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统计代写|多元统计分析作业代写Multivariate Statistical Analysis代考|Comparison of Batches

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统计代写|多元统计分析作业代写Multivariate Statistical Analysis代考|Boxplots

EXAMPLE 1.1 The Swiss bank data (see Appendix, Table B.2) consists of 200 measurements on Swiss bank notes. The first half of these measurements are from genuine bank notes, the other half are from counterfeit bank notes.
The authorities have measured, as indicated in Figure 1.1,
\begin{aligned} &X_{1}=\text { length of the bill } \ &X_{2}=\text { height of the bill (left) } \ &X_{3}=\text { height of the bill (right) } \ &X_{4}=\text { distance of the inner frame to the lower border } \ &X_{5}=\text { distance of the inner frame to the upper border } \ &X_{6}=\text { length of the diagonal of the central picture. } \end{aligned}
These data are taken from Flury and Riedwyl (1988). The aim is to study how these measurements may be used in determining whether a bill is genuine or counterfeit.

The boxplot is a graphical technique that displays the distribution of variables. It helps us see the location, skewness, spread, tail length and outlying points.

统计代写|多元统计分析作业代写Multivariate Statistical Analysis代考|Histograms

Histograms are density estimates. A density estimate gives a good impression of the distribution of the data. In contrast to boxplots, density estimates show possible multimodality of the data. The idea is to locally represent the data density by counting the number of observations in a sequence of consecutive intervals (bins) with origin $x_{0}$. Let $B_{j}\left(x_{0}, h\right)$ denote the bin of length $h$ which is the element of a bin grid starting at $x_{0}$ :
$$B_{j}\left(x_{0}, h\right)=\left[x_{0}+(j-1) h, x_{0}+j h\right), \quad j \in \mathbb{Z},$$

where $[., .)$ denotes a left closed and right open interval. If $\left{x_{i}\right}_{i=1}^{n}$ is an i.i.d. sample with density $f$, the histogram is defined as follows:
$$\widehat{f}{h}(x)=n^{-1} h^{-1} \sum{j \in \mathbb{Z}} \sum_{i=1}^{n} \boldsymbol{I}\left{x_{i} \in B_{j}\left(x_{0}, h\right)\right} \boldsymbol{I}\left{x \in B_{j}\left(x_{0}, h\right)\right} .$$
In sum (1.7) the first indicator function $\boldsymbol{I}\left{x_{i} \in B_{j}\left(x_{0}, h\right)\right}$ (see Symbols \& Notation in Appendix A) counts the number of observations falling into bin $B_{j}\left(x_{0}, h\right)$. The second indicator function is responsible for “localizing” the counts around $x$. The parameter $h$ is a smoothing or localizing parameter and controls the width of the histogram bins. An $h$ that is too large leads to very big blocks and thus to a very unstructured histogram. On the other hand, an $h$ that is too small gives a very variable estimate with many unimportant peaks.

统计代写|多元统计分析作业代写MULTIVARIATE STATISTICAL ANALYSIS代考|Scatterplots

Scatterplots are bivariate or trivariate plots of variables against each other. They help us understand relationships among the variables of a data set. A downward-sloping scatter indicates that as we increase the variable on the horizontal axis, the variable on the vertical axis decreases. An analogous statement can be made for upward-sloping scatters.

Figure $1.12$ plots the 5 th column (upper inner frame) of the bank data against the 6 th column (diagonal). The scatter is downward-sloping. As we already know from the previous section on marginal comparison (e.g., Figure 1.9) a good separation between genuine and counterfeit bank notes is visible for the diagonal variable. The sub-cloud in the upper half (circles) of Figure $1.12$ corresponds to the true bank notes. As noted before, this separation is not distinct, since the two groups overlap somewhat.

This can be verified in an interactive computing environment by showing the index and coordinates of certain points in this scatterplot. In Figure 1.12, the 70th observation in the merged data set is given as a thick circle, and it is from a genuine bank note. This observation lies well embedded in the cloud of counterfeit bank notes. One straightforward approach that could be used to tell the counterfeit from the genuine bank notes is to draw a straight line and define notes above this value as genuine. We would of course misclassify the 70 th observation, but can we do better?

If we extend the two-dimensional scatterplot by adding a third variable, e.g., $X_{4}$ (lower distance to inner frame), we obtain the scatterplot in three-dimensions as shown in Figure 1.13. It becomes apparent from the location of the point clouds that a better separation is obtained. We have rotated the three dimensional data until this satisfactory 3D view was obtained. Later, we will see that rotation is the same as bundling a high-dimensional observation into one or more linear combinations of the elements of the observation vector. In other words, the “separation line” parallel to the horizontal coordinate axis in Figure $1.12$ is in Figure $1.13$ a plane and no longer parallel to one of the axes. The formula for such a separation plane is a linear combination of the elements of the observation vector:
$$a_{1} x_{1}+a_{2} x_{2}+\ldots+a_{6} x_{6}=\text { const. }$$

统计代写|多元统计分析作业代写MULTIVARIATE STATISTICAL ANALYSIS代考|BOXPLOTS

X1= 账单长度  X2= 钞票高度（左）  X3= 钞票高度（右）  X4= 内框到下边框的距离  X5= 内框到上边框的距离  X6= 中央图片的对角线长度。

统计代写|多元统计分析作业代写MULTIVARIATE STATISTICAL ANALYSIS代考|HISTOGRAMS

$$\widehat{f} {h}X=n^{-1} h^{-1} \sum {j \in \mathbb{Z}} \sum_{i=1}^{n} \boldsymbol{I}\left{x_{i} \in B_{j}\左x_{0}, h\右x_{0}, h\右\right} \boldsymbol{I}\left{x \in B_{j}\leftx_{0}, h\右x_{0}, h\右\对} 。$$

统计代写|多元统计分析作业代写MULTIVARIATE STATISTICAL ANALYSIS代考|SCATTERPLOTS

$$a_{1} x_{1}+a_{2} x_{2}+\ldots+a_{6} x_{6}=\text { const. }$$