pca主成分分析_超越普通PCA：⾮线性主成分分析
pca 主成分分析
TL;DR: PCA cannot handle categorical variables becau it makes linear assumptions about them. Nonlinear PCA address this issue by warping the feature space to optimize explained variance. (Key points at bottom.)
TL; DR： PCA⽆法处理分类变量，因为它对它们进⾏了线性假设。 ⾮线性PCA通过扭曲特征空间以优化解释的⽅差来解决此问题。 (关键点在底部。)
Principal Component Analysis (PCA) has been one of the most powerful unsupervid learning techniques in machine learning. Given multi-dimensional data, PCA will find a reduced number of n uncorrelated (orthogonal) dimensions, attempting to retain as much variance in the original datat as possible. It does this by constructing new features (principle components) as linear combinations of existing columns.
主成分分析(PCA)是机器学习中最强⼤的⽆监督学习技术之⼀。 给定多维数据，PCA将发现数量减少的n个不相关(正交)维，尝试在原始数据集中保留尽可能多的⽅差。 它通过将新功能(原理组件)构造为现有列的线性组合来实现。
However, PCA cannot handle nominal — categorical, like state — or ordinal — categorical and quential, like letter grades (A+, B-, C, …) — columns. This is becau a metric like variance, which PCA explicitly attempts to model, is an inherently numerical measure. If one were to u PCA on data with nominal and ordinal columns, it would end up making silly assumptions like ‘California is one-half New Jery’ or ‘A+ minus four equals D’, since it must make tho kinds of relationships to operate.
但是，PCA⽆法处理名义(类别，如状态)或排序(类别和顺序)，如字母等级(A +，B-，C等)的列。 这是因为PCA明确尝试建模的类似⽅差的度量标准是固有的数字度量。 如果要在具有标称和序数列的数据上使⽤PCA，最终将做出愚蠢的假设，例如“加利福尼亚州是新泽西州的⼀半”或“ A +减去四等于D”，因为它必须使这种关系起作⽤。
Rephrad in relation to a mathematical perspective, PCA relies on linear relationships, that is, the assumption that the distance between “strongly disagree” and “disagree” is the same as the differen
ce from “disagree” to “neutral”. In almost every real-world datat, the sorts of linear relationships do not exist for all columns.
从数学⾓度重新描述，PCA依赖于线性关系，即“强烈不同意”和“不同意”之间的距离与“不同意”到“中⽴”的差异相同的假设。 在⼏乎每个现实世界的数据集中，并⾮所有列都存在此类线性关系。
Additionally, using one-hot encoding — that is, converting categorical data into vectors of ones and zeroes — results in an extremely spar and information-parched multidimensional space that PCA cannot perform well on, since veral features contain only two unique values.
此外，使⽤⼀键编码(即将分类数据转换为⼀和零的向量)会导致PCA⽆法很好地执⾏的极为稀疏且信息匮乏的多维空间，因为多个功能仅包含两个唯⼀值。
Nonlinear PCA rectifies this aspect of PCA by generalizing methods to approach dimensionality reduction not only for numerical features, but for categorical and ordinal variables. This is done through categorical quantification.
⾮线性PCA通过泛化⽅法来修正PCA的这⼀⽅⾯，不仅针对数字特征，⽽且针对分类和有序变量也都采⽤降维⽅法。 这是通过分类量化完成的。
Categorical quantification (CQ) is exactly what its name suggests: it attaches a numerical reprentation to each category, converting categorical columns into numerical ones, such that the performance of the PCA model (like explained variance) is maximized. CQ optimally places categories on a numerical dimension instead of making assumptions about them.
分类量化(CQ)正是其名称的含义：它将数字表⽰形式附加到每个类别，将分类列转换为数字列，从⽽使PCA模型的性能(如解释的⽅差)最⼤化。 CQ最佳地将类别放在数字维度上，⽽不是对其进⾏假设。
This information can be very enriching. For instance, we might be able to say that Washington and Idaho have very similar structures in other parts of the data becau they are placed so cloly, or that California and Virginia are nowhere similar becau they are placed far apart. In this n, CQ is not only enriching the PCA model with categorical data but also
giving us a look into the structures of the data by state.
这些信息可以⾮常丰富。 例如，我们也许可以说，华盛顿和爱达荷州在数据的其他部分具有⾮常相似的结构，因为它们放置得太近了；或者，加利福尼亚和弗吉尼亚州的相似之处在于它们的位置很远。 从这个意义上讲，CQ不仅⽤分类数据丰富了PCA模型，⽽且使我们可以按状态查看数据的结构。
An alternative view of CQ is through a line plot. Although in the ca of nominal data, the order of columns is arbitrary and there do not need to be connecting lines, it is visualized in this way to demonstrate the nominal level of analysis. If a feature’s level is specified as nominal, it can take on any numerical value.
CQ的替代视图是通过折线图。 尽管在名义数据的情况下，列的顺序是任意的，并且不需要连接线，但可以通过这种⽅式将其可视化以展⽰名义分析⽔平。 如果将特征级别指定为标称值，则它可以采⽤任何数值。
On the other hand, if a feature level is specified as ordinal, the restriction is that the order must be prerved. For instance, the relation between ‘A’ and ‘B’ in that ‘A’ is better than ‘B’ must be kept, which can be reprented with A=0 and B=5 (assuming 0 is the best) or A=25 and B=26, as long as B is never less than A. This helps retain the structure of ordinal data.
另⼀⽅⾯，如果将要素级别指定为序数，则限制是必须保留顺序。 例如，必须保持“ A”与“ B”之间的关系，因为“ A”优于“ B”，
可以⽤A=0和B=5 (假设0为最佳)或A=25和B=26 ，只要B永远不⼩于A。这有助于保留序数数据的结构。
Note that the next data point is always larger or equal to the previous data point. This is a restriction on ordinal-level features.
请注意，下⼀个数据点始终⼤于或等于前⼀个数据点。 这是对序数级功能的限制。
Like CQ for nominal data, this is tremendously insightful. For instance, we notice that within plus and minus of letter
grades (A+, A, A-), there is not much difference, but the difference between X- and Y+ (X and Y being quential letters)
always leads to a large jump, particularly the difference between D and F. To reiterate the point above — this chart is
generated by finding optimal values for categories such that the PCA model performs best (explained variance is highest).就像名义数据的CQ⼀样，这是⾮常有见地的。 例如，我们注意到在字母等级(A +，A，A-)的正负之间并没有太⼤差异，但是X-和Y +之间的差异( X和Y是顺序字母)总是导致重申以上⼏点-通过查找类别的最佳值以使PCA模型表现最佳(解释⽅差最⾼)来⽣成此图。
Note that becau CQ determines the space between data points (e.g. that the difference between A and A- is much less
than that of D and F), it warps the space in which the points lie. Instead of assuming a linear relati
onship (A and A- are as
clo as D and F), CQ distorts the distances between common intervals — hence, nonlinear PCA.
请注意，由于CQ决定了数据点之间的间隔(例如，A和A-之间的差远⼩于D和F的差)，因此它将扭曲这些点所在的空间。 CQ不会假设线性
关系(A和A-与D和F接近)，⽽是会扭曲公共间隔之间的距离，因此会扭曲⾮线性 PCA。
To give an idea of the nonlinearities that can ari when the distance between quential intervals are altered, here’s a 3 by 3 square in distorted space:
为了让⼈们理解当顺序间隔之间的距离改变时可能出现的⾮线性，这⾥是⼀个3 x 3平⽅的扭曲空间：
By using categorical quantification, the feature space is distorted — in a good way! Intervals are lectively chon such
that the performance of PCA is maximized. Nonlinear PCA, in this n, not only can be thought of as an encoding method for ordinal and nominal variables but also increas the global strength of the PCA model.
通过使⽤分类定量，可以很好地扭曲特征空间！ 选择间隔以使PCA的性能最⼤化。 从这个意义上讲，⾮线性PCA不仅可以看作是序数和名义变量的编码⽅法，⽽且可以提⾼PCA模型的整体强度。
Although the mathematics behind Nonlinear PCA is very rich, generally speaking, NPCA us the same methods as PCA (like eigenvalue solving, etc.), but us CQ to derive the most information and benefit to the model.
尽管⾮线性PCA背后的数学⾮常丰富，但通常来说，NPCA使⽤与PCA相同的⽅法(例如特征值求解等)，但是使⽤CQ来获得最多的信息并从模型中受益。
关键点 (Key Points)
PCA cannot handle nominal (categorical) or ordinal (quential) columns becau it is an inherently
numerical algorithm and makes silly linear assumptions about the types of data.
PCA⽆法处理标称(分类)或序数(顺序)列，因为它是固有的数值算法，并且对这些类型的数据进⾏了愚蠢的线性假设。
Nonlinear PCA us categorical quantification, which finds the best numerical reprentation of unique column values such that the performance (explained variance) of the PCA model using the transformed columns is optimized.
⾮线性PCA使⽤分类量化，它可以找到唯⼀列值的最佳数值表⽰形式，从⽽可以优化使⽤转换列的PCA模型的性能(解释⽅差)。
Categorical quantification is a very insightful data mining method, and can give lots of insight into the structures of the data through the lens of a categorical value. Unfortunately, using Nonlinear PCA means that the coefficients of principal components are less interpretable (but still understandable, just to a less statistical rigor).
分类量化是⼀种⾮常有见地的数据挖掘⽅法，可以通过分类价值的镜头深⼊了解数据的结构。 不幸的是，使⽤⾮线性PCA意味着主成分的系数难以解释(但仍然可以理解，只是对统计的严格要求较低)。
All images created by author.
作者创作的所有图像。
pca 主成分分析
