Kalavri | Boston University 2020 Let h be a hash function that maps each stream element into M = log2N bits, where N is the domain of input elements: For each element x, let rank(x) be the number maximum value of rank(.) seen so far. ̂n = 2R Claim: The maximum observed rank is a good estimate of log2n. In other words, the estimated number of distinct elements is equal to: ??? Vasiliki Kalavri | Kalavri | Boston University 2020 10 Stochastic averaging Use one hash function to simulate many by splitting the hash value into two parts ??? Vasiliki Kalavri | Boston University 2020 10 We split the input

. . . . . . . . . . . . . . . . . . . . . . . 123 10 Group By: split-apply-combine 125 10.1 Splitting an object into groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . stacked=True) 4 Chapter 1. What’s New pandas: powerful Python data analysis toolkit, Release 0.7.3 • Add log x and y scaling options to DataFrame.plot and Series.plot • Add kurt methods to Series and DataFrame attribute-based item access to Panel and add IPython completion (PR563) • Add logy option to Series.plot for log-scaling on the Y axis • Add index and header options to DataFrame.to_string • Can pass multiple DataFrames

. . . . . . . . . . . . . . . . . . . . . . . 237 12 Group By: split-apply-combine 239 12.1 Splitting an object into groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (~series) operators (GH2686) • DataFrame.plot now includes a logx parameter to change the x-axis to log scale (GH2327) • Series arithmetic operators can now handle constant and ndarray input (GH2574) • v.0.7.3 (April 12, 2012) 51 pandas: powerful Python data analysis toolkit, Release 0.12.0 • Add log x and y scaling options to DataFrame.plot and Series.plot • Add kurt methods to Series and DataFrame

. . . . . . . . . . . . . . . . . . . . . . . 115 10 Group By: split-apply-combine 117 10.1 Splitting an object into groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . attribute-based item access to Panel and add IPython completion (PR563) • Add logy option to Series.plot for log-scaling on the Y axis • Add index and header options to DataFrame.to_string • Can pass multiple DataFrames Release 0.7.1 5.2.11 DataFrame interoperability with NumPy functions Elementwise NumPy ufuncs (log, exp, sqrt, ...) and various other NumPy functions can be used with no issues on DataFrame, assuming

. . . . . . . . . . . . . . . . . . . . . . . 115 10 Group By: split-apply-combine 117 10.1 Splitting an object into groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . attribute-based item access to Panel and add IPython completion (PR563) • Add logy option to Series.plot for log-scaling on the Y axis • Add index and header options to DataFrame.to_string • Can pass multiple DataFrames Release 0.7.2 5.2.11 DataFrame interoperability with NumPy functions Elementwise NumPy ufuncs (log, exp, sqrt, ...) and various other NumPy functions can be used with no issues on DataFrame, assuming

. . . . . . . . . . . . . . . . . . . . . . . 319 13 Group By: split-apply-combine 321 13.1 Splitting an object into groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (~series) operators (GH2686) • DataFrame.plot now includes a logx parameter to change the x-axis to log scale (GH2327) • Series arithmetic operators can now handle constant and ndarray input (GH2574) • pandas: powerful Python data analysis toolkit, Release 0.13.1 df.plot(kind=’barh’, stacked=True) • Add log x and y scaling options to DataFrame.plot and Series.plot • Add kurt methods to Series and DataFrame

. . . . . . . . . . . . . . . . . . . . . . . . . . . 313 10 Working with Text Data 317 10.1 Splitting and Replacing Strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432 16 Group By: split-apply-combine 435 16.1 Splitting an object into groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bug in setitem with empty indexer and unwanted coercion of dtypes (GH8669) • Bug in ix/loc block splitting on setitem (manifests with integer-like dtypes, e.g. datetime64) (GH8607) • Bug when doing label

. . . . . . . . . . . . . . . . . . . . . . . . . . . 305 10 Working with Text Data 309 10.1 Splitting and Replacing Strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422 16 Group By: split-apply-combine 425 16.1 Splitting an object into groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bug in setitem with empty indexer and unwanted coercion of dtypes (GH8669) • Bug in ix/loc block splitting on setitem (manifests with integer-like dtypes, e.g. datetime64) (GH8607) • Bug when doing label

. . . . . . . . . . . . . . . . . . . . . . . . . . . 384 11 Working with Text Data 387 11.1 Splitting and Replacing Strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516 17 Group By: split-apply-combine 519 17.1 Splitting an object into groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . DataFrame({'a': np.random.randn(10)}) df.eval("b = sin(a)") The support math functions are sin, cos, exp, log, expm1, log1p, sqrt, sinh, cosh, tanh, arcsin, arccos, arctan, arccosh, arcsinh, arctanh, abs and arctan2

. . . . . . . . . . . . . . . . . . . . . . . 347 13 Group By: split-apply-combine 351 13.1 Splitting an object into groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (~series) operators (GH2686) • DataFrame.plot now includes a logx parameter to change the x-axis to log scale (GH2327) • Series arithmetic operators can now handle constant and ndarray input (GH2574) • pandas: powerful Python data analysis toolkit, Release 0.14.0 df.plot(kind=’barh’, stacked=True) • Add log x and y scaling options to DataFrame.plot and Series.plot • Add kurt methods to Series and DataFrame