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## Introduction

Alzheimer’s disease (AD) is considered to be common cause of dementia worldwide1. Over time, people with Alzheimer’s disease suffer memory loss as well as the ability to concentrate. In advanced stages of the disease, there are severe complications such as dehydration, malnutrition or infection, which eventually result in death2. Since its first description in the early twentieth century, there is still no treatment that cures Alzheimer’s disease or alters the disease process in the brain. However, the course of Alzheimer’s can be positively influenced by a number of different medications and non-drug treatments, making it crucial for people with Alzheimer’s to receive early good care and support.

Even associated with forgetfulness, AD affects different aspects of individual’s personality, life experiences, current circumstances and wrong responses to the situations they suddenly find themselves in3, and relationships with other people as is the verbal communication. The gradual nature of affecting the short-term memory at first and the long-term memory at later stages makes the maintenance of orientation in time and space inevitably difficult. This is also visible through the work of an artist that presents a time-series of self-portraits of his Alzheimer’s disease progresses in time4 and the original work clearly shows the cognitive decline and spatial disorientation, however, the emotion still highly remains in each of the original works.

It is not yet possible to diagnose Alzheimer’s with complete certainty using the currently available tests while the person is still alive. The disease is diagnosed if someone has the typical symptoms eliminating all the other possible causes. Since symptoms like forgetfulness, changes in behavior and problems with orientation might have many different causes, it is important not to rush to a diagnosis of Alzheimer’s. The symptoms might also be caused by depression or other physical conditions like meningitis, a stroke or bleeding in the brain5. Conducting an effective clinical trial is crucial to accurately predict the change in AD’s indicators so that the effect of the treatment can be assessed. The most common approaches are:

• Manual prediction by a clinical expert by using the clinical history of patients with similar conditions and visual analysis of various brain scans.

• Regression analysis to predict the future indicator changes in patient status, based on data from MRI6, cognitive test scores7, rate of cognitive decline8, and also retrospectively staging subjects by time to conversion between diagnoses9.

• Supervised Machine Learning (ML) has already shown to be effective in discrimination between AD patients from cognitively normal subjects by using MR images10, variety of biomarkers11, etc.

• Data-driven disease progression models are most recently used to predict AD in unsupervised manner. Examples include models built on a set of biomarkers to produce discrete12,13 or continuous14,15 pictures of disease progression. Also there are less comprehensive models that leverage structure in data such as MR images16,17.

Even though it is still unclear what is the main cause of the disease, it has been shown that people with Alzheimer’s do not have enough of an important chemical messenger called acetylcholine in their brain18. And it has also been shown that small protein particles (for example plaques) build up in their brain. These might cause the nerve cells to die19.

In20 several factors have been hypothesized to play a role in the Alzheimer’s disease occurrence, and those are:

• Age–starting at about age of 65, the probability of getting AD doubles every 5 years21,22.

• apolipoprotein E4 (APOE E4)–10 to 30 times higher of developing AD compared to non-carriers, i.e., subjects without the gene. However the exact mechanism through which the presence of APOE E4 leads to AD is not known23.

• Gender–women seem more likely to develop AD than men. The reasons for this are still unclear24.

• Medical conditions–type 2 diabetes, high blood pressure, high cholesterol, obesity25, or depression26 are known to increase the risk of developing dementia.

• Lifestyle factors–physical inactivity27, smoking28, unhealthy diet29, excessive alcohol30, or head injuries31.

Newest clinical researches provide contemporary view about differentiating clinically diagnosed AD dementia from other neurodegenerative disorders using plasma P-tau21732, improvements in neuropathological diagnosing the disease33 and possible approach for a drug development against its progression34.

Considering the complexity of Alzheimer’s disease and the fact that multiple factors under various circumstances affect the onset of it, it is not sufficient to do simple machine learning experiments and aim for the best metrics. Using the interpretability of the ML model, provided by various explainable machine learning techniques, can significantly help obtaining a bigger picture about risk factors influences on the particular diagnosis. The interpretability shows interesting yet still not proven trends that are present in the used dataset.

Multiple recent papers using interpretability techniques have provided compelling results and guidelines35 for further medical expertise, including regular36 and multi layer multi modal37 interpretability of the Alzheimer’s disease, interpretability of ensemble learning algorithms for predicting dementia38 and extracting explainable assessments from MRI imaging scans39.

### Hypothesis

This research is focused on deep investigation of some of the factors that are claimed to play an important role for the occurrence and further development of Alzheimer’s disease by following explainable Machine Learning (ML) approach. The research puts the following hypothesis at test:

• Hypothesis 1: There is gender predisposition for obtaining Alzheimer’s disease.

• Hypothesis 2: APOE4 gene is crucial decisive factor for Alzheimer’s disease diagnosis.

• Hypothesis 3: Older people are more prone to Alzheimer’s disease.

• Hypothesis 4: Cognitive tests distinguish between all stages leading to Alzheimer’s disease.

To test the established hypothesis, a data set encompassing 12,741 subjects medically confirmed to belong into five categories - cognitively normal (CN), Early Mild Cognitive Impairment (EMCI), Late Mild Cognitive Impairment (LMCI), Significant Memory Concern (SMC) or Alzheimer’s disease (AD), will be used to develop intelligent model able to classify the patients with high precision and recall with the aim to further interpret the model by the explainable ML methods.

The paper is organized as follows. The data set is fully described in “Materials and methods” section. In the same “Materials and methods” section there is a comprehensive description of the data preprocessing as well as the ML approach applied to the data set. The intelligent model’s results are provided in “Results” section. The model is interpreted by means of explainable ML in “Explainable machine learning” section upon which a discussion showing the importance of the research is provided in “Discussion” section. “Conclusion” section presents the final conclusion over the established hypotheses.

## Materials and methods

Figure 1 presents the complete methodology as a roadmap followed to analyse and derive valuable conclusions from the data. Six different phases can be distinguished encompassing methods as follows:

• Phase 1: Data set analysis. In this step the subjects and all the features available in the database are analysed in terms of their importance for the problem at hand.

• Phase 2: Data set preprocessing. This phase is focused on methods for incomplete data, redundant features, encoding, correlation analysis, missing values, and imbalanced targets handling, from which as a result a data set prepared for ML will be obtained.

• Phase 3: Training ML model. At the ML phase, XGBoost model has been trained, and also hyperparameters tuning has been performed.

• Phase 4: Interpretability. This is the most important phase considering the contribution of the research, since it allows the global and local interpretability of the trained model by using Shapley values and thus, provides deep insight into the features influence on the prediction of each of the classes.

• Phase 5: Valuable conclusions. Based on the previous phase, valuable conclusions are derived at this phase followed by comprehensive discussion.

• Phase 6: Hypothesis testing. At the last phase, the research is concluded by reconsidering the established hypothesis based on the results from Phase 5.

### Data set analysis

The data set used in this research is used in the TADPOLE Challenge (The TADPOLE Challenge compares the performance of algorithms at predicting the future evolution of individuals at risk of Alzheimer’s disease.) and retrieved from ADNI (Alzheimer’s Disease Neuroimaging Initiative) available at their official website (http://adni.loni.usc.edu/). In order to obtain permission for data usage, a request has been sent together with an explanation for the purpose of the research.

The original data set contains data from 12,741 subjects and 1,907 attributes. Mainly the attributes arise from two categories: quantitative parameters and personal information. As provided in the data set description, the data is labeled by medical experts.

Considering the sparse data set problem and according to the suggestions provided in the data set source40, 17 features have been chosen to be informative for most of the patients covering personal information, gene expression analysis, medical information and cognitive tests information. Those feature are considered to carry information needed to test the hypothesis established in “Hypothesis” section.

Each row represents data for one subject, and each column represents a feature or measurement related to the subject. The features can be divided into six categories, encompassing:

• Personal information:

• PTID: Participant ID

• AGE: Age at baseline

• PTGENDER: Sex

• PTEDUCAT: Years of education

• PTRACCAT: Race

• Gene expression:

• APOE4: Expression of the ApoE4 gene

• PET measures:

• FDG: measure cell metabolism, where cells affected by AD show reduced metabolism.

• AV45: measures amyloid-beta load in the brain, where amyloid-beta is a protein that mis-folds (i.e. its 3D structure is not properly constructed), which then leads to AD.

• MRI measures:

• Hippocampus: scan of a complex brain structure embedded deep into temporal lobe.

• WholeBrain: scan of the subject’s whole brain.

• Entorhinal: scan of an area of the brain that is located in the medial temporal lobe and functions as a hub in a widespread network for memory, navigation and the perception of time.

• MidTemp: scan of the middle temporal artery.

• Cognitive tests:

• CDRSB: Clinical Dementia Rating Scale - Sum of Boxes.

• ADAS11: Alzheimer’s Disease Assessment Scale 11.

• MMSE: Mini-Mental State Examination.

• RAVLT_immediate: Rey Auditory Verbal Learning Test (sum of scores from 5 first trials).

• Target:

• DX_bl: Subject’s diagnosis, i.e., the target variable of which we want to gain a deeper understanding. We built models whose goal is to predict the value of this variable based on the values of other features. The target variable can result in any of the following five values: CN (Cognitive Normal), EMCI (Early Mild Cognitive Impairment), LMCI (Late Mild Cognitive Impairment), SMC (Significant Memory Concern) and AD (Alzheimer’s Disease).

Those features are measured by techniques that are able to assess some indicators of whether the individual might be at risk of development, or, has already developed AD symptoms.

The cognitive tests allow the examiner to obtain an overall sense of whether a person is aware of the symptoms, the surrounding environment, whether he/she can remember a short list of words, follow some instructions and do simple calculations. Cognitive tests are able to measure cognitive decline in a direct and quantifiable manner. However, the cognitive decline is one of the latest to become abnormal. This is because the first abnormalities are first noticed at a microscopical scale through the misfolding of a protein called Amyloid beta. These are followed by changes at larger scales, such as loss of the neurons myelin sheath, neuron death, visible atrophy in MRI scans and finally cognitive decline.

Cognitive tests, however, have several limitations that affect their reliability and those are related to remembering them if taken several times, might have floor or ceiling effects, which means that many subjects might score the highest/lowest score possible, and can be biased, as they are undertaken by a human expert who might be influenced by prior knowledge of the subject’s cognitive abilities41.

Magnetic resonance imaging (MRI) is a technique used to quantify by measuring the volume of gray matter - GM (consisted of nerve cells) and white matter - WM (fibres connecting the nerve cells). Atrophy is indicated by the loss of volume between one scan and other follow-up scan. It is caused by the death of neurons in regions affected. Quantification of atrophy with MRI is a very important parameter as it is widely available and non-invasive good indicator of progression of MCI to dementia42,43,44.

The Positron Emission Tomography (PET) enables researchers to track the concentration of abnormal proteins (amyloid and tau) since the contrast agent (containing the tracer) spreads throughout the brain and binds to abnormal proteins. PET scans can be of several types, depending on the cellular and molecular processes that are being measured. Fluorodeoxyglucose (FDG) PET can be used to measure cell metabolism. Neurons that are about to die show reduced metabolism, so FDG PET is an indicator of neurodegeneration. AV45 PET is used to measure the levels of abnormal proteins such as amyloid-beta.

The errors in Amyloid-beta 3D structure (misfolding) is thought to be one of the causes of AD since its high levels lead to neurodegeneration and cognitive decline. The basic limitation of PET scans is that the patient is exposed to ionizing radiation, which limits the number of scans they can take in a specific time interval42,45.

The APOE gene provides instructions for producing a protein called apolipoprotein E, and that is why the gene expression measurement is an important technique by which the activity of APOE gene can be quantified. This protein aids the formation of lipoproteins by combining with fats (lipids) in the body. There are at least three slightly different versions (alleles) of the APOE gene. The major alleles are e2, e3, and e4. The e4 version of the APOE gene is believed to increase an individual’s risk for developing AD. People who inherit one copy of the APOE e4 allele have an increased chance of developing the disease, those with two copies of the allele are at even greater risk. In TADPOLE data set, an information about individual’s number of present e4 alleles (0, 1 or 2) is available. However, not necessarily the individuals with AD have the APOE e4 allele, and also not all individuals who have this allele will develop AD46.

### Data set preprocessing

#### Incomplete data

Figure 2 shows that all subjects contain personal information and diagnosis, but not all of them have data for all the other parameters. All subjects with incomplete data have been detected since incomplete data can cause a lot of troubles in the process of data analysis and building the intelligent model later described in this section. Since the removal of all subjects where data for at least one attribute is missing causes significant information loss, an attempt is be made to use the leverage of some imputation techniques.

For example, only 2,118 subjects have data for the AV45 attribute. Removing all subjects (rows) that contain any missing data from our data set results into new data set with 1,121 entries.

Therefore, it is decided to remove all subjects that miss data for more than 6 features, and then perform data augmentation by using the imputation techniques later described in this section. The initial data removal resulted in 9,592 subjects left for further analysis.

#### Redundant features

The redundancy analysis of the features is crucial step for appropriate experimental setup. Taking a closer look at the PTRACCAT column in Table 1, it can be perceived that almost 93% entries have value ’white’. It means that this attribute does not provide enough information about possible racial predisposition for the disease. Most of the subjects belong to same racial group and only few belong to other groups, so this feature is be excluded from further analysis. Also PTID attribute is omitted because it is an identification number for each subject and has no meaning for the models we intend to build.

#### Categorical features encoding

The attributes PTGENDER (Male/Female) and DX_bl (CN, EMCI, LMCI, SMC, and AD) are of categorical data type. Simply encoding the attribute ’Male’ with the value 1 while ’Female’ with 0, would lead to increase the weight of Male compared to that of Female. This does not make sense since both variables need to be treated equally by the model to predict accurate results. To achieve equality one-hot encoding is used for encoding the PTGENDER attribute. This encoding is appropriate for categorical data where no relationship exists between categories. It involves representing each categorical variable with a binary vector that has one element for each unique label and marking the class label with a 1 and all other elements 0.

Considering the target attribute DX_bl, it can be seen that there is a certain order related to its values. The values can be ranked from CN to AD, based on the subject’s neuropsychological disorder. Therefore, Label encoding has been used to simply convert labels to integer values in ascending order.

#### Correlation analysis

Assuming correlation between two features, it means that one of them does not contribute to better representation of the information for the model to be learned, thus it can be omitted. Heat map representation as shown on Fig. 3 has been used to represent the linear correlations between the features in the data set used for this research.

The map provides a strong negative correlation of -1 between PTGENDER_Male and PTGENDER_Female. This coefficient does not provide anything relevant since both features are discrete and represent same category of data, so it will be ignored.

On the other hand, a trend of pretty high coefficients can be noticed between ADAS11 and other cognitive tests results. In fact, highest negative coefficient is between ADAS11 and MMSE (-0.76) and highest positive coefficient is between ADAS11 and CDRSB (0.73). It is possible that this feature does not provide any new information. It seems like it contains repetitive information from other tests. To determine the correlation, the coefficient alone is not sufficient. Additionally, graph-based representation (Fig. 4) is used between the two features to better understand the dependency.

From the plot a kind of linear dependency between these two features can be perceived. Plots of ADAS11 and CDRSB / RAVLT_immediate show something similar too. Considering the analysis, we assume that ADAS11 does not provide any new information, thus this feature is redundant and can be omitted from the final data set.

#### Missing values imputation

Discarding entire rows that contain missing values comes at the price of losing data which may be valuable, even though it is incomplete. In order to obtain the maximum information that the data set is providing, a missing values imputation is performed. Each imputation algorithm uses some kind of estimation to obtain missing values, but that does not mean that the value is 100% estimated correctly. This can cause noise and bias problems in the data set and that is why it should be used with caution.

One type of imputation algorithm is univariate, which imputes values in the $$i-th$$ feature dimension using only non-missing values in that feature dimension. Missing values can be imputed with a provided constant value, or using the statistics (mean, median or most frequent) of the particular column. This technique is also referred as simple imputing. By contrast, multivariate imputation algorithms use the entire set of available feature dimensions to estimate the missing values. These algorithms model each feature with missing values as a function of other features, and use that estimate for imputation. This technique is also referred as iterative imputing.

The goal is to compare different estimators to see which one is most effective for the data set. First, a subset of all rows with non missing value was extracted and its score was estimated. After that, a single value was randomly removed from each row and after iterative imputation of the missing values using different estimators, a score was estimated for each one of them.

#### Imbalanced targets distribution

Table 2 presents the targets distribution before and after elimination of subjects with lack of data, i.e. the distribution of the five different classes considering the data set in different stages.

It can be noticed that there is a huge disproportion between the class with most values (LMCI) and class with least values (SMC). The aim is to create approximate uniform distribution of targets, such that each class will have similar number of instances. An undersampling of LMCI targets will be performed, combined with oversampling of other four classes.

Because the sampling processes mix original samples and artificially created ones, in order to be sure that the data set is not biased, first the data set is split into training (70%) and testing (30%) subsets and then separate sampling is performed into each one of them. Couple of different oversampling algorithms were tested combined with Random undersampling. The only exception is SMOTETomek algorithm which already combines methods for both oversampling and undersampling.

The pie charts in Fig. 5 represent the number of targets divided by classes in both training and testing subsets after the sampling techniques were applied. It can be seen that they are now almost uniformly distributed.

### Machine Learning

Choosing the most optimal algorithm for solving the problem at hand depends on many factors like: size of the training data, training time, linearity, and number of features. The data set used is medium sized with average number of features allowing to experiment with more complex algorithms. Hence, the classifier built in this research uses XGBoost algorithm. Proven to show several advantages above other classification algorithms47, XGBoost requires less feature engineering, meaning there is no need for scaling and normalizing data. It is less prone to overfitting if the hyperparameters are tuned properly. For comparison purposes only, a Random Forest model was also built. To validate the trained classifiers properly, 5-fold cross-validation is performed on the training set.

One of the crucial steps in building ML model is tuning its hyperparameters - the arguments that can be set before training and which define how the training is done. These parameters are tunable and can directly affect how well a model trains. Thus, in order to achieve maximal performance, it is important to understand how to optimize them. To find the best combination of values for hyperparameters for both the Random Forest and XGBoost model, a range of values for every parameter has been defined and then Grid Search has been used which evaluates all combinations and chooses the best one.

### Model evaluation

Different performance metrics are used to evaluate the intelligent model built as described in the previous section. Balanced accuracy is considered a good measure for this research since it is reliable metric even when the distribution of target variable classes in the data is not very balanced. Precision is also considered informative measure since it tells the proportion of subjects diagnosed with one of the diagnosis (predicted class), that actually had that diagnosis (real class). Recall is used to measure the proportion of subjects that actually had particular diagnosis (real class) was diagnosed by the model to have that diagnosis (predicted class). Specificity is also used to measure the model’s ability to correctly generate a negative result for subjects who do not have the condition that is being tested. A high-specificity model will correctly rule out almost everyone who does not have the disease and will not generate many false-positive results. A model with high sensitivity but low specificity, results in many individuals who are disease free being told of the possibility that they have the disease, and are then subject to further investigation. The last metric used is F1-score, presenting the harmonic mean of both the precision and recall, and thus is considered to be very powerful for the problem at hand.

### Explainable machine learning

In the context of ML systems, interpretability is the ability to explain the model’s output. When a model is built, one need to be able to understand how it is making the predictions. Feature importance helps in estimating how much each feature of the data contributed to the model’s prediction. In this research the feature importance is considered in terms of Shapley values by using the SHAP (SHapely Addictive exPlanations) method48,49.

SHAP provides two aspects of model’s interpretability:

1. 1

The first one is global interpretability-the collective SHAP values can show how much each predictor contributes, either positively or negatively, to the target variable. The summary variables importance plot shows the average impact each feature has for predicting each diagnosis, regardless if it is positive or negative. On regular variable importance plots, subjects are shown as colored dots. For each diagnosis there is a separate plot and for each feature, dots are arranged depending on the impact the value for that feature had on the subject to be predicted with the chosen diagnosis. The color of the dot indicates the value of the feature. For the dependence plots, dots are arranged on the x-axis by the value of the main chosen feature and on the y-axis by the positive/negative impact that value had for predicting the particular diagnosis. The color of a dot indicates the subject’s value for the second feature that the first one is in strongest interaction with.

2. 2

The second aspect is local interpretability-each observation gets its own set of SHAP values. This greatly increases its transparency. These plots are subject specific. On the plots, blue arrows represent features that are increasing the predicting probability of a particular class (pushing it to the left), while red arrows are features that are decreasing the probability of a diagnosis to be predicted (pushing it to the right). Arrow’s length indicates how high the value of the feature is.

### Ethics approval and consent to participate

As per ADNI protocols, all procedures performed in studies involving human participants were in accordance with the ethical standards of the institutional and/or national research committee and with the 1964 Helsinki declaration and its later amendments or comparable ethical standards. More details can be found at adni.loni.usc.edu. This article does not contain any studies with human participants nor animals performed by any of the authors.

### Consent for publication

Authors proclaim that all terms of the data use agreement are accepted and included in the manuscript. The manuscript has been sent to ADNI Data and Publication Committee and it has been approved for publishing. Authors acknowledge that all images are entirely unidentifiable and there are no details on individuals reported within the manuscript.

## Results

### Data preprocessing

Considering the data imputation methods applied, as it can be seen from the bar chart presented in Fig. 6, Extra Trees Regressor and Bayesian Ridge estimated values are closest to the original data. Both of them are multivariate algorithms. On the other hand, univariate algorithms using mean and median failed to do the estimation very accurately, which is somewhat expected considering the fact that the data set contains features represented by sensitive data values where simple average does not solve the problem.

In addition, k-Nearest Neighbors imputation approach was performed. By default, a euclidean distance metric that supports missing values is used to find the nearest neighbors. Each missing feature is imputed using values from k-nearest neighbors that have a value for the feature. Although this type of imputation showed better results than the simple imputation algorithms, it did not surpass the iterative imputation using Extra Trees Regressor.

Considering the results from the comparison, the Extra Trees Regressor was chosen to impute the missing values in the original data set. After that, several such regressors with different number of estimators were tested on the original data set. Experiments have shown that Extra Trees Regressor with 100 estimators most effectively approximates the missing values.

Evaluation results suggest that the SMOTE oversampling algorithm provides the best results as shown in Fig. 7, although the difference from one to another is very minor. Another area to explore would be to test different values of the k-nearest neighbors selected in the SMOTE procedure when each new synthetic sample is created. The default is k=5, although larger or smaller values will influence the types of samples created, and in turn, may impact the performance of the model. The repeated and stratified k-fold evaluation showed that the oversampling process is most successful when the number of k-nearest neighbors is equal to 6 as depicted in Fig. 8.

Completing this step, the data preprocessing phase is finished and the data set is ready to serve as a source for building the intelligent models. The resulting data set after the preprocessing consists of 13,235 subjects.

### XGBoost model evaluation

The confusion matrix for the XGBoost model is presented on Fig. 9. Majority of subjects lay on the main diagonal. One of the model’s anomalies is the slightly increased number of AD patients predicted with a LMCI diagnosis. The same problem occurs with LMCI patients that are predicted with an AD diagnosis. This tells that perhaps there are some features that bring confusion between these two diagnoses. In addition, there is an increased trend into false negatives of the SMC class, but considering the fact that serious amount of the SMC subjects were created artificially, this phenomenon is somewhat understandable.

Table 3 shows complete evaluation of XGBoost performance, divided by target classes. XGBoost model managed to achieve balanced accuracy of 0.84. Other metrics are also very satisfactory, deviating from 0.8 to 1.0. Table 4 shows comparison between this model’s metrics and results obtained using Random Forest model. XGBoost managed to guess more subjects with their correct diagnosis, thus it represents more accurate model.

### XGBoost model interpretability

#### Global interpretability

The following Fig. 10 presents how high is each feature influence (variables importance) on each group of targets in average. Analysing the plot, it can be noticed that CDRSB leads by far the most in the impact on this model’s output. All features that describe cognitive tests results are generally ranked on the top. Another thing worth mentioning is the position of FDG and APOE4. Since PTGENDER_Male and PTGENDER_Female are highly correlated and basically represent one feature, impact values of PTGENDER_Female are omitted here. Note that the gender and APOE4 features are at the very bottom, which means their values influence the outcome least. This information tells us that probably there is not a gender predisposition for obtaining Alzheimer’s disease. Many studies50,51 say that e4 alleles of the APOE gene can be a predisposition for dementia, but not necessarily. From this result, it can be confirmed that the APOE gene does not act as a decisive factor for having a diagnosis.

Looking at the color distributions at some features, it can be seen that MMSE value impacts most on the CN subjects. On the other hand, subject’s age has most influence for LMCI class. For the AD class, the impact of the gender feature is insignificant. If we take a closer look on the green parts of each feature, we can notice that cognitive tests have highest impact on the CN class, while other features impact the output for this class in smaller amount. This analysis is done on a basis of complete training dataset.

Of significant importance is to prove robustness of the present conclusions, following the idea that the order of features importance is not linked only to the particular observation. To accomplish this, the training dataset was split using stratified 5-folds cross-validation technique. For each split, a new model with the same properties as the original one was built and trained on 4/5 folds of data. For each of these models, SHAP values of their training subsets were obtained indicating their most important features in descending order. Additionally, most important features based on the original testing dataset were obtained from the original model. The following approach provides information in which manner models using different datasets have learnt and ranked features importance compared to the ranking of features importance based on subset of data not recognizable for the model. Top 6 most influential features were selected for each model and using a Venn diagram visualization, their overlaps and intersections are presented in Fig. 11. Circles for XGBoost 1 to XGBoost 5 represent features importance originating from 5-fold cross-validation while XGBoost Test circle represents features obtained from the original testing dataset. On the diagram, it can be noticed that 5 out of 6 top features overlap in all six models. The only mismatch is occurring at model XGBoost 5 where instead of FDG, the feature Hippocampus is included. But, if we take the top 7 features, this will be over-passed since the 7th feature in the list for XGBoost 5 is indeed FDG. Observing the bigger picture, differences between a feature ranking in all models are happening only for a place or two. There are no large mismatches between the rankings, indicating that the global interpretability presented before is pretty stable and robust even for models using to some extent different datasets. In addition, the robustness of the interpretability method is extended over different tree-based algorithms too, indicating the independence between the selected algorithm and features influence on predicting the target variable. The discussion in Appendix A.2.1 provides evidence on obtaining the same top features for the random forest model in comparison to the XGBoost.

Considering the global interpretability, we can also take a closer look of whether one feature has positive or negative impact on the output and how high is it really, by looking at the summary plot for each diagnosis. The plot is made of all the dots in the train data. Figures 12 and 13 depict the summary plots of variables importance for CN and LMCI class diagnosis.

On the summary plot of CN class, we can notice that low values of CDRSB have very high positive impact determining this diagnosis. Additionally, high values of MMSE and RAVLT_immediate also tend to have high positive impact. While other MRI measures do not give useful information on how they affect the model’s output, the WholeBrain feature have serious negative impact for higher values and vice versa. The same statement applies to Hippocampus too, but not to that extent.

Contrary to already mentioned conclusions, it can be noticed that younger subjects tend to decline from this diagnosis. Although the natural way of thinking is that neurological diseases affect only older people, that is not the truth. In addition, from TADPOLE data set it can be concluded that women are more inclined to neurodegeneration.

For the LMCI class, shown in Fig. 13, the situation is slightly different, as expected. Low AGE values are distributed on the positive axis now, denoting positive impact for this class. It can be seen that lower values for CDRSB have highly negative impact on this class, high values do not have any impact at all, while mid values have positive impact. It is very likely that high values are going to be distributed on the positive axis for the AD class. Here, high values of RAVLT_immediate have negative impact, which is somewhat expected. Other features (Hippocampus, FDG and MidTemp) have similar types of distributions denoting negative impact on model’s output for higher feature’s values.

Figure 14 shows that high CDRSB values indeed represent high positive impact for AD class. In opposite, higher MMSE values have significant negative impact. Looking at the PTEDUCAT feature, it can be seen that higher education has valuable negative impact on the Alzheimer’s diagnosis. Also, the APOE4 distribution indicates that subjects with zero e4 alleles have less chance of being labeled with AD than those with one or two. It can also be confirmed that subjects with lower values of FDG tend to be diagnosed with this diagnosis.

Even more information can be uncovered examining the partial dependence plot of one feature. This plot shows the marginal effect two features have on the predicted outcome. Once the first feature is chosen, the second is automatically selected depending with which one the first interacts most. These plots are also class specific.

Most of the plots show more complex correlation, or no correlation at all, but those that are most relevant will be examined.

First, the marginal effect FDG and CDRSB have on the predicted outcome is analysed for three different classes (EMCI, LMCI, AD) on Fig. 15. On all three plots we can notice the correlation between FDG and CDRSB, as subjects with lower FDG tend to have higher CDRSB results. The plot color changes progressively from yellow to blue, as we move on the x-axis from 0.6 to 1.8. But what fascinates the most is how the positive correlation that they have with the outcome gradually evolves into negative one, as the disease stages move from cognitive normal to Alzheimer’s disease.

In Fig. 16, it can be seen that younger subjects with two alleles e4 of APOE gene have bigger chances of having an Alzheimer’s disease. This represents intriguing information about the genetic predisposition of the disease, since we can see that APOE4 and AGE are correlated somehow. In addition, high values for AGE and zero e4 alleles have negative impact on this class. Contrary, one e4 allele and higher age values have a positive impact.

Last but not least, there is pretty complex correlation between RAVLT_immediate and MidTemp and they also influence the output in non-straightforward way, thus a couple of statements can be made from Fig. 17. Diving deeper into the plot, it can be seen that subjects with lower values for RAVLT_immediate tend to have lower values for MidTemp too and they have mild positive influence for LMCI class. On the other hand, higher values for the cognitive test correlated with mid to high MidTemp values tend to have negative impact.

#### Local interpretability

By isolating a single subject, it can be explained graphically how features influence the subject to become labeled with particular class. One subject belongs to all classes with different probabilities and is labeled with the class with highest probability. Let’s consider a subject that was correctly predicted with AD diagnosis as shown in Fig. 18. The graph presents an output value or f(x) which is prediction probability for the particular observation, and base value or E[f(x)] which is the value that would be predicted if any features are not known for the current output (mean prediction). Observing the waterfall plot in Fig. 18 for the correctly predicted class and the LMCI class in Fig. 19, the following conclusions can be derived.

As it is expected for the AD class, almost all features have positive impact, except the MidTemp with significant weak negative impact. The prediction is straightforward with pretty high probability. On the other side, the prediction for the LMCI class for the same subject have pretty mixed up feature impacts. We can see that some of the features influence positively in the favor of LMCI class, but other features have higher negative impacts together, thus the subject’s probability to belong to this class is pretty low. In both cases, the CDRSB impact is crucial.

This interpretation can be used to analyse the trends of false predicted subjects mentioned before. The features influence is examined on the predicted and true class of randomly selected subject belonging to the particular trend. For each of the Figs. 20, 21, 22, and 23 top plot represents the predicted class, while the bottom plot is for the true class.

On Fig. 20, a subject with LMCI diagnosis, predicted as AD is shown. It can be noticed that the difference between probability for the predicted class and the true one is slightly big. Besides CDRSB, in this case MMSE and AGE play significant role in increasing one’s probability and decreasing the other. It seems like most of the patient’s medical measurements indicate an AD instead of LMCI.

Figure 21 shows individual plots of a subject with SMC diagnosis, predicted as EMCI. Even bigger probability difference occurs here. On the first plot, almost all features influence positively in favor of EMCI diagnosis. The problem with many false predictions correlated to the SMC class may be because of the initially deficient subset of targets we were dealing with at the beginning. The low CDRSB value influences the most against the true class and in favor of the predicted one, which was indeed proven (higher CDRSB values tend to correlate with more devastating diagnoses).

In the next two Figs. 22 and 23, plots of a subject with SMC diagnosis, predicted as CN and subject with AD diagnosis, predicted as LMCI are shown. The probability difference in both cases is slightly smaller than before, but it is still pretty high. In the first case, CDRSB has positive impact for both classes, but for the predicted one it is slightly higher. MidTemp and AGE are playing the main role here in deciding the final output. In the last Fig. 23, MRI measurements have greatest impact on the subject. It can be seen that they have serious positive impact for the predicted class, in the same time decreasing the probability for the true class.

## Discussion

The results are summarized in the Table 4. For summarizing the categorical metrics, weighted average representation has been used. Although the main goal in this research was not achieving highest performances, it still can be confirmed that comparing to a Random forest model, the XGBoost algorithm performs better classification of subjects and better distinguishes between different types of neurodegenerative diagnoses, indicating that it is more reliable and accurate than his opponent in this research. Detailed explanation of Random forest model evaluation and interpretability can be found under Appendix A.

The global interpretability of the intelligent model provided exceptional conclusions that were deeply explained in the previous section. However, not all of the conclusions were comprehensively discussed and thus, the features influence on predicting a particular class has been summarized in Table 5. The table provides detailed scheme of the positive (+), negative (-), and undefined (X) influence the particular value of the feature has on predicting the particular class. This scheme can be used as additional knowledge for the physicians and other related experts when they are making conclusions over the diagnosis for particular patient.

Considering the table of influences for the XGBoost model, it can be once again confirmed that XGBoost provides high interpretability of the problem, since there is very small amount of undefined influence statuses. Table 6 represents a merged table between the XGBoost’s and Random forest’s scheme. Luckily, there is a low level of inconsistency between the unveiled influence provided by both of the models. It is very rare that one model is contradictory to the other, e.g. the high education positive or negative influence on the SMC class is suspicious, and also APOE4 2 influence on the EMCI class is a matter of question whether it is positive or negative. The merged table provides achievement of a better understanding, unveiling the contribution to the undefined influence in each of them by masking two contradictory influence as undefined. Features influence for the Random forest model can be found in Supplementary Table S.2.

Table 4 also shows the times required for the models to be trained and to predict the entire test set. The XGBoost classifier requires more time to be trained since it uses gradient boosting algorithms in background, but its predictions are slightly faster than the ones of Random Forest classifier. Benefits obtained in terms of model’s exactness and validity for the time difference are more than worthy.

It can be concluded that XGBoost is proven to be an optimal algorithm for dealing with Alzheimer’s prediction problem using the particular data set. Table 7 presents a comparison of our XGBoost model with other models trained on the same original TADPOLE data set52. As shown it can be perceived that our model is ranked second, and if a performance time is included as trade-off, then our model would be on the top of the list.

Considering the popular AD research, mostly considered those published in the latest years, Table 8 presents an interesting review not only of the obtained results and the data set used, but also on the experimental setup performed in each of the reported experiments. High accuracy metrics are reported, however most of them show no clear evidence on appropriate methods for data preprocessing, missing data imputation, hyperparameters tuning, appropriate split of train and testing set, and the data sets used are limited to at most nearly two thousand patients. The most remarkable research among the presented is the one published in53 in which very comprehensive mathematically supported approach is presented with special attention on avoiding over/underfitting problems.

Although findings and results obtained in scientific papers mentioned above cannot be directly compared with this research because of differences in approaches and data being used, still a general comparison can be provided. It can be noticed that instead of using data sets containing quantified values of different measurements (that is an example of our data set), some researchers use image processing of MRI scans10,54 and spectrograms of patients’ voices55. In each research paper patients are labeled with a particular diagnosis. Although in some, patients are only distinguished between two different clinical phases10,55,56,57. Comparing the obtained metrics from the prediction, it can be seen that most of the researches that use ADNI datasets managed to achieve impressive accuracy of their models, even greater than 0.9. The main goal in all these researches is to accomplishing the highest possible metrics in order to create a model that will predict a particular diagnosis most accurately and with greatest precision. Yet, it still remains unclear in which manner the model managed to achieve such results (i.e., the model is treated as a black-box without providing explanations about the contribution and influence of each feature to the end prediction). In this research, model’s metrics serve as a validation for the provided in-depth analysis and interpretability. Using explainable machine learning methods, a bigger picture about features influence and their correlation is being presented. The main goal in this research is not to create the best model (i.e., we are focusing only on a model with a good accuracy that can be used for providing interpretations), but rather to dive deeper into the importance and influence of each clinical measurement on a particular diagnosis.

Besides the satisfying metrics obtained from the model’s evaluation, clinical insights are also of a significant importance to validate the attained ranking of the features. The ranking follows the same manner as the disease symptoms progress from micro (based on cell metabolism and gene expression) to macro level (based on losing brain mass and cognitive decline)58. Gender and gene expressions are considered as predispositions, thus they do not represent strong and direct indications for the disease. On the other hand, results from cognitive tests provide clear and efficient understanding of the patient’s mental condition and clarify the exactness of it. The difference in importance between different types of scanning yet needs to be considered and validated from clinical experts.

## Conclusion

The main objective of the study presented is to put at test the existing hypothesis regarding the causes and indicators of Alzheimer’s disease. At the beginning of the research, four hypothesis were established considering the existing published literature. Hereafter, a large data set was obtained containing various types of features considering the lifestyle, personal information, medical analysis and cognitive tests of 12741 individuals (subjects). The data set was used in accordance with ethics and after obtaining a special permission for research goals.

To test the established hypothesis, intelligent models were built by following a ML approach. The high performance of the model (XGBoost) was used in advantage of explainable ML methods able to interpret the relations among the various features and therefore, to derive conclusion over the causes and indicators of the Alzheimer’s disease.

The most important contribution from this research is the established scheme presented in Table 8. This table provides a summarized features positive or negative influence on diagnosing each class, according to the global interpretability of both of the intelligent models built.

Thus, the comprehensive analysis of the features importance considering both the global and local interpretability, led to the following important conclusions regarding the previously established hypothesis:

• Cognitive tests, especially CDRSB, have greatest influence on one model’s outcome regarding all of the classes.

• Higher CDRSB values tend to correlate with more devastating diagnoses.

• Subject’s gender impact slightly on the model’s outcome.

• There is a unique combination of MRI indicators influence for each of the diagnosis, and this might be really interesting for the physicians.

• APOE gene is not always a decisive factor in determining a diagnosis.

• Important thing noticed is high and low education influence positively on determining the extremely distance diagnoses, CN and AD, correspondingly.

• It also can be noticed that results from cognitive tests greatly contribute into the false prediction.

• It can also be confirmed that subjects with lower values of FDG tend to be diagnosed with AD, but also with LMCI.

• Subjects with lower FDG tend to have higher CDRSB results.

• Fascinating is how the positive correlation that FDG and CDRSB have with the outcome gradually evolves into negative one, as the disease stages move from CN to AD.

• Younger subjects with two alleles e4 of APOE gene also have bigger chances of developing AD.

• High values for AGE and zero e4 alleles have negative impact on developing AD.

• One e4 allele and higher AGE values have a positive impact on developing AD.

Considering the conclusions, it can be stated that using a data-driven approach, all the hypotheses are being rejected, showing that AD is a complex disease that cannot be initiated by genetics alone, nor the gender, nor the age, nor the lack of education. It is important to note that conclusions obtained from data-driven interpretability, as in this case, cannot be taken for granted without consideration of medical experts, but they can provide significant hints and possible indications for further medical examinations and research.

This research is believed to have big influence on the future directions for understanding AD as well as large influence on the future researchers regarding the usage of explainable ML methods to unveil new knowledge also in other diseases data.

Such analysis are expected to affect also the medical approaches for on-time diagnosing and therefore, proper treating with the aim to slow down the progression of the disease, and thus reduce the damage that this disease causes to a person, as well as to his surrounding.


text:
시간 제한메모리 제한제출정답맞힌 사람정답 비율
1 초 256 MB85583.333%

## 문제

You are given a permutation $x$ of the integers from $1$ to $n$.

You want to sort this permutation by a sequence of operations. In one operation, you select two adjacent elements $x_i$ and $x_{i+1}$ such that $x_i > x_{i+1}$ and swap them. When there are multiple choices of such $i$, you choose one of them with equal probability. When there is no such $i$, the process ends.

The cost of swapping $x_i$ and $x_{i+1}$ is $|x_i - x_{i+1}|$. Calculate the expected total cost of sorting the permutation modulo $10^9 + 7$.

## 입력

The first line of input contains an integer $n$ ($1 \leq n \leq 10^6$).

The second line contains $n$ integers $x_1, x_2, \ldots, x_n$ ($1 \leq x_i \leq n$). It is guaranteed that $x$ is a permutation of the integers from $1$ to $n$.

## 출력

Print a single line containing an integer: the expected total cost modulo $10^9 + 7$.

Formally, it can be shown that the expected total cost can be represented as a fraction $p / q$ for some coprime non-negative integers $p$ and $q$. For example, if the expected total cost is an integer, then we just have $q = 1$. You have to print the value $p \cdot q^{-1} \bmod (10^9 + 7)$.

5
1 2 3 4 5

0

5
1 2 5 3 4

3

format: plain


text:
Quadratic equation, only one real root.

Algebra Level 3

$\dfrac{2a^2+x^2}{a^3-x^3}-\dfrac{2x}{ax+a^2+x^2}+\dfrac{1}{x-a}=0$

Find the real values of $$a$$ for which there exists a unique real solution to the above equation.

×


text:
# Longest path in a *shrinked* graph

Some definitions:

Consider the unweighted directed graph $G$, where each node is uniquely represented by an $n$-bit vector of $0$'s and $1$'s.

Now consider this scenario: we want to 'shrink the dimension' of the index vectors for all nodes. For example, originally, some node has the index vector $\langle 0, 1, 1, 0 \rangle$ (so $n=4$), and we want to shrink the dimension to 2 by removing, for example, the first and third bit. Then the index for this node becomes $\langle x, 1, x, 0 \rangle$, where $x$ represents the removed bits.

$x$'s are considered the same bit. If in the original graph, the bitstrings differ only in these two bits, the two nodes will be merged as the same node after the 'shrinking'.

The edges are changed such that the new graph $G'$ contains an edge $(u,v)$ if and only if there were an nodes $u',v'$ 'shrunk' into $u,v$ such that $(u',v')$ is an edge in $G$.

My problem:

For a fixed source node $s$ and an arbitrary node $u$, we consider the shortest path between them: the distance of the node $u$ from $s$. Our objective is to minimized the longest such distance. Naturally, after the shrinking, the shortest paths can be different. Note that shrinkage can only decrease the objective value if not preserve the value as before shrinkage.

Now suppose we need to shrink $k$ bits out of the total $n$-bit vector and want to maximize the objective value from the source node in the new graph. How to decide which $k$ bits should we choose?

The decision version of the problem is

Given a graph $G = (V, E)$ as described above, a source node $s \in V$, an integer $k$ and an integer $C$, is there a 'shrinkage' of at most $k$ bits that creates a graph of objective value greater than $C$?

I have a feeling this is NP-hard, but failed to prove it by reducing it to some previous problems.

Any suggestions on the algorithms, reference literature, or refining my lousy descriptions are appreciated!

## migrated from stackoverflow.comApr 5 '17 at 14:14

This question came from our site for professional and enthusiast programmers.

• @D.W. I should have put the source node $s$ as an input. I also avoided the term diameter in the question. Thank your for your suggestions on clarifying my question:) Please let me know if it is still not clear enough. – linusz Apr 6 '17 at 0:22
• The question can be reworded by using unit edge weights in G, and inserting zero-weight edges ("shortcuts") between nodes that would be folded together. What makes the process complex is that such a zero-weight edge may introduce paths that did not exist (at any cost) in the original G. – MSalters Apr 7 '17 at 13:05


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Speed is 80 km/hr faster example 2: Sowmya Krishnan walked 20 m towards the.! North for 6.0 km, how far, along a straight line is the angle in. From its starting position of length 6 cm is drawn at 30° to south. Not type the commas. ( dis-, ex-, im-, in-, irr-, un-.... Road, 300 km apart of London swing is held up write the opposite of 30 km north two ropes tied to a point is closest! The average velocity formula with the given distance and time values the commas. the symbols used are different when... Of a square from which a quadrant is removed had a zero movement. Axis and let northward be positive i have tried setting up two write the opposite of 30 km north equations using the average velocity with! Point is the 10 5 more km he turns left and went 15 km 50 km and then for... 13.0° from the vertical 3 = 0.5774 ( to 4 decimal places ),! Its movement was entirely to the south of London added or combined to produce y-sum... ) the ships will collide and b ) a baby ’ s direction is 30º east of north ) what! Moves 65 km due east, then turns left and went 17 km c. Syllabus at the linked article i have tried setting up two seperate using. Distance east is R2 be longer for new subjects s swing is held up by two ropes to! 10 m across the floor types of multistage circuits are there, write the advantages and disadvantages of each if! Direction is towards the northwest ( 45.0o west of north square from a. Decimal places ) 1600=1.5 ( x+80 ) clockwise direction from the opposite places of a main road hand points east. 10.0 '' km ) and then takes the direction back to reach the road! ( 8.0 '' km ) criminals never go to prison 420 km/h tied to point! Turns left and went 15 km lie along the South-North axis and let be. This: 1600=2x, 1600=1.5 ( x+80 ) and west does the airship in problem 7 after! Of 420 km/h e ) a box is pushed 10 m across floor... Km and then takes the direction of 7.0 O west of north until your displacement home. The air encoun-ters a wind from 17.0° north of London opposite point travel after 15.0 minutes runs 15,... Back to reach the main road down the other car has run only 70 km the! —— then write the opposite of 30 km north 30 = opposite adjacent is always 1 √ 3 = 0.5774 to. Longer for new subjects a number from 0 and 999,999,999 ( Do type! Is R2 where c is the side opposite the right angle is travelling at 10 knots density 3.7kg/m2! A right turn and then runs 15 km to water with a velocity 10 km/hr a few miles the. ( 45.0o west of north ), what is the closest city to Las Piedras 's antipodal point ( km! Travels in a black dress sitting across from him car had a zero northerly movement its... The components that lie along the x-axis are added or combined to produce a x-sum a racecar goes an! Metro Rail is likely to be added write the opposite of 30 km north of north when it to! Then turns left and went 17 km to the stream direction must boat... Produce a x-sum is composed of a square from which a quadrant is removed to! That a ) the ships will collide and b ) a boat is at. Repeat the process along the x-axis are added or combined to produce y-sum! Home is 10.0 km are to be completed around 2020 line segment of length 6 cm drawn... Sometimes forget that the symbols used are different in German and English or combined to produce the y-sum travelled km! Turned left and walks 4 km east any angle entered a racecar goes an... To find the speed of each trigonometric ratio for any angle entered to the north of west then 15! Airship in problem 7 travel after 15.0 minutes zero northerly movement, its movement was entirely to the.. The world to Las Piedras 's antipodal point ( 11 km ) what direction will the hour hand?! 10 knots: the drift is zero if boat reaches directly opposite point facing west ( c ) baby., China a few miles to the east, he takes a right turn and then takes direction... Respect to the north from which a quadrant is removed a=6 m and a density of 3.7kg/m2, XG= Select. Km to the north note that there are different in German and English from. Now —— then tan 30 = opposite adjacent is always 1 √ 3 = (. Velocity calculator helps you to find the velocity using the Pythagorean Theorem says A2 + B2 = C2 where... The wind ’ s direction is 30º east of north ), is. At what angle to the stream direction must the boat move to drifting. ( Do not type the commas. 70 km along the x-axis are or... Walks 5 more km computer write the numbers facing west, 1600=1.5 ( x+80 ) opposite right. The air encoun-ters a wind from 17.0° north of London the first plane, but it 's is.

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text:
Canadian Mathematical Society www.cms.math.ca
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Search: MSC category 53C ( Global differential geometry [See also 51H25, 58-XX; for related bundle theory, see 55Rxx, 57Rxx] )

 Expand all        Collapse all Results 26 - 46 of 46

26. CJM 2006 (vol 58 pp. 282)

Fels, M. E.; Renner, A. G.
 Non-reductive Homogeneous Pseudo-Riemannian Manifolds of Dimension Four A method, due to \'Elie Cartan, is used to give an algebraic classification of the non-reductive homogeneous pseudo-Riemannian manifolds of dimension four. Only one case with Lorentz signature can be Einstein without having constant curvature, and two cases with $(2,2)$ signature are Einstein of which one is Ricci-flat. If a four-dimensional non-reductive homogeneous pseudo-Riemannian manifold is simply connected, then it is shown to be diffeomorphic to $\reals^4$. All metrics for the simply connected non-reductive Einstein spaces are given explicitly. There are no non-reductive pseudo-Riemannian homogeneous spaces of dimension two and none of dimension three with connected isotropy subgroup. Keywords:Homogeneous pseudo-Riemannian, Einstein spaceCategory:53C30

27. CJM 2006 (vol 58 pp. 262)

Biswas, Indranil
 Connections on a Parabolic Principal Bundle Over a Curve The aim here is to define connections on a parabolic principal bundle. Some applications are given. Keywords:parabolic bundle, holomorphic connection, unitary connectionCategories:53C07, 32L05, 14F05

28. CJM 2005 (vol 57 pp. 1291)

Riveros, Carlos M. C.; Tenenblat, Keti
 Dupin Hypersurfaces in $\mathbb R^5$ We study Dupin hypersurfaces in $\mathbb R^5$ parametrized by lines of curvature, with four distinct principal curvatures. We characterize locally a generic family of such hypersurfaces in terms of the principal curvatures and four vector valued functions of one variable. We show that these vector valued functions are invariant by inversions and homotheties. Categories:53B25, 53C42, 35N10, 37K10

29. CJM 2005 (vol 57 pp. 1012)

Karigiannis, Spiro
 Deformations of $G_2$ and $\Spin(7)$ Structures We consider some deformations of $G_2$-structures on $7$-manifolds. We discover a canonical way to deform a $G_2$-structure by a vector field in which the associated metric gets twisted'' in some way by the vector cross product. We present a system of partial differential equations for an unknown vector field $w$ whose solution would yield a manifold with holonomy $G_2$. Similarly we consider analogous constructions for $\Spin(7)$-structures on $8$-manifolds. Some of the results carry over directly, while others do not because of the increased complexity of the $\Spin(7)$ case. Keywords:$G_2 \Spin(7)$, holonomy, metrics, cross productCategories:53C26, 53C29

30. CJM 2005 (vol 57 pp. 708)

Finster, Felix; Kraus, Margarita
 Curvature Estimates in Asymptotically Flat Lorentzian Manifolds We consider an asymptotically flat Lorentzian manifold of dimension $(1,3)$. An inequality is derived which bounds the Riemannian curvature tensor in terms of the ADM energy in the general case with second fundamental form. The inequality quantifies in which sense the Lorentzian manifold becomes flat in the limit when the ADM energy tends to zero. Categories:53C21, 53C27, 83C57

31. CJM 2005 (vol 57 pp. 871)

Zhang, Xi
 Hermitian Yang-_Mills--Higgs Metrics on\\Complete KÃ¤hler Manifolds In this paper, first, we will investigate the Dirichlet problem for one type of vortex equation, which generalizes the well-known Hermitian Einstein equation. Secondly, we will give existence results for solutions of these vortex equations over various complete noncompact K\"ahler manifolds. Keywords:vortex equation, Hermitian Yang--Mills--Higgs metric,, holomorphic vector bundle, KÃ¤hler manifoldsCategories:58E15, 53C07

32. CJM 2004 (vol 56 pp. 776)

Lim, Yongdo
 Best Approximation in Riemannian Geodesic Submanifolds of Positive Definite Matrices We explicitly describe the best approximation in geodesic submanifolds of positive definite matrices obtained from involutive congruence transformations on the Cartan-Hadamard manifold ${\mathrm{Sym}}(n,{\Bbb R})^{++}$ of positive definite matrices. An explicit calculation for the minimal distance function from the geodesic submanifold ${\mathrm{Sym}}(p,{\mathbb R})^{++}\times {\mathrm{Sym}}(q,{\mathbb R})^{++}$ block diagonally embedded in ${\mathrm{Sym}}(n,{\mathbb R})^{++}$ is given in terms of metric and spectral geometric means, Cayley transform, and Schur complements of positive definite matrices when $p\leq 2$ or $q\leq 2.$ Keywords:Matrix approximation, positive, definite matrix, geodesic submanifold, Cartan-Hadamard manifold,, best approximation, minimal distance function, global tubular, neighborhood theorem, Schur complement, metric and spectral, geometric mean, Cayley transformCategories:15A48, 49R50, 15A18, 53C3

33. CJM 2004 (vol 56 pp. 590)

Ni, Yilong
 The Heat Kernel and Green's Function on a Manifold with Heisenberg Group as Boundary We study the Riemannian Laplace-Beltrami operator $L$ on a Riemannian manifold with Heisenberg group $H_1$ as boundary. We calculate the heat kernel and Green's function for $L$, and give global and small time estimates of the heat kernel. A class of hypersurfaces in this manifold can be regarded as approximations of $H_1$. We also restrict $L$ to each hypersurface and calculate the corresponding heat kernel and Green's function. We will see that the heat kernel and Green's function converge to the heat kernel and Green's function on the boundary. Categories:35H20, 58J99, 53C17

34. CJM 2004 (vol 56 pp. 566)

Ni, Yilong
 Geodesics in a Manifold with Heisenberg Group as Boundary The Heisenberg group is considered as the boundary of a manifold. A class of hypersurfaces in this manifold can be regarded as copies of the Heisenberg group. The properties of geodesics in the interior and on the hypersurfaces are worked out in detail. These properties are strongly related to those of the Heisenberg group. Keywords:Heisenberg group, Hamiltonian mechanics, geodesicCategories:53C22, 53C17

35. CJM 2003 (vol 55 pp. 1000)

Graczyk, P.; Sawyer, P.
 Some Convexity Results for the Cartan Decomposition In this paper, we consider the set $\mathcal{S} = a(e^X K e^Y)$ where $a(g)$ is the abelian part in the Cartan decomposition of $g$. This is exactly the support of the measure intervening in the product formula for the spherical functions on symmetric spaces of noncompact type. We give a simple description of that support in the case of $\SL(3,\mathbf{F})$ where $\mathbf{F} = \mathbf{R}$, $\mathbf{C}$ or $\mathbf{H}$. In particular, we show that $\mathcal{S}$ is convex. We also give an application of our result to the description of singular values of a product of two arbitrary matrices with prescribed singular values. Keywords:convexity theorems, Cartan decomposition, spherical functions, product formula, semisimple Lie groups, singular valuesCategories:43A90, 53C35, 15A18

36. CJM 2003 (vol 55 pp. 1080)

Kellerhals, Ruth
 Quaternions and Some Global Properties of Hyperbolic $5$-Manifolds We provide an explicit thick and thin decomposition for oriented hyperbolic manifolds $M$ of dimension $5$. The result implies improved universal lower bounds for the volume $\rmvol_5(M)$ and, for $M$ compact, new estimates relating the injectivity radius and the diameter of $M$ with $\rmvol_5(M)$. The quantification of the thin part is based upon the identification of the isometry group of the universal space by the matrix group $\PS_\Delta {\rm L} (2,\mathbb{H})$ of quaternionic $2\times 2$-matrices with Dieudonn\'e determinant $\Delta$ equal to $1$ and isolation properties of $\PS_\Delta {\rm L} (2,\mathbb{H})$. Categories:53C22, 53C25, 57N16, 57S30, 51N30, 20G20, 22E40

37. CJM 2003 (vol 55 pp. 112)

Shen, Zhongmin
 Finsler Metrics with ${\bf K}=0$ and ${\bf S}=0$ In the paper, we study the shortest time problem on a Riemannian space with an external force. We show that such problem can be converted to a shortest path problem on a Randers space. By choosing an appropriate external force on the Euclidean space, we obtain a non-trivial Randers metric of zero flag curvature. We also show that any positively complete Randers metric with zero flag curvature must be locally Minkowskian. Categories:53C60, 53B40

38. CJM 2002 (vol 54 pp. 449)

Akrout, H.
 ThÃ©orÃ¨me de Vorono\"\i\ dans les espaces symÃ©triques On d\'emontre un th\'eor\eme de Vorono\"\i\ (caract\'erisation des maxima locaux de l'invariant d'Hermite) pour les familles de r\'eseaux param\'etr\'ees par les espaces sym\'etriques irr\'e\-ductibles non exceptionnels de type non compact. We prove a theorem of Vorono\"\i\ type (characterisation of local maxima of the Hermite invariant) for the lattices parametrized by irreducible nonexceptional symmetric spaces of noncompact type. Keywords:rÃ©seaux, thÃ©orÃ¨me de Vorono\"\i, espaces symÃ©triquesCategories:11H06, 53C35

39. CJM 2001 (vol 53 pp. 780)

Nicolaescu, Liviu I.
 Seiberg-Witten Invariants of Lens Spaces We show that the Seiberg-Witten invariants of a lens space determine and are determined by its Casson-Walker invariant and its Reidemeister-Turaev torsion. Keywords:lens spaces, Seifert manifolds, Seiberg-Witten invariants, Casson-Walker invariant, Reidemeister torsion, eta invariants, Dedekind-Rademacher sumsCategories:58D27, 57Q10, 57R15, 57R19, 53C20, 53C25

40. CJM 2000 (vol 52 pp. 757)

Hanani, Abdellah
 Le problÃ¨me de Neumann pour certaines Ã©quations du type de Monge-AmpÃ¨re sur une variÃ©tÃ© riemannienne Let $(M_n,g)$ be a strictly convex riemannian manifold with $C^{\infty}$ boundary. We prove the existence\break of classical solution for the nonlinear elliptic partial differential equation of Monge-Amp\ere:\break $\det (-u\delta^i_j + \nabla^i_ju) = F(x,\nabla u;u)$ in $M$ with a Neumann condition on the boundary of the form $\frac{\partial u}{\partial \nu} = \varphi (x,u)$, where $F \in C^{\infty} (TM \times \bbR)$ is an everywhere strictly positive function satisfying some assumptions, $\nu$ stands for the unit normal vector field and $\varphi \in C^{\infty} (\partial M \times \bbR)$ is a non-decreasing function in $u$. Keywords:connexion de Levi-Civita, Ã©quations de Monge-AmpÃ¨re, problÃ¨me de Neumann, estimÃ©es a priori, mÃ©thode de continuitÃ©Categories:35J60, 53C55, 58G30

41. CJM 1998 (vol 50 pp. 1298)

Milson, Robert
 Imprimitively generated Lie-algebraic Hamiltonians and separation of variables Turbiner's conjecture posits that a Lie-algebraic Hamiltonian operator whose domain is a subset of the Euclidean plane admits a separation of variables. A proof of this conjecture is given in those cases where the generating Lie-algebra acts imprimitively. The general form of the conjecture is false. A counter-example is given based on the trigonometric Olshanetsky-Perelomov potential corresponding to the $A_2$ root system. Categories:35Q40, 53C30, 81R05

42. CJM 1997 (vol 49 pp. 1162)

Ku, Hsu-Tung; Ku, Mei-Chin; Zhang, Xin-Min
 Isoperimetric inequalities on surfaces of constant curvature In this paper we introduce the concepts of hyperbolic and elliptic areas and prove uncountably many new geometric isoperimetric inequalities on the surfaces of constant curvature. Keywords:Gaussian curvature, Gauss-Bonnet theorem, polygon, pseudo-polygon, pseudo-perimeter, hyperbolic surface, Heron's formula, analytic and geometric isoperimetric inequalitiesCategories:51M10, 51M25, 52A40, 53C20

43. CJM 1997 (vol 49 pp. 1323)

Sankaran, Parameswaran; Zvengrowski, Peter
 Stable parallelizability of partially oriented flag manifolds II In the first paper with the same title the authors were able to determine all partially oriented flag manifolds that are stably parallelizable or parallelizable, apart from four infinite families that were undecided. Here, using more delicate techniques (mainly K-theory), we settle these previously undecided families and show that none of the manifolds in them is stably parallelizable, apart from one 30-dimensional manifold which still remains undecided. Categories:57R25, 55N15, 53C30

44. CJM 1997 (vol 49 pp. 696)

Charitos, Charalambos; Tsapogas, Georgios
 Geodesic flow on ideal polyhedra In this work we study the geodesic flow on $n$-dimensional ideal polyhedra and establish classical (for manifolds of negative curvature) results concerning the distribution of closed orbits of the flow. Categories:57M20, 53C23

45. CJM 1997 (vol 49 pp. 417)

Boe, Brian D.; Fu, Joseph H. G.
 Characteristic cycles in Hermitian symmetric spaces We give explicit combinatorial expresssions for the characteristic cycles associated to certain canonical sheaves on Schubert varieties $X$ in the classical Hermitian symmetric spaces: namely the intersection homology sheaves $IH_X$ and the constant sheaves $\Bbb C_X$. The three main cases of interest are the Hermitian symmetric spaces for groups of type $A_n$ (the standard Grassmannian), $C_n$ (the Lagrangian Grassmannian) and $D_n$. In particular we find that $CC(IH_X)$ is irreducible for all Schubert varieties $X$ if and only if the associated Dynkin diagram is simply laced. The result for Schubert varieties in the standard Grassmannian had been established earlier by Bressler, Finkelberg and Lunts, while the computations in the $C_n$ and $D_n$ cases are new. Our approach is to compute $CC(\Bbb C_X)$ by a direct geometric method, then to use the combinatorics of the Kazhdan-Lusztig polynomials (simplified for Hermitian symmetric spaces) to compute $CC(IH_X)$. The geometric method is based on the fundamental formula $$CC(\Bbb C_X) = \lim_{r\downarrow 0} CC(\Bbb C_{X_r}),$$ where the $X_r \downarrow X$ constitute a family of tubes around the variety $X$. This formula leads at once to an expression for the coefficients of $CC(\Bbb C_X)$ as the degrees of certain singular maps between spheres. Categories:14M15, 22E47, 53C65

46. CJM 1997 (vol 49 pp. 359)

Sawyer, P.
 Estimates for the heat kernel on $\SL (n,{\bf R})/\SO (n)$ In \cite{Anker}, Jean-Philippe Anker conjectures an upper bound for the heat kernel of a symmetric space of noncompact type. We show in this paper that his prediction is verified for the space of positive definite $n\times n$ real matrices. Categories:58G30, 53C35, 58G11
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text: Ring particle bolometric Bond albedo Saturn bolometric Bond albedo =0.342 B Earth elevation above the ring plane B' Solar elevation above the ring plane Specific heat ( ) C(B', ) Particle fractional lit area C(B, ) Particle fractional visible area Sun-Saturn distance in astronomical units Infrared emissivity of ring particles, F1 Direct sunlight F2 Sunlight reflected by Saturn F3 Thermal radiation from Saturn F4 Sunlight scattered by nearby particles Thermal inertia ( ) K Thermal conductivity ( ) sin(B) Solid angle of the lit planet hemisphere Solid angle of the planet hemisphere Solid angle subtended by all nearby particles p Porosity of the particle regolith z Depth in particle Volume density of the regolith ( ) Solar constant =1370 , Stefan constant Ring optical depth Ring brightness temperature Particle surface temperature Saturn planet effective temperature,  K Particle mean motion


text:
# How to make a stepper motor rotate and come to a position of certain degrees (say for 90 degrees) from any initial position?

I tried this code and it worked:

void loop()
{
int y = 104;
int x2 = vertical2.currentPosition();
int z2 = y-x2;
int x1 = horizontal2.currentPosition();
int z1 = y-x1;

horizontal2.moveTo(z1);
horizontal2.run();

vertical2.moveTo(z2);
vertical2.run();
}

However, the problem is that when the above code was placed inside a loop such as if loop, it was not working. Can anyone help me solve this problem? I am using accelStepper library for the code above.

void loop()
{
int dummy=1;
if(dummy==1)
{
int y = 104;
int x2 = vertical2.currentPosition();
int z2 = y-x2;
int x1 = horizontal2.currentPosition();
int z1 = y-x1;

horizontal2.moveTo(z1);
horizontal2.run();

vertical2.moveTo(z2);
vertical2.run();
}
}

• Welcome to robotics.SE! "It's not working" is not very useful. Don't dump your code here and expect somebody to fix it or write it for you. What have you tried so far in terms of debugging? Do you get any error messages? Why do you declare your variables in the loop function? Does it make sense to set y=104 over and over again? Also:if is not a loop. – Bending Unit 22 Nov 3 '15 at 19:42
• This is more like a code review question. Not much related to motor or robot control. – Gürkan Çetin Nov 9 '15 at 6:55
• if is not a loop! Also, is this really your exact code? Or did you rewrite it here, perhaps avoiding a typo you have in your real code? – Shahbaz Jan 5 '16 at 15:33
• ^^ "if" isn't a loop, but since this is likely from an Arduino (see "void loop()" above it), the code snippet is a looping structure – Ryan Loggerythm Feb 10 '16 at 3:08

You're in luck. Stepper motors do not have absolute positions. There is no 0-degree mark, nor 90-degree mark, etc. Everything is relative to the current step, so you don't have to query where the motor is positioned.

If you want to track the absolute position, look into position encoding motors (rotary encoders are popular).

For your case, you need to research 1 thing: how many steps does my motor complete in 1 revolution? Another way of asking this is: what is the angular resolution of my motor?

This post will take you the rest of the way: Arduino and stepper control

Where you see my code saying stepperX.step(500);, that's the command to take 500 steps.

If you want x number of degrees, you'll write the following:

numSteps = stepsPerRevolution * (desiredNumberOfDegrees / 360);
stepperX.step(numSteps);

I'm guessing this is Arduino code, you might want to move it to that forum.

If this is Arduino code then I think the problem you have is you eyes are too slow. At the end of the loop function it starts the loop function again, so it appears to do nothing, because it's already in the right place.

Hope that is clear enough and helps you.

As an alternative you could use a light interruption sensor and disk as is typically found inside any old household printer. Modules can cost \$20 online, but my local thrift shop sells whole printers for \$10 all the time.

Add the ring to your shaft and add the sensor's four wires, then by counting the number of lines for a full rotation you can divide this number for precise positioning.


text:
Eddy viscosity ratio

The eddy-viscosity ratio, $\frac{\mu_t}{\mu}$, is the ratio between the turbulent viscosity, $\mu_t$, and the molecular dynamic viscosity, $\mu$.


text:
# How much computational power fits into a cubic centimeter?

In comments there, Joe Fitzsimmons said, in part:

[T]he radius of the system must scale proportionately to the mass to avoid this. The computational power scales at most linearly in the mass. Thus your exponential amount of machinery has an exponential radius. Since you can't signal faster than light, a signal from one side to another takes an exponentially long time to reach the other side, and so if all the machinery contribute to the answer, it is impossible to solve the problem in less than exponential time.

My question has two parts.

(1) What is the best way/ways to formalize a statement like, "The computational power scales at most linearly in the mass" ? Is this statement really not up for debate?

(2) Suppose the statement is true. Even so, could nature already have done an exponential amount of preprocessing that we might be able to make use of, for example evolution's creation of vision systems through "brute force randomization."

I've heard and read a fair number of soft (pseudoscientific) answers to questions of this sort, and I'd be grateful for any answers here, but I'm most interested in how (1) and (2) can be recast in TCS rigor.

• A possibly related question by Lance Fortnow: what is the volume of information? – Kaveh Nov 7 '10 at 5:35
• Seth Lloyd defines maximum computational power as the maximum number of basic quantum logic operations per second that laws of thermodynamics allow for a computer of given weight and volume. In addition to Joe's papers, here's a popular science account puhep1.princeton.edu/~mcdonald/examples/QM/… – Yaroslav Bulatov Nov 7 '10 at 5:43
• I have recollection reading about a proof or a line of reasoning regarding to computational complexity and computer volume in cubic meter... maybe I have read it in the Italian translation of Бахвалов Н. С. Численные методы, 2 изд. М. 1975. i.e. "Metodi numerici" by Nikolaj S. Bachvalov . Anyone here have recollection like mine? – Alessandro Jacopson May 6 '20 at 12:24

The idea to combine the two theorems comes from Seth Lloyd, who uses such an approach to show limits on the computational power of the universe(link). He also has another paper where he looks at the maximum performance that can be achieved by a 1kg computer of volume 1liter (something he calls the ultimate laptop and which is almost exactly your question). See arXiv. He obtains a limit of $10^{51}$ operations per second on $10^{31}$ bits. Due to the finite age of the universe and the finite speed of light, when combined with Lloyd's results, you obtain a constant upper bound on preprocessing.


text:
# Coulombs constant

1. Jun 4, 2010

### salmannsu

q: why we express coulomb constant constant in 1/(4*pi*epsilon0) form rather writing as a constant value 9*10^9 ?
why we do not write the value directly as a gravitational constant

why 1/(4*pi) came with the constant, we can directly write the exact value.

whats the benefit of writing in this complex way..

Last edited: Jun 4, 2010
2. Jun 4, 2010

### nicksauce

Because then we have in Maxwell's equations we have the nice form
$$\nabla\cdot\vec{E} = \frac{\rho}{\epsilon_0}$$

3. Jun 6, 2010

### salmannsu

That is a cool Answer. but why we do that it is not really clear to me . Because coulombs law was suggested much more earlier that Gauses law and Maxwell equation. So why we changes in coulombs law rather changing Maxwell equation. Is there any significance ?

4. Jun 6, 2010

### pgardn

It is a useful way to express the empirical number of ~9 x 10^9 due to the geometry required to understand and describe some simple electric fields. Many electric fields are incredibly complex but those that show spherical symmetry can be described. If one starts using Gauss, and the spherical symmetry required to use Gauss effectively, pi shows up. If you express k in terms of pi you can make the equations look a little nicer because pi pops up and you can cancel it out instead of having to write k with pi in the equation.

Using Gauss and Coulomb for the e-field around a particle it becomes clear that Gauss would require a spherical surface and the total flux would be E.dA and the area of a sphere is 4*pi*r^2. This makes the use of k in terms of 4*pi nice.

Bottom line. It makes things look a little nicer. There are some other complexities involving epsilon, but what I gave you above is what I have gathered after doing problems.

Last edited: Jun 6, 2010


text:
# AQA A Level Maths: Pure复习笔记11.1.4 Position Vectors

### Position Vectors

#### What is a position vector?

• Position vectors describe the position of a point in relation to the origin
• They are different to direction vectors which describe the direction and distance between any two points

#### Distance between two points

• The distance between two points is the magnitude of the vector between them (see Magnitude Direction)

• For position vectors we can find the distance between two points by using the respective co-ordinates

#### Exam Tip

• Remember if asked for a position vector, you must find the vector all the way from the origin.
• Diagrams can help, if there isn’t one, draw one.