text:
\section{Introduction} \label{sec:introduction}
Increasingly complex machine learning (ML) models are being deployed in industries such as healthcare, cybersecurity, and banking. While industries welcome the performance boost from these ML models, organizations are also starting to require their models to provide clear explanations for their predictions, as necessitated by regulations, such as GDPR~\cite{voigt2017eu}. Thus, explainable artificial intelligence (XAI) techniques are especially relevant, particularly those which provide explanations for individual samples.

There are many techniques which provide \textit{local} explanations for predictive models~\cite{ribeiro2016should}. Given an input $x$ and predictive model $P$, a local explanation method typically generates a real-valued feature vector $e$ which represents the attributions of the input features to the predicted outcome. Since there are many local explanation methods available, it raises an important question -- \textit{how do we assess and compare local explainability frameworks}? One popular method is ablation testing~\cite{hooker2018benchmark}, which assesses the accuracy drops of an input by removing the important features identified by the explanation methods. However, such testing reveals only the ordering of features by their performance impact, not the structure of the explanation space.

In this paper, we propose a simple but approach to globally assess local explanations (GALE). We accomplish this through an analysis of the topological properties of a given explanation space for binary classification problems. Specifically, we model an explanation method as a scalar function that captures the relationship between the explanation space and the class prediction. Using this function, we compute its topological skeleton. This skeleton is used to generate a topological signature which is then used to compare explanation methods. GALE is easy to implement and lacks complex optimizations or parameter tuning. We aim to address the following challenges arising from explanation technique assessments.

\begin{figure*}
    \centering
    \includegraphics[width=\textwidth]{figures/pipeline.png}
    \caption{GALE is a framework which transforms local explanations of a data set into a topological representation which we can use to compare heterogenous explanation methods.}
    \label{fig:pipeline}
\end{figure*}

Prior to comparing multiple local explanation methods, it is important to first understand a given explanation method, and in particular, how it performs on any given data set. Even using a single explanation technique, it is common to observe different outcomes based on different parameters or simply from sampling variability. For example, explanations generated using identical parameters and inputs produced by Local Interpretable Model-Agnostic Explanations (LIME)~\cite{lime}, a popular local explainability method, can produce contradictory feature importance scores for the same input due to the stochastic nature of LIME. Furthermore, many of these explainability methods require user-specified parameters which can further impact the explanation output. While it is easy to observe the differences between multiple local explanations for a given input through visualization, such analysis is only useful for analyzing an explanation method for a single observation. Thus, a visual approach becomes cumbersome when applied to the entire data set. Moreover, it is almost impossible to understand how stable the explanation method is for that data. Doing so would necessitate generating a global measure or signature that can capture the structure of the explanation space with respect to the given data. Our aim is to compute a stable topological signature given an explanation method's output.

While comparing outputs from the same explanation method can be performed by measuring their distances (e.g., Euclidean distance), explanations generated from different methods generally cannot be directly compared since they have different value ranges and dimensions. For example, LIME and SHAP~\cite{shap}, another popular local explanation method, are on different scales. To compare two different explanation methods as a whole, a straightforward approach is to cluster the local explanations and compare the cluster similarities (e.g., Rand index) between them. However, there are two drawbacks to this approach. First, the values of explanations may not provide clear cluster structures, which makes comparisons using only on the explanation values difficult. Second, since explanations are inferred using the predictions of the ML model, the comparisons between different explanation methods are more faithfully measured by the relationships between the model predictions and the explanation output. As we show later, such properties are naturally captured by the shape, or more formally, the topological properties, of the explanation manifold. A crucial advantage of using topology for such an analysis is that it is agnostic to the range and dimensions of the explanation space and thus allows for direct comparison across different explanation methods.
To the best of our knowledge, our approach is the first to use computational topology for comparing XAI methods. We view topology as an intriguing direction for understanding and comparing explanation methods and that GALE, a powerful yet accessible framework, as the first step in that direction. The contributions of this work can be summarized as follows:
\begin{enumerate}[leftmargin=*]
\item We propose GALE (Global Assessing Local Explanations) a topology-based approach to generate a global signature for a given local explanation method's output. By providing a domain-agnostic signature for the explanation techniques, our approach allows comparison across heterogeneous explanation approaches.
\item We demonstrate that GALE is both stable and can elucidate differences between explanations through experiments on gradient-based and surrogate model techniques with real-world and synthetic data.
\item We show that GALE can be used to find appropriate parameters for local explanation methods by comparing topological signatures for different sets of parameters.
\end{enumerate}
\section{Related Work} \label{sec:related_work}
When training models, practitioners can choose from a variety of model architectures, some of which are directly interpretable. For example, in generalized additive models (GAMs), feature attributions are additive and directly comparable from model to model. Chang~et~al.~\cite{DBLP:conf/kdd/ChangTLGC21} find that tree-based GAMs, such as explainable boosting machines~\cite{DBLP:conf/kdd/LouCGH13}, often display a balance of sparsity, fidelity and accuracy. Typically, GAMs are also easy to visualize and provide clear feature attributions to end-users. However, tree-based GAMs can be slow to train and may be outperformed by other models.

To balance the desire for machine learning model performance improvements along with growing calls for explainable model decisions, practitioners are increasingly turning to local, posthoc explainability methods~\cite{DBLP:journals/cacm/WeldB19}. These local explanation methods typically produce a vector representing the attributions of input features, and are generally well-received by users. Jeyakumar~et~al.~\cite{DBLP:conf/nips/JeyakumarNCGS20} find, through surveys with hundreds of non-technical users, that explanation-by-example and LIME were the preferred explanation styles. However, beyond user preferences, it is hard to compare explainability methods. Taxonomies provide some guidance on a unified language around explainability and raise important questions for explainability researchers to address~\cite{doshi2017towards}. Sokol and Flach propose \textit{Explainability Fact Sheets}, a taxonomy that compares explainability methods across their functional, operational, usability and safety properties~\cite{DBLP:conf/fat/SokolF20}.

To assess and compare the attributions from local explanation methods, a straightforward way is to ablate the top K features ranked by the attributions and observe the decrease of the predicted output score. Varying the values of K and recording the output scores results in an ideally downward sloping output score curve. The lower the curve, the better the local explanation method since it shows that the explanation succeeds in identifying the important features.

To avoid simply removing the top K features without considering the correlations among features, we can ablate the center of mass of the input instead~\cite{ghorbani2019interpretation}. Also, to avoid issues of model extrapolation on ablated inputs, we can retrain the model on the ablated data and measure the performance degradation~\cite{hooker2018benchmark}. Furthermore, local explanation methods can also be assessed by comparing their behavior between a randomly parameterized model and a trained model~\cite{adebayo2018sanity}. Besides ablation, we can measure the quality of explanations with metrics such as (in)fidelity and sensitivity under perturbations~\cite{yeh2019fidelity}, or impact score that measures the feature importance on decision marking process~\cite{lin2019explanations}. If the apriori of feature importance of the dataset~\cite{yang2019benchmarking} is known, the feature importance of input across different models can also be assessed.

Topological data analysis (TDA) is beginning to be considered for explainability purposes. For example, Elhamdadi~et~al.~\cite{elhamdadi2021affectivetda} use TDA to study face poses used in affective computing and find that their topology-based approach captures known patterns. Van Veen proposed visual constraints on TDA output to aid interpretability~\cite{van2020novel}, specifically for viewing global, cluster and local explanations. However, there is little work on applying TDA to the outputs of explanation methods to assess the similarity between them.

\section{Methodology} \label{sec:methods}

In this section, we describe the topological concepts we use (Section~\ref{sec:topo-background}), our proposed approach to generate a global signature for local explanation methods  (Section~\ref{sec:our-method}), and our framework for parameter tuning (Section~\ref{sec:hyperparam-tuning}).

\subsection{Topological Background} \label{sec:topo-background}
\subsubsection{Reeb Graphs and the Mapper Algorithm}
Consider a scalar function $f:\mathbb M \rightarrow \mathbb R$, that maps points from a manifold $\mathbb M$ to a real value.
The \textit{level set} $f^{-1}(a)$ at a given scalar value $a$ is the set of all points that have the function value $a$.
The \textit{Reeb graph}~\cite{reeb1946points} of $f$ is computed by contracting each of the connected components of the level sets of $f$ to a single point, resulting in a skeleton-like representation of the input.
Figure~\ref{fig:reeb-graph}(b) shows an example of the Reeb graph of the height function defined on a torus (Figure~\ref{fig:reeb-graph}(a)).

\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{figures/torus_example.png}
\vspace{-0.1cm}
\caption{Approximate Reeb graph of a point cloud.
(a)~Height function defined on a torus. Level sets at two different heights are illustrated.
(b)~Reeb graph computed on the height function defined on the torus.
(c)~A point cloud sampled from the torus.
(d)~The approximate Reeb graph computed using the Mapper algorithm when the height function is divided into 5 intervals as shown in (c).
}
\label{fig:reeb-graph}
\vspace{-0.3cm}
\end{figure}

However, many real-world data sets are available as sets of functions defined on a set of discrete high-dimensional points rather than continuous functions.
The \textit{Mapper} algorithm~\cite{singh2007topological} computes an approximation of Reeb graph of some user-defined function (often called \textit{lens} or \textit{filter} function) of such data.
Essentially, the Mapper algorithm divides the function range into a set of overlapping intervals and approximates the level sets to be the set of points that fall within each of these intervals.
The connected components of these approximate level sets are then computed by clustering the points that are part of a given interval.
Each cluster then forms a node of the approximate Reeb graph, and an edge is present between two nodes if they share one or more input points.

For example, consider the set of points in Figure~\ref{fig:reeb-graph}(c) that are sampled from the torus in Figure~\ref{fig:reeb-graph}(a). Assuming the height function, this is divided into a set of 5 intervals as shown in Figure~\ref{fig:reeb-graph}(c).
The connected components of the points that fall into these intervals are then used to generate the graph as shown in Figure~\ref{fig:reeb-graph}(d).
In this work, we will be using the Mapper algorithm to compute the topological graph to understand and compare the space defined by the different explanation methods.

\subsubsection{Topological Persistence}
Given a scalar value $a$, the sublevel set $f^{-1}((-\infty,a])$ is defined as the set of all points on the domain that have a function value less than or equal to $a$.
Consider a filtration of the input that sweeps the input scalar function $f$ with increasing function values.
As the function value increases, the topology of the sublevel sets changes at the \textit{critical points} of the function (where its gradient is zero), and remains constant at other points.
In particular, at a critical point, either a new topology is created, or some topology is destroyed. Here, topology is quantified by a class of $k$-dimensional cycles (or $k$-cycles). For example, a 0-dimensional cycle represents a connected component, a 1-dimensional cycle is a loop that represents a tunnel, and a 2-dimensional cycle bounds a void.

A critical point is a creator if a new topology appears and a destroyer otherwise.
Given a set of critical points $c_1,c_2,\ldots,c_m$, one can pair up each creator $c_i$ uniquely with a destroyer $c_j$ which destroys the topology created at $c_i$. We say that a topological feature is born at $c_j$ and it dies at $c_j$. The \textit{topological persistence}~\cite{edelsbrunner02} of this topological feature that is created at $c_i$ is defined as $f(c_j)-f(c_i)$, which intuitively indicates the lifetime of this feature in this sweep.

The above notion of persistence allows features to have an ``infinite" persistence, that is, there exist creators that are not paired with any destroyer. The notion of \textit{extended persistence}~\cite{edelsbrunner2010computational,AEHW06} extends the filtration to include a sweep over superlevel sets ($f^{-1}([a,\infty))$, thus allowing pairing of the above mentioned creators. Without loss of generality, we use the term persistence to mean extended persistence for the remainder of this paper.

Since we are working with functions defined over discrete points, we use the graph computed by the Mapper algorithm to compute the topological persistence of the features of the input. Here, the filtration is defined on the nodes and edges of the graph as follows. Each node is assigned a function value equal to the mean of the function values of the clustered points represented by that node. The order of the nodes added during the filtration (or sweep) is defined by the function value of the nodes. An edge is added during the step of the filtration as soon as both its endpoint nodes are added.
Note that the topological features, in this case, correspond to 0- and 1-cycles only.
\subsubsection{Persistence Diagram}
A persistence diagram plots the topological features as a 2-dimensional scatter plot~\cite{DBLP:conf/focs/EdelsbrunnerLZ00}. Each point in the plot corresponds to a single feature and has $x$ and $y$ coordinates equal to its birth and death values respectively obtained from the extended filtration.
The persistence value of each figure is then the height of the corresponding point above (or below) the line $x = y$.
Figure~\ref{fig:reeb-graph}(e) shows the persistence diagram computed using the Mapper graph.
Note that we plot all the features in the same plot, thus resulting in a single persistence diagram.

Persistence diagrams provide a useful mechanism to assess the structure of scalar functions.
Moreover, it has also been shown that persistence diagrams are robust to noise~\cite{CEH07}. In other words, persistence diagrams are stable under small irregular perturbations in the data, and the distance between two such diagrams is bounded. We can also calculate the distance between two persistence diagrams. Hausdorff distance and bottleneck distance are two common measures used to compare two persistence diagrams. In this work, we use the \emph{bottleneck distance}~\cite{CEH07}, which pairs points between two persistence diagrams and then takes the maximum distance from this matching. In Figure~\ref{fig:bottleneck}, we show two explanation spaces generated by LIME and SHAP on a synthetic data set. It is clear that though the two spaces are quite different, and that a simple clustering (e.g., k-means) would return clusters with low cluster similarity, the Mapper outputs are very similar. We can see that in the persistence diagrams that the only features which differ are shown as the two points in the upper right, which have a small distance between each other.

\begin{figure}
    \centering
    \includegraphics[width=\linewidth]{figures/example_bottleneck.png}
    \caption{Comparing two sets of explanations using GALE. First, the explanation spaces are summarized using Mapper. From the Mapper output, we create persistence diagrams. Topological features correspond to points in the persistence diagram. In this example, we see that while the two Mapper outputs are generally similar, the Mapper output from the SHAP explanations is slightly different from that of the LIME explanations (see the node outlined in purple).}
    \label{fig:bottleneck}
\end{figure}

\begin{figure*}
    \centering
    \includegraphics[width=\linewidth]{figures/example_maps.png}
    \caption{End-to-end pipeline showing the initial data, a trained model (decision boundary), LIME and SHAP explanations, and the topological representations for three different 2-dimensional synthetic classification problems with 100 observations each.}
    \label{fig:example_maps}
\end{figure*}

\subsection{GALE: Globally Assessing Local Explanations} \label{sec:our-method}
In this section, we introduce our approach to \textbf{g}lobally \textbf{a}ssesing \textbf{l}ocal \textbf{e}xplanations (GALE). Let $\mathcal{X}\subset\mathbb{R}^n$ be a data set and $P:\mathcal{X}\rightarrow [0,1]$ a binary classification model, where $P(x)\in[0,1]$ is the probability of $x\in\mathcal{X}$ belonging to the ``1'' class. Given $\mathcal{X}$ and the model $P$,
a local explanation method can be seen as a mapping from the data set to the explanation space $E:\mathcal{X}\rightarrow\mathbb{R}^d$, where $E(x)$ provides the ``importance'' of the different attributes for the classification of $x\in\mathcal{X}$.
While $d=n$ for most approaches, $d$ can be greater than $n$ if the explanation method outputs more than a scalar value for each attribute of the input.

The mapping $\mathcal{X}^\prime=E(\mathcal{X})$ gives rise to a point set in $\mathbb{R}^d$, which is the explanation manifold.
We can define a function $f:\mathcal{X}^\prime\rightarrow [0,1]$ as
$f(x^\prime) = P(x)$, where $x^\prime=E(x), x\in\mathcal{X}$.
In GALE, we use the function $f$ as the lens function from which Mapper builds a summary representation $G$. Intuitively, $f$ captures the relationship between the explanations and the classification probabilities. From $G$, we can produce its persistence diagram $D$.

Since the explanation manifold can drastically vary across different methods and parameters, a direct comparison of the geometry of these manifolds is not possible. However, studying the topology of such functions allows us to analyze how explanation methods differ and thus, we can make geometry agnostic comparisons. To do so, we can take the distance between two persistence diagrams arising from two different explanation methods. If the distance is low, then the topologies of the explanation spaces are similar.
Furthermore, each topological feature (a point in the persistence diagram) can be easily mapped back to a set of input data points in $\mathcal{X}$, thus allowing us to also compare how the explanation space is spread across the input data. We show an example of GALE on three synthetic data sets in Figure~\ref{fig:example_maps}. In Figure~\ref{fig:graph_subsets}, we show how the Mapper graph corresponds to the original data and explanation spaces.

\begin{figure}
    \centering
    \includegraphics[width=\linewidth]{figures/mapper_graph_subsets.png}
    \caption{Mapper output for SHAP explanations on a synthetic circles data set with 400 observations. Each quadrant highlights one unique region of the graph, with the lower left of each quadrant showing the selected original data space (bottom) and explanation space (top).}
    \label{fig:graph_subsets}
\end{figure}

\subsection{Tuning MAPPER Parameters} \label{sec:hyperparam-tuning}
The Mapper algorithm used in this work requires three parameters:
1)~the \textit{resolution} $r$ of the lens function that defines the number of intervals into which the scalar function range is divided;
2)~the \textit{gain} $g$, which defines the percentage overlap between successive intervals; and
3)~the \textit{clustering} algorithm used, which may carry its own parameters.
The value of these parameters determines the structure of the resultant graph, and hence the persistence diagram.

In an ideal scenario (e.g., in a dense point set input), increasing the resolution and decreasing the overlap would result in the graph computed using the Mapper algorithm converging to the Reeb graph. However, in real-world data, too high a resolution or too low a gain can result in the graph being a set of disconnected nodes. In our implementation, we use agglomerative clustering, as implemented in the \texttt{sklearn} Python library, which recursively merges pair of clusters. We consider the ``distance threshold'' parameter, which is the linkage distance threshold, above which clusters are not merged. Due to the varying spaces arising from different explanation methods, we set the distance threshold parameter parameter to a fraction of the maximum minus the minimum value of the explanation space.

Depending on the point distribution, small changes in the parameters can drastically change the resultant graph, and hence the persistence diagram. A trivial option would be to use a resolution of one, which would simply be equal to clustering the points. Poor selections of Mapper parameters can produce graphs which are disjointed or unstable under small perturbations to the input. Blaser and Aupetit~\cite{blaser2020research} suggest that direct measures for measuring the quality of Mapper output could be a combination of cluster quality and their consistency, however, they note that they found no previous work in this direction.

We are therefore interested in identifying a set of Mapper parameters that not only produces a ``stable'' graph computation but also produces graphs with clear structure (i.e., a small number of connected components.) When plotting the Mapper graphs, we use the Fruchterman-Reingold force-directed algorithm~\cite{DBLP:journals/spe/FruchtermanR91}.

To measure the aforementioned stability, we use bootstrapping~\cite{chazal2017robust} to compute the confidence intervals for the generated graph in terms of the bottleneck distances. Consider a set of input explanation values $E = \{e_1,e_2,...,e_n\}$ and the persistence diagram $D$ generated by its Mapper graph $G$. $G$ is generated from some parameter set $H$, which contains the resolution, gain and agglomerative clustering distance threshold. Then, for each iteration in the bootstrap, we sample with replacement from $E$ to construct $E^* = \{e^*_1,e^*_2,...,e^*_n\}$, compute Mapper graph $G^*$ and persistence diagram $D^*$ for this input, and calculate the bottleneck distance $d_b(D,D^*)$. Using the distribution of distances created by these iterations, we can then find the value $\hat{b}_\alpha$ such that

\begin{equation}
    P(d_b(D,D^*) \geq \hat{b}_\alpha) = \alpha
\end{equation}

A parameter set which is stable will produce a low value for $\hat{b}_\alpha$. Additionally, for each iteration, we count the number of connected components in $G^*$. We can find the value $\hat{c}_\alpha$ such that
\begin{equation}
    P(\text{connected components of } G^* \geq \hat{c}_\alpha) = \alpha
\end{equation}

To tune the Mapper parameters, we perform a grid search over the resolution, gain and distance threshold parameters. For each parameter combination, we use 100 bootstrap iterations to estimate $\hat{b}_\alpha$ and $\hat{c}_\alpha$ for $\alpha = 0.05$. Using the estimated stability and connected components for every combination of parameters, we then iterate through the parameter set and greedily select the best combination. While this approach does not guarantee an optimal solution (e.g., we could arrive at a parameter set which has a low $\hat{b}_\alpha$ but a high $\hat{c}_\alpha$), we find that a greedy strategy works well and led to interpretable results on a wide array of synthetic and real world data.

\section{Evaluation} \label{sec:results}
In this section, we evaluate GALE by demonstrating that GALE (1) can be used to compare local explanations, (2) generates stable representations and (3) can be used to tune parameters for popular local explanation methods.

\subsection{Comparing Local Explanation Methods} \label{sec:comparison}
\subsubsection{Comparing Baselines in Gradient-Based Methods}
Here, we illustrate how GALE can be used to understand and compare different explanation methods. In particular, we aim to address a common challenge in using local explanation techniques---what should be the baseline for gradient-based explanation methods?
Baselines act as references to compare the relative importance of features for an input so that attributions can be calculated. We provide examples with both synthetic and real data sets to compare the behavior between different baselines. To be specific, we apply three explanation methods---Integrated Gradients~\cite{sundararajan2017axiomatic}, DeepLIFT~\cite{shrikumar2017learning}, and SHAP~\cite{AEHW06} with five different baselines:
(1)~zero baseline (an input with all values being zeros),
(2)~maximum distance baseline~\cite{sturmfels2020visualizing},
(3)~Gaussian baseline~\cite{smilkov2017smoothgrad, sturmfels2020visualizing},
(4)~uniform baseline~\cite{sturmfels2020visualizing}, and
(5)~a trained baseline~\cite{izzo2020baseline},
resulting in 15 tables of explanations for each experiment.

Our goal in this experiment is to illustrate how GALE can be used to identify explanation methods that behave differently. To do so, we generate a synthetic data set with 5 features and determine the labels with an extremely simple logic---the input will be labeled as ``1'' if any of the columns contain a ``zero'' as value. Otherwise, they are labeled as ``0''. When working with real data sets, it is normal for them to contain zero values that may provide important information. Under our construction, only zero values are important to the classification. Thus we expect the zero baseline is the only reference that is not a neutral input to the classifier. We created 20 different synthetic data sets under the aforementioned conditions, where each set varied in its rate of ``0'' labeled instances.

We first calculate a pairwise distance matrix for each of the 20 data sets. Then, we average the distance matrices, and show the result in Figure~\ref{fig:gradient_based_matrix}. We can see that there are three rows and columns which have much brighter colors (i.e. greater average distances) than the others, meaning that these three explanations differ greatly as compared to the other explanations. Such observation is consistent with the fact that zero baselines produce different feature attributions by treating the important values as neutral references. In general, any set of attributions having a significant deviation from the consensus should be treated with caution.

\begin{figure}
    \centering
    \includegraphics[width=\linewidth]{figures/bottleneck_distance_matrix.png}
    \caption{Pairwise distance matrix showing the average bottleneck differences among gradient-based explanation methods across 20 synthetic datasets. We see that the methods which use the zero-baseline have higher bottleneck distances to the explanation topologies using other baselines.}
    \label{fig:gradient_based_matrix}
\end{figure}

When we explore the corresponding topological skeletons, we observe that these explanations typically have a more scattered graph compared with the others. Such visual evidence provide a sense that the explanation values differ greatly among the inputs in the data set. This evidence can be found statistically as well. When computing the variances of the explanation values among different outcomes, the variances are generally high among the outputs with scattered topological graphs. We show an example in Figure~\ref{fig:gradient_based_graphs}. Therefore, the topological representations also provide an overview of the variances of the explanations.

\begin{figure*}
    \centering
    \includegraphics[width=\linewidth]{figures/gradient_based_graph.png}
    \caption{Visualizing the differences between the generated topologies reveals that the zero-baseline topologies contain more, smaller connected components than those from other explanation methods.}
    \label{fig:gradient_based_graphs}
\end{figure*}

\subsubsection{Comparing LIME, SHAP and Explainable Boosting Machines}
Many local explainability methods are run post-hoc, meaning that first a model is trained and subsequently an explainability method is applied to the model's output to produce explanations. However, understanding the computed explanation space is oftentimes unintuitive. Generalized additive models, or GAMs, are a modeling approach whereby each feature contributes additively to a model's prediction. One desirable trait of this approach is that the feature contributions are easily interpretable, and thus present a alternative to LIME and SHAP. Explainable Boosting Machine (EBM) are a popular interpretable tree-based GAM from the \texttt{interpret} Python library~\cite{nori2019interpretml}. While we expect there to be consensus between LIME and Kernel SHAP explanations, since the two methods are similar~\cite{lundberg2017unified}, it is unknown how the explanations from these methods compare to those from EBMs.

We consider both synthetic and real world data sets for our experiments. For our synthetic data sets, we consider the spirals, circles and corners data from Figure~\ref{fig:example_maps}, one linearly-separable 2-dimensional data set, three ``toy'' data sets and the Pima Indians diabetes~\cite{smith1988using} data set. In the toy data sets, we have six features $\{x_{1}, ..., x_{6}\}$, and the target for observation $i$ is determined by $y_i = x_{i,1} + x_{i,2} + x_{i,3} + x_{i,4}$ and each feature is independent from one another. In ``independent'', each feature is weighted equally and is independent from one another. In ``flip'', $y_i = x_{i,1} - x_{i,2} + x_{i,3} - x_{i,4}$. Finally, in ``interaction'', we add equal-weighted interactions in the form of $y_i = \sum_{k=1}^4 x_{i,k} + \sum_{j=2}^4 10 x_{i,1} \cdot x_{i,j}$. For each data set, we train a 2-layer feedforward neural network with 64 hidden units per layer, as well as an Explainable Boosting Machine (EBM). We use the max number of features and a 50-sample neighborhood for our LIME explainer.

\begin{table*}[]
\centering
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
\textbf{Data} & \textbf{Type} & \textbf{Obervations} & \textbf{Features} & \backslashbox{\textbf{LIME}}{\textbf{SHAP}} & \backslashbox{\textbf{LIME}}{\textbf{EBM}} & \backslashbox{\textbf{SHAP}}{\textbf{EBM}} \\ \hline
Spirals            & Synthetic & 100   & 2  & 0.00 & 0.11 & 0.11 \\ \hline
Circles            & Synthetic & 100   & 2  & 0.12 & 0.34 & 0.35 \\ \hline
Corners            & Synthetic & 100   & 2  & 0.08 & 0.15 & 0.15 \\ \hline
Linearly Separable & Synthetic & 100   & 2  & 0.00 & 0.02 & 0.02 \\ \hline
Toy                & Synthetic & 300   & 6  & 0.00 & 0.03 & 0.03 \\ \hline
Toy-Flip           & Synthetic & 300   & 6  & 0.00 & 0.08 & 0.08 \\ \hline
Toy-Interaction    & Synthetic & 300   & 6  & 0.01 & 0.11 & 0.11 \\ \hline
Breast Cancer      & Real      & 569   & 30 & 0.00 & 0.02 & 0.02 \\ \hline
Diabetes           & Real      & 768   & 8  & 0.00 & 0.45 & 0.45 \\ \hline
Sonar              & Real      & 208   & 60 & 0.00 & 0.00 & 0.00 \\ \hline
Banknote           & Real      & 1,372 & 4  & 0.00 & 0.00 & 0.00 \\ \hline
Ionosphere         & Real      & 351   & 34 & 0.00 & 0.01 & 0.01 \\ \hline
Bank Marketing     & Real      & 4,119 & 19 & 0.01 & 0.44 & 0.44 \\ \hline
German Credit      & Real      & 1,000 & 24 & 0.00 & 0.45 & 0.45 \\ \hline
\end{tabular}
\caption{Bottleneck distances between LIME, SHAP and EBM explanation topologies for synthetic and real world data. Intuitively, LIME and SHAP generally have low-bottleneck distances, indicating that their explanation topologies are similar.}
\label{tab:lime-shap-ebm}
\end{table*}

We show the bottleneck distances between the explanations from LIME, SHAP and EBM on each data set in Table~\ref{tab:lime-shap-ebm}. As expected, we see strong consensus between LIME and SHAP explanations, particularly in our real world data sets. However, we see varied consensus between EBM and LIME/SHAP explanations. Noticeably, there was more consensus among the three explanation methods when the prediction problem was ``easy'', such as in the linearly separable and toy synthetic data sets. One possible explanation for the lack of consensus on some data sets is that EBMs, unlike LIME and SHAP, for a $k$-dimensional data set, return explanations of size $j$, where $j \geq k$. This is because EBMs include an interaction component, where the interactions are detected by the EBM algorithm. We can enforce zero interactions when training, so that $j = k$ in the explanation output. However, by enforcing no feature interactions, the effect on explanation consensus was ambiguous, which suggests that the topologies generated by GAMs and LIME/SHAP can differ significantly.

\subsection{Evaluating Topological Stability}
Although the previous method sought to determine appropriate parameters for a local explainability method, many methods, like LIME, rely on stochastic behavior, which can produce different explanations for different runs, even when given the same parameters. It is important that our topological representation is resistant to sampling variability incurred by local explanation methods. In this section, we benchmark GALE's stability with regards to the variation induced by local explainability methods. Additionally, we show how our greedy approach to Mapper parameter selection aids in finding stable topological representations.

To determine the robustness of GALE to randomness induced by a local explanation method, for each synthetic data set and the diabetes data set, we run LIME 30 times. We restricted our experiments to these data sets for computational purposes. For each run, we generate two Mapper outputs: one which uses the Mapper parameters found via a greedy search, as described in section~\ref{sec:our-method}, and one which uses fixed parameters: a resolution of 15, a gain of 0.3 and an agglomerative distance threshold of 0.3 times the range of the explanation space. We then generate two bottleneck distance matrices -- one for distances between the optimized Mappers and another for distances between the fixed parameter Mappers. In Table~\ref{tab:stability}, we report the average row-sum for each of these matrices for each synthetic data set, as well as the average number of connected components from the mappers generated. The higher the row-sum, the less consensus there is among the explanation topologies produced by Mapper. We see that using our greedy Mapper parameter search procedure, our output is resistant to variation in LIME output. Although for some data sets, the fixed parameters returned a lower row-wise sum, we saw that in these cases, the fixed parameters also returned a higher number of connected components.

\begin{table}[]
\centering
\begin{tabular}{|c|cc|cc|}
\hline
\multirow{2}{*}{\textbf{Data}} &
  \multicolumn{2}{c|}{\textbf{Row-Wise Distance}} &
  \multicolumn{2}{c|}{\textbf{\begin{tabular}[c]{@{}c@{}}Connected \\ Components\end{tabular}}} \\ \cline{2-5}
 &
  \multicolumn{1}{c|}{\textit{Greedy}} &
  \textit{Fixed} &
  \multicolumn{1}{c|}{\textit{Greedy}} &
  \textit{Fixed} \\ \hline
Spirals            & \multicolumn{1}{c|}{0.00} & 0.83 & \multicolumn{1}{c|}{2.0} & 4.1  \\ \hline
Circles            & \multicolumn{1}{c|}{1.88} & 2.56 & \multicolumn{1}{c|}{6.1} & 10.5 \\ \hline
Corners            & \multicolumn{1}{c|}{1.22} & 1.61 & \multicolumn{1}{c|}{9.0} & 14.6 \\ \hline
Linearly Separable & \multicolumn{1}{c|}{0.00} & 0.00 & \multicolumn{1}{c|}{4.0} & 4.0  \\ \hline
Toy                & \multicolumn{1}{c|}{0.03} & 0.00 & \multicolumn{1}{c|}{2.0} & 3.0  \\ \hline
Toy-Flip           & \multicolumn{1}{c|}{0.00} & 0.00 & \multicolumn{1}{c|}{2.0} & 2.2  \\ \hline
Toy-Interaction    & \multicolumn{1}{c|}{0.16} & 0.00 & \multicolumn{1}{c|}{2.0} & 5.0  \\ \hline
Diabetes           & \multicolumn{1}{c|}{0.38} & 2.70 & \multicolumn{1}{c|}{4.3} & 36.4 \\ \hline
\end{tabular}
\caption{Average row-wise sum of distance matrix and average connected component counts after using parameters found by our greedy parameter search. We find that our parameter tuning procedure produces more connected and stable Mapper output.}
\label{tab:stability}
\end{table}

\subsection{Using GALE to Tune Explainability Method Parameters} \label{sec:tuning-explain-params}
Many local explainability methods use a variety of parameters which guide the output. For example, two important parameters in LIME are the maximum number of features present in the explanation and the size of the neighborhood to learn the linear model. Likewise, in SHAP, depending on the explanation method, one can specify parameters such as the number of times to re-evaluate the model when explaining each prediction in Kernel SHAP, or the limit on the number of trees used in Tree SHAP. Choosing an appropriate set of parameters is important, as there can be significant variability in explanations coming from even just slightly different parameters~\cite{visani2020statistical}. After establishing that GALE can not only elucidate the differences between explanation sets but is also stable, we now show how to use GALE to tune the parameters of a local explainability methods, in particular, LIME. We consider the toy synthetic data, where each feature is independent of one another, used in Section~\ref{sec:comparison} for our experiment. We train a random forest classifier on this data using the default parameters from \texttt{sklearn}. Our goal is to determine the number of features to use for our LIME explanation, which we know to be four. We set the size of the neighborhood to 50 observations.

Using our Mapper parameter tuning technique to find appropriate parameters, we compute the corresponding graph for explanations using $k$ features, where $k \in \{2, ..., 6\}$. Then, we calculate the bottleneck distance between each of the persistence diagrams to produce a distance matrix. Next, we calculate the row-wise sum of this distance matrix. Persistence diagrams which differ strongly from others will have a high row-wise sum. We plot the row-wise sum in Figure~\ref{fig:lime_feature_selection}. We see that when the number of features is four or higher, the returned topological signatures are identical. Thus, one may conclude that the addition of a fifth or sixth feature in the LIME explanations has no effect on the overall topology of the explanations. Likewise, by only including two or three features in LIME, one may not be estimating appropriate explanations.

\begin{figure}
    \centering
    \includegraphics[width=\linewidth]{figures/lime_parameter_selection.png}
    \caption{Row-wise sum of distance matrix for Mapper graphs computed on LIME explanations of different feature counts. Since the average row-wise distance stays constant when using 4 or more features for LIME, we know that the produced topologies are similar for these parameter values.}
    \label{fig:lime_feature_selection}
\end{figure}

\section{Discussion} \label{sec:discussion}
In this work, we outline and evaluate a methodology to compare local explainability methods through topological data analysis (TDA). While TDA shows promise in tackling challenges encountered in XAI applications, there are still several shortcomings which we plan to address in the future. The main constraint on our current methodology is that we focus on binary classification problems using tabular data. Furthermore, a deeper understanding of the effect and tuning of the Mapper parameters is also warranted.

Extending GALE to handle multi-class classifiers poses an important challenge---modeling a function similar to what is being done results in a multi-variate function, for which defining the appropriate filtration to compute the persistence diagrams becomes non-trivial. One possibility is to transform multi-class probabilities into scalar values. Additionally, future work should be directed towards extending GALE to accommodate prediction problems in the image and sequence spaces. A simple approach could be to reshape image explanations into a vector, and then use the approach we outlined here. While we did not evaluate GALE on regression problems, the evaluation would be similar, except for changing the lens function from the class predictions to the regression output.

Lastly, the output from Mapper is sensitive to the algorithm's parameters. While we introduced a greedy approach to find a good set of parameters which give stable and connected representations, more work is needed to develop alternative ways to tune the parameters. Furthermore, we only considered agglomerative clustering with single linkage. Within agglomerative clustering, there are a variety of linkage criterion to consider, such as Ward or average linkage. Additionally, one may consider other clustering algorithms, such as DBSCAN~\cite{DBLP:journals/tods/SchubertSEKX17}.
\section{Conclusion} \label{sec:conclusion}
In this work, we present GALE, a topology-based framework to globally summarize the outputs of local explanation methods. To do so, we compute a topological skeleton of a scalar function that captures the relationship between the explanation space, such as those created by LIME or SHAP, and a model's predictions. From this skeleton, we compute persistence diagrams. Then, to compare explanations, we compare persistence diagrams. The benefits of our method are that GALE (1) allows for easy comparison of heterogenous local explanations, differing in dimensionality or feature values, (2) is resistant to outliers and variation induced by local explanation method sampling procedures, and (3) can be used to optimize hyperparameters in local explanation methods. We validate our approach through experiments on both real and synthetic data sets. GALE is simple, lacks complex optimizations, and can be broadly applied to many local explanation methods. We believe our method is a fruitful and accessible first step in using topological techniques to understand explainability methods.
\section{Appendix}

\subsection{Part One}

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malesuada, quam in pulvinar varius, metus nunc fermentum urna, id
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vel mollis. Curabitur quis dictum nisl. Phasellus vel semper risus, et
lacinia dolor. Integer ultricies commodo sem nec semper.

\subsection{Part Two}

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ligula, non varius nibh pulvinar semper. Suspendisse nec lectus non
ipsum convallis congue hendrerit vitae sapien. Donec at laoreet
eros. Vivamus non purus placerat, scelerisque diam eu, cursus
ante. Etiam aliquam tortor auctor efficitur mattis.

\section{Online Resources}

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pulvinar ex pretium. Sed interdum orci quis metus euismod, et sagittis
enim maximus. Vestibulum gravida massa ut felis suscipit
congue. Quisque mattis elit a risus ultrices commodo venenatis eget
dui. Etiam sagittis eleifend elementum.

Nam interdum magna at lectus dignissim, ac dignissim lorem
rhoncus. Maecenas eu arcu ac neque placerat aliquam. Nunc pulvinar
massa et mattis lacinia.

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Home > Movie Stars > R > Mercedes Ruehl
Mercedes Ruehl
Mercedes Ruehl began her career as a highly successful New York stage actress, receiving an Obie for the off-Broadway play The Wedding of Betty and Boo, and a Tony Award for her performance in Neil Simon's Lost in Yonkers. She is perhaps best known for her work as Anne Napolitano in 1991's The Fisher King, for which she won an Oscar and a Golden Globe as Best Supporting Actress.
Ms. Ruehl has also appeared in such films as Big, Married to the Mob, and Last Action Hero. In 1995, she had a recurring guest role as station manager Kate Costas on the hit NBC-TV show Frasier. She can be seen this fall in the film What's Cookin', and next spring in the TV mini-series Widows, which co-stars Brooke Shields and Rosie Perez.
Mercedes Ruehl Facts
Occupation Actress
Birthday February 28, 1948 (74)
Sign Pisces
Birthplace Queens, New York, USA
Height 5' 11" (1m80) How tall is Mercedes Ruehl compared to you?
Awards 1992 Academy Awards: Best Supporting Actress (for The Fisher King)
1992 Golden Globe Awards: Best Actress in a Supporting Role - Motion Picture (for The Fisher King)
Peggy Guggenheim: Art Addict
Loving Leah: Hallmark Hall of Fame Gold Crown Collector's Edition
Minus Man
Continue » Find more details on the Mercedes Ruehl Movies page
Bio courtesy Providence Entertainment (01-Jan-2000)

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Grey's Anatomy Jan 11
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'America's Got Talent' Producers and NBC Issue New Statement About Gabrielle Union
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Screenwriters Can Play Key Role in Gender Parity, Actors Say
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Fake German Heiress With 'Not a Cent to Her Name' Set for NYC Trial in Master Swindling Scheme: Prosecutors
Anna Sorokin traveled in celebrity circles and tossed $100 tips — all the more reason to believe she was the German heiress she said she was. But behind the jet-set lifestyle and pricey threads, prosecutors say, was a fraudster who bilked friends, banks and hotels for a taste of the high life.
Michelle Obama in Las Vegas to Drum Up Voter Participation
Former first lady Michelle Obama held a campaign-style rally in Las Vegas on Sunday to urge Nevadans to register to vote and cast ballots this fall, warning them that sitting out means someone else will make decisions for them. "We get the leaders we vote for. We get the policies we vote for. And when we don't vote, that's when...
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Actor Patrick Dempsey says someone is impersonating him online and asking his fans to give money to either him or his Maine-based nonprofit.
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Victoria Mahoney Is 'Star Wars' Movies' First Black Female Director
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Ellen Pompeo Talks Pay Equality and Dempsey's 'Grey's Anatomy' Exit
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text:
Front Porch Punditry » Political News and Views » Breaking Political News » Exclusive — Boehner Has No Plans To Leave Republican Party Despite Newfound Democratic Party Support
#1 | Exclusive — Boehner Has No Plans To Leave Republican Party Despite Newfound Democratic Party Support Fri Mar 06, 2015 2:11 pm
Frank Cannon
Location: Second Circle of Hell
http://www.breitbart.com/big-government/...-party-support/
Thank God. Really hate to lose such a fine man to the other side.....
Despite the fact that Democrats are rushing to protect him from a potential brewing Republican coup attempt, House Speaker John Boehner—currently a Republican—has no intention of leaving the Republican Party to join his new friends in the Democratic Party, his spokesman Michael Steel tells Breitbart News.
"Speaker Boehner is a proud, conservative Republican, and enjoys the strong support of an overwhelming majority of the House Republican Conference," Steel said in an email on Friday morning when asked by Breitbart News if the Speaker plans to switch party affiliations, given that several Democrats now say they'll protect him from a Republican coup.
If Boehner did eventually leave the GOP, he'd be the highest-profile Republican in recent years to do so. Others who have switched to the Democratic Party after abandoning GOP principles as laid out in the party platform include the late Sen. Arlen Specter of Pennsylvania and former Florida Gov. Charlie Crist.
Talk of a coup has blossomed in recent days after Boehner used Democrat votes—and lost a majority, 167 in total, of the Republican conference—to pass a Department of Homeland Security (DHS) funding bill that provides taxpayer dollars to President Barack Obama's executive amnesty.
A new report on Friday from The Hill quotes several Democrats, including some very liberal members, saying they'd vote for Boehner for Speaker to protect him from a Republican affront.
"Tea Party Republicans contemplating a bid to oust Rep. John Boehner (R-Ohio) shouldn't count on Democrats to help them unseat the Speaker," The Hill's Mike Lillis wrote.
And without their support, there is no chance to topple Boehner in this Congress. A number of right-wing Republicans, long-wary of Boehner's commitment to GOP efforts attacking President Obama's policy priorities, have openly considered a coup in an attempt to transfer the gavel into more conservative hands. But Democrats from across an ideological spectrum say they'd rather see Boehner remain atop the House than replace him with a more conservative Speaker who would almost certainly be less willing to reach across the aisle in search of compromise. Replacing him with a Tea Party Speaker, they say, would only bring the legislative process — already limping along — to a screeching halt.
Lillis quotes Democratic Reps. Bill Pascrell (D-NJ) and Raul Grijalva (D-AZ)—both liberal members of Nancy Pelosi's conference—saying they'd back up Boehner in the face of a GOP coup.
"I'd probably vote for Boehner [because] who the hell is going to replace him? [Ted] Yoho?," Pascrell said, adding: "In terms of the institution, I would rather have John Boehner as the Speaker than some of these characters who came here thinking that they're going to change the world."
"Then we would get [Majority Whip Steve] Scalise or somebody? Geez, come on," Grijalva said. "We can be suicidal, but not stupid."
House Minority Whip Steny Hoyer said, according to Lillis' piece, that there are "some disgruntled people who are talking about" a coup against Boehner but if House Freedom Caucus leader Rep. Jim Jordan (R-OH) doesn't lead the coup it won't happen.
"If Jordan's not talking about — he's the head of the Freedom Caucus — it's not going to happen," Hoyer said.
Congressional Black Caucus chairman Rep. G.K. Butterfield (D-NC) said the Republicans should be able to figure out who their own speaker should be.
"If they've got the votes to make it [a coup against Boehner] happen, then they should act accordingly. But I would not want Democrats to be a part of that," Butterfield said. "I would give deference to the choice of the Republicans."
Some other Democrats were unclear on how to move forward.
"I think it would pose a real existential dilemma for us," Rep. Gerry Connolly (D-VA) said. "I mean, on the one hand, if you have a chance to take out a Republican Speaker, why wouldn't you do that? On the other hand, if the obvious alternative is a Tea Party Speaker, now you've got to worry not only about your own political situation but frankly about the institution."
Connolly added that having a conservative Republican speaker is something that would "give very serious pause to the Democrats."
Liberal Rep. Jim McGovern (D-MA) said he's siding with Boehner because Boehner will move big bipartisan big government bills through the House.
"Personally, I don't want to waste two years," McGovern said. "And I think that the crazy Tea Party type would probably not be willing to work with us on anything."
McGovern added that he hopes Boehner learns his lesson about the majority of Republicans after this.
"My hope is that what comes out of this is that Boehner realizes that there are some people in his caucus who are unreasonable, and you can never get them to say 'yes' to anything," McGovern said. "Rather than spending so much time agonizing over how to please them, maybe he just ought to focus on how you build more bipartisan coalitions and actually get some things done."
Romney '16 '20 ..... because we all need a laugh.
#2 | RE: Exclusive — Boehner Has No Plans To Leave Republican Party Despite Newfound Democratic Party Support Fri Mar 06, 2015 2:47 pm
What a waste of Jamesons.
Member of the "Oh, Hell No! Party"
Cyber Liberty
Timothy Murphy
I don't see why either Boehner or the other Dems would want to rock the boat. They're sitting in the catbird seat, getting everything they want and sticking the GOP with the results. It's win-win.
By the way, I'm growing rather weary of the cheap comparisons of Obama with Neville Chamberlain. The British Prime Minister got the biggest issue of the day wrong. But no one ever doubted that he loved his country. That's why, after his eviction from Downing Street, Churchill kept him on in his ministry as Lord President of the Council, and indeed made Chamberlain part of the five-man war cabinet and had him chair it during his frequent absences.
When he [Chamberlain} died of cancer in October 1940, Churchill wept over his coffin.
So please don't insult Neville Chamberlain by comparing him to Obama. -- Mark Steyn"
ThirstyMan
Admiral James O. Richardson - Moderator
Location: Concord MA
Quote: Sanguine wrote in post #2
Jeesh, that's one drink I've never tried.. Better get some tonight! I've heard only good things about it. TM
Daniel Greenfield, January 29, 2015, The Imaginary Islamic Radical
"Our problem is not the Islamic radical, but the inherent radicalism of Islam. Islam is a radical religion. It radicalizes those who follow it. Every atrocity we associate with Islamic radicals is already in Islam. The Koran is not the solution to Islamic radicalism, it is the cause."
Quote: ThirstyMan wrote in post #4
Basic Jamesons is shit. The only good one they make is the 18 year and it is around a buck and a quarter a bottle. Tullimore Dew is what you want. $30 at your local commonwealth package store.
Quote: Frank Cannon wrote in post #5
Tullamore Dew! No way.
Jamesons is the bomb.
FWP
bicinium
Hmmm.... this looks like a genuine difference of opinion here at the Porch....
I have a great idea, I think that TM should buy a bottle of each this evening. Pick one of the two, and drink the whole bottle tonight, and post frequently to let us all know what he thinks of it....
then tomorrow evening, do the same with the other bottle. And then on Sunday he can let us know which he thinks is the better choice!!
Sounds good??
Basically supporting no one at this point!!
#8 | RE: Exclusive — Boehner Has No Plans To Leave Republican Party Despite Newfound Democratic Party Support Sat Mar 07, 2015 12:49 am
What an interesting idea. I'm in!
Now we just have to convince TM.
#9 | RE: Exclusive — Boehner Has No Plans To Leave Republican Party Despite Newfound Democratic Party Support Sat Mar 07, 2015 1:05 am
You can be in. I have such a low regard for you now selling a novice on shitty booze. Come on now. Raise the bar a bit. I bet you think Bankers Club is guest appropriate.
#10 | RE: Exclusive — Boehner Has No Plans To Leave Republican Party Despite Newfound Democratic Party Support Sat Mar 07, 2015 2:23 pm
Hey I was back home after enjoying my very first Jameson, 12 year old that is!
I can't imagine a smoother drink. I had mine straight up, a glass of ice on the side. It was terrific!
Now for stage two of the experiment. Looks like I'm going to have to save up for this stage. Jamesons was only $7 for my glass. Yum!!!
hey! I read that slight. I'm a fast learner my friend.
Bankers Club? I don't know that one. But, I hope it's better than Tullamore Dew.
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Created in forum Politics With an Attitude by Rev 0 Sun Dec 15, 2013 7:32 pm


The Concordat to Support the Career Development of Researchers was launched in 2008. Co-ordinated by Universities UK and Research Councils UK, this document sets out clear, comprehensive expectations and responsibilties for researchers and for managers, funders and employers.
Durham University is committed to the Concordat and supporting its valued Research Staff. This webpage and the ones that follow are designed to inform you about the Concordat and to provide you with a portal uniting all the relevant information pertaining to the terms of the Concordat from within the University's website. Please use the links on the left to find out more.
The Concordat Implementation Group (CIG) oversees the implementation of the Concordat - this has included the production of a Gap Analysis and Action Plan and HR Excellence in Research Report. CIG will monitor the implementation of the action plan and its effectiveness.
To find out more about the Concordat, please contact Joanne Race, Deputy Director of Human Resources, at j.e.race@durham.ac.uk or by telephoning +44 191 33 46500.


Award bookings on Air France/KLM routes have been reduced by 25-50% for travel between August 1 – September 30, 2014.
All destinations below. Note: Everything is to/from Europe.
Book between June 1 – 30, 2014.
This is the now the 4th round of Air France/KLM FlyingBlue's new program for promo awards. They used to publish lists every 2 months, the booking period started immediately and would be valid for appx 2 months. The new plan is that they are publishing new lists every month but the available dates will be 2 months out for a period of 2 months. They promise "more availability" and a "greater choice of routes"–and this current list is looking pretty good and backs up this claim.
The first few rounds were pretty weak with only 15-25 cities and only economy award space available. The last list for the July/August period has probably been the best yet, with 50+ destinations and a lot of 50% off business class space, making it a very tough act to follow. This current round has about 35 destinations with about half priced at the 50% off rate.
Booking for this deal runs from June 1 to June 30, 2014.
You'll still get hit with fuel surcharges!
We've got a How-to Guide for booking flights with Flying Blue points.
BOTTOM LINE: While this list isn't as good as the last one that had tons of business class space, it certainly gets the job done by allowing you to fly from Houston to (just about) ANYWHERE in Europe for only 25,000 miles roundtrip + $200-$350 in taxes. This makes it a great deal in my book.


We have stores in Spain and the Netherlands who will be delighted to help arrange delivery from DFS throughout their regions. We can also arrange deliveries to elsewhere in Europe and worldwide using our trusted third party delivery providers. Alternatively if you have your own transport company that you would like to use for international shipment we can also arrange for your furniture to be delivered to them.
When you've got a new sofa about to arrive, it's easy to forget what you're going to do with your much-loved old one.
for free to help raise funds.


text:
As competições de canoagem nos Jogos Olímpicos de Verão de 2008 foram realizadas no Parque Olímpico Shunyi, na China entre 11 e 23 de agosto. Dezesseis eventos distribuíram medalhas, sendo doze provas para homens e quatro para mulheres.

Calendário

Eventos

Velocidade
 K-1 500 m masculino
 K-1 1000 m masculino
 K-2 500 m masculino
 K-2 1000 m masculino
 K-4 1000 m masculino
 K-1 500 m feminino
 K-2 500 m feminino
 K-4 500 m feminino
 C-1 500 m masculino
 C-1 1000 m masculino
 C-2 500 m masculino
 C-2 1000 m masculino

Slalom
 K-1 masculino
 K-1 feminino
 C-1 masculino
 C-2 masculino

Qualificação

Medalhistas

Velocidade
Masculino

Feminino

Slalom

Masculino

Feminino

Quadro de medalhas

Ligações externas

Eventos da canoagem nos Jogos Olímpicos de Verão de 2008
Canoagem nos Jogos Olímpicos

redpajama_set_name: RedPajamaWikipedia


I used Hot Cocoa Whiskers and a Sweet Sunday Sketch. I stamped the image, colored with Copics, and paper pieced just the cup with a scrap of the same paper. I used a craft knife to cut around Whiskers' hand so that the paper pieced cup would go underneath it.
I added the charm with a double-bow and a sentiment from Papertrey Ink. I hope you like it!
We'd love to have you play along with us! All the details can be found on the La-La Land Crafts blog. TFL!
I stamped Angel Lala, colored her with Copics and paper pieced her dress. I spread Stardust glitter pen all over her halo and wings with a waterbrush, and then glued superfine glitter to just the edges of the halo and wings for a little extra sparkle. Lastly, I cut her out and mounted her foam boosters.
For the frame, I stuck some Basic Grey paper to self-adhesive magnet sheet and cut out the frame using an old Sizzix die. I tied a bit of ribbon around it, stuck a beaded pin in the knot, and added on Lala. I thought this would be the perfect way to stick pictures of your little angel on the fridge!
We hope you'll play along with us! All the details can be found on the La-La Land Crafts blog. TFL!
Pure Innocence Pile It On!
I have a Pile It On! card today and I'm sad to say it's my last one for my term as a DT member. I had a lot of fun and am so happy I got the chance to be on the DT - I am now a huge fan of combining challenges (and of challenges in general!) and I owe it to the PIO! gals Dalis and Vickilyn. Thanks so much for having me on the team!!
I used this adorable winter image from Digi Stamp Characters by Heather Ellis (the Pure Innocence line). I printed her out and colored with Copics. I accented the snow with a little glitter and used some pretty Basic Grey papers from the Eskimo Kisses paper pad. The sentiment is from the Elegant Christmas set from The Cat's Pajamas.
Our Creative Corner - Let it Snow!
I hope you like it and that you'll play along with us! All the details can be found on the Pile It On! blog. TFL!
If you're looking for the La-La Land Crafts' Fairy Tale Release Blog Hop, it's the post directly below this one. Thanks!
I used the set Let It Snow and some super cute papers from My Mind's Eye.
I stamped the snowman twice and colored with Copics. I cut out one full snowman and then just the hat and scarf from the second and layered those pieces on the first with foam boosters. The whole snowman was also popped up on foam boosters. The snowflake was cut from glitter cardstock with a Sizzix die and I tied a button to the center with crochet thread. I distressed all the edges of the papers and did some machine stitching to finish it off. Finished card size is 4.5" x 5.5".
We hope you'll play along with us! One lucky participant will be drawn at random to win the Cupcake Cutie Single Stamp!! You don't have to use YNS images on your card, but if you do you'll get a double entry in the draw!
Also, the challenge hostess will choose her favorite card, and that winner will get the YNS Challenge "Top Pick" blinkie to proudly display on their blog PLUS a $5 gift certificate to spend at the Your Next Stamp store!
Full details can be found at the YNS Challenge Blog. TFL!
Welcome to La-La Land Crafts' Fairy Tale Release Blog Hop! To celebrate the release of our new Princess Rosie stamp, we are giving her the royal treatment that she deserves. Every member of our Design Team and the Guest Designers have made their cards with Princess Rosie which we are now showing on our blogs. To make the Blog Hop even more fun, along with the cards you will find some clues (a secret word or two) on every blog. When you follow the order of the Blog Hop and get to the end, you will have collected all of the words and will be able to put them together into a sentence. When you are done, you will need to go back to the La-La Land Crafts Blog and leave a comment on the Fairy Tale Blog Hop post.
If you came to me from Katie's blog, then you're on the right track! If not, then you'll want to go back and start at the La-La Land Crafts blog and hop to all the stops!
What's even more? While on your treasure hunt for secret words, you might even find some more treats offered by our designers!! And best of all, I have some great news for you!! The Princess Rosie stamp is already available for purchase in our online store!
What do you have to do to join our Blog Hop?
* Final stop is back at the La-La Land Crafts blog where you have to leave a comment on the Blog Hop post. IMPORTANT - Do not reveal your answer in your comment!! We will contact the winner for the answer. If the answer is incorrect we will draw the winning number again.
One entry per person please. We will draw the winning number from all the entries next Wednesday, December 23rd, around 10 am PST. Everyone is welcome to participate. We will ship internationally!!
And now, it's time for my card!
I stamped Rosie twice and colored with Copics. For the bottom layer, I cut her out, leaving out the crown and the hand on the right side (they're so small, they just didn't look right under the top layer). For the top layer, I cut her out, leaving out the bottom ruffle of the dress and her legs and shoes.
I also added a bunch of tiny rhinestones and glitter pen to Rosie and the patterned paper. The papers are from Imaginisce and the sketch is from Unscripted. I hope you like it!
My secret words are "the courage." The next stop on the Hop is the fabulously talented Kristine. Thanks very much for joining us for our Blog Hop!
Hello everybody! Thanks for joining us for another fun and fabulous Your Next Stamp Blog Hop! YNS just released a new Fhiona set (Fitness Fhiona) and a bunch of Cute Little Buggies and Animals so the Starlets (and 2 awesome guest designers!) have whipped up some eye candy using many of the new images.
If you came here by way of Lana's blog, then you're on the right track! If not, please go back and start at Karen's blog because you won't want to miss a stop - one grand prize winner will be randomly drawn to win 6 of the new releases of their choice!! To qualify, you need to comment at all the blogs on the hop with the answer to this question: Which new release stamp/set is your favorite?
One last thing before I get to my projects for today - Karen is offering a super special while the Hop is on (Dec. 10 - 14 at midnight EST) - a FREE stamp or stamp set of your choice from the YNS store with a purchase of $55 or more!!
I colored Sammy with Copics, then cut him and the mat out with Nestabilities. The tree paper is from Upsy Daisy Designs and I made the sentiment myself using 2 different alphabet sets (the smaller one is from Papertrey Ink and the larger is Close To My Heart). I finished it off with some sponging, light distressing, and a couple buttons tied with twine.
I used the new super cute Bee Stings and some really old Imaginisce paper. On the box, I popped up an extra head for the bee for dimension, but I left all the notecard images flat for easy mailing.
I computer generated a sheet with the sayings placed so that they would end up where I wanted them (it took me many tries to get it right, but now I have a template I can use whenever I want!). After printing, I just had to do a little trimming to get them down to the correct size. If you're interested, the sayings are Don't worry, bee happy!, Buzzing by to say hi!, What's the buzz?, and Just bee-cause!
The bees were all colored with Copics and I added Stardust glitter pen on the wings. I embossed the yellow strips with the Birds and Swirls Cuttlebug folder and stitched buttons to the box. Thanks so much to Cindy and Val for the inspiration and especially to Val for the great project and tutorial!
Just a quick reminder - there's still time to enter the YNS Challenge #5 - make a birthday project with hearts. The deadline is December 15 and all the details can be found on the YNS Challenge Blog.
I hope you liked my projects and thanks for hopping along with us! Your next stop is the incredibly talented Maylee.
Yay! Wednesday means it's time for another La-La's Weekly Wednesday Challenge! This week our theme was set by Lisa and is a color challenge: Lisa says, "I'm loving the 'new' Christmas colors of pink and green... so the challenge is to use pink, green and a neutral in your card."
I used Ornament Lala and the Sweet Sunday Sketch.
I stamped Lala and colored her with Copics. For once I didn't cut her out and pop her up with foam boosters (to save some time). I used some old Daisy Bucket papers, a few buttons, ribbon, and a rub-on from Making Memories.
Christmas Michu Pile It On!
I printed Christmas Michu (who can be found in the Softpencil Etsy shop), colored him with Copics, then cut him out and popped him up on foam boosters. I also flocked his hat and coat trim for some extra texture.
For the snowy ground under him, I drew a line with a black liner pen, and then glued on some homemade Flower Soft, that I like to refer to as "Holly Soft" (you can find the tutorial on Holly's blog here), mixed with a bit of iridescent glitter.
Quixotic Challenges - "Be Charming"
Little Red Wagon - "Heavy Metal"
Stamp Something - "Glitzy Somethings"
We'd love to have you play along with us! All the details can be found on the Pile It On! blog. Also, if you'd like to play with a holiday Michu image, Elizabeth has an adorable freebie posted on the Paper Craft Planet group! TFL!
If you're looking for the Your Next Stamp Challenge #5, it's the post directly below this one. Thanks!
Wednesday means it's time for another La-La's Weekly Wednesday Challenge! This week our theme was set by our DT Coordinator Jane, and is Buttons! I love adding buttons to my cards so this was a great challenge for me!
I used Birthday Lala and gave her a haircut again, like I did with this card because I really love the short sassy 'do! I also used a really fun sketch from Sketchy Thursday and some super old papers from a company that's since gone out of business.
Lala was stamped, colored with Copics, cut out and popped up on foam boosters. I did a whole bunch of sewing and added a couple Prima flowers, and of course, buttons! Lastly I stamped a sentiment from Papertrey Ink on a thin strip of cardstock and then snipped down to almost the bottom to make the strip able to follow the curve of the circle (I got the idea from this post by Debbie Olson).
I used Cupcake Cutie Fhiona and a great sketch from PaperPlay Challenges. I also finally used some of this super adorable Doodlebug paper that I've been hoarding for months!
I stamped Fhiona and colored her with Copics. Then I cut her out using a circle Nestabilities die and because I felt the whole thing was too white, I sponged the edges of the circle. Then I sponged the patterned papers to match. I added a couple heart brads, heart buttons, machine stitching, ribbon, and a sentiment from Papertrey Ink.
We hope you'll play along with us! One lucky winner will be drawn at random to win the Shopper Fhiona set!! You don't have to use YNS images on your card, but if you do you'll get a double entry in the draw! Also, we'll have a Top Pick winner (not a random draw) from all the entries that will get a $5 coupon for the YNS store! And last, but certainly not least, if you can't play along, leave a comment anyway because one commenter will be drawn at random to win a $5 coupon to the YNS store! Full details can be found at the YNS Challenge Blog. TFL!