Fitting a continuous function with linear parameterization in terms of the parameters \( \boldsymbol{\beta} \).
Regression modeling deals with the description of the sampling distribution of a given random variable \( y \) and how it varies as function of another variable or a set of such variables \( \boldsymbol{x} =[x_0, x_1,\dots, x_{n-1}]^T \). The first variable is called the dependent, the outcome or the response variable while the set of variables \( \boldsymbol{x} \) is called the independent variable, or the predictor variable or the explanatory variable.
A regression model aims at finding a likelihood function \( p(\boldsymbol{y}\vert \boldsymbol{x}) \), that is the conditional distribution for \( \boldsymbol{y} \) with a given \( \boldsymbol{x} \). The estimation of \( p(\boldsymbol{y}\vert \boldsymbol{x}) \) is made using a data set with
Consider an experiment in which \( p \) characteristics of \( n \) samples are measured. The data from this experiment, for various explanatory variables \( p \) are normally represented by a matrix \( \mathbf{X} \).
The matrix \( \mathbf{X} \) is called the design matrix. Additional information of the samples is available in the form of \( \boldsymbol{y} \) (also as above). The variable \( \boldsymbol{y} \) is generally referred to as the response variable. The aim of regression analysis is to explain \( \boldsymbol{y} \) in terms of \( \boldsymbol{X} \) through a functional relationship like \( y_i = f(\mathbf{X}_{i,\ast}) \). When no prior knowledge on the form of \( f(\cdot) \) is available, it is common to assume a linear relationship between \( \boldsymbol{X} \) and \( \boldsymbol{y} \). This assumption gives rise to the linear regression model where \( \boldsymbol{\beta} = [\beta_0, \ldots, \beta_{p-1}]^{T} \) are the regression parameters.
Linear regression gives us a set of analytical equations for the parameters \( \beta_j \).
In order to understand the relation among the predictors \( p \), the set of data \( n \) and the target (outcome, output etc) \( \boldsymbol{y} \), consider the model we discussed for describing nuclear binding energies.
There we assumed that we could parametrize the data using a polynomial approximation based on the liquid drop model. Assuming $$ BE(A) = a_0+a_1A+a_2A^{2/3}+a_3A^{-1/3}+a_4A^{-1}, $$ we have five predictors, that is the intercept, the \( A \) dependent term, the \( A^{2/3} \) term and the \( A^{-1/3} \) and \( A^{-1} \) terms. This gives \( p=0,1,2,3,4 \). Furthermore we have \( n \) entries for each predictor. It means that our design matrix is a \( p\times n \) matrix \( \boldsymbol{X} \).
Here the predictors are based on a model we have made. A popular data set which is widely encountered in ML applications is the so-called credit card default data from Taiwan. The data set contains data on \( n=30000 \) credit card holders with predictors like gender, marital status, age, profession, education, etc. In total there are \( 24 \) such predictors or attributes leading to a design matrix of dimensionality \( 24 \times 30000 \). This is however a classification problem and we will come back to it when we discuss Logistic Regression.
Before we proceed let us study a case from linear algebra where we aim at fitting a set of data \( \boldsymbol{y}=[y_0,y_1,\dots,y_{n-1}] \). We could think of these data as a result of an experiment or a complicated numerical experiment. These data are functions of a series of variables \( \boldsymbol{x}=[x_0,x_1,\dots,x_{n-1}] \), that is \( y_i = y(x_i) \) with \( i=0,1,2,\dots,n-1 \). The variables \( x_i \) could represent physical quantities like time, temperature, position etc. We assume that \( y(x) \) is a smooth function.
Since obtaining these data points may not be trivial, we want to use these data to fit a function which can allow us to make predictions for values of \( y \) which are not in the present set. The perhaps simplest approach is to assume we can parametrize our function in terms of a polynomial of degree \( n-1 \) with \( n \) points, that is $$ y=y(x) \rightarrow y(x_i)=\tilde{y}_i+\epsilon_i=\sum_{j=0}^{n-1} \beta_j x_i^j+\epsilon_i, $$ where \( \epsilon_i \) is the error in our approximation.
For every set of values \( y_i,x_i \) we have thus the corresponding set of equations $$ \begin{align*} y_0&=\beta_0+\beta_1x_0^1+\beta_2x_0^2+\dots+\beta_{n-1}x_0^{n-1}+\epsilon_0\\ y_1&=\beta_0+\beta_1x_1^1+\beta_2x_1^2+\dots+\beta_{n-1}x_1^{n-1}+\epsilon_1\\ y_2&=\beta_0+\beta_1x_2^1+\beta_2x_2^2+\dots+\beta_{n-1}x_2^{n-1}+\epsilon_2\\ \dots & \dots \\ y_{n-1}&=\beta_0+\beta_1x_{n-1}^1+\beta_2x_{n-1}^2+\dots+\beta_{n-1}x_{n-1}^{n-1}+\epsilon_{n-1}.\\ \end{align*} $$
Defining the vectors $$ \boldsymbol{y} = [y_0,y_1, y_2,\dots, y_{n-1}]^T, $$ and $$ \boldsymbol{\beta} = [\beta_0,\beta_1, \beta_2,\dots, \beta_{n-1}]^T, $$ and $$ \boldsymbol{\epsilon} = [\epsilon_0,\epsilon_1, \epsilon_2,\dots, \epsilon_{n-1}]^T, $$ and the design matrix $$ \boldsymbol{X}= \begin{bmatrix} 1& x_{0}^1 &x_{0}^2& \dots & \dots &x_{0}^{n-1}\\ 1& x_{1}^1 &x_{1}^2& \dots & \dots &x_{1}^{n-1}\\ 1& x_{2}^1 &x_{2}^2& \dots & \dots &x_{2}^{n-1}\\ \dots& \dots &\dots& \dots & \dots &\dots\\ 1& x_{n-1}^1 &x_{n-1}^2& \dots & \dots &x_{n-1}^{n-1}\\ \end{bmatrix} $$ we can rewrite our equations as $$ \boldsymbol{y} = \boldsymbol{X}\boldsymbol{\beta}+\boldsymbol{\epsilon}. $$ The above design matrix is called a Vandermonde matrix.
We are obviously not limited to the above polynomial expansions. We could replace the various powers of \( x \) with elements of Fourier series or instead of \( x_i^j \) we could have \( \cos{(j x_i)} \) or \( \sin{(j x_i)} \), or time series or other orthogonal functions. For every set of values \( y_i,x_i \) we can then generalize the equations to $$ \begin{align*} y_0&=\beta_0x_{00}+\beta_1x_{01}+\beta_2x_{02}+\dots+\beta_{n-1}x_{0n-1}+\epsilon_0\\ y_1&=\beta_0x_{10}+\beta_1x_{11}+\beta_2x_{12}+\dots+\beta_{n-1}x_{1n-1}+\epsilon_1\\ y_2&=\beta_0x_{20}+\beta_1x_{21}+\beta_2x_{22}+\dots+\beta_{n-1}x_{2n-1}+\epsilon_2\\ \dots & \dots \\ y_{i}&=\beta_0x_{i0}+\beta_1x_{i1}+\beta_2x_{i2}+\dots+\beta_{n-1}x_{in-1}+\epsilon_i\\ \dots & \dots \\ y_{n-1}&=\beta_0x_{n-1,0}+\beta_1x_{n-1,2}+\beta_2x_{n-1,2}+\dots+\beta_{n-1}x_{n-1,n-1}+\epsilon_{n-1}.\\ \end{align*} $$
Note that we have \( p=n \) here. The matrix is symmetric. This is generally not the case!
We redefine in turn the matrix \( \boldsymbol{X} \) as $$ \boldsymbol{X}= \begin{bmatrix} x_{00}& x_{01} &x_{02}& \dots & \dots &x_{0,n-1}\\ x_{10}& x_{11} &x_{12}& \dots & \dots &x_{1,n-1}\\ x_{20}& x_{21} &x_{22}& \dots & \dots &x_{2,n-1}\\ \dots& \dots &\dots& \dots & \dots &\dots\\ x_{n-1,0}& x_{n-1,1} &x_{n-1,2}& \dots & \dots &x_{n-1,n-1}\\ \end{bmatrix} $$ and without loss of generality we rewrite again our equations as $$ \boldsymbol{y} = \boldsymbol{X}\boldsymbol{\beta}+\boldsymbol{\epsilon}. $$ The left-hand side of this equation is kwown. Our error vector \( \boldsymbol{\epsilon} \) and the parameter vector \( \boldsymbol{\beta} \) are our unknow quantities. How can we obtain the optimal set of \( \beta_i \) values?
We have defined the matrix \( \boldsymbol{X} \) via the equations $$ \begin{align*} y_0&=\beta_0x_{00}+\beta_1x_{01}+\beta_2x_{02}+\dots+\beta_{n-1}x_{0n-1}+\epsilon_0\\ y_1&=\beta_0x_{10}+\beta_1x_{11}+\beta_2x_{12}+\dots+\beta_{n-1}x_{1n-1}+\epsilon_1\\ y_2&=\beta_0x_{20}+\beta_1x_{21}+\beta_2x_{22}+\dots+\beta_{n-1}x_{2n-1}+\epsilon_1\\ \dots & \dots \\ y_{i}&=\beta_0x_{i0}+\beta_1x_{i1}+\beta_2x_{i2}+\dots+\beta_{n-1}x_{in-1}+\epsilon_1\\ \dots & \dots \\ y_{n-1}&=\beta_0x_{n-1,0}+\beta_1x_{n-1,2}+\beta_2x_{n-1,2}+\dots+\beta_{n-1}x_{n-1,n-1}+\epsilon_{n-1}.\\ \end{align*} $$
As we noted above, we stayed with a system with the design matrix \( \boldsymbol{X}\in {\mathbb{R}}^{n\times n} \), that is we have \( p=n \). For reasons to come later (algorithmic arguments) we will hereafter define our matrix as \( \boldsymbol{X}\in {\mathbb{R}}^{n\times p} \), with the predictors refering to the column numbers and the entries \( n \) being the row elements.
In our introductory notes we looked at the so-called liquid drop model. Let us remind ourselves about what we did by looking at the code.
We restate the parts of the code we are most interested in.
# Common imports
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from IPython.display import display
import os
# Where to save the figures and data files
PROJECT_ROOT_DIR = "Results"
FIGURE_ID = "Results/FigureFiles"
DATA_ID = "DataFiles/"
if not os.path.exists(PROJECT_ROOT_DIR):
os.mkdir(PROJECT_ROOT_DIR)
if not os.path.exists(FIGURE_ID):
os.makedirs(FIGURE_ID)
if not os.path.exists(DATA_ID):
os.makedirs(DATA_ID)
def image_path(fig_id):
return os.path.join(FIGURE_ID, fig_id)
def data_path(dat_id):
return os.path.join(DATA_ID, dat_id)
def save_fig(fig_id):
plt.savefig(image_path(fig_id) + ".png", format='png')
infile = open(data_path("MassEval2016.dat"),'r')
# Read the experimental data with Pandas
Masses = pd.read_fwf(infile, usecols=(2,3,4,6,11),
names=('N', 'Z', 'A', 'Element', 'Ebinding'),
widths=(1,3,5,5,5,1,3,4,1,13,11,11,9,1,2,11,9,1,3,1,12,11,1),
header=39,
index_col=False)
# Extrapolated values are indicated by '#' in place of the decimal place, so
# the Ebinding column won't be numeric. Coerce to float and drop these entries.
Masses['Ebinding'] = pd.to_numeric(Masses['Ebinding'], errors='coerce')
Masses = Masses.dropna()
# Convert from keV to MeV.
Masses['Ebinding'] /= 1000
# Group the DataFrame by nucleon number, A.
Masses = Masses.groupby('A')
# Find the rows of the grouped DataFrame with the maximum binding energy.
Masses = Masses.apply(lambda t: t[t.Ebinding==t.Ebinding.max()])
A = Masses['A']
Z = Masses['Z']
N = Masses['N']
Element = Masses['Element']
Energies = Masses['Ebinding']
# Now we set up the design matrix X
X = np.zeros((len(A),5))
X[:,0] = 1
X[:,1] = A
X[:,2] = A**(2.0/3.0)
X[:,3] = A**(-1.0/3.0)
X[:,4] = A**(-1.0)
# Then nice printout using pandas
DesignMatrix = pd.DataFrame(X)
DesignMatrix.index = A
DesignMatrix.columns = ['1', 'A', 'A^(2/3)', 'A^(-1/3)', '1/A']
display(DesignMatrix)
With \( \boldsymbol{\beta}\in {\mathbb{R}}^{p\times 1} \), it means that we will hereafter write our equations for the approximation as $$ \boldsymbol{\tilde{y}}= \boldsymbol{X}\boldsymbol{\beta}, $$ throughout these lectures.
With the above we use the design matrix to define the approximation \( \boldsymbol{\tilde{y}} \) via the unknown quantity \( \boldsymbol{\beta} \) as $$ \boldsymbol{\tilde{y}}= \boldsymbol{X}\boldsymbol{\beta}, $$ and in order to find the optimal parameters \( \beta_i \) instead of solving the above linear algebra problem, we define a function which gives a measure of the spread between the values \( y_i \) (which represent hopefully the exact values) and the parameterized values \( \tilde{y}_i \), namely $$ C(\boldsymbol{\beta})=\frac{1}{n}\sum_{i=0}^{n-1}\left(y_i-\tilde{y}_i\right)^2=\frac{1}{n}\left\{\left(\boldsymbol{y}-\boldsymbol{\tilde{y}}\right)^T\left(\boldsymbol{y}-\boldsymbol{\tilde{y}}\right)\right\}, $$ or using the matrix \( \boldsymbol{X} \) and in a more compact matrix-vector notation as $$ C(\boldsymbol{\beta})=\frac{1}{n}\left\{\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)^T\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)\right\}. $$ This function is one possible way to define the so-called cost function.
It is also common to define the function \( C \) as $$ C(\boldsymbol{\beta})=\frac{1}{2n}\sum_{i=0}^{n-1}\left(y_i-\tilde{y}_i\right)^2, $$ since when taking the first derivative with respect to the unknown parameters \( \beta \), the factor of \( 2 \) cancels out.
The function $$ C(\boldsymbol{\beta})=\frac{1}{n}\left\{\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)^T\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)\right\}, $$ can be linked to the variance of the quantity \( y_i \) if we interpret the latter as the mean value. When linking (see the discussion below) with the maximum likelihood approach below, we will indeed interpret \( y_i \) as a mean value $$ y_{i}=\langle y_i \rangle = \beta_0x_{i,0}+\beta_1x_{i,1}+\beta_2x_{i,2}+\dots+\beta_{n-1}x_{i,n-1}+\epsilon_i, $$
where \( \langle y_i \rangle \) is the mean value. Keep in mind also that till now we have treated \( y_i \) as the exact value. Normally, the response (dependent or outcome) variable \( y_i \) the outcome of a numerical experiment or another type of experiment and is thus only an approximation to the true value. It is then always accompanied by an error estimate, often limited to a statistical error estimate given by the standard deviation discussed earlier. In the discussion here we will treat \( y_i \) as our exact value for the response variable.
In order to find the parameters \( \beta_i \) we will then minimize the spread of \( C(\boldsymbol{\beta}) \), that is we are going to solve the problem $$ {\displaystyle \min_{\boldsymbol{\beta}\in {\mathbb{R}}^{p}}}\frac{1}{n}\left\{\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)^T\left(\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)\right\}. $$ In practical terms it means we will require $$ \frac{\partial C(\boldsymbol{\beta})}{\partial \beta_j} = \frac{\partial }{\partial \beta_j}\left[ \frac{1}{n}\sum_{i=0}^{n-1}\left(y_i-\beta_0x_{i,0}-\beta_1x_{i,1}-\beta_2x_{i,2}-\dots-\beta_{n-1}x_{i,n-1}\right)^2\right]=0, $$ which results in $$ \frac{\partial C(\boldsymbol{\beta})}{\partial \beta_j} = -\frac{2}{n}\left[ \sum_{i=0}^{n-1}x_{ij}\left(y_i-\beta_0x_{i,0}-\beta_1x_{i,1}-\beta_2x_{i,2}-\dots-\beta_{n-1}x_{i,n-1}\right)\right]=0, $$ or in a matrix-vector form as $$ \frac{\partial C(\boldsymbol{\beta})}{\partial \boldsymbol{\beta}} = 0 = \boldsymbol{X}^T\left( \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right). $$
We can rewrite $$ \frac{\partial C(\boldsymbol{\beta})}{\partial \boldsymbol{\beta}} = 0 = \boldsymbol{X}^T\left( \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right), $$ as $$ \boldsymbol{X}^T\boldsymbol{y} = \boldsymbol{X}^T\boldsymbol{X}\boldsymbol{\beta}, $$ and if the matrix \( \boldsymbol{X}^T\boldsymbol{X} \) is invertible we have the solution $$ \boldsymbol{\beta} =\left(\boldsymbol{X}^T\boldsymbol{X}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y}. $$
We note also that since our design matrix is defined as \( \boldsymbol{X}\in {\mathbb{R}}^{n\times p} \), the product \( \boldsymbol{X}^T\boldsymbol{X} \in {\mathbb{R}}^{p\times p} \). In the above case we have that \( p \ll n \), in our case \( p=5 \) meaning that we end up with inverting a small \( 5\times 5 \) matrix. This is a rather common situation, in many cases we end up with low-dimensional matrices to invert. The methods discussed here and for many other supervised learning algorithms like classification with logistic regression or support vector machines, exhibit dimensionalities which allow for the usage of direct linear algebra methods such as LU decomposition or Singular Value Decomposition (SVD) for finding the inverse of the matrix \( \boldsymbol{X}^T\boldsymbol{X} \).
Small question: Do you think the example we have at hand here (the nuclear binding energies) can lead to problems in inverting the matrix \( \boldsymbol{X}^T\boldsymbol{X} \)? What kind of problems can we expect?
The following matrix and vector relation will be useful here and for the rest of the course. Vectors are always written as boldfaced lower case letters and
matrices as upper case boldfaced letters.
$$
\frac{\partial (\boldsymbol{b}^T\boldsymbol{a})}{\partial \boldsymbol{a}} = \boldsymbol{b},
$$
$$
\frac{\partial (\boldsymbol{a}^T\boldsymbol{A}\boldsymbol{a})}{\partial \boldsymbol{a}} = (\boldsymbol{A}+\boldsymbol{A}^T)\boldsymbol{a},
$$
$$
\frac{\partial tr(\boldsymbol{B}\boldsymbol{A})}{\partial \boldsymbol{A}} = \boldsymbol{B}^T,
$$
$$
\frac{\partial \log{\vert\boldsymbol{A}\vert}}{\partial \boldsymbol{A}} = (\boldsymbol{A}^{-1})^T.
$$
The residuals \( \boldsymbol{\epsilon} \) are in turn given by $$ \boldsymbol{\epsilon} = \boldsymbol{y}-\boldsymbol{\tilde{y}} = \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}, $$ and with $$ \boldsymbol{X}^T\left( \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)= 0, $$ we have $$ \boldsymbol{X}^T\boldsymbol{\epsilon}=\boldsymbol{X}^T\left( \boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\right)= 0, $$ meaning that the solution for \( \boldsymbol{\beta} \) is the one which minimizes the residuals. Later we will link this with the maximum likelihood approach.
Let us now return to our nuclear binding energies and simply code the above equations.
It is rather straightforward to implement the matrix inversion and obtain the parameters \( \boldsymbol{\beta} \). After having defined the matrix \( \boldsymbol{X} \) we simply need to write
# matrix inversion to find beta
beta = np.linalg.inv(X.T.dot(X)).dot(X.T).dot(Energies)
# and then make the prediction
ytilde = X @ beta
Alternatively, you can use the least squares functionality in Numpy as
fit = np.linalg.lstsq(X, Energies, rcond =None)[0]
ytildenp = np.dot(fit,X.T)
And finally we plot our fit with and compare with data
Masses['Eapprox'] = ytilde
# Generate a plot comparing the experimental with the fitted values values.
fig, ax = plt.subplots()
ax.set_xlabel(r'$A = N + Z$')
ax.set_ylabel(r'$E_\mathrm{bind}\,/\mathrm{MeV}$')
ax.plot(Masses['A'], Masses['Ebinding'], alpha=0.7, lw=2,
label='Ame2016')
ax.plot(Masses['A'], Masses['Eapprox'], alpha=0.7, lw=2, c='m',
label='Fit')
ax.legend()
save_fig("Masses2016OLS")
plt.show()
We can easily test our fit by computing the \( R2 \) score that we discussed in connection with the functionality of Scikit-Learn in the introductory slides. Since we are not using Scikit-Learn here we can define our own \( R2 \) function as
def R2(y_data, y_model):
return 1 - np.sum((y_data - y_model) ** 2) / np.sum((y_data - np.mean(y_data)) ** 2)
and we would be using it as
print(R2(Energies,ytilde))
We can easily add our MSE score as
def MSE(y_data,y_model):
n = np.size(y_model)
return np.sum((y_data-y_model)**2)/n
print(MSE(Energies,ytilde))
and finally the relative error as
def RelativeError(y_data,y_model):
return abs((y_data-y_model)/y_data)
print(RelativeError(Energies, ytilde))
Normally, the response (dependent or outcome) variable \( y_i \) is the outcome of a numerical experiment or another type of experiment and is thus only an approximation to the true value. It is then always accompanied by an error estimate, often limited to a statistical error estimate given by the standard deviation discussed earlier. In the discussion here we will treat \( y_i \) as our exact value for the response variable.
Introducing the standard deviation \( \sigma_i \) for each measurement \( y_i \), we define now the \( \chi^2 \) function (omitting the \( 1/n \) term) as $$ \chi^2(\boldsymbol{\beta})=\frac{1}{n}\sum_{i=0}^{n-1}\frac{\left(y_i-\tilde{y}_i\right)^2}{\sigma_i^2}=\frac{1}{n}\left\{\left(\boldsymbol{y}-\boldsymbol{\tilde{y}}\right)^T\frac{1}{\boldsymbol{\Sigma^2}}\left(\boldsymbol{y}-\boldsymbol{\tilde{y}}\right)\right\}, $$ where the matrix \( \boldsymbol{\Sigma} \) is a diagonal matrix with \( \sigma_i \) as matrix elements.
In order to find the parameters \( \beta_i \) we will then minimize the spread of \( \chi^2(\boldsymbol{\beta}) \) by requiring $$ \frac{\partial \chi^2(\boldsymbol{\beta})}{\partial \beta_j} = \frac{\partial }{\partial \beta_j}\left[ \frac{1}{n}\sum_{i=0}^{n-1}\left(\frac{y_i-\beta_0x_{i,0}-\beta_1x_{i,1}-\beta_2x_{i,2}-\dots-\beta_{n-1}x_{i,n-1}}{\sigma_i}\right)^2\right]=0, $$ which results in $$ \frac{\partial \chi^2(\boldsymbol{\beta})}{\partial \beta_j} = -\frac{2}{n}\left[ \sum_{i=0}^{n-1}\frac{x_{ij}}{\sigma_i}\left(\frac{y_i-\beta_0x_{i,0}-\beta_1x_{i,1}-\beta_2x_{i,2}-\dots-\beta_{n-1}x_{i,n-1}}{\sigma_i}\right)\right]=0, $$ or in a matrix-vector form as $$ \frac{\partial \chi^2(\boldsymbol{\beta})}{\partial \boldsymbol{\beta}} = 0 = \boldsymbol{A}^T\left( \boldsymbol{b}-\boldsymbol{A}\boldsymbol{\beta}\right). $$ where we have defined the matrix \( \boldsymbol{A} =\boldsymbol{X}/\boldsymbol{\Sigma} \) with matrix elements \( a_{ij} = x_{ij}/\sigma_i \) and the vector \( \boldsymbol{b} \) with elements \( b_i = y_i/\sigma_i \).
We can rewrite $$ \frac{\partial \chi^2(\boldsymbol{\beta})}{\partial \boldsymbol{\beta}} = 0 = \boldsymbol{A}^T\left( \boldsymbol{b}-\boldsymbol{A}\boldsymbol{\beta}\right), $$ as $$ \boldsymbol{A}^T\boldsymbol{b} = \boldsymbol{A}^T\boldsymbol{A}\boldsymbol{\beta}, $$ and if the matrix \( \boldsymbol{A}^T\boldsymbol{A} \) is invertible we have the solution $$ \boldsymbol{\beta} =\left(\boldsymbol{A}^T\boldsymbol{A}\right)^{-1}\boldsymbol{A}^T\boldsymbol{b}. $$
If we then introduce the matrix $$ \boldsymbol{H} = \left(\boldsymbol{A}^T\boldsymbol{A}\right)^{-1}, $$ we have then the following expression for the parameters \( \beta_j \) (the matrix elements of \( \boldsymbol{H} \) are \( h_{ij} \)) $$ \beta_j = \sum_{k=0}^{p-1}h_{jk}\sum_{i=0}^{n-1}\frac{y_i}{\sigma_i}\frac{x_{ik}}{\sigma_i} = \sum_{k=0}^{p-1}h_{jk}\sum_{i=0}^{n-1}b_ia_{ik} $$ We state without proof the expression for the uncertainty in the parameters \( \beta_j \) as (we leave this as an exercise) $$ \sigma^2(\beta_j) = \sum_{i=0}^{n-1}\sigma_i^2\left( \frac{\partial \beta_j}{\partial y_i}\right)^2, $$ resulting in $$ \sigma^2(\beta_j) = \left(\sum_{k=0}^{p-1}h_{jk}\sum_{i=0}^{n-1}a_{ik}\right)\left(\sum_{l=0}^{p-1}h_{jl}\sum_{m=0}^{n-1}a_{ml}\right) = h_{jj}! $$
The first step here is to approximate the function \( y \) with a first-order polynomial, that is we write $$ y=y(x) \rightarrow y(x_i) \approx \beta_0+\beta_1 x_i. $$ By computing the derivatives of \( \chi^2 \) with respect to \( \beta_0 \) and \( \beta_1 \) show that these are given by $$ \frac{\partial \chi^2(\boldsymbol{\beta})}{\partial \beta_0} = -2\left[ \frac{1}{n}\sum_{i=0}^{n-1}\left(\frac{y_i-\beta_0-\beta_1x_{i}}{\sigma_i^2}\right)\right]=0, $$ and $$ \frac{\partial \chi^2(\boldsymbol{\beta})}{\partial \beta_1} = -\frac{2}{n}\left[ \sum_{i=0}^{n-1}x_i\left(\frac{y_i-\beta_0-\beta_1x_{i}}{\sigma_i^2}\right)\right]=0. $$
For a linear fit (a first-order polynomial) we don't need to invert a matrix!! Defining $$ \gamma = \sum_{i=0}^{n-1}\frac{1}{\sigma_i^2}, $$ $$ \gamma_x = \sum_{i=0}^{n-1}\frac{x_{i}}{\sigma_i^2}, $$ $$ \gamma_y = \sum_{i=0}^{n-1}\left(\frac{y_i}{\sigma_i^2}\right), $$ $$ \gamma_{xx} = \sum_{i=0}^{n-1}\frac{x_ix_{i}}{\sigma_i^2}, $$ $$ \gamma_{xy} = \sum_{i=0}^{n-1}\frac{y_ix_{i}}{\sigma_i^2}, $$
we obtain $$ \beta_0 = \frac{\gamma_{xx}\gamma_y-\gamma_x\gamma_y}{\gamma\gamma_{xx}-\gamma_x^2}, $$ $$ \beta_1 = \frac{\gamma_{xy}\gamma-\gamma_x\gamma_y}{\gamma\gamma_{xx}-\gamma_x^2}. $$
This approach (different linear and non-linear regression) suffers often from both being underdetermined and overdetermined in the unknown coefficients \( \beta_i \). A better approach is to use the Singular Value Decomposition (SVD) method discussed below. Or using Lasso and Ridge regression. See below.
Before we continue, let us introduce yet another example. We are going to fit the nuclear equation of state using results from many-body calculations. The equation of state we have made available here, as function of density, has been derived using modern nucleon-nucleon potentials with the addition of three-body forces. This time the file is presented as a standard csv file.
The beginning of the Python code here is similar to what you have seen before, with the same initializations and declarations. We use also pandas again, rather extensively in order to organize our data.
The difference now is that we use Scikit-Learn's regression tools instead of our own matrix inversion implementation. Furthermore, we sneak in Ridge regression (to be discussed below) which includes a hyperparameter \( \lambda \), also to be explained below.
# Common imports
import os
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
import matplotlib.pyplot as plt
import sklearn.linear_model as skl
from sklearn.metrics import mean_squared_error, r2_score, mean_absolute_error
# Where to save the figures and data files
PROJECT_ROOT_DIR = "Results"
FIGURE_ID = "Results/FigureFiles"
DATA_ID = "DataFiles/"
if not os.path.exists(PROJECT_ROOT_DIR):
os.mkdir(PROJECT_ROOT_DIR)
if not os.path.exists(FIGURE_ID):
os.makedirs(FIGURE_ID)
if not os.path.exists(DATA_ID):
os.makedirs(DATA_ID)
def image_path(fig_id):
return os.path.join(FIGURE_ID, fig_id)
def data_path(dat_id):
return os.path.join(DATA_ID, dat_id)
def save_fig(fig_id):
plt.savefig(image_path(fig_id) + ".png", format='png')
infile = open(data_path("EoS.csv"),'r')
# Read the EoS data as csv file and organize the data into two arrays with density and energies
EoS = pd.read_csv(infile, names=('Density', 'Energy'))
EoS['Energy'] = pd.to_numeric(EoS['Energy'], errors='coerce')
EoS = EoS.dropna()
Energies = EoS['Energy']
Density = EoS['Density']
# The design matrix now as function of various polytrops
X = np.zeros((len(Density),4))
X[:,3] = Density**(4.0/3.0)
X[:,2] = Density
X[:,1] = Density**(2.0/3.0)
X[:,0] = 1
# We use now Scikit-Learn's linear regressor and ridge regressor
# OLS part
clf = skl.LinearRegression().fit(X, Energies)
ytilde = clf.predict(X)
EoS['Eols'] = ytilde
# The mean squared error
print("Mean squared error: %.2f" % mean_squared_error(Energies, ytilde))
# Explained variance score: 1 is perfect prediction
print('Variance score: %.2f' % r2_score(Energies, ytilde))
# Mean absolute error
print('Mean absolute error: %.2f' % mean_absolute_error(Energies, ytilde))
print(clf.coef_, clf.intercept_)
# The Ridge regression with a hyperparameter lambda = 0.1
_lambda = 0.1
clf_ridge = skl.Ridge(alpha=_lambda).fit(X, Energies)
yridge = clf_ridge.predict(X)
EoS['Eridge'] = yridge
# The mean squared error
print("Mean squared error: %.2f" % mean_squared_error(Energies, yridge))
# Explained variance score: 1 is perfect prediction
print('Variance score: %.2f' % r2_score(Energies, yridge))
# Mean absolute error
print('Mean absolute error: %.2f' % mean_absolute_error(Energies, yridge))
print(clf_ridge.coef_, clf_ridge.intercept_)
fig, ax = plt.subplots()
ax.set_xlabel(r'$\rho[\mathrm{fm}^{-3}]$')
ax.set_ylabel(r'Energy per particle')
ax.plot(EoS['Density'], EoS['Energy'], alpha=0.7, lw=2,
label='Theoretical data')
ax.plot(EoS['Density'], EoS['Eols'], alpha=0.7, lw=2, c='m',
label='OLS')
ax.plot(EoS['Density'], EoS['Eridge'], alpha=0.7, lw=2, c='g',
label='Ridge $\lambda = 0.1$')
ax.legend()
save_fig("EoSfitting")
plt.show()
The above simple polynomial in density \( \rho \) gives an excellent fit to the data.
We note also that there is a small deviation between the standard OLS and the Ridge regression at higher densities. We discuss this in more detail below.
It is normal in essentially all Machine Learning studies to split the data in a training set and a test set (sometimes also an additional validation set). Scikit-Learn has an own function for this. There is no explicit recipe for how much data should be included as training data and say test data. An accepted rule of thumb is to use approximately \( 2/3 \) to \( 4/5 \) of the data as training data. We will postpone a discussion of this splitting to the end of these notes and our discussion of the so-called bias-variance tradeoff. Here we limit ourselves to repeat the above equation of state fitting example but now splitting the data into a training set and a test set.
import os
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from sklearn.model_selection import train_test_split
# Where to save the figures and data files
PROJECT_ROOT_DIR = "Results"
FIGURE_ID = "Results/FigureFiles"
DATA_ID = "DataFiles/"
if not os.path.exists(PROJECT_ROOT_DIR):
os.mkdir(PROJECT_ROOT_DIR)
if not os.path.exists(FIGURE_ID):
os.makedirs(FIGURE_ID)
if not os.path.exists(DATA_ID):
os.makedirs(DATA_ID)
def image_path(fig_id):
return os.path.join(FIGURE_ID, fig_id)
def data_path(dat_id):
return os.path.join(DATA_ID, dat_id)
def save_fig(fig_id):
plt.savefig(image_path(fig_id) + ".png", format='png')
def R2(y_data, y_model):
return 1 - np.sum((y_data - y_model) ** 2) / np.sum((y_data - np.mean(y_data)) ** 2)
def MSE(y_data,y_model):
n = np.size(y_model)
return np.sum((y_data-y_model)**2)/n
infile = open(data_path("EoS.csv"),'r')
# Read the EoS data as csv file and organized into two arrays with density and energies
EoS = pd.read_csv(infile, names=('Density', 'Energy'))
EoS['Energy'] = pd.to_numeric(EoS['Energy'], errors='coerce')
EoS = EoS.dropna()
Energies = EoS['Energy']
Density = EoS['Density']
# The design matrix now as function of various polytrops
X = np.zeros((len(Density),5))
X[:,0] = 1
X[:,1] = Density**(2.0/3.0)
X[:,2] = Density
X[:,3] = Density**(4.0/3.0)
X[:,4] = Density**(5.0/3.0)
# We split the data in test and training data
X_train, X_test, y_train, y_test = train_test_split(X, Energies, test_size=0.2)
# matrix inversion to find beta
beta = np.linalg.inv(X_train.T.dot(X_train)).dot(X_train.T).dot(y_train)
# and then make the prediction
ytilde = X_train @ beta
print("Training R2")
print(R2(y_train,ytilde))
print("Training MSE")
print(MSE(y_train,ytilde))
ypredict = X_test @ beta
print("Test R2")
print(R2(y_test,ypredict))
print("Test MSE")
print(MSE(y_test,ypredict))
The Boston housing data set was originally a part of UCI Machine Learning Repository and has been removed now. The data set is now included in Scikit-Learn's library. There are 506 samples and 13 feature (predictor) variables in this data set. The objective is to predict the value of prices of the house using the features (predictors) listed here.
The features/predictors are
import numpy as np
import matplotlib.pyplot as plt
import pandas as pd
import seaborn as sns
and load the Boston Housing DataSet from Scikit-Learn
from sklearn.datasets import load_boston
boston_dataset = load_boston()
# boston_dataset is a dictionary
# let's check what it contains
boston_dataset.keys()
Then we invoke Pandas
boston = pd.DataFrame(boston_dataset.data, columns=boston_dataset.feature_names)
boston.head()
boston['MEDV'] = boston_dataset.target
and preprocess the data
# check for missing values in all the columns
boston.isnull().sum()
We can then visualize the data
# set the size of the figure
sns.set(rc={'figure.figsize':(11.7,8.27)})
# plot a histogram showing the distribution of the target values
sns.distplot(boston['MEDV'], bins=30)
plt.show()
It is now useful to look at the correlation matrix
# compute the pair wise correlation for all columns
correlation_matrix = boston.corr().round(2)
# use the heatmap function from seaborn to plot the correlation matrix
# annot = True to print the values inside the square
sns.heatmap(data=correlation_matrix, annot=True)
From the above coorelation plot we can see that MEDV is strongly correlated to LSTAT and RM. We see also that RAD and TAX are stronly correlated, but we don't include this in our features together to avoid multi-colinearity
plt.figure(figsize=(20, 5))
features = ['LSTAT', 'RM']
target = boston['MEDV']
for i, col in enumerate(features):
plt.subplot(1, len(features) , i+1)
x = boston[col]
y = target
plt.scatter(x, y, marker='o')
plt.title(col)
plt.xlabel(col)
plt.ylabel('MEDV')
Now we start training our model
X = pd.DataFrame(np.c_[boston['LSTAT'], boston['RM']], columns = ['LSTAT','RM'])
Y = boston['MEDV']
We split the data into training and test sets
from sklearn.model_selection import train_test_split
# splits the training and test data set in 80% : 20%
# assign random_state to any value.This ensures consistency.
X_train, X_test, Y_train, Y_test = train_test_split(X, Y, test_size = 0.2, random_state=5)
print(X_train.shape)
print(X_test.shape)
print(Y_train.shape)
print(Y_test.shape)
Then we use the linear regression functionality from Scikit-Learn
from sklearn.linear_model import LinearRegression
from sklearn.metrics import mean_squared_error, r2_score
lin_model = LinearRegression()
lin_model.fit(X_train, Y_train)
# model evaluation for training set
y_train_predict = lin_model.predict(X_train)
rmse = (np.sqrt(mean_squared_error(Y_train, y_train_predict)))
r2 = r2_score(Y_train, y_train_predict)
print("The model performance for training set")
print("--------------------------------------")
print('RMSE is {}'.format(rmse))
print('R2 score is {}'.format(r2))
print("\n")
# model evaluation for testing set
y_test_predict = lin_model.predict(X_test)
# root mean square error of the model
rmse = (np.sqrt(mean_squared_error(Y_test, y_test_predict)))
# r-squared score of the model
r2 = r2_score(Y_test, y_test_predict)
print("The model performance for testing set")
print("--------------------------------------")
print('RMSE is {}'.format(rmse))
print('R2 score is {}'.format(r2))
# plotting the y_test vs y_pred
# ideally should have been a straight line
plt.scatter(Y_test, y_test_predict)
plt.show()
Many Machine Learning problems involve thousands or even millions of features for each training instance. Not only does this make training extremely slow, it can also make it much harder to find a good solution, as we will see. This problem is often referred to as the curse of dimensionality. Fortunately, in real-world problems, it is often possible to reduce the number of features considerably, turning an intractable problem into a tractable one.
Later we will discuss some of the most popular dimensionality reduction techniques: the principal component analysis (PCA), Kernel PCA, and Locally Linear Embedding (LLE).
Principal component analysis and its various variants deal with the problem of fitting a low-dimensional affine subspace to a set of of data points in a high-dimensional space. With its family of methods it is one of the most used tools in data modeling, compression and visualization.
Before we proceed however, we will discuss how to preprocess our data. Till now and in connection with our previous examples we have not met so many cases where we are too sensitive to the scaling of our data. Normally the data may need a rescaling and/or may be sensitive to extreme values. Scaling the data renders our inputs much more suitable for the algorithms we want to employ.
Scikit-Learn has several functions which allow us to rescale the data, normally resulting in much better results in terms of various accuracy scores. The StandardScaler function in Scikit-Learn ensures that for each feature/predictor we study the mean value is zero and the variance is one (every column in the design/feature matrix). This scaling has the drawback that it does not ensure that we have a particular maximum or minimum in our data set. Another function included in Scikit-Learn is the MinMaxScaler which ensures that all features are exactly between \( 0 \) and \( 1 \). The
The Normalizer scales each data point such that the feature vector has a euclidean length of one. In other words, it projects a data point on the circle (or sphere in the case of higher dimensions) with a radius of 1. This means every data point is scaled by a different number (by the inverse of it’s length). This normalization is often used when only the direction (or angle) of the data matters, not the length of the feature vector.
The RobustScaler works similarly to the StandardScaler in that it ensures statistical properties for each feature that guarantee that they are on the same scale. However, the RobustScaler uses the median and quartiles, instead of mean and variance. This makes the RobustScaler ignore data points that are very different from the rest (like measurement errors). These odd data points are also called outliers, and might often lead to trouble for other scaling techniques.
# Common imports
import os
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
import sklearn.linear_model as skl
from sklearn.metrics import mean_squared_error
from sklearn.model_selection import train_test_split
from sklearn.preprocessing import MinMaxScaler, StandardScaler, Normalizer
# Where to save the figures and data files
PROJECT_ROOT_DIR = "Results"
FIGURE_ID = "Results/FigureFiles"
DATA_ID = "DataFiles/"
if not os.path.exists(PROJECT_ROOT_DIR):
os.mkdir(PROJECT_ROOT_DIR)
if not os.path.exists(FIGURE_ID):
os.makedirs(FIGURE_ID)
if not os.path.exists(DATA_ID):
os.makedirs(DATA_ID)
def image_path(fig_id):
return os.path.join(FIGURE_ID, fig_id)
def data_path(dat_id):
return os.path.join(DATA_ID, dat_id)
def save_fig(fig_id):
plt.savefig(image_path(fig_id) + ".png", format='png')
def FrankeFunction(x,y):
term1 = 0.75*np.exp(-(0.25*(9*x-2)**2) - 0.25*((9*y-2)**2))
term2 = 0.75*np.exp(-((9*x+1)**2)/49.0 - 0.1*(9*y+1))
term3 = 0.5*np.exp(-(9*x-7)**2/4.0 - 0.25*((9*y-3)**2))
term4 = -0.2*np.exp(-(9*x-4)**2 - (9*y-7)**2)
return term1 + term2 + term3 + term4
def create_X(x, y, n ):
if len(x.shape) > 1:
x = np.ravel(x)
y = np.ravel(y)
N = len(x)
l = int((n+1)*(n+2)/2) # Number of elements in beta
X = np.ones((N,l))
for i in range(1,n+1):
q = int((i)*(i+1)/2)
for k in range(i+1):
X[:,q+k] = (x**(i-k))*(y**k)
return X
# Making meshgrid of datapoints and compute Franke's function
n = 5
N = 1000
x = np.sort(np.random.uniform(0, 1, N))
y = np.sort(np.random.uniform(0, 1, N))
z = FrankeFunction(x, y)
X = create_X(x, y, n=n)
# split in training and test data
X_train, X_test, y_train, y_test = train_test_split(X,z,test_size=0.2)
clf = skl.LinearRegression().fit(X_train, y_train)
# The mean squared error and R2 score
print("MSE before scaling: {:.2f}".format(mean_squared_error(clf.predict(X_test), y_test)))
print("R2 score before scaling {:.2f}".format(clf.score(X_test,y_test)))
scaler = StandardScaler()
scaler.fit(X_train)
X_train_scaled = scaler.transform(X_train)
X_test_scaled = scaler.transform(X_test)
print("Feature min values before scaling:\n {}".format(X_train.min(axis=0)))
print("Feature max values before scaling:\n {}".format(X_train.max(axis=0)))
print("Feature min values after scaling:\n {}".format(X_train_scaled.min(axis=0)))
print("Feature max values after scaling:\n {}".format(X_train_scaled.max(axis=0)))
clf = skl.LinearRegression().fit(X_train_scaled, y_train)
print("MSE after scaling: {:.2f}".format(mean_squared_error(clf.predict(X_test_scaled), y_test)))
print("R2 score for scaled data: {:.2f}".format(clf.score(X_test_scaled,y_test)))
Video of Lecture and handwritten notes
More material will be added here, see handwritten notes also.


--- layout: default group: 'navigation' title: About bodyclass: about headline: 'About SPIFFE' ---
SPIFFE (Secure Production Identity Framework For Everyone) is an open-source framework that aims to provide a secure identity (and a means to prove that identity) to every workload in a modern production environment, no matter where it is running.
Distributed design patterns and practices such as microservices, container orchestrators, and cloud computing have led to production environments that are increasingly dynamic and heterogenous. Conventional security practices (such as network policies that only allow traffic between particular IP addresses) struggle to scale under this complexity. A first-class identity framework for workloads in an organization becomes necessary.
Further, modern developers are expected to understand and play a role in how applications are deployed and managed in production environments. Operations teams require deeper visibility into the applications they are managing. As we move to a more evolved security stance, we must offer better tools to both teams so they can play an active role in building secure, distributed applications.
The SPIFFE project will provide both a specification for, and open-source reference implementation of a framework for bootstraping and issuing identity to services accross a heterogeneous hosting environment.
Flexible bootstrap. Identity is about trust, and that trust must be bootstrapped. The system should support multiple mechanisms as they make sense, including hooking into hardware security mechanisms (such as trusted platform modules (TPMs)).
Easy to use. Cryptography is difficult to use and even more difficult to use correctly. SPIFFE will provide automated, API-driven mechanisms for automatically issuing, verifying, and rotating identity documents.
Broad integration. The more frameworks and tools that build in support for SPIFFE, the more useful it will be. Built-in support in RPC/micro-service frameworks, storage systems, and orchestrators will multiply its usefulness.
Reliable. The single points of failure (SPOF) in the system will be minimized, and the system will degrade gracefully when any SPOF is down. All “steady state” operations should not have requirements from a specific node. This precludes any centralized signing/verification service.
Scoped trust roots. While public key infrastructure may used as part of the solution, there will be no hardcoded, global trust roots as we see in the web browser world. Users must explicitly decide which roots to trust for which identities. Typically, these roots will be set per organization.
Federatable. Identity mechansims can be used across organizations. However, federation trust must be explicitly configured. The nitty gritty of how to establish this trust may be out of scope for the first cut of these specifications.
Legacy friendly. SPIFFE can be applied without re-writing applications or moving them to a new management system. It will be possible to adapt this system as separable infrastructure to secure existing systems with little or no code change.
Container friendly. The system should work well in dynamically orchestrated container environments.
Minimal knowledge. A compromised host will only expose secrets for workloads running only on that host.
SPIFFE allows organizations to employ more robust secuity tooling by ensuring all running workloads within an organization are issued an identity and the means to prove that identity to other running workloads.
SPIFFE's initial release focuses on issuing SPIFFE Verifiable Identity Documents (SVID). In practice SVIDs are specially-formatted X.509 leaf certificates that allow a workload to assert identity, and set of root or intermediate certificates to allow for third-party verification.
Enable two (2) workloads to establish a mTLS connection
Provide PKI certificates to be consumed by service proxies such as Envoy, Linkerd, and nginx
Provide PKI certificates to allow a service to communicate with a central secret store (such as HashiCorp Vault)
Audit exactly what systems are running on which nodes in a production infrastructure
SPIFFE's specification and reference implementation are being developed by engineers from organizations such as Docker, Dropbox, Google, and Twillio. Read more on current docs and designs, and learn how to get involved.


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Preparation time
less than 30 mins
Cooking time
10 to 30 mins
Serves
Serves 4
This spicy one-pot dish can be ready in under 30 minutes - perfect when you want a quick and hearty meal without any fuss. This is designed to be a low cost recipe.
Heat a large saucepan with a lid over a medium heat. Add the olive oil and, once hot, fry the onion with a pinch of salt for 5-6 minutes or until cooked through and lightly golden-brown. Add the garlic, cumin seeds and chilli and continue to cook for two minutes, or until aromatic.
Add the beef mince to the pan and cook for 4-5 minutes, or until all the meat is brown rather than raw.
Add the butter and, once melted, stir in the rice, mixing over the heat for two minutes so that the rice begins to absorb the melted butter and any juices in the pan. Pour in enough of the stock to cover the rice by 1cm/½in. Stir once, cover with a tight fitting lid, reduce the heat to medium-low and leave for five minutes.
After five minutes remove the lid, add the peas and stir through. Poke some holes into the rice to allow any steam to escape and continue to cook with the lid off until all the liquid is absorbed and the rice is cooked through.
Serve warm, dividing up the crisp rice at the bottom of the pan.
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Browning meat


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--- layout: post title: Graduated custom_css: true custom_js: true date: 2018-04-30 23:16:30 -5000 ---
After the long 8 years Bachelor's path from Bunker hill where I got my associates and transferred to Wentworth Institute of Technology, I can finally say I am done. The ceremony was interesting I had really no idea what was going on. Apparently, I was the only one from my major graduating that day.
They told us to come at 12:30, but it seemed to drag on waiting for the commencement at 2 PM. I managed to fill myself up on chewy Nature's Valley bars and chips just before we had to go. We filed in right after the bagpipe marching band. Everything seemed to go smoothly after that.
I graduated cum laude with a Bachelors in Computer Networking and a minor in computer science. Great job by Wentworth to power through 300+ graduates in a relatively short period of time. My wife, Melania, my wife's friends, my parents, and brother came to the ceremony. Seats were not a problem since there was a low amount of graduates at the graduation.
They had two speakers before the diplomas. The first speaker was a master student graduating with special honors. She was mostly talking to the women of the audience and told them to stop caring because it was being a roadblock in their careers. The second speaker was more interesting. He moved to the US from Lebanon in the 80's and basically only knew 'hi' and 'bye'. He worked his way up and by accident landed himself a public sector job at Massport. He talked about some of his struggles as a Muslim immigrant after 9/11 and his successes at work. He was given an honorary degree from Wentworth.
Afterwards, my family left for the afternoon, and my wife and I went home with her friends and had some Monte Cristo's to celebrate. I thought the Russians understood what fajitas were and they ordered two big meals for themselves. I just went with the tried and trusted burrito. They were so surprised at the amount of food and the three of them were unable to finish. Rest of the night was spent with well-deserved relaxation.


Q1 : Can I return my product if I dislike its look after purchasing? A1 : No, you are able to return the product if there are any recognizable faults within a time period of 14 days. Q2 : How long would it take for my money to be returned to me after i place return request? A2 : It will take 3-5 days for the delivery company to come pick up your package, afterwards it will go through our quality assurance check and if a defect is detected you will recieve your money back within 4-6 days of our check


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v4.5.0
- Added "reviewcount" filter to filter on the number of reviews on an item.
- Fix slight horizontal scroll on inventory view.
- On mobile, tapping outside of dialogs and dropdowns to dismiss them now works.
- The item detail popup now does a better job of fitting itself onto the screen - it may appear to the left or right of an item now!
- Press on a talent grid node to read its description. The same goes for the stats under your character.
- Subclasses now have the correct elemental type in their header color.
- Drag and drop should be much smoother now.
- You can select Destiny 2 accounts from the account dropdown now - but won't do much until Destiny 2 is released and we have a chance to update DIM to support it!
View the changelog for
the full history.
Follow us on:


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Although I've always had an interest for programming and technology, I've only just recently started taking a deep dive into it all. Being torn between my need for creativity and technical problem solving, I'd say I have found the best field to focus on. Being obsessed with learning about everything, my interests include game development and design, task automation and real world applications, full stack development and security.


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Qualifications and Awards
- Member of team NAIST-RITS-Panasonic at Nara Institute of Science and Technology
- Participated in the World Robot Summit 2020, Future Convenience Store Challenge (Nagoya, Japan):
-
Overall winner prize and 1st place in two of three main tasks using UR5e platform.
- Participated in the World Robot Summit 2018, Future Convenience Store Challenge (Tokyo, Japan):
-
2nd place in one of three main tasks using using UR5e platform.
- April 2020 - March 2023
- Member of team marketing at Thai Students’ Association in Japan under the Royal Patronage
-
Communicated with companies and organizations located in Thailand and Japan using Thai and English languages
- April 2018 - March 2019
- Interns at CT Asia Robotics (Bangkok, Thailand)
-
Implemented a human’s face recognition program on a Dinsaw Mini service robot
- June 2016 - August 2016
- Member of team Skuba and Skuba-JR at Kasetsart University
- RoboCup Asia Pacific 2017, RoboCup@Home Education League (Bangkok, Thailand):
-
2nd place in RoboCup@Home Education using Turtle bot platform
- RoboCup Japan Open 2017, RoboCup@Home Education League (Nagoya, Japan):
-
2nd place in RoboCup@Home Education using Turtle bot platform
- June 2015 - June 2017
- President of student council at Computer Engineering Department, Kasetsart University
-
Managed internal events for computer engineering students and collaborate with faculty members
-
Arranged social events such as BarCamp Bangkhen
- April 2015 - March 2016


-
peter speaks of mary magdalene
Make your own ringtones online | How To - CNET
Sep 16, 2009 . Finding ringtones is easy, but if you want to create your own, you might be happy to know that there are some great resources that will help you .
http://howto.cnet.com/8301-11310_39-10355022-285/make-your-own-ringtones-online/
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How to: create your own iPhone ringtones | Ars Technica
Feb 28, 2012 . Some iPhone users have been creating their own ringtones for years, which
http://arstechnica.com/apple/guides/2012/02/how-to-create-your-own-iphone-ringtones.ars
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How to Make Your Own Ringtones for TracFones | Tech Tips - Salon ...
You can add downloaded ringtones to the TracFone. You can also make your own ringtones through online ringtone-making software applications.
http://techtips.salon.com/make-own-ringtones-tracfones-1847.html
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Make your own ringtone, create ringtones from your music
our online ringtone maker lets you create your own ringtone in a couple of easy steps - Free.
http://www.cellmind.com/make_ringtones/
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10 Tools For Creating Your Own Ringtones
Mar 13, 2008 . Changing ringtones often, and you can never find that perfect one? There is a wide selection of tools on the web now for you to chop up your .
http://mashable.com/2008/03/13/create-your-own-ringtone/
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iTunes 11: How To Create Free Ringtone and Text Tone - SimonBlog
Dec 1, 2012 . It turns out the feature is still here and you're allowed to create ringtone with 40- second in length. If you want to create your own ringtone or text .
http://www.simonblog.com/2012/12/01/itunes-11-how-to-create-free-ringtone-and-text-tone/
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CyberNotes: Create Free iPhone Ringtones Using iTunes in Windows
One of the things that Apple hasn't made easy for iPhone owners is providing a way to create custom ringtones in Windows. If you're a Mac owner you can use .
http://cybernetnews.com/cybernotes-create-free-iphone-ringtones-using-itunes-in-windows/
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How to Make Custom iPhone Ringtones for Free | eHow.com
There are plenty of songs in your music library, but no way to make ringtones from . you can personalize the unit is to add your own hand-selected ringtones.
http://www.ehow.com/how_2160460_custom-iphone-ringtones-free.html
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How to make your own iPhone ringtone with Xilisoft iPhone ...
Free download the smart iPhone ringtone maker and try it to help you make your own iPhone ringtone with easy-to-use Xilisoft iPhone Ringtone Maker software.
http://www.xilisoft.com/iphone-ringtone-maker/how-to-make-your-own-iphone-ringtone.html
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How to Create your own ringtone on an iPhone « Smartphones
If you're tired of the boring old ringtones on your iPhone but don't want to shell out extra dough to buy a new one, here's an easy way to make your own.
http://smartphones.wonderhowto.com/how-to/create-your-own-ringtone-iphone-279247/
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How to Quickly Make a Ringtone in iTunes | Mac|Life
Jul 10, 2012 . To create the 30-second version of the song, right-click on the song and select . To sync the custom ringtone, select your device in the iTunes .
http://www.maclife.com/article/howtos/how_quickly_make_ringtone_itunes
-
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How to Create Your Own Free Ringtone for your Apple iPhone
How to Create Your Own Free Ringtone for your Apple iPhone. We've all been in the same situation, there is a song that you very much like and just can't get out .
http://www.wikihow.com/Create-Your-Own-Free-Ringtone-for-your-Apple-iPhone
-
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Ringtone Maker - Android Apps on Google Play
Mar 11, 2013 . Make unlimited free ringtones with the music on your Android device with . Create unlimited ringtones, custom lengths... up to 40-seconds!
https://play.google.com/store/apps/details?id=com.mobile17.maketones.android.free&hl=en
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How do I create a custom ringtone for my iPhone? - Ask Different
Jul 30, 2010 . This will create a new file in your iTunes library. Delete it from your iTunes library now, but be sure to check "keep file" in the delete dialog so .
http://apple.stackexchange.com/questions/1884/how-do-i-create-a-custom-ringtone-for-my-iphone
-
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A Handy Tool to Create Your Own Ringtones on Your Android ...
Jul 3, 2012 . One of the best and most popular free apps not to miss out and absolutely a handy tool to make your own great ringtones right on your Android .
http://www.techsupportalert.com/content/handy-tool-create-your-own-ringtones-your-android-phone.htm
-
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How to add custom ringtones to your iPhone?
Jul 31, 2009 . All you need to get this done is an iPhone, Connector Cable and iTunes. Follow these (simple) steps to add a custom ringtone to your iPhone, .
http://www.demogeek.com/2009/07/31/how-to-add-custom-ringtones-to-your-iphone/
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All about iPhone Ringtone Format (iPhone 5 included)
Jan 7, 2013 . Part 1: Purchase iPhone ringtone on iTunes store; Part 2: Create iPhone ringtone with your own song for free; Part 3: Activate iPhone ringtone .
http://www.iskysoft.com/topic-ringtone/whats-format-of-iphone-ringtone.html
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How to set YOUR OWN music as ringtones on iPhone (tutorial ...
May 2, 2011 . How to set your own music as ringtones on iPhone Download . Sign in to add channels to your guide and for great recommendations! Sign in .
http://www.youtube.com/watch?v=mCLKHKrAZcQ
-
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How to Create Your Own Free Ringtone for your Apple iPhone
How to Create Your Own Free Ringtone for your Apple iPhone. We've all been in the same situation, there is a song that you very much like and just can't get out .
http://www.wikihow.com/Create-Your-Own-Free-Ringtone-for-your-Apple-iPhone
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AVS Ringtone Maker - create your own ringtones and realtones from ...
AVS Ringtone Maker. Create ringtones and realtones from audio files.
http://www.avs4you.com/AVS-Ringtone-Maker.aspx


Links
Room to window calculation
DQB Discord Bot (WIP)
1.7.10 minecraft Modpack
GAM-gen Project
RJS Codeworks


THE NORTH SIDE OF A CHURCH. [a]
STRONG prejudice has long existed against burying on, the northern side of the church. In many churchyards the southern side will be found full of graves, with scarcely any on the northern side.
I have sought to discover, if possible, the origin of this prejudice, but I have not been able to trace it to any well-defined feeling. I have been answered, "Oh, we like to bury a corpse where the sun will shine on the grave ;" and, "The northern graveyard is in the shadow, and cold ;" but beyond this I have not advanced.
We may infer that this desire to place the remains of our friends in earth on which the sun shines, is born of that love which, forgetting mortality, lives on the pleasant 'memories of the past, hoping for that meeting beyond the grave which shall know no shadow. The act of planting flowers, of nurturing an evergreen tree, of hanging "eternals" on the tomb, is only another form of the same sacred feeling.
[a] See Appendix DD.


DIRECTOR, PROJECT IMPLEMENTATION FEDERAL MINISTRY OF HEALTH AND SOCIAL SERVICES, DAKAR SENEGAL, TEL:008821655515306, EMAIL<EMAIL_ADDRESS>STRICTLY CONFIDENTIAL Dear Sir/Madam, TRANSFER OF US$25.5 MILLION INTO A PERSONAL/COMPANY'S OFFSHORE ACCOUNT Based on the information gathered from the Ministry of Trade and Industry,we intend to solicit your assistance on this transaction with you on the assumption that you will not disappoint us. We have Twenty Five Million, Five Hundred Thousand United States Dollars (US$25,500,000.00) which we made over time from over inflated contracts in my Ministry (Federal Ministry of Health and Social Services). We are seeking your assistance and permission to remit this amount into your account or any other nominated account you can provide for us. Your commission will be 20% of the total sum, 10% for expenses and the remaining 70% is for my colleagues and myself. Could you please notify me of your acceptance to carry out this transaction urgently by e-mail only on the receipt of this message. Kindly, acknowledge the receipt of this letter by sending to me by email a copy of this letter with your private Tel. And Fax number. I shall in turn inform you of the modalities for a formal application to secure the necessary approvals for the immediate release of this fund into your account. Thanks for your co-operation.


my digital and very public journal...
I write about my experience and process learning something new. Want to read about the technical side of my learning? You are in the right place. Maybe you want to know me as a learner, as a thinker, as a former math teacher turned web developer. Got those, too. Check out the cultural posts.


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VisuaLynk IoT Sensor Monitoring System
Demo integration of < VisuaLynk_> for YIT Hack for IoT data transfer.
12.0
0.7
1
Demo integration of < VisuaLynk_> for YIT Hack for IoT data transfer.


Download and explore datasets from UCSC Xena data hubs, which are a collection of UCSC-hosted public databases such as TCGA, ICGC, TARGET, GTEx, CCLE, and others. Databases are normalized so they can be combined, linked, filtered, explored and downloaded. This package is based on 'XenaR' package.
| Version: | 0.2.6 |
| Depends: | R (≥ 2.10) |
| Imports: | httr, readr, methods, utils, dplyr, shinydashboard, shiny, DT |
| Suggests: | knitr, rmarkdown, prettydoc |
| Published: | 2018-12-09 |
| Author: | Shixiang Wang [aut, cre], Martin Morgan [aut] |
| Maintainer: | Shixiang Wang <w_shixiang at 163.com> |
| BugReports: | https://github.com/ShixiangWang/UCSCXenaTools/issues |
| License: | GPL-3 |
| URL: | https://github.com/ShixiangWang/UCSCXenaTools |
| NeedsCompilation: | no |
| Materials: | README NEWS |
| CRAN checks: | UCSCXenaTools results |
| Reference manual: | UCSCXenaTools.pdf |
| Vignettes: |
download publical data in batch quickly and easily |
| Package source: | UCSCXenaTools_0.2.6.tar.gz |
| Windows binaries: | r-devel: UCSCXenaTools_0.2.6.zip, r-release: UCSCXenaTools_0.2.6.zip, r-oldrel: UCSCXenaTools_0.2.6.zip |
| OS X binaries: | r-release: UCSCXenaTools_0.2.6.tgz, r-oldrel: UCSCXenaTools_0.2.6.tgz |
| Old sources: | UCSCXenaTools archive |
Please use the canonical form https://CRAN.R-project.org/package=UCSCXenaTools to link to this page.


Leka - The Amazing Online Store
Contrary to popular belief, Lorem Ipsum is not simply random text.  Richard McClintock, a Latin professor at Hampden-Sydney College in Virginia,
It is a long established fact that a reader will be distracted by the readable content of a page when looking at its layout. The point of using Lorem Ipsum is that it has a more-or-less normal distribution of letters, as opposed to using 'Content here, content here', making it look like readable English. Many desktop publishing packages and web page editors now use Lorem Ipsum as their default model text, and a search for 'lorem ipsum' will uncover many web sites still in their infancy. Various versions have evolved over the years, sometimes by accident, sometimes on purpose (injected humour and the like).
Happy Fashion Designer
It is a long established fact that a reader will be distracted by the readable content of a page when looking at its layout. The point of using Lorem Ipsum is that it has a more-or-less normal distribution of letters,
as opposed to using 'Content here, content here', making it look like readable English. Many desktop publishing packages and web
We Make Your Success
It is a long established fact that a reader will be distracted by the readable content of a page when looking at its layout. The point of using Lorem Ipsum is that it has a more-or-less normal distribution of letters, as opposed to using 'Content here, content here', making it look like readable English. Many desktop publishing packages and web page editors now use Lorem Ipsum as their default model text, and a search for 'lorem ipsum' will uncover many web sites still in their infancy. Various versions have evolved over the years, sometimes by accident, sometimes on purpose (injected humour and the like).
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"The Mexican" – Super Cast Sizzles, Story Fizzles by Homer Yen (c) 2001
The good news is that for two hours, "The Mexican" brings together actors who are currently at the pinnacle of their careers and popularity. Though longevity in Hollywood is frequently ephemeral, heartthrob Brad Pitt is still the recipient of much adoration from moviegoers. Effervescent Julia Roberts is the hands-down favorite to walk away with her first Oscar as Best Actress for her recent work as the sassy "Erin Brokovich." And burly James Gandolfini is known as everyone's favorite Mob boss in HBO's incredibly edgy drama, "The Soprano's." The 'star-studded' wattage generated by these three could solve California's utility crisis.
The bad news is that for most of the film, this quirky romantic comedy separates Pitt and Roberts, who play Jerry and Samantha, two star-crossed lovers with their own agendas. In fact, they each go their own way about 15 minutes into the film. It's a move that, unfortunately, hinders the film's romantic cuteness from really taking off.
In one of the first scenes, for example, Pitt and Roberts are discussing their divergent priorities. He's an accident-prone bagman for a crime boss and has promised Samantha that he won't take any more jobs. Samantha wants to go to Vegas to become a croupier.  There's zest in their emotions and these two make an adorable couple.
Samantha leaves for Vegas while Jerry heads to Mexico. Well, maybe these two stars are so big that they just couldn't fit on the screen at the same time. But it does give an opportunity for both of these stars to flex their comedic muscles.
Jerry's final job takes him to some pallid village so that he can pick up "The Mexican," a fabled antique gun. All he has to do is collect the antique and then go home. This fabulous weapon has quite a bit of interesting lore, which is told through sepia-toned flashbacks interspersed throughout the film. This is actually a nicely presented sequence, which tells a tale that is fantastical and preternatural.
Poor Jerry. Our hapless gringo is mired with bad luck. His rental car gets stolen, he gets arrested, while allies become enemies. He can only raise his arms in disbelief. We may raise our arms in disbelief too. This portion never has few precious funny moments; and the bizarre violence isn't consistent enough to add pep. Jerry is probably just as annoyed with his lack of luck as we are with the accompanying harmonica and banjo soundtrack that makes it feel like a bad Spaghetti Western.
Samantha represents the better half of this relationship as well as the more enjoyable portion of the story. On her way to Vegas, she is intercepted by a hitman (Gandolfini) who uses her as a hostage to make sure that Jerry succeeds. During the course of her ordeal, they establish a connection with each other. She talks about Jerry; he talks about his love woes. Their dialogue scenes add a needed sense of character dimension. It's actually the most entertaining element of this film.
Pitt, Roberts, and Gandolfini give fine performances. But it's a bit hard for us to stay in synch as the momentum shifts between the two stories. Overall, "The Mexican" is sometimes cute, sometimes funny, and offers a few neat surprises. But its offbeat qualities are uneven.
Grade: B- S: 1 out of 3 L: 2 out of 3 V: 2 out of 3
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Test Case for encoded ampersand in URL
Open Firebug and enable the Net panel.
Click on the
Execute Test
button below.
Expand the new entry and select
Params
tab.
Verify the content. It should be following:
value1 1 value2 2 value3 3
Execute Test
Ampersands in the URL
&
should be properly escaped.


Wayne Rooney, Jimmy Carr and other stars win £263m HMRC battle over North Tyneside developments
As a result of the ruling, some of the investors will be able to claim money back
by Kieran Murray, Jeremy ArmstrongWayne Rooney is among a host of stars who have beaten HMRC in a fight over a £263million investment scheme which built developments in Tyneside.
The taxman argued the England legend and hundreds of celebrities had benefited excessively from tax relief for the project to build two data centres at Cobalt Business Park in North Tyneside , the Mirror reports.
The centres are now trading under the name Stellium Datacenters.
But the High Court ruled against HMRC so some of the investors who have already paid tax demands will be able to claim money back.
Rooney is among 675 investors, including comedian Jimmy Carr, Arsene Wenger, Roy Hodgson, the late Status Quo guitarist Rick Parfitt, and Lady Elizabeth-Ann Redgrave, wife of Olympic legend Sir Steve.
In 2011 they put in £79m – an average of £117,000 each – towards the £263m Cobalt Data Centres 2 and 3 in North Tyneside , which were unused for years.
But they got back £131m in tax relief via the scheme to start economic growth in deprived areas.
It is not clear how much each celebrity put in.
There was no suggestion of wrongdoing by any of the investors.
An HMRC spokesman said: “We are disappointed with the decision and are considering an appeal.”
Carr’s tax affairs were scrutinised after he was named among investors in a tax avoidance scheme.
In 2012, he admitted making a “terrible error of judgement”.
In 2016, according to Companies House, he resigned from the Cobalt investment.
That year, some investors received accelerated payment notices, which meant they had to pay HMRC first. Some then appealed to a tax tribunal.
A spokesman for Hodgson and Wenger said then they were “pleased to invest in Government-backed programmes which will hopefully help to generate new jobs.”
Lady Redgrave also said she had done so “in good faith on advice of my financial adviser through an investment scheme”.
Last year Rooney’s spokesman said he always paid “all UK taxes in full and in accordance with the law of the time”.
On Monday he declined to comment on the ruling but scheme backer Harcourt Capital LLP was said to be “very pleased.”
A spokesman said it showed, based on the law and custom and practice, that the HMRC had been wrong to deny enterprise zone allowances to their investors.


Across: 1. Experts consulted about very young Storm Boy: is leg broken? (13) 8. To be young: to be limited and do what you're told! (4) 9. Orientation of ghastly new building (10) 10,2dn. 8 a "turn" direction on track by cave 13 or claypool group (6 3) 12. Ongoing dung - nine motions! (8) 14. 16 crashed car (5) 15. A broad range, but lacking one misfortune (9) 18.  Pants after rum based cocktail (8) 24. No gear so wildly coloured as 13's 7 (6) 25. Cryptic may take one serious time input (10) 27. Method working for Keith with sticks (4) 28. Easy to recall work of 13 (13) Down: 1,26. Second person easily wrapped up in number by 13 (11 3) 2. see 10ac 3. Loudly made vocal call (hence heard ultimately) (6) 4. Lazy way of speaking with era lost to ego (7) 5. Her great activity being complement to hunter (8) 6. Put into bed, thus married (5) 7. Heaven could be a bit lean without pub (3) 11. Places where you pass water - they lack fashionable range in Russia (5) 13. Singer's tone lacking variation (3 4 4) 16. Cosmic 17 through South American mountains, heading north (5) 17. Perhaps gypsy queens at first stick with magic (8) 19. Residing in metropolis, so metro-flexible (7) 21. What holds in most of inner group? (6) 23. Theme recurring regularly - mood thief! (5) 26. see 1 27. In the crowd, someone fallen into drink (3)


N64 Region Differences
NTSC-U was the first version released and thus is the fastest with the most glitches. NTSC-J and PAL were released later with PAL being the last released with the most patches. All tricks patched on JP were carried over to PAL with PAL having more patches.
Patches in PAL / JP
When taking damage trying to do a clockwork warp it will fail unless you do it on 1 specific frame, making them not viable. However clockwork warps where you break the clockwork with the grenade still work, an example is the Oogle Boogle warp. These warps have been dubbed “PAL warps”
This Switch is a solo Banjo switch as that is the intended way to get this jiggy, however it can be opened as Banjo-Kazooie on NTSC-U and J. On PAL it only opens with solo Banjo as it was intended and thus is necessary to take Solo Banjo across to get the jiggy as Egg barging was patched.
almost all forms of collecting items through walls through various means, (Egg shot, Egg barge, Skew, Swimming (In the case of Weldar’s Toxic Waste Jiggy)) The exception to this is Pecking or Beak barging notes/empty Honeycombs, those still work. Do note you can still get skew, but it is useless.
The height gain trick of rubbing against slopes in water does not work.
When leaving bottles house there is a small cutscene that shows Jiggywiggy’s temple. These versions actually ALLOW you to skip it which is good, saving about 10 seconds.
After seeing Mumbo the first time where you have to go to the top level to speak to him, every other Mumbo visit after that you can speak to him from the bottom level on NTSC-U, this was patched, you must go to the top every time.
On NTSC-U after you screw the first and last bolts for platforms in Grunty Industries there is a cutscene that shows the platform shaking and then falling down. When you do this you can move and jump to prepare to unscrew the next bolt, however this was changed and the game freezes you before these cutscenes play.
When doing this glitch you do not get a cursor to shoot on these versions.
Some loading zone properties were changed and the Triple Banjo glitch doesn’t work.
PAL Quirks
Due to PAL region hardware differences and TV standards during the late 90s, PAL copies of N64 games run at 25FPS instead of the faster NTSC copies which run at 30FPS. In order to compensate for this, Banjo games compensate for any lag you experience until the frame rate reaches 20FPS. Once you create enough lag for the game to run at 19FPS or less, you will end up losing real time to this lag.
Due to lag calculations on PAL Banjo jumps slightly lower (~10 units or equal to about 1 pixel) This makes doing some precise beak busts to skip grabbing ledges difficult, but they are all possible. The hardest beak bust is plateau early which only works if you turn the camera a certain way to increase lag.
NTSC-J Quirks
Text is a little bit slower on NTSC-J. In the Hag 1 fight for example which is full of unskippable text you lose about 5 seconds.
Xbox Differences
There is only 1 Xbox version of Banjo-Tooie that is based off the PAL version of the game in terms of glitches, however the frame rate consistently runs at 30FPS so you save time over all versions of N64 but especially PAL N64. (ironic because this version is based off PAL) Loading times are also much faster with most being instant.
It is currently unknown if this version uses the same distance calculation coded that makes the height go slightly lower.
In terms of text the Japanese text is actually faster for Xbox. All skippable text that uses L + R + B does not work, only text skipped with just B still works (jamjars except for the last, Dragon Bros, Jongo, Mumbo, Wumba etc.) as the L button was not mapped to the N64 L Button.
Patches in the Xbox Release
This version inherited all patches from the PAL version, hitbox extensions, detonator levitator, regular clockwork warps etc, but also has one new patch.
Xbox cannot get in and out of the jelly castle as the Bee, you must use the intended method (Shack Pack).
Summary
N64 NTSC-U is the fastest for speedrunning every Banjo-Tooie category except for Any%, Cheato%, and Low%, which are faster on NTSC-J. The reason is because of the ability to skip the JiggyWiggy cutscene saving 10 seconds, however because of the slower Japanese text you lose 5 seconds in Hag 1, saving 5 seconds total. None of the tricks patched are used in this run so JP works.


Reception
Counting
Number Names
Addition up to 20
Comparing Numbers
Naming Shapes
Grade 1
Counting to 100
Skip Counting (2, 5, 10)
Comparing Numbers
Basic Addition and Subtraction
Tell The Time
Grade 2
Skip Counting (2, 3, 5, 10)
Comparing Numbers to 1000
Intro To Multiplication
Intro To Division
Basic Patterns
Grade 3
Odd and Even Numbers
Comparing to 10000
Multiplication
Hard Addition
Hard Patterns
Grade 4
Numbers to 100000
20x20 Multiplication
4 Division
Simplify Fractions
Fractions To Decimals
Grade 5
Expert Multiplication
Adding Fractions
Comparing Decimals
Converting Time
Multiplication (calculator allowed)
Grade 6
Arithmetic Test
Comparing Fractions
Add And Subtract Decimals
Multiply Decimals
Fractions to Percentages
Reception
Counting
Help dumb receptions become child prodigies in this fun counting game.
This game will make counting fun, with engaging graphics and fun sounds. 🙂
Play
Number Names
Receptions can learn how to make connections between numerals and words in this intuitive game.
Make Learning fun with vibrant colours and animations.
Play
Addition up to 20
Receptions can learn to add numbers between 0 and 20 in this fun game.
Play
Comparing Numbers
Receptions can learn how to find the value of numbers by finding the greater number.
Play
Naming Shapes
Receptions can learn basic geometry in this game. They answer the number of sides given shapes have.
Play
Grade 1
Counting to 100
Grade 1s can further their counting knowledge in this game.
Play
Skip Counting (2, 5, 10)
Grade 1s can further their counting knowledge in this game.
Play
Comparing Numbers
Grade 1s can further their number knowledge in this game.
Play
Basic Addition and Subtraction
Grade 1s can further their addition and subtraction knowledge in this game.
Play
Basic Addition and Subtraction
Grade 1s can further their addition and subtraction knowledge in this game.
Play
Tell The Time
Grade 1s can learn how to tell the time in this game.
Play
Grade 2
Skip Counting (2, 3, 5, 10)
Grade 2s can learn how to tell the time in this game.
Play
Comparing Numbers to 1000
Grade 2s can learn how to compare numbers in this game.
Play
Intro To Multiplication
Grade 2s can learn how to multiply in this game.
Play
Intro To Division
Grade 2s can learn how to divide in this game.
Play
Basic Patterns
Grade 2s can learn how to recognise patterns in this game.
Play
Grade 3
Odd and Even Numbers
Grade 3s can learn how to recognise odd and even numbers in this game.
Play
Comparing to 10000
Grade 3s can learn how to compare numbers up to 10,000 in this game.
Play
Multiplication
Grade 3s can learn how to multiply in this game.
Play
Hard Addition
Grade 3s can learn how to add ans subtract in this game.
Play
Hard Patterns
Grade 3s can learn patterns in this game.
Play
Grade 4
Numbers to 100000
Grade 4s can learn numbers in this game.
Play
20x20 Multiplication
Grade 4s can learn multiplication in this game.
Play
4 Division
Grade 4s can learn division in this game.
Play
Simplify Fractions
Grade 4s can learn to simplify fractions in this game.
Play
Fractions To Decimals
Grade 4s can learn to convert fractions to decimals in this game.
Play
Grade 5
Expert Multiplication
Grade 5s can learn to become multiplication experts in this game.
Play
Adding Fractions
Grade 5s can learn to add fractions in this game.
Play
Comparing Decimals
Grade 5s can learn to compare decimals in this game.
Play
Converting Time
Grade 5s can learn to convert from 12 to 24 hour time in this game.
Play
Multiplication (calculator allowed)
Grade 5s can learn to use a calculator in this game.
Play
Grade 6
Arithmetic Test
Grade 6s can learn to become arithmetic experts in this game.
Play
Comparing Fractions
Grade 6s can learn to compare fractions in this game.
Play
Add And Subtract Decimals
Grade 6s can learn to add and subtract decimals in this game.
Play
Multiply Decimals
Grade 6s can learn to multiply decimals decimals in this game.
Play
Fractions to Percentages
Grade 6s can learn to convert fractions to percentages in this game.
Play


This is an homage to this absolute work of art by Shesh Babu.
What and why§
Rust's strong typing and fearless concurrency means we can skip virtual DOM differencing.
In the JavaScript world, avoiding the vDOM is the bread and butter of Svelte started by Rich Harris, which uses compile-time code generation to assist.
When I can't have static typing I love Clojure, and have spent some time reviewing and playing with the stupendous full-stack Clojure SPA framework Fulcro by Tony Kay (and associated back-end enabler Pathom by Wilker Lucio). It does use vDOM, leveraging Clojure immutable/concurrent data structures for time travel superpowers.
For me, the major innovation (among many) in Fulcro is the use of a browser-side normalised database which is queried to populate properties for components. This means that updating (mutating) a uniquely-keyed item results in the update trivially propagating to any and all components referencing the data through that identifier. In Shesh Babu's language: all state is App state.
This article, or series of articles, is going to share my findings and thinking on the state of the nation in Rust front-end frameworks which are avoiding the vDOM strategy.
There are actually two Rust front-end frameworks with significant progress already, and they are awesome:
Our lense: the React tutorial TicTacToe§
The official React site offers a guided introduction by progressively implementing a client-side tic-tac-toe game (also known as noughts and crosses). They don't explore a back-end, routing or forms, or many of the other SPA complexities.
For us, it is just enough to highlight the potential of the two Rust frameworks above and paint a picture of how those advanced extensions can be easily incorporated, and gives us a solid reference point from the old world.
Our approach§
This is the first in a series of articles showing how the two frameworks might attack the example application, and then show some code which extends the frameworks to incorporate Fulcro-like app state.
My code will concentrate on the ergonomics of the frameworks from the perspective of the SPA-writer.
Next up, I'll show an implementation of the game in Valerie.


>From MrI.Kamara TEL +27 83 486 2761 DEAR , This letter was borne out of my sincere desire to establish a business/mutual relationship with you. I got your contact through the international trade and commerce (ITCC). My name is Ibram Kamara the son of late Chief Abraham P.Kamara (the former deputy minister of finance under the ousted civilian government) who was killed and mutilated by the military junta led by major, Paul Koroma after overthrowing the elected government of president Tejan. Through, I do not know to what extent you are familiar with events and disturbance in Sierra-Leone but the pressure of war drove my mother and I out of Sierra -Leone into exile in South Africa where we have been living under political asylum for years. Sadly, my mother died of cancer 2 months ago and was buried here in South Africa . Prior to her death, she handed me over a certificate meant for a secret deposit, which my father made in a security/finance company in europe the deposit that worth US$25,000,000 ( Twenty five million U.S dollars) this was the money paid to his corporation by it's overseas customers in the heat of the conflict. He made the deposit in his name and also deposited it as a family tresures, with the hope of converting it to his personal use at the end of the war but he was killed when the conflict intensified as a result of his opposition to the rebel forces. I have contracted the Security/finance Company to confirm the deposit and establish ownership. Due to the death of my Mother, I have decided to solicit for the participation of an honest and trustworthy person or company that will assist in acting as my representative who will jointly signed the agreement for the safe keeping of the money in the security/finance company. The security/finance company don't know the content of the consignment, they believe that it is African artefact. As contained in 2 trunk boxes. Unfortunately, My mum who is surpose to assist me in releasing the consignment has died of cancer that is why I am looking for your assistance. I cannot do it alone due to my present social status and total ignorance of the business world. I shall also give you 20% of the funds for your assistance to relocate this fund to your country and for re-investment while the rest will be kept in any account of your choice for inverstmant purpose. i would have done the transfer myself as my name is written as the next of kin,but because of my present status in this country as a refugee i will not be allowed to handle such money or else it will be confisticated by the goverment if found in my possetion,and also i will not be allowed to open an account as a refugee not even to talk of inversting such an amount of money,these are the reasons why i need your assistance,If you are interested in helping me out of this my present situation, contact me immediately through my email and i will funish you with other details you need to conclude this transaction.this transaction is absolutely risk free and of biliteral benefits as i have all neccesray legal documents backing it up.please get back to me with your full name,tel.no fax no and also your home addres so as to enable me introduce you to the security/finance company handling the consignment as my representative.after all these you will then make an arrangement on how we are going to meet face to face and discuss in details. Meanwhile, I cannot travel out because i have no travelling documents for now,i also hope that after everything you will secure a visa for me to join you in your country so that i can continue my studies over there. I hope to hear from you soon Best regards, MR IBRAM KAMARA _________________________________________________________________ Cut through the clutter - search faster with MSN Toolbar! http://toolbar.msn.co.za?DI=1054&XAPID=2083


The Campus Parking Project story began in December of 2019. Frustrated by the parking management methods used at the University of Southern Maine, Carter Haid and Connor Black set out to create an open source parking management system that could be used by anyone.
Since 2019 Carter has continued to develop the project while Connor has taken an advisory role... mainly focusing on keeping the HTML/CSS within material design guidelines. Several other developers have written small scripts for the project all under the management of Carter.
If you'd like to collaborate on the project feel free to make a pull request on GitHub. To collaborate in a different way please feel free to reach out via our contact form. We're looking forward to the new year and to the future of Campus Parking Project. Have a great and safe 2021.


abstract fun digest(): Bytes
Create a Bytes from the digested data contained in this DigestBuilder so far.
See Also
update


%page expression_filter="h"/>
<%inherit file="main.html" />
<%namespace name='static' file='static_content.html'/>
<%!
from django.utils.translation import ugettext as _
from django.urls import reverse
from openedx.core.djangolib.markup import HTML, Text
%>
<%include file="${courses_list}" />
% if show_homepage_promo_video:
% endif
<%block name="js_extra">
%block>
% if show_signup_immediately is not UNDEFINED:
## NOTE: This won't work in production, because anonymous views are cached, so it will
## show either with or without this extra js for 3 minutes at a time.
% endif


N1QL Scan Consistency Using the Java SDK with Couchbase Server
You can balance performance against consistency in N1QL queries via the Couchbase Java client and the STATEMENT_PLUS (AtPlus) option.
N1QL provides a couple of options for scan consistency, represented in Java by the ScanConsistency
enum:
NOT_BOUNDED REQUEST_PLUS STATEMENT_PLUS
"Not bounded" being the default, the query engine will grab and return what is in the index right now. It is the fastest option, but not suitable for "read your own writes" since they may not yet be indexed.
At the other end of the scale there is "request plus" which will — when a query is performed — wait until all of the indexes used have caught up to their highest internal sequence numbers. This will make sure that you read your own writes, but it might take some time if there is a write-heavy workload. Also sometimes you are only interested in the last mutation(s) you performed, so waiting for everything else in the index to be updated can mean unnecessary latency.
This is where "statement plus" comes in, also named "at plus". This option will send some information previously retrieved at mutation time to the query engine which will then make sure to only wait for (at least) those mutations to have been indexed.
If statement_plus
is used, mutation tokens need to be enabled in the environment:
CouchbaseEnvironment env = DefaultCouchbaseEnvironment .builder() .mutationTokensEnabled(true) .build();
There is a downside: from now on every mutation will return a couple more bytes of metadata, so there is slightly more overhead over the network.
Note though that this mechanism is also used for better handling of durability requirements (ReplicateTo
/PersistTo
), so the user might have it already enabled in their environment.
Now when a mutation is performed the MutationToken
is part of the response.
Thus:
JsonDocument written = bucket.upsert(JsonDocument.create("mydoc", JsonObject.empty())); System.err.println(written.mutationToken());
Which will print:
mt{vbID=55, vbUUID=166779084590420, seqno=488, bucket=travel-sample}
This information can then be passed in at query time:
bucket.query( N1qlQuery.simple("select count(*) as cnt from `travel-sample`", N1qlParams.build().consistentWith(written)) );
If you enable trace logging, you can see that on the wire the appropriate token vector is passed along to the query engine, which will handle the rest:
[cb-io-1-2] 2018-07-09 13:28:01 TRACE LoggingHandler:94 - [id: 0x439d3980, L:/<IP_ADDRESS>:53732 - R:localhost/<IP_ADDRESS>:8093] WRITE: 231B +-------------------------------------------------+ | 0 1 2 3 4 5 6 7 8 9 a b c d e f | +--------+-------------------------------------------------+----------------+ |00000000| 7b 22 73 74 61 74 65 6d 65 6e 74 22 3a 22 73 65 |{"statement":"se| |00000010| 6c 65 63 74 20 63 6f 75 6e 74 28 2a 29 20 61 73 |lect count(*) as| |00000020| 20 63 6e 74 20 66 72 6f 6d 20 60 74 72 61 76 65 | cnt from `trave| |00000030| 6c 2d 73 61 6d 70 6c 65 60 22 2c 22 63 6c 69 65 |l-sample`","clie| |00000040| 6e 74 5f 63 6f 6e 74 65 78 74 5f 69 64 22 3a 22 |nt_context_id":"| |00000050| 66 32 32 64 31 36 36 61 2d 35 38 30 35 2d 34 30 |f22d166a-5805-40| |00000060| 62 36 2d 62 31 65 63 2d 66 62 62 35 61 66 36 33 |b6-b1ec-fbb5af63| |00000070| 30 37 37 63 22 2c 22 73 63 61 6e 5f 76 65 63 74 |077c","scan_vect| <---- |00000080| 6f 72 73 22 3a 7b 22 74 72 61 76 65 6c 2d 73 61 |ors":{"travel-sa| <---- |00000090| 6d 70 6c 65 22 3a 7b 22 35 35 22 3a 5b 34 38 38 |mple":{"55":[488| <---- |000000a0| 2c 22 31 36 36 37 37 39 30 38 34 35 39 30 34 32 |,"16677908459042| <---- |000000b0| 30 22 5d 7d 7d 2c 22 74 69 6d 65 6f 75 74 22 3a |0"]}},"timeout":| <---- |000000c0| 22 37 35 30 30 30 6d 73 22 2c 22 73 63 61 6e 5f |"75000ms","scan_| |000000d0| 63 6f 6e 73 69 73 74 65 6e 63 79 22 3a 22 61 74 |consistency":"at| |000000e0| 5f 70 6c 75 73 22 7d |_plus"} | +--------+-------------------------------------------------+----------------+