Processing and Analytics Vasiliki (Vasia) Kalavri
 vkalavri@bu.edu Spring 2020 4/23: Cardinality and frequency estimation ??? Vasiliki Kalavri | Boston University 2020 Counting distinct elements 2 University 2020 LogLog algorithm Input: stream S, array of m counters, hash fiction h Output: cardinality of S for j=0 to m-1 do: COUNT[j] = 0 for x in S do: i = h(x) j = getLeftBits(i, p) r = Boston University 2020 26 • Query approximation error • Error probability Guarantee: The estimation error for frequencies will not exceed with probability • A higher number of hash functions

Release 0.12.0 Example above is the same as previous except the plot is set to kernel density estimation. This shows how easy it is to have different plots for the same Trellis structure. In [12]: plt Python data analysis toolkit, Release 0.12.0 Above is a similar plot but with 2D kernel desnity estimation plot superimposed. In [24]: plt.figure() In [25]: plot relative weightings (viewing EWMA as a moving average) bias : boolean, default False Use a standard estimation bias correction Returns y : type of input argument 472 Chapter 25. API Reference pandas: powerful

satisfied Feng Li (SDU) Regularization and Bayesian Statistics September 20, 2023 11 / 25 Parameter Estimation in Probabilistic Models Assume data are generated via probabilistic model d ∼ p(d; θ) p(d; θ): Regularization and Bayesian Statistics September 20, 2023 12 / 25 Maximum Likelihood Estimation (MLE) Maximum Likelihood Estimation (MLE): Choose the parameter θ that maximizes the probability of the data, given parameter estimation θMLE = arg max θ ℓ(θ) = arg max θ m � i=1 log p(d(i); θ) Feng Li (SDU) Regularization and Bayesian Statistics September 20, 2023 13 / 25 Maximum-a-Posteriori Estimation (MAP)

having absolute values which are greater than one, and/or a non-invertible covariance matrix. See Estimation of covariance matrices for more details. In [8]: frame = DataFrame(randn(1000, 5), columns=[’a’ Release 0.14.0 Example above is the same as previous except the plot is set to kernel density estimation. This shows how easy it is to have different plots for the same Trellis structure. In [12]: plt Python data analysis toolkit, Release 0.14.0 Above is a similar plot but with 2D kernel desnity estimation plot superimposed. In [24]: plt.figure() Out[24]: In

Release 0.13.1 Example above is the same as previous except the plot is set to kernel density estimation. This shows how easy it is to have different plots for the same Trellis structure. In [12]: plt Python data analysis toolkit, Release 0.13.1 Above is a similar plot but with 2D kernel desnity estimation plot superimposed. In [24]: plt.figure() Out[24]: In relative weightings (viewing EWMA as a moving average) bias : boolean, default False Use a standard estimation bias correction Returns y : type of input argument Notes Either center of mass or span must

having absolute values which are greater than one, and/or a non-invertible covariance matrix. See Estimation of covariance matrices for more details. In [8]: frame = DataFrame(randn(1000, 5), columns=[’a’ at 0xa42e05cc> Example above is the same as previous except the plot is set to kernel density estimation. This shows how easy it is to have different plots for the same Trellis structure. In [196]: plt Out[207]: Above is a similar plot but with 2D kernel density estimation plot superimposed. In [208]: plt.figure() Out[208]:

having absolute values which are greater than one, and/or a non-invertible covariance matrix. See Estimation of covariance matrices for more details. In [8]: frame = DataFrame(randn(1000, 5), columns=[’a’ at 0xa473c24c> Example above is the same as previous except the plot is set to kernel density estimation. This shows how easy it is to have different plots for the same Trellis structure. In [196]: plt Out[207]: Above is a similar plot but with 2D kernel density estimation plot superimposed. In [208]: plt.figure() Out[208]:

having absolute values which are greater than one, and/or a non-invertible covariance matrix. See Estimation of covariance matrices for more details. In [8]: frame = pd.DataFrame(np.random.randn(1000, 5) especially true for text data columns with relatively few unique values (commonly referred to as “low-cardinality” data). By using more efficient data types, you can store larger datasets in memory. In [6]: ts powerful Python data analysis toolkit, Release 1.0.0 • ‘box’ : boxplot • ‘kde’ : Kernel Density Estimation plot • ‘density’ : same as ‘kde’ • ‘area’ : area plot • ‘pie’ : pie plot • ‘scatter’ : scatter

having absolute values which are greater than one, and/or a non-invertible covariance matrix. See Estimation of covariance matrices for more details. In [8]: frame = pd.DataFrame(np.random.randn(1000, 5) especially true for text data columns with relatively few unique values (commonly referred to as “low-cardinality” data). By using more efficient data types, you can store larger datasets in memory. In [6]: ts powerful Python data analysis toolkit, Release 1.0.5 • ‘box’ : boxplot • ‘kde’ : Kernel Density Estimation plot • ‘density’ : same as ‘kde’ • ‘area’ : area plot • ‘pie’ : pie plot • ‘scatter’ : scatter

having absolute values which are greater than one, and/or a non-invertible covariance matrix. See Estimation of covariance matrices for more details. In [8]: frame = pd.DataFrame(np.random.randn(1000, 5) especially true for text data columns with relatively few unique values (commonly referred to as “low-cardinality” data). By using more efficient data types, you can store larger datasets in memory. In [6]: ts powerful Python data analysis toolkit, Release 1.0.4 • ‘box’ : boxplot • ‘kde’ : Kernel Density Estimation plot • ‘density’ : same as ‘kde’ • ‘area’ : area plot • ‘pie’ : pie plot • ‘scatter’ : scatter