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“Leads are dropping faster than water poured into a leaky bucket” Khan exclaimed to his business partner. He was visibly frustrated. And it was true. A close look at their Hotjar dashboard reveals that they attract visitors to their website all right. But they hit a snag when it comes to converting those users to paying customers. Ouch. Khan and his partner aren’t the only ones battling this problem, however. In fact, according to , a massive 79% of leads are not converted to sales. Marketing Sherpa If you’re losing that many leads, then it’s time to take a step back and analyze the path your customers take to conversion. This is so you can uncover friction in your customer journey that reduces the efficiency of your SaaS sales funnels. A confusing task if you don’t know where to begin. But it doesn’t have to be. This article will show you 4 tactics for optimizing your SaaS sales funnel. But before we jump into that, let’s be clear on what a sales funnel is. What exactly is a SaaS sales funnel? Imagine all the steps your prospects go through on their way to becoming paying customers. The moment they discovered your brand How they decided your product is worth a trial When they decided they’ve found the real deal and became your customer And started spreading the good word about your product All of these stages are what make up the SaaS sales funnel. Source: Klozers Think of it as a multi-staged selling process that begins the moment your prospect first becomes aware of your brand and continues even after the prospect becomes a paying customer. How do you know it’s time to optimize your funnel? Serial entrepreneur turned VC, David Skok, gave a warning in . In his words: his brilliant article “No matter how large or successful your business is, you will have at least one place that is a blockage point in your customer acquisition funnel. This is the point where the conversion rates from one stage to another are not satisfactory or you have a scaling problem…” What does David mean? He’s saying it’s always time to optimize your funnel. A perfect funnel — one that is 100% efficient and requires no fine-tuning — isn’t achievable. There’ll always be stages in your customers’ journey where the conversion rates are less than ideal. And the onus falls on you as the business owner to find these blockages and clear them. There are three main places to look at when searching for funnel blockages. . Are your leads stuck on top of your sales funnel? This means you have no problem driving visitors to your website. But they do not seem interested enough in your offering and messaging to move on to the next stage. Top of the funnel . Are your leads stuck in the middle of your funnel? It means you’re generating leads. But they’re not just qualified. Qualified leads show a clear indication that they’re interested in your business. Middle of the funnel . Are your leads stuck at the bottom of the funnel? This means you’re having no sales. Your leads show intent but you can’t convert them into paying customers. Bottom of the funnel Once you’re able to pinpoint where these funnel blockages are, the next logical question becomes: “How do I eliminate them?” These are 4 tactics for you. 4 tactics for optimizing your sales funnel Create content that solves the specific needs of your prospects It’s not just enough to create content for SEO's sake. Your content has to appeal to prospects at specific stages in their buyer journey. That’s how they get enough information to move from one funnel stage to the other. This stat by the emphasizes that. According to the research, 95% of buyers choose SaaS companies that provide them with sufficient content to navigate the buying process. Demand Gen report Wondering what type of content you should produce at each stage of the funnel? This illustration by Myk Pono provides the answer. It also shows how each content type fits into your customer acquisition framework. Source: Myk Pono Shorten your sales process Have you ever forfeited your spot on a queue because it was too long? At that moment, your time suddenly becomes so valuable that you’d rather go home without what you came for. Or find somewhere else to get it. That’s how your prospects feel when you make them go through multiple (unnecessary) steps on their way to conversion. Time is money. The shorter your sales cycle is, the more optimized your funnel is. Not to mention your customer acquisition costs rise with every increase in your sales cycle. Source: Zendesk Here’s one effective way to figure out where the time traps in your sales cycle are. Revisit your interactions with existing customers. What specific issues extended your sales process? Depending on what funnel stage this issue occurs, create content that provides a solution to your customers. Measure your funnel metrics Remember the saying “what can’t be measured, can’t be managed.” It holds true here. The only way to improve your SaaS funnel is if you have data on what’s wrong. Not everyone realizes this though. This surprisingly revealed that 68% of companies have not identified or attempted to measure their sales funnel. research If you’re wondering what sales funnel metrics you should keep tabs on, these are the most useful ones. Leads Conversion rates Acquisition costs Customer lifetime value Average deal time Cycle time Total sales This by David Skok gives a deep dive into understanding and calculating SaaS funnel metrics. video Do not forget your old customers Often when founders think of their sales funnel, they only think of how new prospects can become new customers. But what about your old customers? 80% of your company’s profit will come from 20% of your existing customers, according to this research by . This underlines the fact that you should include existing customers in your funnel optimization plans. Customer Thermometer How do you do that? One way is to roll out a referral program. Dropbox used this tactic to improve the efficiency of its sales funnel. And , the company’s co-founder, referrals increased their signups by 60%. according to Drew Houston by Avoma gives actionable tips on how to set up a SaaS referral program. This article Conclusion To grow your SaaS business and drive predictable revenue, understanding how your customers become customers is vital. That’s how you get information on how to get new customers to grow your customer base. With this in mind, the importance of a SaaS sales funnel in mapping out the journey your customers take becomes important. What’s also important is optimizing your funnel so that the customers’ journey turns out to be a smooth ride. The four tips detailed in this article will show you how to do that. lead Image Also published . here

4 Tips SaaS Founders can use to Optimize their Sales Funnel

We've made incredible leaps since the days of wireless radio. Once, families sat huddled around their radios in the living room, waiting impatiently for their favorite shows. It's unbelievable to compare that time period with the broadcasting technology of today. Think about the last time you listened to a podcast. How many options did you have to choose from? Did you pause for any incoming calls, or to watch a quick Instagram reel? Needless to say, things have changed – but not just in the way we listen. The nature of we listen to has evolved as well. what Today, I want to touch on something super relevant to those starting their business podcast journey: the art of creating something truly valuable. Because although making a podcast is relatively simple in 2022, making something worth listening to is a far greater challenge – and for every expertly crafted podcast episode on the block, there are ten more that could use a bit more TLC. Let's dive in. Today's guest: Yaagneshwaran Ganesh What sparked my interest in this issue was an interview I had with Yaagneshwaran Ganesh, award-winning author and Director of Marketing at Avoma. Yaagneshwaran is a highly intelligent, highly successful individual – and one who places a tremendous emphasis on the quality of the content he produces. If you haven't seen the on my channel yet, here's a rundown of Yaagneshwaran's career and achievements thus far: interview He's an award-winning marketer, an author, and a TEDx speaker, and has spoken at other business forums like Performance Marketing Moscow, Chamber of Commerce Netherlands, and The World Marketing Congress With more than 12 years of marketing experience in the MarTech space, he successfully produces industry-specific books, podcasts, and other content He is recognized by The World Marketing Congress as one of the Marketing Mavericks From all of that, I don't need to tell you that Yaagneshwaran knows a thing or two about marketing. What I do want to share, however, is his immense wisdom in the art of producing value. What is the meaning of 'value'? In the content creation industry, this concept of 'value' is something we hear and use often. Of course, it's a standard we all aspire to meet in the podcasts we publish and the posts we write – but what does it actually mean? I posed this question to Yaagneshwaran during our interview. As the successful creator of the top podcast for B2B professionals – The AMB Conversations Podcast – I was keen to know how Yaagneshwaran defines this all-important term. "Value is a word that is so commonly used, and so commonly misused. Everybody says but nobody tells you what value actually is." deliver value! He's right, of course. A simple Google search digs up thousands of articles, but they're mostly focused on product or service value and calculating utility. "To me, what I mean by value here is this: somebody gives you 45 minutes to one hour of their time. And if you don't deliver , they can clearly see that." something that is useful to them I love Yaagneshwaran's explanation. It's so simple, but so clear – deliver something that your audience will find useful. It also serves as a reminder that time is a commodity, and if people are willing to use their time to listen to your podcast, you'd better make sure it's worth their while. Let's dig into that a little further. What's the opposite of value? Despite its ambiguity, I'm sure you can recognize value when you see it. Value is a blog post you read until the very end and a book you can't stop talking about. It's the video you watch twice and the podcast episode you listen to over and over. By the same token, you can identify the absence of value when you experience it. A blog post that's full of fluff and awkwardly-placed keywords, or a podcast that would be better described as one long-winded advertisement – these are the things that lack value. These are also the things we want to avoid. An SEO-optimized blog post will get you good rankings, affiliate commissions, and possibly advertisement clicks, but if you don't balance it with real value, you'll lose the most important thing: trust. Why create valuable content? Audience trust and loyalty are one of the most valuable assets a content creator can have. It's what keeps people coming back for more, even when there are so many other things competing for their attention. These aren't just my shower thoughts – research actually shows that over everything else when deciding whether or not to consume media. consumers value trustworthiness This makes total sense. Would you go back to a blog if the first article you read delivers zero value? If you spent an hour listening to a useless podcast, would you then subject yourself to another hour of the same? Probably not. We're all busy people, and we want to make sure that the time we spend consuming content is time well spent. This is why creating valuable content is so important. It's what will set you apart from the competition and make your audience come back for more. Why podcasts must have value When content creators write blog posts, it's preferable that they write something valuable to their audience – but if not, they've got backup. Think affiliate links, sponsorships, advertisements, and so on. Not so with podcasts. There are millions of people tuning in to podcasts on a regular basis, but their attention is split between the thousands of options at their disposal – and research shows that if your podcast is sub-par, they'll ditch you like a hot rock. Take a look at these stats: of listeners will drop off within the first five minutes of a podcast episode. 20 to 35 percent The average podcast will only receive per episode 27 listens Apple Podcasts hosts almost (so there's a lot of competition to cut through) 50 million podcast episodes In other words: if your podcast doesn't hold immense value for listeners, they won't stick around. So how do you make sure your podcast is valuable? Expert advice on delivering valuable podcasts In 2019, Yaagneshwaran launched his highly successful podcast for B2B marketing professionals. He didn't set out to be a content creator – you can hear more about his career journey in – but he quickly learned the importance of delivering value through his podcast episodes. the full interview And it worked, evidently. Fast forward to 2022, and the AMB Conversations podcast is now rated among the top 1% of millions of podcasts. Knowing this, I was excited to pose the question: how do you ensure your podcast provides immense value to listeners? Here's what he had to say. Tip 1: Ask the right questions "The difference between a good podcast episode and a bad podcast episode is purely on the host. That's what I believe – it's about the questions that you ask." I love this pointer from Yaagneshwaran. As interviewers, there's no such thing as blaming the guest for a lack of value in an episode – it's on us to ask the right questions. This requires preparation, research, and knowing one's audience intimately. You might be thinking, "But what if the guest is rude? What if they refuse to cooperate? What if they just don't have anything valuable to say?" Of course, there will be times when things don't go as planned, and guests aren't always predictable. But if you've been diligent in every episode, and you've built a loyal following as a result, those occasional bumps in the road aren't going to kill your show. As Yaagneshwaran pointed out: "There will be guests who answer you in a Tweet format – maybe one or two lines – and there are also going to be people who give you a half-hour answer for a single question. The beauty is that you bring in your own curiosity." Tip 2: Know your guest If you are going to host a podcast based on interviews, it's crucial to know the person you are interviewing. There's no need to be best friends with them – but if you at least have a cursory understanding of their work and what they stand for, it will show in your questioning and the conversation as a whole. "I generally take at least two days to go back and do my research about a particular person," Yaagnewshwaran said. "I listen to at least ten of their episodes, or anywhere they've spoken. Any time I go on a jog, I'm listening to their shows." Now, commitment. that's What you gain by doing this extra prep work is the ability to frame questions in a way that draws out interesting stories and insights from your guests. You'll also be able to better understand their work, which will give you ammunition for follow-up questions. Tip 3: Bring your energy and curiosity You'll never create a valuable podcast by asking the same old questions that have been asked a million times before. Yaagnewshwaran approaches his interviews with a curious and proactive attitude. "I go and look at whatever they've written," Yaagnewshwaran said. "What are the different places that they have spoken? What have they accomplished? Is there anything relevant to what I'm trying to say through this podcast? When you look at those details, you end up with a set of things that you want to learn." This is the difference between someone who simply asks questions and someone who has a desire to learn. There's no point feigning interest – your listeners aren't going to be fooled. But if you bring genuine interest and energy to each conversation, your guests will feed off of that and the episode will be more valuable as a result. "Because you are not the best person at [the interview topic], it makes you all the more curious and wanting to learn. And when you show that energy – that curiosity – that also gets the other person going, because it's a very honest conversation." Here's the conclusion that I've reached from these three (highly valuable!) tips: if you want to create value for your listeners, you need to truly That means doing your homework, being curious and energetic, and asking interesting questions. value your guests. "When a guest has given you one hour of their time, the one way that you can show them the respect that you have for them is by going in prepared and asking the right questions. It's showing them that you value their time." Final thoughts on the art of value We've focused on podcast creation today, but in my opinion, the value of creating value can't be emphasized enough – no matter which medium you work with. The easiest way to understand this concept is by reflecting on your own habits as a media consumer. What do your favorite podcasts have in common? If a blog post is just an advertisement thinly disguised as "life advice", how quickly do you click onto something more personable? This is the power of providing value. It's what makes your audience feel like they're getting something special, and it's a key ingredient in building an audience that trusts you and wants to come back for more. When it comes to podcasts, remember that you've got access to the ears of your listeners. Out of millions of podcast episodes, they've chosen you; amid the mad rush of their day, they've made time for – so take the time to create content that's valuable, interesting, and worth their time. I think you'll find that their loyalty pays you back in full. you An enormous thank you to Yaagneshwaran for his insights and advice – be sure to check out for more marketing goodness. his podcast Also Published Here

The Art of Creating Valuable Podcasts

**GRAND CAYMAN, Cayman Islands, December 12th, 2024/Chainwire/--**Sui, the Layer 1 blockchain that delivers unparalleled scalability and speed, and Atoma Network, a next generation decentralized AI engine, today announced that Atoma’s fully decentralized AI layer, first announced earlier this year, is live on Sui’s Mainnet. The team also offered reporters a first glimpse into the potential of decentralized AI with a decentralized AI Chat app (code named “Utopia”) built on Sui that enables users to have full ownership of their private data while still providing a user experience on par with centralized alternatives. With its integration currently live in its alpha version on Sui’s Mainnet, Atoma is actively expanding its decentralized ecosystem with new nodes and application builders. For these developers, building on Atoma is an easy transition because Atoma’s API endpoints are directly analogous to those provided by the OpenAI API, which many of them are already familiar with. Then, with Atoma, they can create personalized applications like chatbots connected to social media, finances, or health applications that still maintain individuals’ control over their data. One of the unique strengths of Atoma’s structure is that all of its AI processing occurs in secure areas within a processor — called Trusted Execution Environments, or TEEs. As a result, AI computations are isolated from the rest of the system and AI inputs and outputs are fully private and beyond access by AI companies. Unlike centralized AI applications like ChatGPT, Claude, or Perplexity, where the companies that created the technology can associate inputs with individual accounts, Atoma protects users with complete privacy and censorship resistance. At the same time, by leveraging one of the world’s fastest blockchains, Atoma can offer a user experience with speed and efficiency on par with the centralized versions. While the AI chat app Atoma created as an initial demonstration represents one application of its technology, Atoma’s infrastructure empowers developers to create private AI agents that perform a multitude of tasks while protecting individual user data. Because the private AI agents on Atoma can access and process personal information without exposing user data, they are ideal for applications that require personalization and confidentiality. For example, Atoma-powered decentralized AI can: In DeFi: Manage portfolios, balance assets, and mitigate risks automatically.
Execute trading strategies in real-time on market conditions.
Interact securely with smart contracts for yield farming, lending, and staking. In Social Media: Automate content moderation, curation, and community management.
Deliver personalized recommendations for content, connections, and ads.
Verify interactions and maintain user privacy. Private AI agents can also provide significant value across areas like personal assistants, shopping, gaming, and decentralized science (DeSci). Atoma leverages Sui-related technology platforms to provide its best-in-class service. Atoma relies on the Sui blockchain for settlement, managing payments, and validating transactions, building on Sui’s industry-leading sub-second transaction finality, support for high volumes, and predictable, low-cost transactions. Atoma also integrates Walrus, a decentralized storage solution built on Sui for fast, secure access to private data. This integration ensures that sensitive information is securely stored and easily accessible, supporting AI models in retrieving external data for context-rich insights. By automatically managing data retrieval during long interactions, Walrus enhances memory, contextual understanding, and workflow efficiency, enabling more accurate AI responses. “Decentralized AI requires a blockchain with high degrees of scalability, lightning fast transaction speeds, and low, predictable transaction costs and on these metrics, Sui and Walrus really shine,” ​​said Jorge Antonio, CTO and Co-founder at Atoma. “Sui and Walrus offer a combination of performance, cost effectiveness, reliability, and ease of integration that is unique in our industry and we are thrilled to build on the foundation they provide.” "AI technology is advancing at an unprecedented pace, transforming industries and reshaping how we interact with the world. As adoption accelerates, decentralized and scalable solutions will be critical to safeguarding user control over personal information," said Jameel Khalfan, Head of Ecosystem Development at the Sui Foundation. "Atoma's work, built on the highly scalable platforms of Sui and Walrus, ensures that users can leverage the transformative power of AI without compromising the security and privacy of their data." Contact Sui Foundation media@sui.io This story was distributed as a release by Chainwire under HackerNoon’s Business Blogging Program. Learn more about the program here **GRAND CAYMAN, Cayman Islands, December 12th, 2024/Chainwire/--**Sui, the Layer 1 blockchain that delivers unparalleled scalability and speed, and Atoma Network, a next generation decentralized AI engine, today announced that Atoma’s fully decentralized AI layer, first announced earlier this year, is live on Sui’s Mainnet. The team also offered reporters a first glimpse into the potential of decentralized AI with a decentralized AI Chat app (code named “Utopia”) built on Sui that enables users to have full ownership of their private data while still providing a user experience on par with centralized alternatives. With its integration currently live in its alpha version on Sui’s Mainnet, Atoma is actively expanding its decentralized ecosystem with new nodes and application builders. For these developers, building on Atoma is an easy transition because Atoma’s API endpoints are directly analogous to those provided by the OpenAI API, which many of them are already familiar with. Then, with Atoma, they can create personalized applications like chatbots connected to social media, finances, or health applications that still maintain individuals’ control over their data. One of the unique strengths of Atoma’s structure is that all of its AI processing occurs in secure areas within a processor — called Trusted Execution Environments, or TEEs. As a result, AI computations are isolated from the rest of the system and AI inputs and outputs are fully private and beyond access by AI companies. Unlike centralized AI applications like ChatGPT, Claude, or Perplexity, where the companies that created the technology can associate inputs with individual accounts, Atoma protects users with complete privacy and censorship resistance. At the same time, by leveraging one of the world’s fastest blockchains, Atoma can offer a user experience with speed and efficiency on par with the centralized versions. While the AI chat app Atoma created as an initial demonstration represents one application of its technology, Atoma’s infrastructure empowers developers to create private AI agents that perform a multitude of tasks while protecting individual user data. Because the private AI agents on Atoma can access and process personal information without exposing user data, they are ideal for applications that require personalization and confidentiality. For example, Atoma-powered decentralized AI can: In DeFi: Manage portfolios, balance assets, and mitigate risks automatically. Execute trading strategies in real-time on market conditions. Interact securely with smart contracts for yield farming, lending, and staking. Manage portfolios, balance assets, and mitigate risks automatically. Execute trading strategies in real-time on market conditions. Interact securely with smart contracts for yield farming, lending, and staking. In Social Media: Automate content moderation, curation, and community management. Deliver personalized recommendations for content, connections, and ads. Verify interactions and maintain user privacy. Automate content moderation, curation, and community management. Deliver personalized recommendations for content, connections, and ads. Verify interactions and maintain user privacy. Private AI agents can also provide significant value across areas like personal assistants, shopping, gaming, and decentralized science (DeSci). Atoma leverages Sui-related technology platforms to provide its best-in-class service. Atoma relies on the Sui blockchain for settlement, managing payments, and validating transactions, building on Sui’s industry-leading sub-second transaction finality, support for high volumes, and predictable, low-cost transactions. Atoma also integrates Walrus, a decentralized storage solution built on Sui for fast, secure access to private data. This integration ensures that sensitive information is securely stored and easily accessible, supporting AI models in retrieving external data for context-rich insights. By automatically managing data retrieval during long interactions, Walrus enhances memory, contextual understanding, and workflow efficiency, enabling more accurate AI responses. “Decentralized AI requires a blockchain with high degrees of scalability, lightning fast transaction speeds, and low, predictable transaction costs and on these metrics, Sui and Walrus really shine,” ​​said Jorge Antonio, CTO and Co-founder at Atoma. “Sui and Walrus offer a combination of performance, cost effectiveness, reliability, and ease of integration that is unique in our industry and we are thrilled to build on the foundation they provide.” “Decentralized AI requires a blockchain with high degrees of scalability, lightning fast transaction speeds, and low, predictable transaction costs and on these metrics, Sui and Walrus really shine,” ​​said Jorge Antonio, CTO and Co-founder at Atoma. “Sui and Walrus offer a combination of performance, cost effectiveness, reliability, and ease of integration that is unique in our industry and we are thrilled to build on the foundation they provide.” "AI technology is advancing at an unprecedented pace, transforming industries and reshaping how we interact with the world. As adoption accelerates, decentralized and scalable solutions will be critical to safeguarding user control over personal information," said Jameel Khalfan, Head of Ecosystem Development at the Sui Foundation. "AI technology is advancing at an unprecedented pace, transforming industries and reshaping how we interact with the world. As adoption accelerates, decentralized and scalable solutions will be critical to safeguarding user control over personal information," said Jameel Khalfan, Head of Ecosystem Development at the Sui Foundation. "Atoma's work, built on the highly scalable platforms of Sui and Walrus, ensures that users can leverage the transformative power of AI without compromising the security and privacy of their data." "Atoma's work, built on the highly scalable platforms of Sui and Walrus, ensures that users can leverage the transformative power of AI without compromising the security and privacy of their data." Contact Sui Foundation media@sui.io This story was distributed as a release by Chainwire under HackerNoon’s Business Blogging Program. Learn more about the program here This story was distributed as a release by Chainwire under HackerNoon’s Business Blogging Program. Learn more about the program here here here

Atoma Launches Private, Decentralized AI On Sui

Atoma, Sui-də Şəxsi, Mərkəzləşdirilməmiş AI-ni işə salır

Atoma lansira privatnu, decentraliziranu umjetnu inteligenciju na Sui

Atoma führt private, dezentrale KI auf Sui ein

Atoma lanza una IA privada y descentralizada en Sui

Atoma が Sui でプライベートな分散型 AI をリリース

Atoma Sui-де жеке, орталықтандырылмаған AI іске қосады

Atoma e Thakgola AI ya Poraebete, ye e Arotšwego Ka Sui

اتوما په Sui کې خصوصي، غیر متمرکز AI په لاره واچوي

Атома покреће приватну, децентрализовану вештачку интелигенцију на Суи

Atoma AI-и хусусӣ ва ғайримарказонидашударо дар Sui оғоз мекунад

Atoma, Sui'de Özel, Merkezi Olmayan Yapay Zeka Başlatıyor

Atoma 在 Sui 上推出私有、去中心化的人工智能

Authors:
(1) Guillaume Staerman, INRIA, CEA, Univ. Paris-Saclay, France;
(2) Marta Campi, CERIAH, Institut de l’Audition, Institut Pasteur, France;
(3) Gareth W. Peters, Department of Statistics & Applied Probability, University of California Santa Barbara, USA. Table of Links Abstract and 1. Introduction 2. Background & Preliminaries 2.1. Functional Isolation Forest 2.2. The Signature Method 3. Signature Isolation Forest Method 4. Numerical Experiments 4.1. Parameters Sensitivity Analysis 4.2. Advantages of (K-)SIF over FIF 4.3. Real-data Anomaly Detection Benchmark 5. Discussion & Conclusion, Impact Statements, and References Appendix A. Additional Information About the Signature B. K-SIF and SIF Algorithms C. Additional Numerical Experiments B. K-SIF and SIF Algorithms This section provides the algorithms for the two proposed methods, Kernel Signature Isolation Forest and Signature Isolation Forest. The steps for each algorithm are described and presented below, introducing the input of each procedure and the steps followed to construct the nodes of the partition trees and the children subsets and datasets. Finally, the output of each method is given. The details of these procedures are provided in the main paper in Section 3. Note that the ω parameter, corresponding to the number of split windows used for the signature, is considered as input since it must be chosen for the procedures to advance. Still, it is hidden within the presentation of the algorithms. We further provide the following remark to explain what is the main link between the two proposed algorithms. Remark B.1 (LINK BETWEEN K-SIF AND FIF). The first order coefficients of the signature on an interval [s, t] ⊂ [0, 1] represent the displacement of the function: This paper is available on arxiv under CC BY 4.0 DEED license. Authors: (1) Guillaume Staerman, INRIA, CEA, Univ. Paris-Saclay, France; (2) Marta Campi, CERIAH, Institut de l’Audition, Institut Pasteur, France; (3) Gareth W. Peters, Department of Statistics & Applied Probability, University of California Santa Barbara, USA. Authors: Authors: (1) Guillaume Staerman, INRIA, CEA, Univ. Paris-Saclay, France; (2) Marta Campi, CERIAH, Institut de l’Audition, Institut Pasteur, France; (3) Gareth W. Peters, Department of Statistics & Applied Probability, University of California Santa Barbara, USA. Table of Links Abstract and 1. Introduction Abstract and 1. Introduction 2. Background & Preliminaries 2. Background & Preliminaries 2.1. Functional Isolation Forest 2.1. Functional Isolation Forest 2.2. The Signature Method 2.2. The Signature Method 3. Signature Isolation Forest Method 3. Signature Isolation Forest Method 4. Numerical Experiments 4. Numerical Experiments 4.1. Parameters Sensitivity Analysis 4.1. Parameters Sensitivity Analysis 4.2. Advantages of (K-)SIF over FIF 4.2. Advantages of (K-)SIF over FIF 4.3. Real-data Anomaly Detection Benchmark 4.3. Real-data Anomaly Detection Benchmark 5. Discussion & Conclusion, Impact Statements, and References 5. Discussion & Conclusion, Impact Statements, and References Appendix A. Additional Information About the Signature A. Additional Information About the Signature B. K-SIF and SIF Algorithms B. K-SIF and SIF Algorithms C. Additional Numerical Experiments C. Additional Numerical Experiments B. K-SIF and SIF Algorithms This section provides the algorithms for the two proposed methods, Kernel Signature Isolation Forest and Signature Isolation Forest. The steps for each algorithm are described and presented below, introducing the input of each procedure and the steps followed to construct the nodes of the partition trees and the children subsets and datasets. Finally, the output of each method is given. The details of these procedures are provided in the main paper in Section 3. Note that the ω parameter, corresponding to the number of split windows used for the signature, is considered as input since it must be chosen for the procedures to advance. Still, it is hidden within the presentation of the algorithms. We further provide the following remark to explain what is the main link between the two proposed algorithms. Remark B.1 (LINK BETWEEN K-SIF AND FIF). The first order coefficients of the signature on an interval [s, t] ⊂ [0, 1] represent the displacement of the function: Remark B.1 This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv

Two Algorithms, One Goal: Changing the Face of Anomaly Detection with KIF and SIF 

Authors:
(1) Guillaume Staerman, INRIA, CEA, Univ. Paris-Saclay, France;
(2) Marta Campi, CERIAH, Institut de l’Audition, Institut Pasteur, France;
(3) Gareth W. Peters, Department of Statistics & Applied Probability, University of California Santa Barbara, USA. Table of Links Abstract and 1. Introduction 2. Background & Preliminaries 2.1. Functional Isolation Forest 2.2. The Signature Method 3. Signature Isolation Forest Method 4. Numerical Experiments 4.1. Parameters Sensitivity Analysis 4.2. Advantages of (K-)SIF over FIF 4.3. Real-data Anomaly Detection Benchmark 5. Discussion & Conclusion, Impact Statements, and References Appendix A. Additional Information About the Signature B. K-SIF and SIF Algorithms C. Additional Numerical Experiments A. Additional Information About the Signature In this section, we provide additional information regarding the signature. Firstly, the formula for the coordinate signature is provided, followed by some further information about the truncated signature and the link of the signature with moments. We then introduce references and further discussion for the computation of the signature, the linear-closed form utilised and, lastly, a review of the inner product of tensors required for the signature method is presented. Computation of the Signature. The signature is usually computed on the linear paths reconstructed from observations (Lyons and McLeod, 2022; Fermanian, 2021) thanks to the two properties listed below. First, the signature benefits from a closed form when computed on linear paths. Second, a concatenation formula has been provided by Chen (1958), where the coordinate signature of two concatenate segments is given by a tensor product of the specific coordinate signatures on each segment. These two properties are described formally below. In practice, the signature is usually computed as follows. First, a linear path is reconstructed by interpolation from the observed data. In a second step, the signature of each segment of the linear path are calculated separately thanks to the closed-form (2). Then the signature of the entire path is computed recursively using the Chen’s identity property. Definition of Inner Product of Tensors for Signature Method. Following Chevyrev and Oberhauser (2022) and Lee and Oberhauser (2023), we introduce the inner product of the tensors required to compute the truncated signature and its kernel. This paper is available on arxiv under CC BY 4.0 DEED license. Authors: (1) Guillaume Staerman, INRIA, CEA, Univ. Paris-Saclay, France; (2) Marta Campi, CERIAH, Institut de l’Audition, Institut Pasteur, France; (3) Gareth W. Peters, Department of Statistics & Applied Probability, University of California Santa Barbara, USA. Authors: Authors: (1) Guillaume Staerman, INRIA, CEA, Univ. Paris-Saclay, France; (2) Marta Campi, CERIAH, Institut de l’Audition, Institut Pasteur, France; (3) Gareth W. Peters, Department of Statistics & Applied Probability, University of California Santa Barbara, USA. Table of Links Abstract and 1. Introduction Abstract and 1. Introduction 2. Background & Preliminaries 2. Background & Preliminaries 2.1. Functional Isolation Forest 2.1. Functional Isolation Forest 2.2. The Signature Method 2.2. The Signature Method 3. Signature Isolation Forest Method 3. Signature Isolation Forest Method 4. Numerical Experiments 4. Numerical Experiments 4.1. Parameters Sensitivity Analysis 4.1. Parameters Sensitivity Analysis 4.2. Advantages of (K-)SIF over FIF 4.2. Advantages of (K-)SIF over FIF 4.3. Real-data Anomaly Detection Benchmark 4.3. Real-data Anomaly Detection Benchmark 5. Discussion & Conclusion, Impact Statements, and References 5. Discussion & Conclusion, Impact Statements, and References Appendix A. Additional Information About the Signature A. Additional Information About the Signature B. K-SIF and SIF Algorithms B. K-SIF and SIF Algorithms C. Additional Numerical Experiments C. Additional Numerical Experiments A. Additional Information About the Signature In this section, we provide additional information regarding the signature. Firstly, the formula for the coordinate signature is provided, followed by some further information about the truncated signature and the link of the signature with moments. We then introduce references and further discussion for the computation of the signature, the linear-closed form utilised and, lastly, a review of the inner product of tensors required for the signature method is presented. Computation of the Signature. The signature is usually computed on the linear paths reconstructed from observations (Lyons and McLeod, 2022; Fermanian, 2021) thanks to the two properties listed below. First, the signature benefits from a closed form when computed on linear paths. Second, a concatenation formula has been provided by Chen (1958), where the coordinate signature of two concatenate segments is given by a tensor product of the specific coordinate signatures on each segment. These two properties are described formally below. Computation of the Signature. In practice, the signature is usually computed as follows. First, a linear path is reconstructed by interpolation from the observed data. In a second step, the signature of each segment of the linear path are calculated separately thanks to the closed-form (2). Then the signature of the entire path is computed recursively using the Chen’s identity property. Definition of Inner Product of Tensors for Signature Method. Following Chevyrev and Oberhauser (2022) and Lee and Oberhauser (2023), we introduce the inner product of the tensors required to compute the truncated signature and its kernel. Definition of Inner Product of Tensors for Signature Method. This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv

Unlocking the Power of Signatures in Anomaly Detection

How K-SIF and SIF Revolutionize Anomaly Detection in Complex Datasets

Authors:
(1) Guillaume Staerman, INRIA, CEA, Univ. Paris-Saclay, France;
(2) Marta Campi, CERIAH, Institut de l’Audition, Institut Pasteur, France;
(3) Gareth W. Peters, Department of Statistics & Applied Probability, University of California Santa Barbara, USA. Table of Links Abstract and 1. Introduction 2. Background & Preliminaries 2.1. Functional Isolation Forest 2.2. The Signature Method 3. Signature Isolation Forest Method 4. Numerical Experiments 4.1. Parameters Sensitivity Analysis 4.2. Advantages of (K-)SIF over FIF 4.3. Real-data Anomaly Detection Benchmark 5. Discussion & Conclusion, Impact Statements, and References Appendix A. Additional Information About the Signature B. K-SIF and SIF Algorithms C. Additional Numerical Experiments 4.3. Real-data Anomaly Detection Benchmark To evaluate the effectiveness of the proposed (K-)SIF algorithms and provide a comparison with FIF, we perform a comparative analysis using ten anomaly detection datasets constructed in Staerman et al. (2019) and sourced from the UCR repository (Chen et al., 2015). In contrast to Staerman et al. (2019), we do not use a training/test part since the labels are not used for the training and train and evaluate models on the training data only. We evaluate the algorithms’ performance by quantifying the AUC under the ROC curves. On one hand, Figure 4 illustrates the performance disparity between FIF and K-SIF using the Brownian dictionary. Notably, K-SIF exhibits a significant performance advantage over FIF. This observation underscores the effectiveness of the signature kernel in improving FIF’s performance across most datasets, emphasizing the advantages of utilizing it over a simple inner product. On the other hand, considering the intricacy of functional data, no unique method is expected to outperform others universally. However, SIF demonstrates strong performance in most cases, achieving the best results for five datasets. In contrast to FIF and K-SIF, it shows robustness to the variety of datasets while not being drastically affected by the choice of the parameters involved in FIF (dictionary and inner product) and K-SIF (dictionary). This paper is available on arxiv under CC BY 4.0 DEED license. Authors: (1) Guillaume Staerman, INRIA, CEA, Univ. Paris-Saclay, France; (2) Marta Campi, CERIAH, Institut de l’Audition, Institut Pasteur, France; (3) Gareth W. Peters, Department of Statistics & Applied Probability, University of California Santa Barbara, USA. Authors: Authors: (1) Guillaume Staerman, INRIA, CEA, Univ. Paris-Saclay, France; (2) Marta Campi, CERIAH, Institut de l’Audition, Institut Pasteur, France; (3) Gareth W. Peters, Department of Statistics & Applied Probability, University of California Santa Barbara, USA. Table of Links Abstract and 1. Introduction Abstract and 1. Introduction 2. Background & Preliminaries 2. Background & Preliminaries 2.1. Functional Isolation Forest 2.1. Functional Isolation Forest 2.2. The Signature Method 2.2. The Signature Method 3. Signature Isolation Forest Method 3. Signature Isolation Forest Method 4. Numerical Experiments 4. Numerical Experiments 4.1. Parameters Sensitivity Analysis 4.1. Parameters Sensitivity Analysis 4.2. Advantages of (K-)SIF over FIF 4.2. Advantages of (K-)SIF over FIF 4.3. Real-data Anomaly Detection Benchmark 4.3. Real-data Anomaly Detection Benchmark 5. Discussion & Conclusion, Impact Statements, and References 5. Discussion & Conclusion, Impact Statements, and References Appendix A. Additional Information About the Signature A. Additional Information About the Signature B. K-SIF and SIF Algorithms B. K-SIF and SIF Algorithms C. Additional Numerical Experiments C. Additional Numerical Experiments 4.3. Real-data Anomaly Detection Benchmark To evaluate the effectiveness of the proposed (K-)SIF algorithms and provide a comparison with FIF, we perform a comparative analysis using ten anomaly detection datasets constructed in Staerman et al. (2019) and sourced from the UCR repository (Chen et al., 2015). In contrast to Staerman et al. (2019), we do not use a training/test part since the labels are not used for the training and train and evaluate models on the training data only. We evaluate the algorithms’ performance by quantifying the AUC under the ROC curves. On one hand, Figure 4 illustrates the performance disparity between FIF and K-SIF using the Brownian dictionary. Notably, K-SIF exhibits a significant performance advantage over FIF. This observation underscores the effectiveness of the signature kernel in improving FIF’s performance across most datasets, emphasizing the advantages of utilizing it over a simple inner product. On the other hand, considering the intricacy of functional data, no unique method is expected to outperform others universally. However, SIF demonstrates strong performance in most cases, achieving the best results for five datasets. In contrast to FIF and K-SIF, it shows robustness to the variety of datasets while not being drastically affected by the choice of the parameters involved in FIF (dictionary and inner product) and K-SIF (dictionary). This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv

How (K-)SIF Outperforms FIF in Real-Data Anomaly Detection

Hoe (K-)SIF beter as FIF presteer in Real-Data Anomalie Detection

كيف يتفوق (K-)SIF على FIF في اكتشاف الشذوذ في البيانات الحقيقية

Wie (K-)SIF FIF bei der Erkennung von Anomalien in Echtdaten übertrifft

Cómo (K-)SIF supera a FIF en la detección de anomalías de datos reales

実データ異常検出において (K-)SIF が FIF を上回る理由

Hoe (K-)SIF FIF overtreft bij het detecteren van anomalieën in echte data

Imaynatataq (K-)SIF FIFmanta aswan allinta ruwan Chiqa-Dato Anomalía Detección nisqapi

Cum (K-)SIF depășește FIF în detectarea anomaliilor de date reale

Si (K-)SIF tejkalon FIF në zbulimin e anomalive të të dhënave reale

Како (К-)СИФ надмашује ФИФ у откривању аномалија у реалним подацима

Чӣ тавр (K-)SIF дар ошкоркунии аномалияи воқеии маълумот аз FIF бартарӣ дорад

(K-)SIF vượt trội hơn FIF như thế nào trong việc phát hiện dị thường dữ liệu thực

Authors:
(1) Guillaume Staerman, INRIA, CEA, Univ. Paris-Saclay, France;
(2) Marta Campi, CERIAH, Institut de l’Audition, Institut Pasteur, France;
(3) Gareth W. Peters, Department of Statistics & Applied Probability, University of California Santa Barbara, USA. Table of Links Abstract and 1. Introduction 2. Background & Preliminaries 2.1. Functional Isolation Forest 2.2. The Signature Method 3. Signature Isolation Forest Method 4. Numerical Experiments 4.1. Parameters Sensitivity Analysis 4.2. Advantages of (K-)SIF over FIF 4.3. Real-data Anomaly Detection Benchmark 5. Discussion & Conclusion, Impact Statements, and References Appendix A. Additional Information About the Signature B. K-SIF and SIF Algorithms C. Additional Numerical Experiments 3. Signature Isolation Forest Method With Kernel Signature Isolation Forest (K-SIF), we aim to leverage the truncated kernel signature (Kiraly and Oberhauser, 2019) to overcome the linearity constraint imposed by the inner product in FIF. In contrast to FIF, which explores only one function characteristic at each node using a unique inner product with the function sampled in the dictionary, K-SIF captures significantly more information. This is achieved by computing several coefficient signatures, summarizing multiple data attributes at each node. The ability to explore non-linear directions in the feature space through the kernel signature makes the algorithm more efficient for an equivalent computational cost (refer to Figure 9 in the Appendix With Signature Isolation Forest (SIF), the aim is to leverage the coordinate signature and remove any restrictions imposed by FIF and K-SIF. We replace the use of a dictionary D and rely on the coordinate signature only to detect anomalies. The intuition is to provide a more straightforward, entirely data-driven solution and remove any a priori required choice. The construction procedures of (K-)SIF are now described. Splitting Criterion. The difference between a ksi-tree and a si-tree lies in the splitting criterion computed at each node of both trees. Below, we detail the splitting procedures occurring at a particular internal node for both algorithms. The following step is to choose uniformly and at random a split value γ such that The children datasets are given as A ksi-tree is built by iterating this procedure until all training data curves are isolated. Signature Isolation Forest. In contrast to ksi-trees, the definition of a si-tree will rely on using a different split criterion, expressed according to the following procedure. Once again, the depth of the truncation level k has to be chosen a priori, but no dictionaries nor distribution has to be set. Here, at each internal node (p, q), a coordinate (i1, . . . , iℓ) is chosen randomly and uniformly in the set The children datasets are given as A si-tree is also built by iterating this procedure until all training data curves are isolated. The two algorithms of K-SIF and SIF are summarized in the Appendix B. where c(m) is the average path length of unsuccessful searches in a binary search tree and m the size of the subsample linked to each tree. Parameters of K-SIF and SIF. For both algorithms, key parameters typical of isolation forest-based methods, such as the number of trees N or the subsample size m, must be pre-selected (Liu et al., 2008). The truncated level k of the signature depth must also be chosen. In the case of K-SIF, similar to FIF, selecting a dictionary D and a distribution ν is required. We opt to implement three standard dictionaries (also utilized in FIF): ‘Brownian’, representing a traditional Brownian motion path; ‘Cosine’, employing a cosine basis; and ‘Gaussian wavelets’, utilizing a Gaussian wavelets basis. In contrast, SIF does not require any of these sensitive parameters. We follow the approach outlined in Morrill et al. (2019) to enhance the algorithms’ performance by introducing a split window parameter ω. This implies that, at each tree node, the truncated signature is computed on a randomly selected portion of the functions with a size of ⌊p/ω⌋. 4. Numerical Experiments This section presents a series of numerical experiments supporting the proposed class of AD methods. We organize the experiments into three categories. First, a parameter sensitivity analysis sheds light on the behavior of the parameters algorithms and provides insights for achieving optimal performances. Subsequently, we compare (K-)SIF and FIF, illustrating the validity of such methods and showcasing the power of the signature in the context of AD solutions. Finally, we perform a benchmark that compares several existing AD methods on various real datasets. This benchmark aims to demonstrate the results in real applications and assess the overall performance of these algorithms relative to other AD methods documented in the literature. This paper is available on arxiv under CC BY 4.0 DEED license. Authors: (1) Guillaume Staerman, INRIA, CEA, Univ. Paris-Saclay, France; (2) Marta Campi, CERIAH, Institut de l’Audition, Institut Pasteur, France; (3) Gareth W. Peters, Department of Statistics & Applied Probability, University of California Santa Barbara, USA. Authors: Authors: (1) Guillaume Staerman, INRIA, CEA, Univ. Paris-Saclay, France; (2) Marta Campi, CERIAH, Institut de l’Audition, Institut Pasteur, France; (3) Gareth W. Peters, Department of Statistics & Applied Probability, University of California Santa Barbara, USA. Table of Links Abstract and 1. Introduction Abstract and 1. Introduction 2. Background & Preliminaries 2. Background & Preliminaries 2.1. Functional Isolation Forest 2.1. Functional Isolation Forest 2.2. The Signature Method 2.2. The Signature Method 3. Signature Isolation Forest Method 3. Signature Isolation Forest Method 4. Numerical Experiments 4. Numerical Experiments 4.1. Parameters Sensitivity Analysis 4.1. Parameters Sensitivity Analysis 4.2. Advantages of (K-)SIF over FIF 4.2. Advantages of (K-)SIF over FIF 4.3. Real-data Anomaly Detection Benchmark 4.3. Real-data Anomaly Detection Benchmark 5. Discussion & Conclusion, Impact Statements, and References 5. Discussion & Conclusion, Impact Statements, and References Appendix A. Additional Information About the Signature A. Additional Information About the Signature B. K-SIF and SIF Algorithms B. K-SIF and SIF Algorithms C. Additional Numerical Experiments C. Additional Numerical Experiments 3. Signature Isolation Forest Method With Kernel Signature Isolation Forest (K-SIF), we aim to leverage the truncated kernel signature (Kiraly and Oberhauser, 2019) to overcome the linearity constraint imposed by the inner product in FIF. In contrast to FIF, which explores only one function characteristic at each node using a unique inner product with the function sampled in the dictionary, K-SIF captures significantly more information. This is achieved by computing several coefficient signatures, summarizing multiple data attributes at each node. The ability to explore non-linear directions in the feature space through the kernel signature makes the algorithm more efficient for an equivalent computational cost (refer to Figure 9 in the Appendix With Signature Isolation Forest (SIF), the aim is to leverage the coordinate signature and remove any restrictions imposed by FIF and K-SIF. We replace the use of a dictionary D and rely on the coordinate signature only to detect anomalies. The intuition is to provide a more straightforward, entirely data-driven solution and remove any a priori required choice. The construction procedures of (K-)SIF are now described. Splitting Criterion. The difference between a ksi-tree and a si-tree lies in the splitting criterion computed at each node of both trees. Below, we detail the splitting procedures occurring at a particular internal node for both algorithms. Splitting Criterion. The following step is to choose uniformly and at random a split value γ such that The children datasets are given as A ksi -tree is built by iterating this procedure until all training data curves are isolated. ksi Signature Isolation Forest. In contrast to ksi-trees, the definition of a si-tree will rely on using a different split criterion, expressed according to the following procedure. Once again, the depth of the truncation level k has to be chosen a priori, but no dictionaries nor distribution has to be set. Here, at each internal node (p, q), a coordinate (i1, . . . , iℓ) is chosen randomly and uniformly in the set The children datasets are given as A si-tree is also built by iterating this procedure until all training data curves are isolated. The two algorithms of K-SIF and SIF are summarized in the Appendix B. where c(m) is the average path length of unsuccessful searches in a binary search tree and m the size of the subsample linked to each tree. c(m) m Parameters of K-SIF and SIF. For both algorithms, key parameters typical of isolation forest-based methods, such as the number of trees N or the subsample size m, must be pre-selected (Liu et al., 2008). The truncated level k of the signature depth must also be chosen. In the case of K-SIF, similar to FIF, selecting a dictionary D and a distribution ν is required. We opt to implement three standard dictionaries (also utilized in FIF): ‘Brownian’, representing a traditional Brownian motion path; ‘Cosine’, employing a cosine basis; and ‘Gaussian wavelets’, utilizing a Gaussian wavelets basis. In contrast, SIF does not require any of these sensitive parameters. We follow the approach outlined in Morrill et al. (2019) to enhance the algorithms’ performance by introducing a split window parameter ω. This implies that, at each tree node, the truncated signature is computed on a randomly selected portion of the functions with a size of ⌊p/ω⌋. Parameters of K-SIF and SIF. 4. Numerical Experiments This section presents a series of numerical experiments supporting the proposed class of AD methods. We organize the experiments into three categories. First, a parameter sensitivity analysis sheds light on the behavior of the parameters algorithms and provides insights for achieving optimal performances. Subsequently, we compare (K-)SIF and FIF, illustrating the validity of such methods and showcasing the power of the signature in the context of AD solutions. Finally, we perform a benchmark that compares several existing AD methods on various real datasets. This benchmark aims to demonstrate the results in real applications and assess the overall performance of these algorithms relative to other AD methods documented in the literature. This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv

Redefining Anomaly Detection with Signature Isolation Forests

Authors:
(1) Guillaume Staerman, INRIA, CEA, Univ. Paris-Saclay, France;
(2) Marta Campi, CERIAH, Institut de l’Audition, Institut Pasteur, France;
(3) Gareth W. Peters, Department of Statistics & Applied Probability, University of California Santa Barbara, USA. Table of Links Abstract and 1. Introduction 2. Background & Preliminaries 2.1. Functional Isolation Forest 2.2. The Signature Method 3. Signature Isolation Forest Method 4. Numerical Experiments 4.1. Parameters Sensitivity Analysis 4.2. Advantages of (K-)SIF over FIF 4.3. Real-data Anomaly Detection Benchmark 5. Discussion & Conclusion, Impact Statements, and References Appendix A. Additional Information About the Signature B. K-SIF and SIF Algorithms C. Additional Numerical Experiments 2. Background & Preliminaries 2.1. Functional Isolation Forest Consider H the functional Hilbert space equipped with a inner product ⟨., .⟩H such that any x ∈ H is a real function defined on [0, 1]. A Functional Isolation Forest is created through an assembly of functional isolation trees (F-itrees). Each F-itree is constructed via a series of random splits from a subsample (of size m) of Xn. The abnormality score for an observation x is then determined as a monotonically decreasing transformation of x’s average depth across the trees. The core concept lies in the randomness of the splits, where an observation markedly different from others is more likely to be isolated from Xn, appearing at shallower levels in the F-itrees. The F-itrees are built based on a predetermined dictionary D ⊂ H, encompassing both deterministic and/or stochastic functions capturing pertinent data properties, which may also be a subset of Xn. Before each random univariate split, all node observations are projected onto a line defined by a randomly selected element from the dictionary D. The selection of a suitable dictionary plays a pivotal role in shaping the FIF score construction. The projection criterion at each node of each F-itree is defined as: This paper is available on arxiv under CC BY 4.0 DEED license. Authors: (1) Guillaume Staerman, INRIA, CEA, Univ. Paris-Saclay, France; (2) Marta Campi, CERIAH, Institut de l’Audition, Institut Pasteur, France; (3) Gareth W. Peters, Department of Statistics & Applied Probability, University of California Santa Barbara, USA. Authors: Authors: (1) Guillaume Staerman, INRIA, CEA, Univ. Paris-Saclay, France; (2) Marta Campi, CERIAH, Institut de l’Audition, Institut Pasteur, France; (3) Gareth W. Peters, Department of Statistics & Applied Probability, University of California Santa Barbara, USA. Table of Links Abstract and 1. Introduction Abstract and 1. Introduction 2. Background & Preliminaries 2. Background & Preliminaries 2.1. Functional Isolation Forest 2.1. Functional Isolation Forest 2.2. The Signature Method 2.2. The Signature Method 3. Signature Isolation Forest Method 3. Signature Isolation Forest Method 4. Numerical Experiments 4. Numerical Experiments 4.1. Parameters Sensitivity Analysis 4.1. Parameters Sensitivity Analysis 4.2. Advantages of (K-)SIF over FIF 4.2. Advantages of (K-)SIF over FIF 4.3. Real-data Anomaly Detection Benchmark 4.3. Real-data Anomaly Detection Benchmark 5. Discussion & Conclusion, Impact Statements, and References 5. Discussion & Conclusion, Impact Statements, and References Appendix A. Additional Information About the Signature A. Additional Information About the Signature B. K-SIF and SIF Algorithms B. K-SIF and SIF Algorithms C. Additional Numerical Experiments C. Additional Numerical Experiments 2. Background & Preliminaries 2.1. Functional Isolation Forest Consider H the functional Hilbert space equipped with a inner product ⟨., .⟩H such that any x ∈ H is a real function defined on [0, 1]. A Functional Isolation Forest is created through an assembly of functional isolation trees (F-itrees). Each F-itree is constructed via a series of random splits from a subsample (of size m) of Xn. The abnormality score for an observation x is then determined as a monotonically decreasing transformation of x’s average depth across the trees. The core concept lies in the randomness of the splits, where an observation markedly different from others is more likely to be isolated from Xn, appearing at shallower levels in the F-itrees. The F-itrees are built based on a predetermined dictionary D ⊂ H, encompassing both deterministic and/or stochastic functions capturing pertinent data properties, which may also be a subset of Xn. Before each random univariate split, all node observations are projected onto a line defined by a randomly selected element from the dictionary D. The selection of a suitable dictionary plays a pivotal role in shaping the FIF score construction. The projection criterion at each node of each F-itree is defined as: This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv

How Functional Isolation Forest Detects Anomalies

Anomaly Detection: The Dark Horse of Fraud Detection

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