24000000000000005, 0.24000000000000005, 0.24000000000000005, 0.24000000000000005, 0.24000000000000005]
```
Can you show if there is any performance difference before and after this change? > Can you show if there is any performance difference before and after this change?

@mroeschke there is a difference but not sure if it matters since the previous implementation wasn't correct

| Change   | Before [ebca6dfb] <main>   | After [a695dc48] <fix-rolling-std-custom-weights>   |   Ratio | Benchmark (Parameter)                           |
|----------|----------------------------|-----------------------------------------------------|---------|-------------------------------------------------|
| +        | 2.18±0.02ms                | 764±4ms                                             |  351.17 | rolling.WeightedVar.time_method(1000, 'boxcar') |
| +        | 2.25±0.02ms                | 768±4ms                                             |  341.83 | rolling.WeightedVar.time_method(1000, 'triang') |
| +        | 2.24±0.03ms                | 73.6±0.3ms                                          |   32.83 | rolling.WeightedVar.time_method(100, 'boxcar')  |
| +        | 2.36±0.3ms                 | 74.6±0.1ms                                          |   31.57 | rolling.WeightedVar.time_method(100, 'triang')  |
| +        | 2.96±0.9ms                 | 5.09±0.06ms                                         |    1.72 | rolling.WeightedVar.time_method(10, 'boxcar')   |
| +        | 3.07±0.9ms                 | 5.07±0.04ms                                         |    1.65 | rolling.WeightedVar.time_method(10, 'triang')   |

---

```python
class WeightedVar:
    params = (
        [10, 100, 1000],
        ["triang", "boxcar"]
    )
    param_names = ["window", "win_type"]

    def setup(self, window, win_type):
        N = 10**5
        arr = np.random.random(N)
        obj = pd.Series(arr)
        self.window = obj.rolling(window, win_type=win_type)

    def time_method(self, window, win_type):
        self.window.var() the new implementation is O^2 
this is not good at all  > the new implementation is O^2 this is not good at all

@jreback I don't think there is a better solution. Let me explain what is wrong with the previous implementation and what this implementation does using an example.

Let's say we have:
```python
values = [0, 1, 2, 3]
weights = [10, 20]
```
We are supposed to update the sum of weights, mean and t each time a value and a weight is added (or removed) to the window as per the [paper](https://dl.acm.org/doi/pdf/10.1145/359146.359153). After each pair is added (or removed) to the window we can calculate weighted variance.

The previous implementation follows the steps below:
```python
var, w = 0, 10
add(var, w)
output[0] = calculate()

var, w = 1, 20
add(var, w)
output[1] = calculate()

var, w = 2, 10
add(var, w)
var_pre, w_pre = 0, 10
remove(var_pre, w_pre)
output[2] = calculate()

var, w = 3, 20
add(var, w)
var_pre, w_pre = 1, 20
remove(var_pre, w_pre)
output[3] = calculate()
```
Single pass is possible if adding the head pair (or removing the tail) to the window is enough but it is not. Since all of the pairings are different, we need to first remove all the pairs that are previously added (or start from scratch) and then add the pairs we want.


What the new implementation does is:
```python
var, w = 0, 20
add(var, w)
output[0] = calculate()

reset_t_sumw_mean()
var, w = 0, 10
add(var, w)

var, w = 1, 20
add(var, w)
output[1] = calculate()

reset_t_sumw_mean()
var, w = 1, 10
add(var, w)

var, w = 2, 20
add(var, w)
output[2] = calculate()

reset_t_sumw_mean()
var, w = 2, 10
add(var, w)

var, w = 3, 20
add(var, w)
output[3] = calculate()
``` @jreback @WillAyd @mroeschke any other comments/concerns? Is there a way to fix this while keeping the current algorithm? The slowdown is concerning for more common cases where this was already correct > Is there a way to fix this while keeping the current algorithm?

I don't think there is. I tried to explain what was wrong with it (and why it is faster) in the previous comment. I think it should be incorrect almost all the time because it is calculating variance for the wrong pairs of weights and values From the paper linked:

> It is frequently convenient to calculate these quanti-
ties by a method that, unlike direct implementation of
the above definitions, requires only one pass through the
set of data pairs (X~, W~). Such a method is also useful
where it is necessary tc, update (rather than recalculate
ab initio) the mean and variance after the data set has
been extended to include another data pair.

But isn't this algorithm requiring multiple passes instead of simply updating records as the window expands? The scaling and performance at 1000 rows makes this pretty impractical to use for usual datasets I think > The scaling and performance at 1000 rows makes this pretty impractical to use for usual datasets I think

@WillAyd It is not the performance at 1000 rows. The number of rows is constant at 100,000 and the varying one is the length of weights. 

> But isn't this algorithm requiring multiple passes instead of simply updating records as the window expands?

It still calculates variance in one pass (over the same value and weight pairs). The problem is nothing here is expanding really. We **have to** recalculate variance for each window because pairs are different in each window. I have an excalidraw board below showing how this works using the same example. Hope this makes sense.

![Untitled-2023-09-30-1837](https://github.com/pandas-dev/pandas/assets/13166114/274ed80e-632b-475b-8dee-88a956445c7b)


By the way, both weighted rolling mean and sum have the same complexity O(wn) where w is the number of weights and n is the number of rows.
https://github.com/pandas-dev/pandas/blob/6f0cd8d9ed251aa80f7415c565176fc33487110a/pandas/_libs/window/aggregations.pyx#L1495-L1500
 This pull request is stale because it has been open for thirty days with no activity. Please [update](https://pandas.pydata.org/pandas-docs/stable/development/contributing.html#updating-your-pull-request) and respond to this comment if you're still interested in working on this. Thanks for the PR here but it has appeared to gone stale. I think the discussion on the PR is better suited for the original issue(s) so let's move the discussion there before proceeding with this