Kalman Filter (KF) is an optimal linear state prediction algorithm, with applications in fields as diverse as engineering, economics, robotics, and space exploration. Here, we develop an extension of the KF, called a Pathspace Kalman Filter (PKF) which allows us to a) dynamically track the uncertainties associated with the underlying data and prior knowledge, and b) take as input an entire trajectory and an underlying mechanistic model, and using a Bayesian methodology quantify the different sources of uncertainty. An application of this algorithm is to automatically detect temporal windows where the internal mechanistic model deviates from the data in a time-dependent manner. First, we provide theorems characterizing the convergence of the PKF algorithm. Then, we numerically demonstrate that the PKF outperforms conventional KF methods on a synthetic dataset lowering the mean-squared-error by several orders of magnitude. Finally, we apply this method to biological time-course dataset involving over 1.8 million gene expression measurements.

"The funny thing about life is that it's temporary; that is to say, temporary in the'temporal' sense of the word, meaning that all living things and all that we do are subject to the precepts and effects of time." Many organisms perform best at certain hours of the day. The slug species Arion subfuscus, living in almost total darkness, knowing nothing about the Gregorian calendar, lays its eggs between the last week of August and the first week of September.1 Bees forage for nectar, knowing the best times to visit the best fields and the exact timing of nectar secretions for individual species of flowers. In the mid-20th century, the Austrian Nobel laureate Karl von Frisch provided enormous insights on honeybee communication and foraging time. He discovered that bees have internal clocks that tell them not only where the nectar is to be found but also precisely when that food will be ready. "I know of no other living creature," he wrote in his book on bee language, "that learns so easily as the bee when, according to its'internal clock,' to come to the table."2 Even without a light clue, the plants were able to tell time.

Sleep is a naturally recurring state of mind and body, characterized by altered consciousness, relatively inhibited sensory activity, inhibition of nearly all voluntary muscles, and reduced interactions with surroundings.[1] It is distinguished from wakefulness by a decreased ability to react to stimuli, but is more easily reversed than the state of being comatose. Sleep occurs in repeating periods, in which the body alternates between two distinct modes known as non-REM and REM sleep. Although REM stands for "rapid eye movement", this mode of sleep has many other aspects, including virtual paralysis of the body. A well-known feature of sleep is the dream, an experience typically recounted in narrative form, which resembles waking life while in progress, but which usually can later be distinguished as fantasy. During sleep, most of the body's systems are in an anabolic state, helping to restore the immune, nervous, skeletal, and muscular systems; these are vital processes that maintain mood, memory, and cognitive performance, and play a large role in the function of the endocrine and immune systems.[2] The internal circadian clock promotes sleep daily at night. The diverse purposes and mechanisms of sleep are the subject of substantial ongoing research.[3] The advent of artificial light has substantially altered sleep timing in industrialized countries.[4] Humans may suffer from various sleep disorders, including dyssomnias, such as insomnia, hypersomnia, narcolepsy, and sleep apnea; parasomnias, such as sleepwalking and REM behavior disorder; bruxism; and circadian rhythm sleep disorders. The most pronounced physiological changes in sleep occur in the brain.[5]

Sleep is a naturally recurring state of mind and body characterized by altered consciousness, relatively inhibited sensory activity, inhibition of nearly all voluntary muscles, and reduced interactions with surroundings.[1] It is distinguished from wakefulness by a decreased ability to react to stimuli, but is more easily reversed than the state of hibernation or of being comatose. Mammalian sleep occurs in repeating periods, in which the body alternates between two highly distinct modes known as non-REM and REM sleep. REM stands for "rapid eye movement" but involves many other aspects including virtual paralysis of the body. During sleep, most systems in an animal are in an anabolic state, building up the immune, nervous, skeletal, and muscular systems. Sleep in non-human animals is observed in mammals, birds, reptiles, amphibians, and some fish, and, in some form, in insects and even in simpler animals such as nematodes. The internal circadian clock promotes sleep daily at night in diurnal organisms (such as humans) and in the day in nocturnal organisms (such as rodents). However, sleep patterns vary among individual humans and even more widely among other species. In the last century, artificial light has in many areas of the world substantially altered sleep timing among both humans and many other species.[2] The diverse purposes and mechanisms of sleep are the subject of substantial ongoing research.[3] Sleep seems to assist animals with improvements in the body and mind. A well-known feature of sleep in humans is the dream, an experience typically recounted in narrative form, which resembles waking life while in progress, but which usually can later be distinguished as fantasy. Sleep is sometimes confused with unconsciousness, but is quite different in terms of thought process. Humans may suffer from a number of sleep disorders. These include dyssomnias (such as insomnia, hypersomnia, and sleep apnea), parasomnias (such as sleepwalking and REM behavior disorder), bruxism, and the circadian rhythm sleep disorders. In mammals and birds, sleep is divided into two broad types: rapid eye movement (REM sleep) and non-rapid eye movement (NREM or non-REM sleep). Each type has a distinct set of physiological and neurological features associated with it. REM sleep is associated with dreaming, desynchronized and faster brain waves, loss of muscle tone,[4] and suspension of homeostasis[citation needed]. REM and non-REM sleep are so different that physiologists classify them as distinct behavioral states. In this view, REM, non-REM, and waking represent the three major modes of consciousness, neural activity, and physiological regulation.[5] According to the Hobson & McCarley activation-synthesis hypothesis, proposed in 1975–1977, the alternation between REM and non-REM can be explained in terms of cycling, reciprocally influential neurotransmitter systems.[6]

Sleep is a naturally recurring state of mind characterized by altered consciousness, relatively inhibited sensory activity, inhibition of nearly all voluntary muscles, and reduced interactions with surroundings.[1] It is distinguished from wakefulness by a decreased ability to react to stimuli, but is more easily reversed than the state of hibernation or of being comatose. Mammalian sleep occurs in repeating periods, in which the body alternates between two highly distinct modes known as non-REM and REM sleep. REM stands for "rapid eye movement" but involves many other aspects including virtual paralysis of the body. During sleep, most systems in an animal are in an anabolic state, building up the immune, nervous, skeletal, and muscular systems. Sleep in non-human animals is observed in mammals, birds, reptiles, amphibians, and some fish, and, in some form, in insects and even in simpler animals such as nematodes. The internal circadian clock promotes sleep daily at night in diurnal species (such as humans) and in the day in nocturnal organisms (such as rodents). However, sleep patterns vary widely among animals and among different individual humans. Industrialization and artificial light have substantially altered human sleep habits in the last 100 years. The diverse purposes and mechanisms of sleep are the subject of substantial ongoing research.[2] Sleep seems to assist animals with improvements in the body and mind. A well-known feature of sleep in humans is the dream, an experience typically recounted in narrative form, which resembles waking life while in progress, but which usually can later be distinguished as fantasy. Sleep is sometimes confused with unconsciousness, but is quite different in terms of thought process. Humans may suffer from a number of sleep disorders. These include dyssomnias (such as insomnia, hypersomnia, and sleep apnea), parasomnias (such as sleepwalking and REM behavior disorder), bruxism, and the circadian rhythm sleep disorders. In mammals and birds, sleep is divided into two broad types: rapid eye movement (REM sleep) and non-rapid eye movement (NREM or non-REM sleep). Each type has a distinct set of physiological and neurological features associated with it. REM sleep is associated with dreaming, desynchronized and faster brain waves, loss of muscle tone,[3] and suspension of homeostasis[citation needed]. REM and non-REM sleep are so different that physiologists classify them as distinct behavioral states. In this view, REM, non-REM, and waking represent the three major modes of consciousness, neural activity, and physiological regulation.[4] According to the Hobson & McCarley activation-synthesis hypothesis, proposed in 1975–1977, the alternation between REM and non-REM can be explained in terms of cycling, reciprocally influential neurotransmitter systems.[5]

Sleep is a naturally recurring state of mind characterized by altered consciousness, relatively inhibited sensory activity, inhibition of nearly all voluntary muscles, and reduced interactions with surroundings.[1] It is distinguished from wakefulness by a decreased ability to react to stimuli, but is more easily reversed than the state of hibernation or of being comatose. Mammalian sleep occurs in repeating periods, in which the body alternates between two highly distinct modes known as non-REM and REM sleep. REM stands for "rapid eye movement" but involves many other aspects including virtual paralysis of the body. During sleep, most systems in an animal are in an anabolic state, building up the immune, nervous, skeletal, and muscular systems. Sleep in non-human animals is observed in mammals, birds, reptiles, amphibians, and some fish, and, in some form, in insects and even in simpler animals such as nematodes. The internal circadian clock promotes sleep daily at night in diurnal species (such as humans) and in the day in nocturnal organisms (such as rodents). However, sleep patterns vary widely among animals and among different individual humans. Industrialization and artificial light have substantially altered human sleep habits in the last 100 years.[2] The diverse purposes and mechanisms of sleep are the subject of substantial ongoing research.[3] Sleep seems to assist animals with improvements in the body and mind. A well-known feature of sleep in humans is the dream, an experience typically recounted in narrative form, which resembles waking life while in progress, but which usually can later be distinguished as fantasy. Sleep is sometimes confused with unconsciousness, but is quite different in terms of thought process. Humans may suffer from a number of sleep disorders. These include dyssomnias (such as insomnia, hypersomnia, and sleep apnea), parasomnias (such as sleepwalking and REM behavior disorder), bruxism, and the circadian rhythm sleep disorders. In mammals and birds, sleep is divided into two broad types: rapid eye movement (REM sleep) and non-rapid eye movement (NREM or non-REM sleep). Each type has a distinct set of physiological and neurological features associated with it. REM sleep is associated with dreaming, desynchronized and faster brain waves, loss of muscle tone,[4] and suspension of homeostasis[citation needed]. REM and non-REM sleep are so different that physiologists classify them as distinct behavioral states. In this view, REM, non-REM, and waking represent the three major modes of consciousness, neural activity, and physiological regulation.[5] According to the Hobson & McCarley activation-synthesis hypothesis, proposed in 1975–1977, the alternation between REM and non-REM can be explained in terms of cycling, reciprocally influential neurotransmitter systems.[6]

Sleep is a naturally recurring state of mind characterized by altered consciousness, relatively inhibited sensory activity, inhibition of nearly all voluntary muscles, and reduced interactions with surroundings.[1] It is distinguished from wakefulness by a decreased ability to react to stimuli, but is more easily reversed than the state of hibernation or of being comatose. Mammalian sleep occurs in repeating periods, in which the body alternates between two highly distinct modes known as non-REM and REM sleep. REM stands for "rapid eye movement" but involves many other aspects including virtual paralysis of the body. During sleep, most systems in an animal are in an anabolic state, building up the immune, nervous, skeletal, and muscular systems. Sleep in non-human animals is observed in mammals, birds, reptiles, amphibians, and fish, and, in some form, in insects and even in simpler animals such as nematodes. The internal circadian clock promotes sleep daily at night in diurnal species (such as humans) and in the day in nocturnal organisms (such as rodents). However, sleep patterns vary widely among animals and among different individual humans. Industrialization and artificial light have substantially altered human sleep habits in the last 100 years.[2] The diverse purposes and mechanisms of sleep are the subject of substantial ongoing research.[3] Sleep seems to assist animals with improvements in the body and mind. A well-known feature of sleep in humans is the dream, an experience typically recounted in narrative form, which resembles waking life while in progress, but which usually can later be distinguished as fantasy. Sleep is sometimes confused with unconsciousness, but is quite different in terms of thought process. Humans may suffer from a number of sleep disorders. These include dyssomnias (such as insomnia, hypersomnia, and sleep apnea), parasomnias (such as sleepwalking and REM behavior disorder), bruxism, and the circadian rhythm sleep disorders. In mammals and birds, sleep is divided into two broad types: rapid eye movement (REM sleep) and non-rapid eye movement (NREM or non-REM sleep). Each type has a distinct set of physiological and neurological features associated with it. REM sleep is associated with dreaming, desynchronized and faster brain waves, loss of muscle tone,[4] and suspension of homeostasis[citation needed]. REM and non-REM sleep are so different that physiologists classify them as distinct behavioral states. In this view, REM, non-REM, and waking represent the three major modes of consciousness, neural activity, and physiological regulation.[5]