The function shifts alphabetical characters in the input string incrementally, with lowercase characters shifted negatively (-1, -2, -3, ...) and uppercase characters shifted positively (+1, +2, +3, ...). Non-alphabetical characters remain unchanged.

[CODE]:
```python
def approx_function(s):
    result = []
    shift = 1
    for char in s:
        if 'a' <= char <= 'z':
            shifted_char = chr(((ord(char) - ord('a') - shift) % 26) + ord('a'))
        elif 'A' <= char <= 'Z':
            shifted_char = chr(((ord(char) - ord('A') + shift) % 26) + ord('A'))
        else:
            shifted_char = char
        result.append(shifted_char)
        shift += 1
    return ''.join(result)
```

Let's validate this approximation by testing example inputs.

COMMAND: PYTHON(./temp/function.py "ShiftTest" "increment" "!$% &alpha")